Space Collision Avoidance using Interval Analysis by Bart Romgens

Space Collision Avoidance using Interval Analysis A Literature Survey

Bart RÂ¨ omgens

Space Collision Avoidance using Interval Analysis A Literature Survey

Supervisor: dr.ir. E. Mooij

Author: Bart RÂ¨omgens

October 6, 2009

Contents Contents

Preface

List of Symbols

List of Abbreviations

VIII

1 Introduction 2 Collisions in Space 2.1 Objects in Space . . . . . . . 2.1.1 Active Satellites . . . 2.1.2 Manned Missions . . . 2.1.3 Space Debris . . . . . 2.1.4 Asteroids, Comets and 2.2 Collisions in Earth Orbit . . . 2.2.1 History of Collisions . 2.2.2 Collision Effects . . . 2.2.3 Future of Collisions . 2.3 Collision Avoidance Systems . 2.3.1 Orbit Determination . 2.3.2 Collision Prediction . 2.3.3 Maneuver Strategies . 2.3.4 Space Debris Removal

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3 4 4 5 8 15 18 18 20 21 22 23 24 24 26

3 Astrodynamics 3.1 N-body Problem . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Equations of Motion . . . . . . . . . . . . 3.1.2 Relative Equations of Motion . . . . . . . . . . . . 3.1.3 Constants of Motion . . . . . . . . . . . . . . . . . 3.1.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Transformations between Reference Frames . . . . 3.3 Perturbing Forces . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Earthâ&#x20AC;&#x2122;s Gravity Field . . . . . . . . . . . . . . . . 3.3.2 Gravitational Attraction by other Celestial Bodies 3.3.3 Aerodynamic Forces . . . . . . . . . . . . . . . . . 3.3.4 Radiation Pressure . . . . . . . . . . . . . . . . . .

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28 28 28 30 31 32 35 36 37 37 40 40 41

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41 42 42 42 43 44 44 47 47 47 48 48

4 Unified State Model 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Phase Space and Velocity Hodograph . . . . . . . 4.1.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Transformations between Rotation Representations 4.2 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . 4.3 State Description . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Cartesian Coordinates . . . . . . . . . . . . . . . . 4.3.2 Kepler Elements . . . . . . . . . . . . . . . . . . . 4.3.3 Unified State Model Variables . . . . . . . . . . . . 4.4 Variation of Unified State Variables . . . . . . . . . . . . . 4.5 Perturbing Forces in the Unified State Model . . . . . . .

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50 50 50 52 55 57 58 59 59 60 64 65

5 Conventional ODE Integration 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Numerical Methods . . . . . . . . . . . 5.2.1 Taylor Series . . . . . . . . . . 5.2.2 Euler Methods . . . . . . . . . 5.2.3 Runge-Kutta Methods . . . . . 5.2.4 Multistep Methods . . . . . . . 5.3 Errors . . . . . . . . . . . . . . . . . . 5.4 Accuracy Properties . . . . . . . . . . 5.4.1 Convergence and Order . . . . 5.4.2 Stability and Stiffness . . . . . 5.5 Random Error Propagation . . . . . . 5.5.1 Monte Carlo . . . . . . . . . . 5.5.2 Covariance Matrix Propagation

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68 68 70 70 71 71 72 73 75 75 75 76 77 77

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80 81 82 83 83 84 84 86 86

3.4

3.5

3.3.5 Electromagnetic Forces . . . 3.3.6 Relativistic Effects . . . . . . 3.3.7 Thrust . . . . . . . . . . . . . 3.3.8 Enclosing Perturbing Forces . Perturbation Methods . . . . . . . . 3.4.1 Method of Encke . . . . . . . 3.4.2 Variation of Orbital Elements 3.4.3 Singularities . . . . . . . . . . Uncertainties . . . . . . . . . . . . . 3.5.1 Physical Models . . . . . . . 3.5.2 Physical Constants . . . . . . 3.5.3 Parameters . . . . . . . . . .

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6 Interval ODE Integration 6.1 Interval Analysis . . . . . . . . . . . . . . . . . 6.1.1 Interval Arithmetic . . . . . . . . . . . . 6.1.2 Interval Algebraic Properties . . . . . . 6.1.3 Inclusion Monotonic Interval Extensions 6.1.4 The Dependency Problem . . . . . . . . 6.1.5 Interval Intersections . . . . . . . . . . . 6.2 Interval Taylor Series Methods . . . . . . . . . 6.2.1 Bounding Taylor Series . . . . . . . . .

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6.3

6.4 6.5 6.6

6.7

6.8

6.2.2 A Priori Enclosure (Verifying Existence and Uniqueness) . 6.2.3 Interval Integration using Bounded Taylor Series . . . . . . 6.2.4 Reducing the Wrapping Effect . . . . . . . . . . . . . . . . Taylor Model Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Taylor Models . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Verified ODE Integration . . . . . . . . . . . . . . . . . . . 6.3.3 Reducing the Wrapping Effect . . . . . . . . . . . . . . . . 6.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Interval Evaluation of Taylor Model Solutions . . . . . . . . Interval Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . Bounding Rounding Errors . . . . . . . . . . . . . . . . . . . . . . Automatic Interval Differentiation . . . . . . . . . . . . . . . . . . 6.6.1 Finite Difference Numerical Differentiation . . . . . . . . . 6.6.2 Symbolic Differentiation . . . . . . . . . . . . . . . . . . . . 6.6.3 Interval Taylor Coefficients using Automatic Differentiation Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Automatic Differentiation . . . . . . . . . . . . . . . . . . . 6.7.3 Interval ODE Integration . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Thesis Assignment 7.1 Research Question . . . . . . . . . . 7.2 Objectives . . . . . . . . . . . . . . . 7.3 Computer Implementation . . . . . . 7.3.1 Software Tools and Libraries 7.3.2 Software Requirements . . . . 7.3.3 Units . . . . . . . . . . . . . 7.3.4 Interfaces . . . . . . . . . . . 7.3.5 Unit Tests . . . . . . . . . . . 7.3.6 Integration . . . . . . . . . . 7.3.7 Verification . . . . . . . . . . 7.3.8 Parallel Computing . . . . . . 7.4 Simulation . . . . . . . . . . . . . . . 7.5 Tasks and Schedule . . . . . . . . . .

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Appendix A - Transformations

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88 92 95 102 103 103 104 105 108 109 111 112 112 113 113 118 118 119 119 123 123 127

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129 129 129 131 131 131 131 134 135 136 137 137 138 138 140

Bibliography

144

III

Preface This literature survey is part of the MSc. in Aerospace Engineering, profile Astrodynamics and Space Missions, at Delft University of Technology. It is the preparatory work for the final MSc. thesis that completes the MSc. degree. I want to thank my supervisor Erwin Mooij for letting me free in my own research and learning process while guiding me when necessary. I also want to thank the one and almighty Flying Spaghetti Monster for supporting me during rough times. This is the digital PDF version of this report. It has a few features to improve the reading experience. The chapters and sections in the table of contents are linked to the corresponding chapters and sections; simply click on the chapter or section you want to read. A table of contents is also available in the sidebar of your PDF reader. Furthermore, the chapter, section, figure, table, algorithm and equation numbers mentioned in the text are linked to the actual chapter, section, figure, etc. The same goes for the references, click the reference number between square brackets to move to that particular reference in the bibliography.

List of Symbols Astrodynamics r v a F m G µ H V C e a i Ω ω θ E M τ ω R U Jn Jn,m φ Λ ρ ρ0 CD S W CR S∗ c

position vector velocity vector acceleration vector a force vector mass the gravitational constant standard gravitational parameter angular momentum vector orbital speed (size of velocity vector) orbital energy eccentricity (Kepler Element) semi-major axis inclination longitude of the ascending node argument of periapsis true anomaly eccentric anomaly mean anomaly time of pericenter passage angular velocity vector planet radius gravitational potential represents the influence of deviations in the mass density distribution in North-South direction the influence of mass distribution in the North-South and East-West direction latitude longitude atmospheric density atmospheric density at mean sea level drag coefficient frontal satellite area power density of incoming sunlight satellite reflectivity effective surface area to the Sun speed of light

B g1 q m ˙ Isp g0 ρ αi fS fN fW p u

magnetic induction of Earth’s magnetic field geomagnetic parameter electric charge of the satellite mass flow specific impulse gravitational acceleration at the Earth’s surface reference orbit position vector nr i of six Kepler elements force component in the radial direction force component perpendicular to the radius vector in the orbital plane force component perpendicular to the orbital plane semi-latus rectum argument of latitude

Unified State Model Tib q q4 q1 , q2 , q3 (g1 , g2 , g3 ) (f1 , f2 , f3 ) (e1 , e2 , e3 ) C Rf 1 Rf 2 vr ωe M

transformation matrix from frame b to i a quaternion Euler parameter defining the scalar quaternion part Euler parameters defining the vectorial quaternion part planetocentric inertial reference frame unit vectors intermediate reference frame unit vectors rotating body frame unit vectors USM variable related to angular momentum USM variable related to angular and radial momentum USM variable related to angular and radial momentum relative velocity to the atmosphere planetary rate of rotation Mach number

Conventional ODE Integration x˙ Rn h CP ei+1 Φ p

first time derivative Taylor series remainder term of order n stepsize covariance matrix local truncation error at step n state transition matrix order

Interval ODE Integration [x] x ¯ x ¯ w([x]) m([x]) Rn (t) h [ˆ x] [˜ x] x(t) [c] J Aj R Q P [R] A()

interval variable x lowest endpoint of interval [x] highest endpoint of interval [x] width of interval [x] midpoint of interval [x] Taylor series remainder term, local truncation error stepsize a priori interval enclosure guessed interval enclosure true solution interval parameter (constant) Jacobian transformation matrix triangular matrix orthogonal matrix polynomial part of a Taylor Model interval part of a Taylor Model Banach operator

VII

List of Abbreviations AD CA CAM CAS ECIRF ECEF GEO GPS HEO ICAS ICRF ISS ITS IVP LEO MEO ODE PDF RK RPL TM UCS USM US SSN

Automatic Differentiation Conjunction Assessment Collision Avoidance Maneuver Collision Avoidance System Earth Centered Inertial Reference Frame Earth Centered Earth Fixed Geostationary Orbit Global Positioning System High Elliptical Orbit Interval Collision Avoidance System International Celestial Reference Frame International Space Station Interval Taylor Series Initial Value Problem Low Earth Orbit Medium Earth Orbit Ordinary Differential Equation Probability Density Function Runge Kutta Reactor Pumped Laser Taylor Models Union of Concerned Scientists Unified State Model United States Space Surveillance Network

VIII

Chapter 1

Introduction Two active satellites, Iridium 33 and Kosmos-2251, collided with a relative speed of 11.7 km/s at 790 km above Siberia on February 10, 2009. It was the first accidental collision between two intact satellites. Both were completely destroyed and created two large debris clouds orbiting the Earth. Collisions between Earth satellites are rare, but will increase with increasing number of satellites and space debris. A single collision has a large impact on the amount of space debris, because space debris is primarily created by collisions. More space debris in turn creates more collisions, which creates more space debris. In orbits where drag is too small to remove debris faster than it is created, those regions will end up full of very small debris. Affordable space tourism is still fiction, but may finally become reality in the coming decades. With more humans in space, the probability of fatal collisions increases. A Space Collision Avoidance System (CAS) may be necessary to safe lives and keep a safe record for the space tourism industry. Such a CAS is already used for the Space Shuttle and ISS, but is only based on a collision probability estimate. The methods discussed in this survey could lead to a more secure CAS that can rule out collisions. On a larger scale, outside Earth orbits, there are asteroids orbiting the Sun that may collide with Earth. Although a collision between a large asteroid and Earth is very rare, the results of such an impact are disastrous; most life on Earth may be killed, depending on the size and speed of the asteroid. This literature survey is the preparatory work for a research into the feasibility of guaranteed collision avoidance; a secure space traffic control system. It combines and orders all necessary information for the computer implementation of a method that simulates satellite and asteroid orbits in such a way that it is possible to rule out collisions, or change the orbit so that collisions can be ruled out. A complete CAS consists of three components; orbit determination, orbit propagation (simulation), collision determination and avoidance maneuvers. Orbit determination provides the necessary input, position and velocity, for orbit propagation. Orbit propagation predicts future positions of satellites and checks whether they may collide. If a collision may occur, one or both satellites will have to change their orbit by means of an avoidance maneuver. Although all three components are interlinked, this research focusses on the orbit propagation component of the CAS. The problem with conventional orbit propagation methods is that they only yield an approximation; the integration method is not exact. Uncertainties in the modelâ&#x20AC;&#x2122;s parameters and uncertain position and velocity from orbit determination, make the predicted orbit an approximation for which the error is not known. These unknown errors are acceptable for

Chapter 1. Introduction most space applications, but cannot be ignored for guaranteed collision avoidance. To solve this problem, a different type of propagation method is investigated, Interval Integration. This method produces a region around the â&#x20AC;?nominalâ&#x20AC;? orbit that contains all possible errors and uncertainties. This means that the orbit can be guaranteed, by mathematical proof, to be within that region, taking all errors and uncertainties into account (the satellite is assumed to be within a certain region at the moment of orbit determination). The regions around the nominal orbit can be represented by intervals, a set of real numbers that extend single values to a continuous range of values, represented by two numbers, the upper and lower bound. This makes it easy to check whether two interval regions intersect. If they do not intersect, collision is ruled out. The problem with this new interval propagation (integration) method is that the size of the region tends to grow very fast with increasing propagation time. Therefore, a few different orbital models that may reduce this growth will be investigated. One of these models is called the Unified State Model, which has only once been applied after its development in 1972 by Altman [1] . The structure of this survey is as follows. Chapter 2 gives an overview of objects in space, collisions between objects and current ways to prevent these collisions. This chapter provides the context for the other, more mathematical and physical, chapters. Chapter 3 presents the conventional differential equations that model the motion of a satellite. It also presents models for perturbation forces that may act on a satellite. Examples are atmospheric drag and solar radiation pressure. Chapter 4 presents the earlier mentioned Unified State Model and provides all information for its first application. Chapter 5 provides a brief overview of conventional integration (propagation) methods that yield a single line (no region) without errors bounds. Chapter 6 presents the new interval integration methods that produce a region in which the satellite is guaranteed to be. This is the main topic of research and provides all information, including software tools, to apply the method to collision avoidance. The literature survey ends with Chapter 7, an outline and schedule for the actual implementation of all the methods in this survey into a single CAS based on interval integration.

Chapter 2

Collisions in Space The Solar System is mostly empty space, only 0.25% of its mass resides outside the Sun, where collisions between two objects seem very unlikely. However, by looking at planets and their moons, we can see from their surfaces that many collisions have occurred since the creation of the solar system, 4 billion years ago. A recent collision between two man-made communication satellites in Earth orbit notified us about the possibility of satellite collisions, even while there are only about 900 active satellites in the vast space around Earth. Craters on planets and moons, e.g. Mercury and the Moon, are the result of high velocity collisions with meteoroids, asteroids and comets. The surface of Earth was, and still is, bombarded with the same objects. Most craters are not visible any more due to the dynamics of Earthâ&#x20AC;&#x2122;s surface, which removes evidence of impacts. There are, nevertheless, a few large craters on Earth that indicate that we were and still are at risk of high energy collisions with asteroids. The largest recognizable crater on Earth is the Vredefort crater in South Africa with a diameter of 300 km, created 2 billion years ago. A more recent and smaller (1.2 km in diameter) crater is the Barringer crater in Arizona (USA), created 50,000 years ago. Some asteroids are predicted to pass Earth at very close distance in the coming decades. One asteroid that gained a lot of attention from media and the scientific community is Apophis, which is predicted to pass within geosynchronous orbit in 2029 and maybe again in 2036 [2] . The number of active satellites in Earth orbit increased every year since the first satellite, Sputnik 1, was launched in Earth orbit by the Soviet Union in 1957. Most satellite launches do not only bring a satellite into orbit, but also leave part of the rocket in orbit around Earth. At the same time, objects in low Earth orbits do lose energy due to atmospheric drag and burn up in the lower and denser atmosphere after a few months to years. Objects in higher orbits, however, will stay there for hundreds of years or longer. Most satellites are not removed from orbit at the end of their operational life and are left in orbit as inactive satellites; they become space debris. Space debris is all non-active material in Earth orbit. Some satellites or rocket engine components explode or break down, and create more and smaller space debris. This space debris can collide with active satellites, manned missions or other space debris, and causes satellite damage and even more space debris. This chapter gives an overview of the causes and probabilities of collisions in space. The focus is on collisions in Earth orbit, but collisions with Earth are also discussed. The objects that orbit Earth and may cause collisions are discussed in Section 2.1. The history, future and effects of collisions are discussed in Section 2.2. The chapter ends with a discussion about collision avoidance methods in Section 2.3.

2.1. Objects in Space

Chapter 2. Collisions in Space

Elliptical 4%

Other/Unknown 1% Technology Development 4% Space Science 6%

M EO 6%

Reconnaissance 6%

Navigation 8%

LEO 49%

GEO 40%

Communications 62%

Earth Science 14%

Figure 2.1: Left: Active satellite orbit type pie chart. Right: Active satellite type pie chart. Data source: Union of Concerned Scientists [3] .

2.1

Objects in Space

We distinguish three types of objects that orbit Earth: active satellites, manned vehicles, and artificial and natural space debris. There are also three types of objects in orbit around the Sun that may collide with Earth or Earthâ&#x20AC;&#x2122;s satellites; asteroids, comets and meteoroids. A few spacecraft orbit the Sun as well, but they can be seen as asteroids and will not be treated separately in this report. The following subsections give some details about numbers, size and mass of the named objects in space that may cause a collision with Earth, manned missions or active satellites.

2.1.1

Active Satellites

According to the Union of Concerned Scientists (UCS) satellite database [3] , 905 active satellites are orbiting Earth as of January 2009, supporting a wide range of scientific, commercial and military purposes. The first satellite, Sputnik 1, was launched by the Soviet Union in 1957. Over 4,500 additional launches have taken place since the launch of Sputnik. Most satellites are not active any more and have completely burned in the atmosphere, crashed on Earth or are still orbiting Earth as a form of space debris. The 905 cataloged satellites have an average mass of 2,050 kg. 49% have a Low Earth Orbit (LEO), they orbit at an altitude between 0 and 2,000 km. LEO satellites are used for Earth observation, communication and military reconnaissance. 40% have a Geostationary Orbit (GEO). GEO satellites are mainly used for communication. 6% have a Medium Earth Orbit (MEO). MEO satellites are mainly used for global navigation (GPS, Glonass, Galileo) and generally have an altitude around 20,000 km. And 4 % have a Highly Elliptical Orbit (HEO). HEO satellites are used for communication, they can be designed to stay above a point on Earth for a large part of one orbital period, and for science missions. Figure 2.1 shows pie graphs of the orbit type and purpose of satellites currently in orbit. [3] Most satellites have almost circular orbits and can be found in specific altitude regions. This causes the collision probability to be higher in these regions of space. The launch rate of new satellites peaked during the space race between the USA and the Soviet Union, 1957-1975. Although the launch rate declined somewhat after the space race, it did not continue to decline. An average of 61 satellites were launched every year over the

2.1. Objects in Space

Chapter 2. Collisions in Space

Othe r 2% Other 153

Civil 6%

ESA 11 Luxe m bourg 14 Canada 15 Germ any 15 India 19

USA 432

M ilitary 24%

Com m ercial 42%

United Kingdom 21 Japan 36 M ultinational 48 China (PR) 51 Russ ia 90

Gove rnm e nt 25%

Figure 2.2: Left: Satellite distribution per country. Right: Satellite owner distribution. Data source: Union of Concerned Scientists [3] . past 10 years [3] . The distribution of satellites per country is shown in Figure 2.1. The number of active satellites in Earth orbit depends on the launch rate and the lifetime of satellites. The lifetime of satellites may slightly increase due to improved and matured technology, but is probably not going to change very much in the coming 10-50 years. The number of new satellites mainly depends on the cost of launch and the demand for science, commercial and military satellites. The cost of launch is dropping steadily, although no great breakthroughs in launchers have yet made space accessible for ordinary people. The launch price per kg to LEO is between 3,000 and 15,000 USD. While the average satellite is rather heavy, the launch of much smaller and lighter satellites has gained popularity in recent years. An example are the student satellites launched by Indiaâ&#x20AC;&#x2122;s PSLV launcher. These 1 kg, 1000 cm3 mini-satellites are launched together in one rocket. Sensors and electronics continue to decrease in size while performance is often increased. This, combined with the relatively cheap launch, makes satellites possible for smaller groups of people or communities like students, space enthusiasts, earth scientists and other amateur or professional scientist. Once electronics become even smaller and lighter and launch becomes cheaper and cheaper, an increase in launch of small satellites can be expected. However, to create a true revolution in satellite launch rate, the cost has to drop to a few hundred dollars per kg.

2.1.2

Manned Missions

Only 4 years after the launch of the first satellite, the Soviet Union brought the first human in Earth orbit in 1961. The USA followed only a year later. Humans have been in space every year since 1961. The first man set foot on the Moon in 1969 and the Soviet Union launched the first Space Station, the Salyut 1, in 1971. The USA followed with Skylab in 1973. China became the third country that put a human in space in 2003. India, Japan and Europe (ESA) have plans to develop their own facilities to train, launch and keep humans in space. All humans who went to space were part of a governmental prestige or science mission, but this is likely to change in the coming decade. Commercial space tourism spacecraft are in development by several companies. [4]

2.1. Objects in Space

Chapter 2. Collisions in Space

63 59

number of humans in space

48 48

22 19

16 13 8

5 2

3 3

10 7

10 6 6

9 8

62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 2000 1961 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

year

Figure 2.3: Number of humans in space since the launch of the first satellite. Data source: Verger et al [4] . Prestige, Science and Technology Missions The first efforts to put man in space were prestige projects in a space race between the USA and the Soviet Union. Later, manned space flight focused more on science and technology. The USA had only one space station before the International Space Station (ISS) while Soviet Union had 8 different space stations; Salyut 1-7 and Mir. The USA, however, has the reusable Space Shuttle, which it often uses for short manned space flights. The only space station currently in orbit is the ISS, which is used by many nation for scientific research. The ISS is not yet finished and was only able to house 3 crew members until May 2009, after which 6 crew members permanently work in the ISS. Humans have almost permanently been in space since the launch of the Mir, although always in small numbers. Different countries have plans for human science and technology missions in Earth orbit, on the Moon or even Mars. There is no reason to expect a great increase in the number of prestige, science and technology missions. If an increase of humans in space would happen, it will probably be the result of space tourism. Space Tourism Only 6 people have made paid space flights, all of them with the Soyuz spacecraft to the ISS. They paid over $20 million for their 10 day visit to the ISS. Although this can be seen as space tourism, it is not supported by a commercial launchers, spacecraft and space station and requires long and extensive training and actual work in the ISS. However, there are fully commercial launchers, spacecraft and space stations in development. The first space tourism will be short trips straight up to a high altitude and down again, orbital flights require much

2.1. Objects in Space

Chapter 2. Collisions in Space

Figure 2.4: Space Stations. Top left: Salyut 7. Bottom left: Mir. Top right: Skylab. Bottom right: the International Space Station. Image source: Wikimedia Commons [5] .

2.1. Objects in Space

Chapter 2. Collisions in Space

more propellant and a way to slow down during re-entry, which is more difficult. [6] The Ansari X Prize, started in 1996, can be seen as the unofficial start of the private space tourism industry. ”The Ansari X Prize was a space competition in which the X PRIZE Foundation offered a $10 million prize for the first non-government organization to launch a reusable manned spacecraft into space twice within two weeks. The prize was won on October 4, 2004, by the Tier One, using the experimental spaceplane SpaceShipOne. $10 million was awarded to the winner, but more than $100 million was invested in new technologies in pursuit of the prize.” [7] SpaceShipTwo is a suborbital spacecraft under development and based on SpaceShipOne. It will bring 2 pilots and 6 passengers to 110 km and then directly back to Earth. The first 100 tickets are sold for $200,000, but prices are predicted to drop to $20,000 after a few successful years. According to Virgin Galactic, the owner of SpaceShipOne, more than 65,000 people have registered their interest and 200 seats were already reserved by making a payment deposit in 2006. EADS Astrium in Europe [8] , XCOR Aerospace [9] and other companies are developing similar suborbital services and will compete with SpaceShipTwo in the coming decade. Orbital flights require more complex, expensive and larger launchers and spacecraft, but are the next logical step after suborbital space tourism. SpaceX is a private space company which develops their own rockets for orbital payloads. Their largest rocket, the Falcon 9, is scheduled to launch in 2009 and can bring between 9,900 kg and 27,500 kg to LEO for a price starting at $27 million [10] . A possible destination for a manned capsule launched by the Falcon 9, is a space hotel in development by Bigelow Aerospace. Two prototype inflatable space hotels, Genesis I and II [11] , are already orbiting Earth. These examples show that serious efforts and progress is made to make space tourism reality. Initially, space tourism will be expensive, but scaling, competition and mass production will likely make space tourism cheaper if there is a demand for these expensive flights. The high demand for expensive suborbital flights shows a promising demand for orbital space vacations. High demand and recent developments suggest that the space tourism industry may experience rapid evolution in the coming decades. While we cannot predict the future, it is certain that the safety of launch, re-entry and stay in space will be very important to keep demand up. Collision of a space hotel with space debris, active satellites or other hotels will not only kill people, but will also be a blow for space tourism and the private space industry. A demand for a collision avoidance system will therefore not only come from public demand for safety and governmental regulations, but also from the private space industry itself. As a drawback, the private space race will create a lot of new space debris. Space tourism greatly increases the number of launches and will therefore also create more space debris coming from launch vehicles. Hotels and spacecraft need to be tested and will probably fail, explode or break up during tests. To be competitive, private companies have to try new and cheaper technology.

2.1.3

Space Debris

Space debris, also called orbital debris, is human-generated and orbits Earth. All nonfunctional objects in Earth orbit are considered to be debris. ”This debris can be anything from a piece of paint that has flaked off of a rocket or a spacecraft, to fragments of an exploded rocket upper stage, or an entire derelict spacecraft” [13] .

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Figure 2.5: Space Tourism. Top left: SpaceShipOne. Bottom left: Falcon 9 launcher. Top right: EADS spaceplane (artistic render). Bottom right: Genesis II. Image source: Wikimedia Commons [12] .

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Figure 2.6: Number of objects orbiting Earth versus time. Image source: Leleux [16] .

Quantity, Orbit and Size The largest catalog of space debris is compiled by the U.S. Space Surveillance Network (US SSN). â&#x20AC;?As a rule of thumb, low altitude cataloged debris are assessed to be larger than 10 cm in diameter. At higher altitudes objects less than 1 m in diameter may be undetectable. Individual object sensitivities may vary dramatically from this simple generalization.â&#x20AC;? [14] . The US SSN catalog contained orbit element information of over 12,146 Earth orbiting objects in 2008 [14] [15] , most larger than 5-10 cm. Figure 2.6 shows the number of cataloged debris from 1957 to 2000. Small particles come in higher numbers; there are approximately 350,000 objects larger than 1 cm and 300 million particles larger than 1 mm [15] . Most debris is orbiting in LEO or at GEO as a result of fragmentation of satellites in those orbits. The space debris density as function of altitude is shown in Figure 2.7. This graph is made with a 2006 catalog and does not yet show a new peak at 890 km due to the intentional destruction of a Fengyun 1C derelict weather spacecraft [14] . In addition, Figures 2.8 and 2.9 show the 3D distribution of cataloged space debris in Earth orbit. The debris and meteoroid flux as function of particle size in ISS orbit can be seen in Figure 2.10 (meteoroids are discussed in Section 2.1.4). Origin 4500 space missions have flown since 1957. Only 11 out of those 4500 missions are responsible for 32% of all cataloged space debris. Moreover, the source of 9 of these 11 fragmentations were discarded rocket bodies that had operated as designed, but broke up later. The primary factors affecting the growth of the satellite population are the international space launch rate, satellite fragmentation and solar activity. [14] Almost half of the cataloged satellite/debris population was determined to be fragmentation debris, as can be seen in Figure 2.11. A total of 194 satellite fragmentations and 51 anomalous events have been identified since 1957. The primary cause of satellite breakups are propulsion related events and collisions, mostly deliberate, with other objects in Earth

2.1. Objects in Space

Chapter 2. Collisions in Space

Figure 2.7: Object density around Earth as function of altitude. Image source: Rossi [17] .

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Figure 2.8: Space debris distribution around Earth as seen from a polar view. Outer band is GEO. Image source: NASA [18] .

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Figure 2.9: Space debris distribution around Earth as seen from a 3D perspective and a close-up of LEO. Image source: NASA [18] .

Figure 2.10: Debris and meteoroids flux against diameter in the ISS orbit according to different ESA and NASA models. Image source: Fukushige [15] .

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unknown 2%

mission related debris 12%

rocket bodies 13% breakup 48%

payloads 25%

Figure 2.11: Causes of space debris. Data source: Johnson [14] .

orbit. Payloads (the actual satellites) make up 25% of all large debris. It should be noted that almost three out of every four payloads are no longer operational. The two other causes of debris are rocket bodies and mission related debris. Mission related debris results from the intentional release of objects, usually small numbers, during normal on-orbit operations. [14] According to Rossi â&#x20AC;?collisions are going to become the most important source of debris in a not too distant future.â&#x20AC;? [17] Evolution of Debris Clouds Most space debris is the result of satellite fragmentation. When a satellite explodes or collides, it creates a debris cloud of hundreds to thousands of fragments. These fragments do not stay close together in orbit, as the simulation results in Figure 2.12 show; the debris spreads out along the orbit and in the long run over a shell around Earth. This is the result of different orbital velocities after fragmentation and collisions between fragments. If the debris density in a certain orbit altitude region becomes large enough, those regions of space can become supercritical. This means that collisions between objects are frequent enough to create more additional debris than atmospheric drag removes from the region. The new debris increase the collision rate, which in turn creates more debris. This chain reaction is a runaway reaction, similar to the reaction in nuclear weapons, above a certain supercritical density. On the other hand, while the number of debris particles increases, the size of the particles decreases. [19] A study by NASAâ&#x20AC;&#x2122;s orbital debris program office in 2006, before the Chinese collisions test, showed that parts of space have already reached supercritical debris densities. For example, in the heavily used altitude band from 900-1000 km, the number of debris fragments larger than 10 cm is expected to more than triple in the next 200 years, even if no additional objects are launched into that region. The same study estimates that the large debris population in LEO will increase by 40% in 200 years, again assuming no new launches into LEO. [19] [20] [21] If the current intact satellite population is allowed to increase in the high density regions, the rate of collisions can increase significantly. In practice, after some period of time,

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Chapter 2. Collisions in Space

Figure 2.12: Simulated evolution of debris larger than 10 cm, following the breakup of a satellite after collision. Image source: Wright, 2007 [19] .

the intact population would be difficult to maintain because the region would become too hazardous to continue space operations in LEO. [20] Future Even if we stop launching new satellites, the orbital debris population in the most popular LEO regions will triple in 200 years. In reality, with the launch of new satellites, the situation will be worse. Serious effort is required to prevent a fast runaway debris reaction that will make LEO and GEO unusable for satellites and manned missions. Even launch to higher altitudes may be in danger when there is too much debris in LEO. Satellite fragmentation has to be minimized by reducing explosion risk, de-orbiting a satellite after operation and avoiding collisions with other objects. Even in the optimistic case when new satellites do not add to the current debris population, debris and collisions will increase and may never reach an equilibrium above 800 km. Moreover, the NASA breakup model shows that the catastrophic breakup of a single satellite of 5-10 tons would roughly double the amount of debris larger than 1 mm currently in LEO [19] . Finally, a pessimistic scenario is a war in space; nations destroy their enemyâ&#x20AC;&#x2122;s communication, navigation and reconnaissance satellites which creates enormous amounts of new debris. Such a space war may make space almost inaccessible in a very short time and should be avoided at all cost.

2.1.4

Asteroids, Comets and Meteoroids

Asteroids, Comets and Meteoroids are all small solar system bodies, smaller than planets, in orbit around the Sun. Some asteroids, comets and meteoroids have orbits that cross Earthâ&#x20AC;&#x2122;s orbit or come close to Earth, they are called Near-Earth objects [22] . It is possible that these bodies collide with Earth or satellites in Earth orbit. Asteroids â&#x20AC;?Asteroids, sometimes called minor planets or planetoids, are small Solar System bodies in orbit around the Sun, especially in the inner Solar System; they are smaller than planets

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Figure 2.13: Lifetime of orbital debris for different LEO altitudes for (a) maximum solar activity and (b) for minimum solar activity. Image source: Wright [19] .

but larger than meteoroids.” [24] Most asteroids orbit within the main asteroid belt between Mars and Jupiter, see Figure 2.14. This belt contains an estimated 1.2-2 million asteroids larger than 1 km in diameter and millions of smaller ones [25] . The total mass of all asteroids is estimated to be 4% of the mass of the Moon. Ceres is the largest asteroid and takes up 32% of all asteroid mass. About 6000 asteroids larger than 50 m are known to have a near-Earth orbit, that crosses or comes close (< 0.3 AU) to Earth’s orbit. The largest is Ganymed with a diameter of 32 km. There are more than 500-1000 near-Earth asteroids with a diameter larger than 1 km. Near Earth asteroids survive for a few million years. They die by crashing into the Sun, Planets or are ejected from the solar system due to fly-bys of planets. New asteroids are constantly moved into near-Earth orbits. It is believed that this happens through orbital perturbation of Jupiter that have a resonating effect for some asteroids. It is not known whether the current near-Earth asteroid population is stable, increasing or decreasing. [26] [22] Comets ”A comet is a Small Solar System Body (neither planets nor dwarf planets) that orbits the Sun. Comet nuclei are themselves loose collections of ice, dust and small rocky particles, ranging from a few kilometers to tens of kilometers across. Comets are distinguished from asteroids by the presence of a coma or tail.” [27] Comets have a wide variety of different orbits that can be put in two groups; short (up to 200 years) and long period comets. Short period comets orbit in the Kuiper Belt, beyond Neptune. Long period comets are believed to come from the hypothetical Oort Cloud, 2,00050,000 AU away from the Sun. They are sometimes inserted into the inner-planet region by perturbations, fly-by’s or collisions with foother comets. Near-Earth comets are rare compared to near-Earth asteroids. Only 82 near-Earth comets had been discovered as of December 2008. Most near-Earth comets end by falling into the Sun or other Planets. Collisions with comets did occur in the Solar System. Some craters on the Moon, for example, are thought to be a result of comet impact. Comets impacts on Earth are a popular explanation for the origin of water on Earth. [26]

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Figure 2.14: Plot of all known asteroids for 2006 May 3, within 6 AU from the Sun. The dense disk is the â&#x20AC;?main beltâ&#x20AC;? between Jupiter and Mars, and the two blobs to the bottom and left are the Trojans in the Langrage points of Jupiter. Image source: Murray and Dermott, 2000 [23]

2.2. Collisions in Earth Orbit

Chapter 2. Collisions in Space

Meteoroids ”A meteoroid is defined by the International Astronomical Union as ”a solid object moving in interplanetary space, of a size considerably smaller than an asteroid and considerably larger than an atom.” [28] Meteoroids are small asteroids or comets with basically the same origin, composition and orbits. They have a low mass but come high in numbers; millions of meteoroids are vaporized in Earth’s atmosphere each day. A typical encounter with Earth’s atmosphere occurs at speeds of 15 km/s for asteroidal and 30 km/s for cometal meteoroids, but top speeds reach 73 km/s. At these speeds, they have enough kinetic energy per unit mass to completely vaporize when slowed down. Most of them vaporize in Earth’s atmosphere, but some make it to the ground. Meteoroids that hit earth at hypervelocity (> 3000 m/s) are typically larger than 30 m before encounter with Earth’s atmosphere and are rare. Although most meteoroids pose no threat to Earth’s surface, satellites operate at altitudes where meteoroids have not yet vaporized. [22] [26]

2.2

Collisions in Earth Orbit

Collisions in space have already happened and will happen again. Different types of collisions are possible; debris or meteoroids can collide with other debris and with active satellites. Two satellites can collide as well. This Section gives a short overview of known historic collisions, the result of a collision and the predicted future of collisions in space.

2.2.1

History of Collisions

Collisions in space are only identified if the generated debris cloud is large enough to be tracked or when active satellites fail and investigation of the failure is possible. This means that collisions between small debris particles cannot be noticed or studied and that collisions with satellites may cause satellite failure while we do not know this cause. Collisions can also be the result of deliberate actions; testing debris cloud evolution or, more realistic, an anti-satellite weapon test or real space war. Deliberate Collisions The USA, Russia and China all successfully conducted anti satellite tests. Space weapon development and testing has so far not been done by other space exploring nations. The USA conducted two deliberate satellite collisions. One as an anti satellite weapon test and one supposed safety destruction of a satellite that would fall back to Earth. The anti satellite weapon test was conducted in 1985 by an ASM-135 anti satellite missile, launched from a F-15A making a steep climb at Mach 0.9. The target, the 907 kg Solwind P78-1 satellite flying at 555 km, was successfully destroyed [30] . The second deliberate collision by the USA was the destruction of an American military spy satellite, named USA 193. This satellite had to be destroyed because it would return to Earth with dangerous hydrazine in its fuel tank and contained spy satellite technology. A SM-3 missile was fired from the Ticonderoga class missile cruiser USS Lake Erie, and intercepted USA 193 about 247 km above the Pacific Ocean. The Department of Defense expressed a ”high degree of confidence” that the fuel tank was hit and destroyed. The satellite’s remnants were expected to burn up over the course of the next 40 days, with most of the satellite’s mass re-entering the atmosphere 24 to 48 hours after the missile strike.” [31]

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Figure 2.15: Left: Debris cloud just 5 minutes after the deliberate destruction of a defunct Chinese weather satellite. Right: Chinese weather satellite debris cloud (in red) after 6 months. Image source: Kelso [29] .

Russia conducted most anti satellite weapon tests, all during the cold war. Its anti satellite system is a co-orbital system in which a missile armed with conventional explosives is launched into the same orbit as the target. It then makes a close approach on the target in one or two orbits and detonates its explosives to destroy the target with shrapnel fragments from the explosion. It is thought to be effective when detonated within a kilometer of the target. Russia performed 20 launches, seven interceptions and five actual detonations. The tests were successful enough to make the system operational in 1972. How much debris was created and whether this debris is still in orbit is unknown. [32] China conducted a direct-ascent anti satellite test against one of their own defunct weather satellites, the Fengyun 1C, in 2007. The test produced over 2000 pieces of trackable debris and NASA Orbital Debris Program Office estimated it generated over 35,000 pieces of debris down to 1 cm in size. Figure 2.15 shows the evolution of the debris cloud created by the Chinese satellite test. [29] Accidental Collisions Only a few large accidental collisions have been detected in space since the launch of Sputnik. One between two intact satellites, two between an operational satellite and large space debris and one between two rocket bodies. Collisions between inactive satellites are hard to detect and can only be reconstructed from observed debris. Smaller collisions occur more often, but are also more difficult to detect. However, the Space Shuttle provides good evidence for smaller collisions since it returns to Earth and can therefore be inspected. The largest and most notable collision in Earth orbit was the collision between two intact satellites, the operational American communication satellite, Iridium 33, and the inactive Russian navigation satellite, Cosmos 2251, in 2009. Both satellites were completely destroyed. The incident was observed by the U.S. SSN, above Siberia at an altitude of 790 km, which later tracked two large clouds of debris. [33] The operational French satellite Cerise was hit, at a relative velocity of 14.77 km/s by a fragment of about 10 cm2 , coming from the 1986 explosion of an Ariane 1 rocket upper

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stage. The debris hit Cerise’s tether and left it tumbling and uncontrolled. [34] [17] [35] In 2005, a 31 year old American rocket body was hit by a fragment from the third stage of a Chinese CZ-4 launch vehicle, which had exploded in March 2000. The impact happened in a 885 km high circular orbit above the southern polar region. Three large debris pieces were cataloged from the impact. [17] Another collision in LEO was discovered while analysing old tracking data. A Russian inactive navigation satellite, Cosmos 1934, was hit by a piece of debris from the similar Cosmos 926 in 1991. [17] Collisions between operational spacecraft and small (< 1 cm) particles occur more often since there are much more small particles in orbit around Earth (space debris) and in orbit around the Sun (meteoroids). For example, the windows in the Skylab Apollo Command Modules were found in 1974 to have been struck by small hypervelocity particles, the orbital or meteoroid origin of some of the craters was not realized until 1980 [36] . The first Space Shuttle mission in 1983 confirmed the frequent impact of small particles, when three days into the mission an astronaut noticed a small pit in one of the windows of the crew cabin. Spectrographic analysis of the residue left in this tiny pit revealed the presence of titanium and aluminium, suggesting that the orbiter had been hit by a chip of paint that had flaked off of some unknown spacecraft or rocket body. This was one of the first indications that orbital debris might pose a hazard to the Space Shuttle. Window impacts had increased dramatically by 1995 and the debris risk caused future Shuttle missions to change. Not only the windows were hit by small particles, but the entire Space Shuttle. In 1995 and 1996, for example, significant impacts occurred on the Space Shuttle’s payload bay door and rudder speed brakes, as well as on the tethered satellite pallet. Space Shuttle external surfaces have experienced impacts from particles on every shuttle mission. The outer thermal panes of the crew cabin windows have sustained one or more impact pits greater than 0.25 mm in diameter on most flights. [13]

2.2.2

Collision Effects

Collisions in space are extremely energetic due to the high velocity of meteoroids, debris and satellites. The velocity of objects in circular LEO is 7.8 km/s, the velocity at GEO is 3.1 km/s and average mass of satellites is 2000 kg. Collisions between solid objects at such high speeds are difficult to model and expensive to test. Little detailed collision information is gathered from real high velocity collisions in space because one can only observe the large debris pieces that result from the collision. However, high impact models are made and actual tests are done. These tests and models give us the best predictions of a high speed collision. An object striking a spacecraft at 10 km/s can cause several types of damage. Impacts can crater or perforate surfaces, create holes and cracks, or cause the back surfaces of walls to fracture, sending material from the into the spacecraft’s internal structure. If an object penetrates the wall of a spacecraft, its fragments will travel into the spacecraft and deposit over an area significantly larger than the impact hole, as can bee seen in Figure 2.16. [37] [13] . To highlight the danger posed by smaller particles, it should be kept in mind that the average impact velocity in LEO is about 8-10 km/s. This means that a 1 mm particle is able to penetrate the reinforced carbon tiles on the Space Shuttle wing’s leading edges; this event may cause a loss of the Shuttle during re-entry. A particle of 5 mm is able to penetrate the Shuttle cabin. [17] Shielding is possible against debris sizes of up to 1 cm. Shielding against larger debris is almost impossible and would require excessive shielding mass. [35]

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Chapter 2. Collisions in Space

Figure 2.16: High speed images of an aluminium sphere (3.18 mm diameter) impacting a 1 mm thick aluminium plate with a speed of 5.00 km/s. Image source: Chi et al [37] .

According to the NASA model, a collision between a large object and a smaller one will be catastrophic if the ratio of the impact kinetic energy of the smaller object to the mass of the larger object is greater than 40,000 J/kg. That condition implies that an interceptor of 20 kg striking a large satellite at 7.5 km/s could completely fragment a satellite with a mass up to about 14 tons. [19] When the shuttle is in a 51.6 degree inclination 400 kilometer altitude orbit, NASAâ&#x20AC;&#x2122;s model of the debris environment predicts an average collision velocity of 9 km/s for orbital debris with a diameter of 1 cm or more. [13] . Astronauts are also at risk from meteoroids and orbital debris during space walks. The most vulnerable parts of their space suit are the soft areas; the arms, gloves and lower torso. The secondary oxygen pack of the space suite is made to provide astronauts with a 30 minute supply of oxygen in case of a 4 mm puncture in the suit. This should be sufficient time for an injured astronaut to be assisted back to the pressurized spacecraft. NASA estimates that a 2 mm particle could cause a 4 mm hole, and a 0.1 mm particle or larger could cause a puncture. [13]

2.2.3

Future of Collisions

The number of future space collisions greatly depends on the number of satellite launches, collision mitigation policies and possible deliberate satellite destructions (anti satellite weapons test or real attacks). Space is mainly used for scientific, communication, navigation and reconnaissance satellites today. An great increase in launches, spacecraft and satellites can be expected when space tourism becomes affordable for a larger group of people. When the current rate of launches is used for collision predictions, 60 catastrophic collisions will kill operational satellites in the next 100 years, according to Monte Carlo simulations done by A. Rossi, 2006 [17] (see Figure 2.18). The collision rate will increase and will continue to increase after 2100 for this scenario. New space debris from large collisions is not taken into account in this model. Since one large collision can double the amount of space debris, this is a large inaccuracy in the model; the actual number of collisions will probably be higher in the business-as-usual scenario. The number of collisions in the coming century will greatly increase if space tourism becomes reality and no good collision avoidance systems are active. This may not only endanger expensive satellites, but also human lives. Space tourism will start with sub-orbital

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Chapter 2. Collisions in Space

Kinetic Energy (J)

15000

10 2000

1500

10000 1000

5000

500

0 Speed (m/s)

Mass (kg)

Figure 2.17: Kinetic energy as function of mass and speed.

flights in the coming years. These sub-orbital flights are also at collision risk, because they do enter space at altitudes where debris and active satellites orbit. Once orbital space tourism becomes affordable for a few thousand Earth inhabitants, many shuttle launches to LEO space hotels may greatly increase the debris created by launcher upper stages, depending on their design. New and cheaper techniques have to be tested to be competitive in a global space tourism market, without strict regulations this will almost certainly lead to a significant increase in space debris.

2.3

Collision Avoidance Systems

Collision Avoidance Systems (CAS) are systems that try to prevent collisions of operational spacecraft in space. They generally consist of orbit determination, orbit prediction and maneuver strategies. Orbit determination is needed to determine the location and velocity of objects in Earth orbit. With these position and velocity data, a prediction of future positions of the objects can be made by orbit prediction models. When a probable collision is noticed from the prediction, a third system has to determine if a maneuver is possible and which spacecraft will have to perform the maneuver. It should also come up with a optimal avoidance maneuver. These three major components have to work together quickly. More accurate observations will be required when a probable collision is noticed, while the maneuver strategy will need the orbit prediction system to check whether the maneuver is safe and does not increase risk of future predictions. An indirect way of collision avoidance is done by carefully planning satellite orbits and removing space debris. CASs are needed to prevent loss of operational satellites, spacecraft and human lives, as well as to prevent future space debris that will increase the future collision rate and may

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Chapter 2. Collisions in Space

Figure 2.18: Cumulative number catastrophic collisions for (sold line) a ”business-asusual” and (dashed line) mitigation scenario, coming from Monte Carlo simulations. Image source: Rossi [17] .

even make space unusable due to a runaway collision reaction in parts of space that reach supercritical density (Section 2.1.3). Current CAS are almost only used for manned spaceflight with the Space Shuttle, and for the ISS. No well integrated single collision avoidance system is currently available, however, collisions and collision avoidance gained more attention since the recent collision between Cosmos and Iridium (Section 2.2.1). The three main components of a CAS; orbit determination, collision prediction and maneuver strategies; are discussed in the following subsections.

2.3.1

Orbit Determination

”Orbit determination of an artificial satellite requires as input measurements that are related to the satellite’s position or velocity. These data are collected by a satellite tracking system that measures the properties of electromagnetic wave propagation between the transmitter and the receiver. The transmitter as well as the receiver may either be a ground station or a satellite.” [38] Position and velocity observations are made using radar, laser or Global Positioning System (GPS). Only operational satellites can use GPS for position determination, laser tracking requires retro reflectors on the satellite and radar or optical observations can be used for all objects large enough to be detected. The US SSN uses 25 ground based radar and optical sensors to track space objects. It has the largest and most complete database of large space objects. Most objects with a diameter larger than 10 cm have been cataloged. Other space exploring nations like Japan and Europe also have tracking sensors and cooperate with the US SSN and use the US SSN catalog to prevent collisions of their operational satellites. [39] [40]

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The position of most operational satellites can be determined very accurately, in the meter or even centimeter range, if it has a laser reflector, GPS receiver or radar transponder. However, precise orbit determination of satellites or space debris without a specific orbit determination system on board is difficult. NASA has requested that objects down to 1 cm have to be tracked as part of efforts to protect the International Space Station. Should this be done, the catalog of man-made objects in Earth orbit would grow from its current set of 10,000 objects to well over 100,000. [40] â&#x20AC;?In addition to the detection sensitivity limits of the SSN, the quality of tracking data, atmospheric drag, and the effects of gravitational and solar radiation pressure perturbations all combine to degrade the effectiveness of the conjunction assessment (CA) process.â&#x20AC;? [41]

2.3.2

Collision Prediction

Collision prediction of cataloged objects is based on the orbit prediction of tracked space objects and calculating distances of closest approach or collision probability. Orbit predictions are made using force models and analytical or numerical integration of the satellite trajectory. Analytical methods are fast, but are based on the simplified two body problem and cannot take other forces into account. Numerical integration is computationally more intensive, but generates more accurate orbit predictions. The dynamics and integration methods for trajectory simulation are treated in Chapter 3. Flux or collision probability models are used for smaller debris and meteoroids that are currently not cataloged. These models can only help during spacecraft structural and orbit design, and cannot predict specific collisions. Collision avoidance maneuvers are therefore not performed for small debris and meteoroids. Both ESA and NASA have developed meteoroid and small debris flux models; MASTER2005 and ORDEM2000 [15] (see Figure 2.9). Currently, no global collision prediction system that checks for possible collisions of all operational satellites with tracked space debris or with other operational satellites. Manned Space Shuttle and ISS missions do use the US SSN catalog and orbit prediction methods to warn for possible collisions [41] . ESA also uses the US SSN catalog to check for possible collisions with some of its operational satellites like Envisat and ERS-2 [35] . The initial conditions (position and velocity) for the numerical or analytical orbit prediction come from orbit determination methods. Different initial conditions result in a different predicted trajectory, accurate orbit determination and uncertainty propagation are therefore extremely important for the quality and usability of the predicted orbit. A Monte Carlo (random brute force technique) simulation or covariance propagation can be used to analyse the effects of uncertainties in the initial conditions. The Space Shuttle used a Monte Carlo approach before 2001, but implemented a covariance propagation method, already in use by the ISS since 2001. The old method was based on an alert box around the Space Shuttle while the new method is based on collision probability. [16] [35]

2.3.3

Maneuver Strategies

Once an estimate of the collision probability is made, a decision has to be made about possible maneuvers. The estimated probability risk may be too low for a maneuver. But when the decision is made to make a Collision Avoidance Maneuver (CAM); an efficient, effective and safe maneuver has to be designed and performed by the spacecraft.

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Chapter 2. Collisions in Space

Figure 2.19: Velocity increment required to lower collision probability below 1 per million for a GEO satellite. Image source: Ailor [42] .

Satellites Not all satellites have a propulsion system to perform CAM, for example, the Hubble Space Telescope, hundreds of Russian Parus and Tsikada navigation and Strella communications satellites, and virtually all the new small satellites (minisats, microsats, nanosats and picosats) [41] . Most satellites, however, have a propulsion system for small orbit corrections that could also perform CAM. Only a few science satellites actually try to prevent collisions by using the US SSN catalog to perform collision predictions [35] . The operators of the Iridium constellation said, after the fatal collision of one of the communication satellites, that there are hundreds of close approaches each day and that itâ&#x20AC;&#x2122;s not worth making maneuvers based on the uncertain collision predictions. Communication constellation satellites are less expensive and often redundant, this in contrast with expensive single science satellites for which collision avoidance is of higher priority. CAM are most fuel efficient when performed long before a possible collision, as can be seen in Figure 2.19. The sinusoidal shape of the graph is the result of the effect of a thrust force on the orbit. An impulsive force at one point will cause the largest differences in the original orbit at 180 degrees from the propulsion position. Since GEO satellites have a period of one day, the maneuver has the largest effect 0.5 (or 1.5, 2.5, etc.) days in advance. A collision maneuver at least 1.5 days before collision is ideal from a fuel perspective. The accuracy of the observation and prediction can, however, be increased when closer to time of collision. A higher accuracy may mean that a maneuver is not required any more, which also saves fuel. In practice it is therefore better to wait for more accurate observations and predictions and maneuver only a few hours before a possible collision. [42] Maneuvers will generally be performed by a small thrust in the along track direction, because it is the most efficient way to change the orbit enough to avoid collision. The current collision avoidance systems were build and are mainly used for manned missions (Space Shuttle and ISS). Both the Space Shuttle and ISS have made CAM in the

2.3. Collision Avoidance Systems

Chapter 2. Collisions in Space

past and will do so in the future. Normal satellites will perform avoidance maneuvers in the future when more accurate collision prediction systems are available or when collision risk becomes higher due to more objects in space. Manned Missions Current CAM are designed to prevent collisions with the ISS and Space Shuttle. It is difficult to place a monetary value on human life, but even if we do not take the loss of life into account, it is very important for future manned missions that no accidents with humans occur in space, and notably for two reasons. First, the general public (the taxpayer who finances science missions) does not allow frequent casualties, even while astronauts know the risk and are willing to take that risk. Second, it is important for commercial space tourism to have a reliable and safe record. During realtime Shuttle operations, U.S. Space Command (USSPACECOM) screens the most current satellite catalog every 8 hours to determine if there may be a close approach between any object and the Shuttle within the following 36 hours [16] . In addition, launchers in the United States are currently required to assure that their launch will not get too close to the Space Shuttle and International Space Station [40] . The ISS program requires the space station to conduct a CAM whenever the estimated probability of collision exceeds 1 in 10000, unless such a maneuver would impact primary missions or payload objectives. Whenever the estimated probability of collision exceeds 1 in 1000 an avoidance burn is performed unless it would result in Shuttle re-flight, hardware damage, additional space walks or additional risk for crew or vehicle. Typically, the ISS conducts a CAM once or twice a year. [41] [16]

2.3.4

Space Debris Removal

Besides avoiding collisions with space debris, we can also remove the space debris from space. This means that the space debris has to be tracked and then vaporized (in the atmosphere or by active heating). There is currently no technically feasible and economically viable method that can remove a large part of the debris [43] [41] . A few options that have been researched for feasibility are ground and space based laser, tether drag, tether catch, large foamy balls and micro-satellites. A ground based laser can change the trajectory of space debris by hitting the debris with the laser. This laser has to be very powerful, because of the great distance and atmospheric interference. Monroe calculated in 1993 that to de-orbit 1 kg debris from 500 km altitude, one needs a 5 MW reactor-pumped laser (RPL) with a 10 m diameter beam director and adaptive optics [44] . Space based lasers do not need to be as massive as ground based lasers, because they can get closer to their target and do not need to penetrate an atmosphere. Space based lasers, however, need to be low mass and low power. Small debris, 100 g, can be completely vaporized from a distance of 70 km with a 100 kW pulsed laser. Larger debris can be deflected. [45] Tethers can be attached to debris and inactive satellites to generate electrodynamic drag. This drag lowers the orbit and ensures that the debris is burned in the atmosphere. A special spacecraft will need to rendez-vous with the debris to attach the tether. The tether itself is a few kilometers long and causes a temporarily increase probability of collision, especially with other tethered systems. [46] [47] A tether can also be used to power a small satellite that gathers debris and brings it, propelled by an ion engine, to a lower orbit where aerodynamic drag lowers the orbit even

2.3. Collision Avoidance Systems

Chapter 2. Collisions in Space

further until it vaporizes in Earthâ&#x20AC;&#x2122;s atmosphere [48] . This concept can also use the tether to actually grab the debris, so that it does not need very large orbit changes to get close to the debris [41] . JAXA studied the concept of a micro-satellite that acts as a vacuum cleaner in space. Small satellites fly an efficient route to gather debris, which is then brought to a lower or higher orbit [49] . Other less feasible concepts include large foamy balls, up to 2 km across, which would decelerate small particles enough when they travel through the ball so that they fall back to Earth when they exit the ball. The problem with the ball is that the ball itself will easily fall back to Earth and will also collide with active satellites. [43]

Chapter 3

Astrodynamics Astrodynamics (or Celestial Mechanics) is the application of Newtonian mechanics to the motion of celestial bodies (i.e. satellites and planets) under the force of gravity and possible perturbing forces. To model the trajectory of a celestial body, we have to know the forces and direction of these forces that act on the satellite at every position along the trajectory. With the use of Newtonâ&#x20AC;&#x2122;s laws we can model the trajectory with differential equations; the equations of motion. For efficient and accurate simulation and general analysis, different coordinate sets, reference frames and equations of motion have to be used. All force models and equations of motion required to model a satellite orbiting a planet (or a planet orbiting the Sun) are given in this chapter. Chapter 5 gives the methods to solve the equations of motion. This chapter is, when not specified, based on the books on astrodynamics by Wakker [50] and Battin [51] . Section 3.1 describes the dynamics of n-bodies under the influence of their mutual gravitational attraction. Section 3.2 the commonly used reference frames and the transformations between reference frames and coordinate sets. Section 3.3 the perturbing forces that may act on an orbiting body. Section 3.4 methods that analyse and incorporate the perturbing forces in the 2-body problem, this section is of particular interest to the interval integration methods in Chapter 6. At last, Section 3.5 discusses the uncertainties in the models used in this chapter.

3.1

N-body Problem

The n-body problem is the problem of finding the motion of n point masses under their mutual attraction. The oldest n-body problem arose from the attempts to predict the motion of the planets around the Sun. It is the problem of the motion of radially symmetrical bodies with mutual gravitational attraction; every body exerts a force on the other bodies, in the direction of the line between the bodies and with a size proportional to the product of the masses of the bodies and inversely proportional to the square root of the distance between the bodies.

3.1.1

General Equations of Motion

Equations of motion are the differential equations that describe the behaviour of a system as a function of time. The equations of motion of the n-body problem are the differential equations that describe the motion of n mutual attracting bodies. These equations of motion

3.1. N-body Problem

Chapter 3. Astrodynamics

Figure 3.1: Position vectors in the n-body problem with respect to an inertial reference frame with origin O. Image based on: Wakker [50] . are the result of a combination of Newton’s law of universal gravitation and Newton’s laws of motion. Newton’s laws of motion are three physical laws that form the basis of classical mechanics. The first law states that objects only change velocity with respect to an inertial reference frame if an external force is acting on it. The second law states that the net force, F, on a body is proportional to the time rate of change of its linear momentum, mv, F=

d(mv) dt

(3.1)

or more commonly known (for constant mass) as, F = ma

(3.2)

And Newton’s third law states that for every force there is an opposite and equal reaction force; ”action is reaction”. Newton’s laws define the motion of a particle in an inertial reference frame in case the size and direction of the net force on the particle are known. In the case of the gravitational n-body problem, the direction and size of the gravitational forces are given by Newton’s law of universal gravitation. For the force of body 1 on body 2 we have, m1 m2 F12 = −G 3 r12 (3.3) r12 where G is the universal gravitation constant, mi the mass of body i, rij the position vector from body i to body j and rij the magnitude of the position vector rij . d2 r dv Assuming that the body’s masses stay constant and using 2 = = a, Newton’s law dt dt of universal gravitation (3.3) and Newton’s second law of motion (3.1), the motion of body

3.1. N-body Problem

Chapter 3. Astrodynamics

Figure 3.2: Position vectors for the relative motion of body j with respect to the noninertial reference frame with origin at body k. O is the origin of the inertial reference frame. Image based on: Wakker [50] . 2 under the attraction of body 1, with respect to an inertial reference frame, can be written as, d2 r 2 m1 m2 m2 2 = −G 3 r12 (3.4) dt r12 where the position vectors ri and rij are defined as in Figure 3.2. The force exerted on body i by all other n − 1 bodies, is the sum of the individual forces of all bodies on body i. The equations of motion of body i, for the general n-body problem, can thus be written as, ∗ X d2 ri mi mj mi 2 = G 3 rij (3.5) dt rij j where ∗ indicates that the summation is taken from j = 1 to j = n, but excluding j = i since body i does not exert a force on itself.

3.1.2

Relative Equations of Motion

The previous subsection derived the equations of motion with respect to an inertial reference frame. In practice, however, one is often more interested in the motion of a body with respect to another body. Examples are the motion of the Moon or an artificial satellite orbiting Earth. The equations of motion of bodies n relative to body k can be directly derived from the general equations of motion (3.5) by a translation of the coordinate origin to the center of body k. This new origin does therefore not rotate with respect to the old inertial reference frame. The derivation can be found in literature, see for example [50] and [38] . The resulting

3.1. N-body Problem

Chapter 3. Astrodynamics

equations of motion of body i relative to body k are, ∗

X (mi + mk ) d2 ri = −G r + G mj i dt2 ri3 j

3.1.3

rj rj − ri − 3 3 rij rj

! (3.6)

Constants of Motion

The gravitational n-body problem has some properties that are constant. These constants are called ”integrals of motion” or ”constants of motion” and can be used for orbit analysis, to estimate errors in numerical methods and as orbital elements in a method called variation of parameters (see Section 3.4.2) or Unified State Model (see Chapter 4). There are six integrals of motion with relation to the center of mass (velocity and position), three for the components of the total angular momentum and one for the total energy of the system. Other constants of motion depending on these constants can, of course, be defined, which may be useful for numerical integration. Center of Mass The position of the center of mass, with respect to an inertial reference frame, of the system is given by, P mi ri rcm = Pi (3.7) i mi Assuming no external forces on the system, the acceleration of the center of mass is zero. Combining this with the equations of motion of the system and integrating twice yields the following integration constants for the velocity and initial position of the center of mass, d(xcm ) = a1 dt xcm = a1 t + b1

d(ycm ) = a2 dt ycm = a2 t + b2

d(zcm ) = a3 dt zcm = a3 t + b3

(3.8) (3.9)

Total Angular Momentum The total angular momentum is defined by, H=

mi ri ×

dri dt

The three components of H are three integrals of motion, X Hx = mi (yi z˙i − zi y˙ i ) = c1

(3.10)

(3.11)

Hy =

mi (zi x˙ i − xi z˙i ) = c2

(3.12)

mi (xi y˙ i − yi x˙ i ) = c3

(3.13)

Hz =

X i

It should be noted that these constants are not constant when external forces are exerted on the system, like drag or radiation pressure.

3.1. N-body Problem

Chapter 3. Astrodynamics

Energy An energy integral for the system can be derived from the equations of motion (3.5), ∗

1 X X mi mj mi Vi2 − G 2 2 rij i i

X1 i

(3.14)

where the left term is the total kinetic energy of the system and the right term the total potential energy, Ekin + Epot = Etotal (3.15)

3.1.4

Solutions

While the formulation of the n-body problem is rather simple, solving the non-linear differential equations of a general n-body problem, (3.5) is difficult. The two body problem can be solved analytically, but for three and more bodies the motion becomes seemingly chaotic and no general analytical solutions exist. A new way to solve the equations of motion arose with the invention of the computer in the 20th century: numerical integration. Some basic numerical methods were already developed and used by human calculators, but the computer immensely increased the number of possible computations and opened a whole new field of methods and applications of numerical integration. All problems that involve more than two bodies, or other forces, cannot be solved exactly in an analytical way. Analytical Many mathematicians and physicists of the previous centuries, after Newton, have tried to solve the n-body problem. The focus on the n-body problem triggered many new mathematical studies, methods and theories. Euler, Laplace, Lagrange, Jacobi and Hamilton are a few historic mathematicians who worked on this problem and by doing so invented new methods that are still used today in a large variety of science and engineering fields. Even Newton’s laws of motion are a result of his effort to mathematically derive Kepler’s laws (the two body problem), which were previously only based on observations. The two body problem, also called Kepler problem, can be solved analytically. This problem can be separated into two independent one-body-problems about their common center of mass (barycenter), which does not experience acceleration with respect to an inertial reference frame. The solution to this one-body-problem is a conical section, also called Kepler orbit. Six Kepler elements completely define the position of the body in three dimensional space. Two keplerian orbital elements define the shape and size of the ellipse: e, the eccentricity a, the semimajor axis Three elements define the orientation of the ellipse: i, the inclination Ω,the longitude of the ascending node ω, and the argument of periapsis The last element defines the position of the orbiting body along the ellipse as a function of time:

3.1. N-body Problem

Chapter 3. Astrodynamics

Figure 3.3: Definition of the eccentricity, e, and semi major axis, a. Wakker [50] .

Image source:

Figure 3.4: Definition of the Kepler elements i, ω, Ω and θ. Image source: Wakker [50] . θ (sometimes ν), the true anomaly or, E, the eccentric anomaly or, M , the mean anomaly or,

3.1. N-body Problem

Chapter 3. Astrodynamics

Figure 3.5: Definition of the true anomaly, θ (sometimes ν), eccentric anomaly, E, and mean anomaly, M . Image source: Wakker [50] . τ , the time of pericenter passage. M is a function of time, t and the time of pericenter passage, τ , r µ M= (τ − t) a3

(3.16)

M and E are related by, M = E − e · sin E

(3.17)

Figure 3.3 shows the definition of a and e, Figure 3.4 shows the definition of the three elements (Euler angles) defining the orientation, and Figure 3.5 shows the definition of the true, eccentric and mean anomaly. While the two body problem results in a simple geometric and repeating motion which can be solved analytically, this does not hold for the general (n ≥ 3)-body problem. Problems with more than two bodies can generally not be solved analytically, although analytical solutions do exist for some specific geometries and mass distributions of the three bodies. Because the Sun has very little influence on the position of satellites relative to Earth, the motion of satellites around Earth and Earth around the Sun, for example, can be approximated by the superposition of two 2-body problems. General solutions to the n ≥ 3-body problem can only be approximated using numerical methods.

3.2. Reference Frames

Chapter 3. Astrodynamics

Numerical Numerical methods use truncated (finite) Taylor series expansions of the force function to approximate the solution of differential equations. Numerical solutions are only an approximation of the real solutions because it uses finite steps and finite digit numbers to approximate the infinite and continuous differential operators and infinite digit numbers. In addition, it provides only a single solution for some initial values and does not give a symbolic solution that provides insight into the problem. With the increasing computation power it is, however, possible to get insight into the general behaviour of a system by generating solutions for many initial values; i.e. use brute force to generate data. A more detailed discussion of numerical methods and their application to the n-body problem is given in Chapter 5.

3.2

Reference Frames

A reference frame is a ”realization of a reference system through accessible physical objects along with their coordinates” [52] . A reference system defines origin, axis and methods to transform between observables and reference data. Physical objects are benchmark objects in the real world with precise coordinates for each. This section presents two important reference frames; the International Celestial Reference Frame (ICRF) and the Earth Centered Inertial Reference Frame (ECIRF). Transformations between coordinate sets are given in Appendix A. Transformations between different origins and axis are discussed after the ICRF and ECIRF. International Celestial Reference Frame The International Celestial Reference Frame (ICRF) is an inertial reference frame with origin and the barycenter of the Solar System. It is defined by the position of 212 extragalactic sources, mainly quasars. These sources are so far away that their is no significant angular motion. It is the standard reference frame used to define the position of bodies orbiting the Sun. [53] [54] The origin is at the center of mass of the Solar System, the barycenter, which is generally assumed to not experience acceleration (galactic rotation is insignificant). The epoch (time of measurement) is usually 12h (Terrestrial Time) TT on 1 January 2000 (J2000.0). The xy-plane is the plane of Earth’s orbit at the reference epoch (J2000.0). The x-axis out along the ascending node of the instantaneous plane of Earth’s orbit and Earth’s mean equator. And the z-axis perpendicular to the xy-plane in the directional (+ or -) sense of Earth’s north pole at the reference epoch. JPL HORIZONS ephemeris provides high precision and accessible Cartesian coordinates of all major solar system bodies. Their reference frame is only slightly different from the ICRF, which is more precise. [55] Earth Centered Inertial Reference Frame Earth-Centered Inertial Reference Frames (ECIRF) are a group of non rotating coordinate frames with their origins at the center of mass of the Earth. ECIRF frames are called inertial in contrast to the Earth-centered, Earth-fixed (ECEF) frames which rotate in inertial space. Although the ECIRF does not rotate, it is not a true inertial reference frame, because it accelerates with Earth around the Sun.

3.2. Reference Frames

3.2.1

Chapter 3. Astrodynamics

Transformations between Reference Frames

If we know the position of a body in one reference frame, it may be useful to transform this to the position in a second reference frame. The same goes for the velocity and acceleration.

Figure 3.6: Motion of body P in the non-inertial reference frame xyz within the inertial reference frame XYZ. Source: T¨or¨ok [56] . Figure 3.6 shows the position vectors of the xyz reference frame with respect to the inertial XYZ frame, and the position of body P with respect to the xyz and XYZ reference frame. The xyz reference frame is only an inertial reference frame if the xyz is in pure translation (linear motion) with respect to XYZ and the local unit vectors remain unchanged (no rotation). In case the xyz rotates with velocity, ω, the position, velocity and acceleration of body P with respect to the XYZ frame are given by, rP = rB + rrel

(3.18)

vP = vB + (ω × rrel ) + vrel

(3.19)

aP = aB + (ω˙ × rrel ) + ω × (ω × rrel ) + 2(ω × vrel ) + arel

(3.20)

Derivation of these equations can be found in T¨or¨ok [56] . These equations can be reduced for the ECIRF and ICRF because their local unit vector remains unchanged, ω = 0, so that (3.18-3.20) reduce to, rP = rB + rrel (3.21) vP = vB + vrel

(3.22)

aP = aB + arel

(3.23)

The ECIRF is a pseudo-inertial reference frame where the simplified equations above hold, because the local unit vectors in the ECIRF remain unchanged with respect to an inertial reference frame (for example, ICRF). The ECIRF rotates (accelerates) around the Sun due to the gravitational attraction between the Earth and the Sun. It is therefore not an inertial

3.3. Perturbing Forces

Chapter 3. Astrodynamics

reference frame; bodies on which no forces act do not move in straight lines in these reference frames. However, the gravitational acceleration of the Sun is almost equal for bodies close to Earth. Because all objects close to the origin of the ECIRF experience almost the same gravitational acceleration, bodies in the frame act as if it is almost inertial (pseudo-inertial). The mathematical representation of the orientation (rotation) of a reference frame and transformations between different representations is given in Section 4.1.2.

3.3

Perturbing Forces

The n-body problem is a simplified model of the motion of planets and satellites in a star system. It assumes radially-symmetrical rigid bodies and no other forces than gravity. However, in reality other forces than gravity act on planets and satellites. And bodies are not completely rigid and radially-symmetrical. Moreover, the model is based on Newton’s laws of motion, while Einstein’s general relativity (unification of Newton’s law of motion and special relativity) is the more accurate model. Forces other than gravitational forces are atmospheric drag, radiation pressure (photons from the Sun or Planets) and electrical forces. Accurate force models can be made to model these forces, if enough information about the environment is available from observations. Figure 3.7 shows the size of the largest perturbing forces for different orbit altitudes. Non-radially symmetrical planets cause the gravity vector to point away from the center of mass of bodies. When the complete gravity field of a planet is known, a small correcting force can be calculated. This slightly off center gravitational force only significantly affects masses close to the body, because the mass becomes more like a point mass for distant bodies (the off center angel becomes smaller). Non-rigid bodies can exchange momentum. This exchange of momentum can be modelled for, but is difficult and detailed information about the composition of a body is required. Luckily, this moment exchange is generally very small and only has a noticeable effect over a long time. A correcting perturbing force can also be calculated for the relativistic effects. For long time periods, long distances and high velocities this can may have a significant effect on the Newtonian orbit. The gravitational force of other planets and the Sun on Earth satellites has a small effect on the motion of the satellite relative to Earth. This makes it possible to model an Earth satellite orbit as a 2-body problem with the gravitational attraction of other celestial bodies as perturbing forces. A force model for a man-made thrust force is necessary to model rockets or satellites with propulsion systems. This size and direction of the thrust force follows from the rocket engine specifications and settings, and is generally known. The next paragraphs give the force models for the perturbing forces on an Earth orbiting satellite, these can be used in the perturbation methods presented in Section 3.4 and apply to all perturbed n-body systems. They describe and model the small perturbing forces that cause a slight disturbance in the almost Keplerian orbits of satellites and planets.

3.3.1

Earth’s Gravity Field

The gravitational potential, or gravity field, outside a body with a radially symmetrical mass density distribution is equal to the potential of a point mass at the center of mass

3.3. Perturbing Forces

Chapter 3. Astrodynamics

Figure 3.7: Order of magnitude of various perturbing forces for a satellite orbit versus orbit altitude. Image source: Montenbruck et al. [38] 38

3.3. Perturbing Forces

Chapter 3. Astrodynamics

of the body with the same mass as the body. However, celestial bodies are not perfectly symmetrical spheres and for some problems this has a significant effect on the orbit. Orbiting bodies can assumed to be perfectly symmetrical density spheres and can thus be modelled as point masses when high accuracy and precision are not essential or when distances are large. This is, for example, the case for planets orbiting the Sun or for a first-order approximation of Earth satellite trajectories. For bodies orbiting at close distances to their main body, the real shape and density distribution of the main body can have a significant effect on the orbit. This is especially true when the gravity field disturbance forces act on the satellite in a periodic way or in the same direction, thereby creating a resonance which can yield large deviations over long time periods. The gravitational potential of a body with arbitrary mass density distribution, at a point outside the body, can be modelled by, µ U =− r

1−

∞ X n=2

Re r

n Pn (sin φ) −

∞ X n X

Jn,m

n=2 m=1

Re r

n Pn,m (sin φ)(cos m(Λ − Λn,m ))

(3.24) where, for the Earth, r, φ, Λ are the spherical coordinates of the point that is considered with respect to the geocentric rotating reference frame, Re is the mean equatorial radius. The factors Jn,m and Λn,m are constants of the gravity field. Jn (m = 0) represents the influence of deviations in the mass density distribution in North-South direction, called zonal harmonics. And Jn,m the influence of mass distribution in the North-South and East-West direction, called tesseral (m 6= n) and sectorial harmonics (m = n). The Jn,m and Λn,m coefficients up to order and degree 360 (EGM96) and 2159 (EGM2008) are determined from the orbit determination of Earth satellites and gravimeter (accelerometer) measurements. Digital tables for the EGM96 and EGM2008 coefficients can be found on-line [57] [58] . The terms Pn (sin φ) and Pn,m (sin φ) are Legendre polynomials and Legendre functions of the first kind, for which the following expressions hold,

Pn (x) =

x = sin φ

(3.25)

P0 (x) = 1

(3.26)

P1,0 (x) = x

(3.27)

1 dn (1 − x2 )n (−2)n n! dxn

(3.28)

dm Pn (x) (3.29) dxm Legendre polynomials can be computed recursively for computational efficiency. A recursive method can be found in Balmino, 1991 [59] . The magnitude of the perturbing acceleration f can be found from, µ (3.30) f = −∇ U + r m

Pn,m (x) = (1 − x2 ) 2

3.3. Perturbing Forces

Chapter 3. Astrodynamics

When we consider only the J2 term, we may write in Cartesian variables, 2 z 1 R2 f = −∇ µJ2 3 3 2 − 1 2 r r 3 z2 R2 fx = − µJ2 5 x 1 − 5 2 2 r r 3 z2 R2 fy = − µJ2 5 y 1 − 5 2 2 r r z2 3 R2 fz = − µJ2 5 z 3 − 5 2 2 r r

3.3.2

(3.31)

(3.32)

(3.33)

(3.34)

Gravitational Attraction by other Celestial Bodies

The gravitational attraction by other celestial bodies follows from (3.6), where it is the last term, ! ∗ X rj − ri rj − 3 (3.35) mj Fbodies = G 3 rij rj j In case of an Earth orbiting satellite, the gravitational attraction from the Sun and the Moon have the largest influence of all disturbing bodies. Figure 3.7 gives an indication of the relative sizes of the disturbance forces of the Sun, Moon and other planets on the satellite.

3.3.3

Aerodynamic Forces

The Earth and some other planets have an atmosphere which may extend to high altitudes above the surface. The atmosphere of Earth consists mostly of nitrogen and oxygen gas. Interaction (collisions) between the gas particles and a satellite cause a change of momentum of the satellite. This change of momentum (force) is called drag and is related to the number of particle collisions and the speed of the collisions. In the macro world this means it is related to the density of the gas and velocity of the satellite with respect to the atmosphere. The atmospheric density decreases with increasing altitude and can be assumed non existent at large distance, higher than 800 km for Earth. A very rough estimate of the density, at altitudes up to 100 km, is given by the exponential atmosphere model, ρ = ρ0 e−h/H0

(3.36)

where h is the altitude, ρ0 the density at sea level and H0 the density scale height, which is 7.9 km at sea level. For higher altitudes, where most satellites orbit, the MSIS-E-90 [60] atmospheric model can be used. This model is available on the world wide web and can generate a table of densities at different positions. A local polynomial fit of this table creates a continuous approximation of the local densities. The drag force can be modelled by the following equation, 1 Fdrag = − CD ρSV V 2

(3.37)

3.3. Perturbing Forces

Chapter 3. Astrodynamics

where CD is the drag coefficient of the satellite that has to be determined from experiment, a typical value is 2.3 [38] . ρ the density of the atmosphere, S the frontal surface area of the satellite and V the velocity of the spacecraft with respect to the atmosphere. The density of the atmosphere changes with position and time and is primarily a function of altitude and temperature.

3.3.4

Radiation Pressure

Radiation pressure is the result of photons hitting the satellite. Radiation from the Sun, directly or reflected from Earth, is the main radiation pressure source for an Earth satellite. The radiation power directly from the Sun is almost constant, while the radiation reflected from Earth depends on the reflectivity of the Earth’s surface in view of the satellite. One should not forget that the radiation from the Sun and Earth in the shadow of Earth are completely different. These variations makes modelling of the radiation pressure rather complex. For The radiation force can be modelled by, Fsol = CR

W S∗ c

(3.38)

where CR is the reflectivity, between 0 and 1. W the power density of the incoming light, 1366 W/m2 from the Sun. S ∗ the effective surface area, and the speed of light c = 299, 792, 458 m/s.

3.3.5

Electromagnetic Forces

An electromagnetic perturbing force acts on the satellite because the atmosphere at high altitudes is ionized. This causes the satellite to become electrically charged; it obtains a negative potential with respect to its environment. The interaction between the charged satellite and Earth’s magnetic field causes a Lorenz force to act on the satellite. The Lorenz force is expressed as, Fem = qV × B (3.39) where q is the electric charge of the satellite, B the magnetic induction of Earth’s magnetic field and V the velocity of the satellite. The external magnetic field of Earth can be approximated by a dipole coinciding with the Earth’s axis of rotation. This approximation yields the following magnetic potential, Udip = g1

R3 sin φ r2

(3.40)

where g1 is a geomagnetic parameter, R the mean radius of Earth, r the magnitude of the radius vector and φ the latitude of the satellite. The magnetic induction of this dipole potential field is given by, B = −∇Udip The Lorenz force on the satellite can now be written as, sin φ Fem = −(qg1 R3 ) · V × ∇ r2

(3.41)

(3.42)

3.3. Perturbing Forces

Chapter 3. Astrodynamics

The order of the electromagnetic force is much smaller than, for example, the atmospheric drag and gravity field perturbations. Its exact values are therefore not required for collision avoidance and this force will be included in a force that encloses all small perturbation forces and a uncertainty margin.

3.3.6

Relativistic Effects

Newton’s theory of gravity and motion were not able to fully explain the motion of all planets in the solar system. Especially Mercury had a measurable different orbit than predicted by Newton’s theories. A more accurate model came in the form of Einstein’s general theory of relativity, which unified his special theory of relativity with Newton’s law of universal gravity. Using general relativity instead of Newton’s theories is much more complicated and not necessary for most orbit predictions, since the perturbing acceleration due to relativistic effects is in the order of 3 · 10−10 m/s2 for a typical Earth orbiting satellite. A first order post-Newtonian relativity correction is given by, µ V2 V2 µ 4 2 − 2 er + 4 2 (er · ev )ev (3.43) frel = − 2 r c r c c where er and ev are the unit position and velocity vector, c the speed of light and µ the standard gravitational parameter of the main body.

3.3.7

Thrust

Most spacecraft have a propulsion system that exerts a force on the spacecraft, to change the orbit, by expelling mass. This force is generally known for a specific propulsion system from tests on Earth. Otherwise, this force can be modelled in terms of mass flow and specific impulse, Fthrust = g0 mI ˙ sp

(3.44)

where g0 is the gravitational acceleration at the Earth’s surface, Isp the specific impulse in vacuum and m ˙ the mass flow of the propellant. Isp depends on the rocket engine used and is between 250 and 3,000 seconds. The specific impulse and mass flow are also known from tests or theoretical predictions.

3.3.8

Enclosing Perturbing Forces

For Interval Integration discussed in Chapter 6, all possible values of the perturbing forces have to be enclosed in an interval (a set defined by upper and lower bounds). Table 3.3.8 gives intervals that enclose all possible perturbing forces. They are based on the typical parameter values given in the previous subsections and are enlarged to make sure the force is in the interval. While many of the perturbing forces act in known directions, the given intervals can be used in all directions for simplicity, while making sure that all possible perturbing forces are included. The perturbing forces that are proportional to the satellite mass are given as acceleration, the other as force. More accurate, and thus smaller, interval enclosures can be obtained for a specific orbit with the parameter values and models presented in the previous subsections.

3.4. Perturbation Methods

Chapter 3. Astrodynamics

Table 3.1: Interval enclosures of perturbing forces or acceleration for near circular orbits between 200 km and 50,000 km above Earth’s surface. The intervals are enlarged to be sure to contain all possible forces. Real forces are most likely smaller and depend on the orbit. Force Type Drag Earth’s Gravity Field Other Planets and Sun Radiation Pressure Relativistic Effects Other

3.4

Acceleration Interval [km/s2 ] S m [−30, 30] [−10−8 , 10−8 ] [−10−7 , 10−7 ] S −7 , 10−7 ] m [−10 −9 [−10 , 10−9 ] [−10−9 , 10−9 ]

Perturbation Methods

The motion of a satellite with respect to a non-rotating Earth centred reference frame when perturbing forces are taken into account can be described by, µ d2 r + 3 r = −∇R + f 2 dt r

(3.45)

where R is the disturbing potential that describes all perturbing forces that can be expressed as potential function, and f are all other perturbing accelerations. Equation (3.45) cannot be solved analytically and has to be solved using analytical approximations, called general perturbation methods, or using numerical integration techniques, called special perturbation methods. General perturbation methods symbolically expand the perturbing accelerations in truncated series that are integrated analytically. General perturbation methods are very fast once the series expansion of the perturbing forces is known. The other main advantage of general perturbation methods is that it can reveal the source of perturbations from orbital data. The disadvantage is its low accuracy compared to special perturbation methods. Special perturbation methods are all methods that use perturbing forces and numerical integration methods to model the trajectory of a specific body. Three commonly used methods are Cowell’s method, Encke’s method and the method of variation of orbital elements. Cowell’s method is the simplest method that uses rectangular coordinates and numerically integrates all forces, main gravitational and perturbing. No use is made of the fact that analytical solutions of the unperturbed two body problem exist. Both the method of Encke and the method of variation of orbital elements use an analytically computed reference orbit. Only deviations from this orbit are integrated numerically. The method of variation of orbital elements is most advanced because it uses the almost constant Kepler elements. However, Encke’s method is easier to implement because it uses the orthogonal Cartesian coordinates. While the method of variation of orbital elements is more computational efficient (or more accurate for equal computation times), it is also more complex than Encke’s or Cowell’s method. The increase in computational power and the need for non-astronomers, lacking specialized knowledge, made the method of Encke and later Cowell most popular.

3.4. Perturbation Methods

3.4.1

Chapter 3. Astrodynamics

Method of Encke

The Method of Encke, introduced by J.F. Encke in 1852, uses a reference orbit and only numerically integrates the deviations from this reference orbit. Any orbit can be used as reference orbit, but for a near two body problem a Kepler reference orbit is a logical choice because the unperturbed orbit is Keplerian. For the Kepler reference orbit we write, µ d2 ρ + 3ρ = 0 dt2 ρ

(3.46)

where ρ is the position vector of the satellite if no perturbing forces would act on it. At some instant of time, the position and velocity of the real and reference satellite are set equal (the constant reference orbit is defined), t = t0

(3.47)

r=ρ

(3.48)

dρ dr = (3.49) dt dt For the deviation of the acutal trajectory from the reference orbit at time t, we write, ∆r = r − ρ

(3.50)

By differentiating this relation twice with respect to time and substitution of (3.45) and (3.46) into , we obtain, ρ r d2 ∆r =µ − 3 − ∇R + f (3.51) dt2 ρ3 r The reference orbit, ρ, can be computed analytically, and the deviations, ∆r, from this reference orbit numerically. The actual position and velocity follow from the addition of ρ dρ d∆r and ∆r, and and . dt dt (3.51) can be rewritten in a better form for numerical computation, d2 ∆r µ = 3 [(ρ + ∆r)qf (q) − ∆r] − ∇R + f dt2 ρ

(3.52)

2q 1 √ qf (q) = 1+ 1 + 2q 1 + 2q + 1 + 2q

(3.53)

where,

and, q=

3.4.2

∆r · (ρ + 21 ∆r) ρ2

(3.54)

Variation of Orbital Elements

The orbital elements a, e, i, Ω, ω en τ (see Section 3.1.4) are constants of motion for the unperturbed problem and completely determine the orbit of the body. For a certain position and velocity of a body, one corresponding set of orbital elements can be determined, this is a fictitious momentary Kepler orbit; a Kepler orbit that corresponds with the instantaneous position and velocity but does not stay constant due to perturbing forces. The fictitious momentary Keplerian orbit is therefore the orbit the body would follow if from that moment

3.4. Perturbation Methods

Chapter 3. Astrodynamics

all perturbing forces become zero. Such an instantaneous Kepler orbit can be found at every moment in time, therefore, one may consider the motion of a satellite as a continuous transition between instantaneous orbits. This can also be seen as orbital elements that continuously change with time. The method of variation of orbital elements uses mathematical relations between the time rate of change of the orbital elements and the perturbing forces. It consists of a set of six first-order differential equations for the variation of the ”osculating” orbital elements. These differential equations can be integrated, like the more common equations of motion in rectangular coordinates, to yield the orbital elements and thus the position and velocity at any moment in time. The advantage of this method is that the change of orbital elements is small for small perturbing forces, which makes it possible to use large time-steps in numerical integration and decrease the integration errors described in Section 3.1.4 and 5.3. Another advantage may be its use in interval (or verified) integration of satellite orbits (see Chapter 6), since the variations in the orbital elements are much smaller than the variations in Cartesian coordinates. Different sets of constants of motion (orbital elements) can be found and used in the method of variation of orbital elements. Hybrid methods that use constants of motion and other convenient variables also exist. One such method is the Unified State Model which is discussed in detail in Chapter 4. The classical differential equations for the Kepler elements are Lagrange’s Planetary Equations. These equations take only forces that can be described by a perturbing potential R in account. More general equations are Gauss’ form of the planetary equations, which can take any force into account. Lagrange’s Planetary Equations J.L. Lagrange developed his method of variation of orbital elements in 1780. The equations only describe the change of the elements due to potential forces. This means that forces like drag and radiation pressure cannot be applied in this method. The derivation of the Lagrange’s Planetary Equations can be found in [50] and [51] and is not discussed here. The six variational equations are, a2 ∂R da = −2 dt µ ∂τ s de a(1 − e2 ) ∂R 1 1 − e2 ∂R =− − dt µe ∂τ e µa ∂ω di cot i ∂R 1 ∂R =p −p 2 2 dt ∂ω µa(1 − e ) µa(1 − e ) sin i ∂Ω s dω 1 1 − e2 ∂R cot i ∂R = −p 2 dt e µa ∂e µa(1 − e ) ∂i

(3.55)

(3.56) (3.57)

(3.58)

dΩ 1 ∂R =p 2 dt µa(1 − e ) sin i ∂i

(3.59)

dτ a2 ∂R a(1 − e2 ) ∂R =2 + dt µ ∂a µe ∂e

(3.60)

3.4. Perturbation Methods

Chapter 3. Astrodynamics

When the perturbing potential R is a known function of x, y, z, then the partial derivatives ∂R can be determined from, ∂αi ∂R ∂x ∂R ∂y ∂R ∂z ∂R = + + (3.61) ∂αi ∂x ∂αi ∂y ∂αi ∂z ∂αi The differential equations (3.60) can be solved numerically using this relation. It should be noted that it was never assumed that the perturbing forces were small in the derivation of the variational equations. The equations are therefore generally valid although it loses its advantage over the simpler Cowell’s and Encke’s methods when perturbing forces become comparable to the main gravitational force. Gauss’ form of Lagrange’s Planetary Equations J.C.F Gauss elaborated the partial derivatives in equations (3.61) in 1813 to include non potential forces. He wrote the differential equations in terms of the perturbing accelerations in the radial direction, as , perpendicular to the radius vector, an , and a component perpendicular to the orbital plane, aw . The resulting set of differential equations is called Gauss’ form of Lagrange’s Planetary Equations, a derivation can be found [51] , a2 h pi da = 2√ fS e sin θ + fN (3.62) dt µp r r p de = [fS sin θ + fN (cos E + cos θ)] (3.63) dt µ di r = fW √ cos(ω + θ) dt µp r dω p 1 r r =− fW cot i sin u + fS cos θ − fN 1 + sin θ dt u p e p r dΩ = fW √ sin u dt µp sin i r r dM 2r 1 − e2 a 1 − e2 a r = n − fS √ − cos θ − fN 1+ sin θ dt µa e µ e µ p

(3.64) (3.65) (3.66)

(3.67)

where, r n=

µ a3

(3.68)

M = n(t − tperigee )

(3.69)

p = a(1 − e2 )

(3.70)

p (3.71) 1 + e cos θ These equations can be used to predict the perturbed orbit of a satellite around Earth or a planet around the Sun. It is valid for small and large perturbing forces and can take all known forces on the orbiting body into account. r=

3.5. Uncertainties

3.4.3

Chapter 3. Astrodynamics

Singularities

From the relations (3.55-3.60) and (3.62-3.67) we can see that problems arise in Langrange’s and Gauss’ form of the planetary equations if e → 0 or sin i = 0 because e and sin i occur in the denominator. This means that singularities occur in the variational equations for near circular or equatorial orbits. These singularities are a result of the chosen coordinate set (Kepler elements) and are not a physical phenomena. This problem can be explained by the fact that the argument of perigee, ω, is not defined for circular orbits (e = 0) and the argument of perigee makes discontinuous jumps of 90 degree because the major and minor axis change or are not defined. For circular and equatorial orbits, the Kepler elements have to be replaced by a different set of elements to remove the singularity. One common replacement in orbital elements for near-circular orbits is replacing e, ω and τ with the following elements, l = e cos ω

m = e sin ω

χ = ω − nτ

(3.72)

The resulting variational equations can be found in [50] . For the qualitative analysis of the interval integration methods discussed in Chapter 6, these equations are not necessary since the methods can be analysis without using orbits with zero eccentricity and inclination. A different set of orbital elements, called Unified State Variables, also remove this singularity and is described in more detail in Chapter 4. These Unified State Variables are defined by Altman in 1972, but have only been applied to once to a navigation problem [61] .

3.5

Uncertainties

Very accurate orbit predictions can be made using the methods described in this Chapter. The predictions will, however, rarely correspond exactly with reality. This is due to truncation and rounding errors of numerical integration methods, but also due to the models itself and the uncertainty in the constants and parameters used in the models. Furthermore, the initial position and velocity (initial values) are not precisely known and contain a certain uncertainty. Truncation and rounding errors were already discussed in Section 3.1.4 (and in more detail in Section 5.3). The other causes of uncertainties in the predictions are discussed in the following subsections.

3.5.1

Physical Models

Physical models (or laws, or theories) try to predict a large field of phenomena in an accurate way. The fact that something is called a physical law, model or theory does not say anything about the ultimate truth. A scientific theory never claims to be 100% accurate and never claims to describe how the universe really works; it is only a model, even if it is accurate up to current measurement precision. [62] Newton’s theories of motion and gravitation (1687) described a wide range of physical phenomena; from an apple falling to the ground to galaxies orbiting other galaxies. It also did this with great accuracy. The existence, for example, of Neptune was predicted by the motion of Uranus, using Newton’s theories. But at the same time, the orbit of Mercury showed slight perturbations that could not entirely be accounted for by Newton’s theory. Einstein’s General Theory of Relativity (1915) was the new theory of gravitation and was able to account for the complete motion of

3.5. Uncertainties

Chapter 3. Astrodynamics

Mercury. It had long been believed that Newton had found the true nature of the universe, but Einstein showed that his theory was only a very accurate model of reality. Although Einstein’s theory is accurate up to current measurement precision of dynamics on the large scale, it does not mean that Einstein’s theory is exactly right and it does not claim to be. Furthermore, Einstein’s General Theory of Relativity produces impossible results when applied to microscopic problems where gravity becomes as strong as the other fundamental forces (the electromagnetic, strong nuclear and weak nuclear force); for example in black holes and the big bang. This shows that Einstein’s more accurate theory is not a perfect model and should not be seen as the ultimate truth. Nevertheless, Newton’s theory was accurate enough to land a man on the Moon and Einstein’s theory is more accurate and precise, on solar-system scales or larger, than we can measure. Newton’s theory is much easier to apply and is still used to predict orbits. A correction for the General Relativity effects can be applied to correct for the small differences between Newton and Einstein. When these corrections are applied, the uncertainties resulting from the accuracy of the model, assuming exact physical constants and parameters, is insignificant and not measurable. [63] [64]

3.5.2

Physical Constants

A physical constant is a quantity that is believed, or assumed, to be universal and constant in time. Physical constants can be seen as the scale or ratio of relations between physical m3 concepts like mass and force. For example, the constant of gravitation, G in kgs 2 , in Newton’s law of gravitation (3.3) can be seen to provide the scale of the relation between the gravitational force and the mass and distance of two bodies. Physical constants can be defined exactly in the basic units when they define the size of, for example, a meter or a second. However, most constants are determined from physical measurements. These measurements have a certain limited accuracy and precision and the physical constants therefore also have a limited accuracy and precision. This means that when we apply Newton’s law of gravitation, the precision of the gravitational constant, among others, limits the precision of our results. The Newtonian constant of gravitation, G, and the speed of light, c are the only physical constants used in the models described in this Chapter. The value of G is 6.67428 · 10−11 m3 kg −1 s−2 with a standard uncertainty of 0.00067 · 10−11 m3 kg −1 s−2 [65] . The standard uncertainty is the estimated standard deviation of G, where the measurement uncertainty is approximately normal distributed. This implies that it is believed with an approximate level of confidence of 68 % that G is greater than or equal to G − 0.00067 · 10−11 , and is less than or equal to G + 0.00067 · 10−11 , which is commonly written as G = 6.67428 · 10−11 ± 0.00067 · 10−11 m3 kg −1 s−2 . The speed of light, c, in ms−1 is exact (or defined) because it defines the size of one meter. The speed of light is exactly 299, 792, 458ms−1 .

3.5.3

Parameters

Besides physical constants, there are some parameters in the force models described in this Chapter that are assumed to be constant or a function of the state variables, but are problem specific and not universal. The physical parameters used in this Chapter are the drag coefficient, CD ; the effective surface area, S; the atmospheric density, ρ; the reflectivity, CR ; the power density, W ; mean equatorial radius, Re ; gravity factors, Jn , Jn,m and Λn,m ; the electric charge, q; the geomagnetic potential, g1 ; the mass flow, m; ˙ and specific impulse, Isp .

3.5. Uncertainties

Chapter 3. Astrodynamics

CD , CR and S depend on the orientation of the orbiting body with respect to its flight direction. The values for a specific orientation can be measured very precisely on Earth, but are very uncertain for objects that cannot be measured on Earth. The uncertainty for these parameters varies per object and should be defined for every specific object. ρ, the atmospheric density, varies with altitude, position and time. The uncertainty in the atmospheric density is the largest cause of uncertainties in orbit predictions of low earth orbits. Time of the day, the season and solar activity all cause a local change of atmospheric temperature which makes the atmosphere expand or contract. This causes large difference in local atmospheric densities at high altitudes and is difficult to model. Atmospheric density may vary with a factor 100 at altitudes above 300 km where solar activity has the greatest influence on the density. W , the power density, from the Sun is almost constant and can be easily and accurately measured. Solar radiation reflected from Earth depends on the surface of Earth in view of the satellite, but is also accurately known from measurements. The position of Eearth, Sun and satellite are required to determine whether the satellite is illuminated by the Sun, or in Earth’s shadow. Larger uncertainties in the radiation pressure force come from the direction of this force, depending on the aberration of the incoming light and the angle of reflection. Fem , the electromagnetic force, is modelled by (3.42) by using a simplified magnetic potential. This simplification together with the uncertain values of q and g1 result in a large relative uncertainty in the electromagnetic force. This force is usually very small compared to other perturbing forces, and therefore does not contribute much to the uncertainty of the total perturbing force. Upper and lower bounds to the electromagnetic force can be determined from observations of satellite orbits. Jn , Jn,m and Λn,m model the gravity field of a planet. In case of Earth, these parameters are determined from the precise orbit determination of satellites. The terms are known up to degree and order (n and m) 360 [38] . Precision (standard error) of the individual values are provided by the different gravity models and depend on the gravity model used. m ˙ and Isp , the parameters defining the thrust of a rocket engine, are often precisely known from rocket engine tests on Earth. The uncertainties of the thrust force on the satellite orbit prediction are the result of the timing of the rocket engine and the direction of the thrust force. Since the thrust force is often large compared to the other perturbing forces, a slight misalignment of the thrust vector can result in large variations in the new orbit. The uncertainties for a satellite maneuver are different for different satellites, rocket engines and maneuver type.

Chapter 4

Unified State Model The Unified State Model (USM) is a combination of coordinate and attitude variable sets, corresponding equations of motion and reference frames developed by Altman in 1972 [1] [66] . It is a type of perturbation method (see Section 3.4) that uses the velocity hodograph as reference â&#x20AC;?orbitâ&#x20AC;? and quaternions to define the orientation of the orbital plane. This novel approach may increase computational efficiency and removes singularities in rotation. However, the qualities of the USM are not yet known because there is no literature on the application of the USM to spacecraft trajectory propagation. This chapter gives an introduction to the USM and defines all necessary information to apply the method to trajectory propagation. Equations of motion and variables for the attitude model are not discussed, but can be found in [1] . Section 4.1 gives a short introduction to the velocity hodograph and quaternions. Section 4.2 provides the reference frames used in the USM. Section 4.3 defines the USM variables and Section 4.4 the equations of motion. The chapter ends with Section 4.5, the perturbation forces expressed in the USM reference frame.

4.1

Introduction

This section introduces the concepts of phase space, velocity hodograph and mathematical ways to represent rotations. These concepts are essential to understand the Unified State Model and its advantages and disadvantages.

4.1.1

Phase Space and Velocity Hodograph

A phase space is a space in which all possible states of a system are represented; every state of the system represents one point in the phase space. For two dimensional systems, the phase space can be shown in a two dimensional diagram. The evolution of the state through time is a trajectory (or orbit) in this diagram. Phase space diagrams give insight in the behaviour of dynamical systems. It can show stability, periodicity, equilibrium points, limit behaviour and is the starting point for a wide range analysis of dynamical systems. [67] As a planet or satellite moves along a closed elliptical orbit or along an open parabolic or hyperbolic trajectory, rotation of the velocity vector is non-uniform, and both the direction and magnitude of the vector change. However, these variations occur in such a way that the end of the velocity vector generates a circle in velocity phase space (the Velocity Hodograph, see Figure 4.1) whose centre is not

4.1. Introduction

Chapter 4. Unified State Model

Figure 4.1: Postition and Velocity phase space (Velocity Hodograph). Source: Altman [1] .

Figure 4.2: Phase spaces of a Keplerian orbit. Source: Altman [1] .

4.1. Introduction

Chapter 4. Unified State Model

at the origin. In other words, the hodograph of the velocity vector for an arbitrary Keplerian motion is a circle, even for hyperbolic orbits. Figure 4.1 shows the definition and construction of a velocity hodograph (velocity phase space). A Keplerian orbit, but not the position in the orbit, can thus be described by two constants (states); the position of the centre of the circle and the radius of the circle in the velocity phase space. This interesting property is ignored in almost all textbooks on mechanics that treat the orbital motion. The Unified State Model uses these constants as state variables in a dynamical model of (perturbed) Keplerian motion. [68] [50] Figure 4.2 shows the position, velocity and acceleration spaces of different orbits around a single spherical mass. One can see that only the velocity phase space geometry remains invariant; a circle. This means that a differential formulation of the orbital trajectory dynamics, due to perturbing forces, will not encounter singularities in the state variables; these velocity parameters are regularized. [1]

4.1.2

Rotations

The orientation of a rigid body or plane (e.g., a Keplerian orbit) in three dimensional space can be described by a rotation of the body fixed reference frame with respect to the another reference frame. The orientation can thus be represented by rotation(s). Rotations in three dimensional space can be defined in different mathematical forms. Different forms have different properties and qualities. Euler Angles and Quaternions keep track of rotations, while Rotation Matrices provide a convenient way to transform between different reference frames. A short description of Euler Angles, Rotation Matrices and Quaternions will be given in the following paragraphs. Euler Angles Three dimensional rotations are often represented by three Euler angles, see Figure 4.3. These three angles describe, for example, the orientation of a Keplerian orbit with respect to an inertial frame or the rotation of a reference frame with respect to another reference frame. Euler angles were already used in astrodynamics as three of the six Kepler elements before Euler started to use them in general mechanics. The three angles represent three successive rotations about the coordinate axis. The order of the three rotations, the direction of the rotation and the axis of rotation all influence the final rotation. A clear definition of the angles and sequence of rotation is therefore always required when using Euler angles. Different conventions exist for different fields of research and application. In this chapter we use the three Kepler rotation elements as Euler angles. Euler angles are easy to use, intuitive and can often be directly determined from observations, but they degenerate (lose a degree of freedom, also called â&#x20AC;?gimbal lockâ&#x20AC;?) for some Euler angles and the conversion to rotational transformation matrices may result in singular matrices (determinant is zero, inverse does not exist). Quaternions are introduced in the Section 4.1.2 to solve these problems of Euler angles by using four parameters to describe the rotation. Rotation Matrix A rotation matrix is a square matrix that defines a rotation, for example, the orientation of reference frame F1 with respect to reference frame F2 . A transformation matrix can be constructed from Euler angles or any other parameter set that defines the orientation of a rigid body. Rotation matrices can also be converted back to Euler angles or any other

4.1. Introduction

Chapter 4. Unified State Model

Figure 4.3: Euler Angles α, γ and β. Source: WikiMedia [69] .

Figure 4.4: According to Euler’s rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices D, C, and B, then a general rotation A can be written as A = BCD. Source: Wolfram [70] . orientation parameter set. A rotation matrix does not contain information about possible translation between reference frames. Any vector that is expressed in one reference frame can be expressed in a second reference frame, with same origin, by multiplication with the rotation matrix. For example, rotation matrix Tib rotates from reference frame b to i, xi = Tib xb

(4.1)

where xi and xb are the state vector in the inertial and body fixed reference frame (in vector space). As example, we examine the two dimensional rotation of coordinate axis in the plane by an angle θ, as seen in Figure 4.5. From basic trigonometry, we can see that, x2 = x1 cos θ + y1 sin θ

(4.2)

y2 = −x1 sin θ + y1 cos θ

(4.3)

which we can write in matrix form, the rotation matrix, cos θ sin θ T= − sin θ cos θ

(4.4)

4.1. Introduction

Chapter 4. Unified State Model

Figure 4.5: Point P in reference frame XY and frame X’Y’, rotated over an angle θ with respect to frame XY. We want rotation matrices to be orthogonal such that the inverse of the matrix is simply its transpose, T T−1 (4.5) ab = Tab = Tba A matrix is orthogonal if it has the following properties [71] , • all eigenvalues are 1. one eigenvalue is 1 and the other two are -1. one eigenvalue is 1 and the other two are complex conjugates. • the determinant of the transformation matrix must be equal to 1, to preserve the size of the vectors. Applying this property to (4.1) yields, xb = Tbi xi

(4.6)

xi = TTbi xb

(4.7)

When Euler angles are used to construct the rotation matrix, the rotation matrix can be defined as three subsequent multiplications of transformation matrices. Every transformation matrix is the rotation about one of the fixed axis. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out, the direction of positive rotation and the sequence of rotations [70] . Rotation matrices are useful to transform (rotate) vectors between different reference frames in a fast and efficient way that is suitable for use on computers [72] . Quaternions The product of two complex numbers can be interpreted as the rotation of vectors in a plane. Quaternions, or hyper-complex numbers of rank 4, are an analogous relation in three dimensional space and use four parameters. Quaternions were first used and published by

4.1. Introduction

Chapter 4. Unified State Model

William Hamilton in 1847 [73] , after his realization that three parameters were not enough for a complete non-singular description of rotation. Four parameters are the minimal set for defining a non-singular mapping between the parameters and their corresponding transformation matrix [74] . Quaternions provide this improvement over Euler angles. [75] [76] Part of Hamiltonâ&#x20AC;&#x2122;s quaternion definition were the following rules [73] , i2 = j2 = k2 = ijk = â&#x2C6;&#x2019;1 ij = k

jk = i

ki = j

ji = -k

kj = -i

ik = -j

(4.8) (4.9) (4.10)

With these relations, we now define the quaternion, q, as a number with a scalar and a three dimensional vector part, q = q4 + q1 i + q2 j + q3 k

(4.11)

where i, j and k are the standard orthonormal basis vector in R3 and q4 , q1 , q2 and q3 are called Euler parameters. Quaternions satisfy all algebraic properties (associative and commutative) under the operations of addition and multiplication, except for the commutative law for multiplication. They can be used in a wide range of applications and mathematical fields. However, in this section we focus on the quaternionâ&#x20AC;&#x2122;s ability to represent a rotation in three dimensions. For use in a three dimensional rotation, quaternions have to satisfy the following constraint to make the degrees of freedom equal to three instead of four, q42 + q12 + q22 + q32 = 1

(4.12)

Since every rotation in three dimensions can be represented by a rotation angle, Î¸, about a unit vector [Ë&#x2020; x0 yË&#x2020;0 zË&#x2020;0 ]T , this rotation angle and vector can be represented by a quaternion (or a quaternion can be written in terms of an angle and spatial vector), q4 = cos

Î¸ 2

q1 = x Ë&#x2020;0 sin

Î¸ 2

q2 = yË&#x2020;0 sin

Î¸ 2

q3 = zË&#x2020;0 sin

Î¸ 2

(4.13)

We can thus easily compute the rotation angle and axis from the quaternion. This quaternion or rotation angle and vector representation of a rotation does not cause the gimbal lock problem because the rotation is described in a single move (independent of order). This is an advantage over the order dependent successive rotations of Euler angles. The conversion from quaternion to rotation matrix and Euler angles is given in the next section.

4.1.3

Transformations between Rotation Representations

The following transformations can be used to convert between the different rotation representations; Euler angles, rotation matrices and quaternions. Quaternions to Rotation Matrix The rotation matrices that rotate from reference frame i to the rotated frame b are given by [1] , ďŁŽ ďŁš 11 12 13 Tbi = ďŁ° 21 22 23 ďŁť (4.14) 31 32 33

4.1. Introduction

Chapter 4. Unified State Model

TTbi

ďŁŽ 11 = ďŁ° 12 13

ďŁš 31 32 ďŁť 33

21 22 23

(4.15)

where ii are given in terms of the Euler parameters by, 11 = 1 â&#x2C6;&#x2019; 2(q22 + q32 )

(4.16)

22 = 1 â&#x2C6;&#x2019; 2(q12 + q32 )

(4.17)

q22 )

(4.18)

12 = 2(q1 q2 + q3 q4 )

(4.19)

13 = 2(q1 q3 â&#x2C6;&#x2019; q2 q4 )

(4.20)

21 = 2(q1 q2 â&#x2C6;&#x2019; q3 q4 )

(4.21)

23 = 2(q2 q3 + q1 q4 )

(4.22)

31 = 2(q1 q3 + q2 q4 )

(4.23)

32 = 2(q2 q3 â&#x2C6;&#x2019; q1 q4 )

(4.24)

33 = 1 â&#x2C6;&#x2019;

We can also write the rotation matrices ďŁŽ 1 â&#x2C6;&#x2019; 2(q22 + q32 ) Tbi = ďŁ°2(q1 q2 â&#x2C6;&#x2019; q3 q4 ) 2(q1 q3 + q2 q4 )

2(q12

directly in terms of the quaternions, ďŁš 2(q1 q2 + q3 q4 ) 2(q1 q3 â&#x2C6;&#x2019; q2 q4 ) 1 â&#x2C6;&#x2019; 2(q12 + q32 ) 2(q2 q3 + q1 q4 )ďŁť 2(q2 q3 â&#x2C6;&#x2019; q1 q4 ) 1 â&#x2C6;&#x2019; 2(q12 + q22 )

(4.25)

Euler Angles to Rotation Matrix The rotation between two reference frames can be performed in 12 different ways. Three consecutive rotations are performed about one of the axis of the body-fixed reference frame. No consecutive rotations about the same axis are allowed. This results in 12 possible sequences to perform the three dimensional rotation. [77] If the rotation sequence is z â&#x2020;&#x2019; y â&#x2020;&#x2019; x, with Îą, Î˛, Îł the Euler angles about the x, y, z axis, the complete rotation is descbibed by, xa = Tab xb = Tx (Îą)Ty (Î˛)Tz (Îł)

(4.26)

where, ďŁŽ 1 ďŁŻ0 Tx (Îą) = ďŁ° 0

ďŁš 0 sin Îą ďŁş ďŁť

0 cos Îą â&#x2C6;&#x2019; sin Îą

ďŁŽ ďŁš cos Î˛ 0 â&#x2C6;&#x2019; sin Î˛ ďŁŻ ďŁş 1 0 ďŁť Ty (Î˛) = ďŁ° 0 sin Î˛ 0 cos Î˛ ďŁŽ ďŁš cos Îł sin Îł 0 ďŁŻ ďŁş Tz (Îł) = ďŁ°â&#x2C6;&#x2019; sin Îł cos Îł 0ďŁť 0

(4.27)

cos Îą

(4.28)

(4.29)

4.2. Reference Frames

Chapter 4. Unified State Model

Quaternions to Euler Angles The Euler angels in terms of the Euler parameters are given by,     4 q1 +q2 q3 ) arctan 2(q α 1−2(q12 +q22 )  β  =  arcsin(2(q4 q2 − q3 q1 )) 2(q q +q q ) 4 3 1 2 γ arctan 1−2(q2 +q2 ) 2

(4.30)

√ 1

For q1 = q2 = 2 2, the arctangent contains a division by zero. This problem can be solved by using the atan2 function in computer implementations. Atan2(1,0), for example, correctly yields π2 , while arctan( 10 ) is not defined. Euler Angles to Quaternions Quaternion in terms of Euler Angles (Kepler rotation angles); i, Ω and u (u = θ + ω),     sin 2i cos Ω−u q1 2 q2   sin i sin Ω−u  2 2   = (4.31) q3   cos i sin Ω+u  2 2 i Ω+u q4 cos 2 cos 2 For orbits with zero eccentricity, e = 0, the argument of periapsis, ω, and therefore the argument of latitude, u, are undefined. This is a problem of Kepler elements and means that circular orbits can not be converted to and from Kepler elements.

4.2

Reference Frames

In this section we define the reference frames that are used in the USM. The USM uses three orthogonal reference frames to describe the motion of an orbiting body. The three reference frames are visualized in Figure 4.6 and explained below. • (Fg ), the planetocentric inertial reference frame • (Ff ), an intermediate frame • (Fe ), the rotating body frame The axis of the reference frames are defined by the following unit vectors, • (g1 , g2 , g3 ) • (f1 , f2 , f3 ) • (e1 , e2 , e3 ) g1 and g2 lie in the equatorial plane and are fixed in inertial space. g1 can, for example, be defined to point towards the mean vernal equinox at noon on January 1, 2000. g3 is directed along the spin axis of the planet and completes the right handed system. The intermediate frame Ff is formed by rotating the inertial frame about the line of nodes. f1 and f2 lay in the orbital plane, f3 is normal to the orbital plane. e1 is directed along the position vector, e2 in the orbital perpendicular to the position vector, e3 is completes the right handed system. e3 and f3 have the same direction.

4.3. State Description

Chapter 4. Unified State Model

Figure 4.6: Reference frames and angles used in the USM. Source: Chodas [61] based on Altman, 1972 [1] . The position and velocity of the rotating body reference frame with respect to the inertial reference frame are given by,     r X  Y  = TTbi 0 (4.32) 0 Z     X ve1 d   Y = TTbi ve2  (4.33) dt Z 0 where TTbi is the transpose of rotation matrix (4.14). And r, ve1 and ve2 can be determined from the Unified State Variables, which is presented in the next section.

4.3

State Description

The state vector of an orbiting body consists of variables (called ”dynamical variables” or ”coordinate sets”) that define the state of the body such that past and future states are uniquely defined. An example of a state vector is a combination of the Cartesian position and velocity vector. However, the state can also be described in variables that describe the geometry of the (periodic) motion, rotations and in variables with a less clear physical representation. Examples of state variables are Cartesian position and velocity and Kepler elements. Many alternative sets of variables are available to describe the state of a moving point mass in three dimensions. The minimum number of variables is six, resulting from the six

4.3. State Description

Chapter 4. Unified State Model

first order differential equations that have to be solved, but more can be used. Different sets have different analytical and computational properties, but none is best or most efficient for all problems. [50] The Cartesian coordinates, for example, are easy to visualize, understand, transform and model with. They do, however, change rapidly for satellite orbits and do not use the simple geometry of orbits to increase computational efficiency in trajectory propagation. Kepler elements, on the other hand, contain five constants. The five constant Kepler elements completely describe the geometry of a Keplerian orbit, its orientation in three dimensional space. The sixth element describes its position on the ellipse as function of time. Kepler elements can be used for analytical orbit analysis and more accurate and efficient orbit simulations. The use of (almost) constant variables also has advantages for interval integration, this will be explained in detail in Chapter 6. However, changing Kepler elements are more difficult to implement and transform. The USM variables also contain constants, not related to the shape of the orbit, but to the shape of de velocity hodograph (see Section 4.1.1), which is always a circle for Keplerian orbits. For the orientation of the orbit, the Unified State Model uses a quaternion (four parameters) instead of the three Euler angles in the Kepler elements. This removes the gimbal lock problem with Euler angles and guarantees transformation matrices to have an inverse. To show the similarities and differences between Cartesian, Kepler and USM state descriptions, a short overview of the Cartesian state vector and Kepler elements will be given in the next subsections, before we present the full USM state description.

4.3.1

Cartesian Coordinates

Cartesian position (r = [x y z]T ) and velocity (v = [vx vy vz ]T ) vectors are widely used as state variables in dynamical modelling. When these two vectors are known at a certain time, they can serve as initial values in an initial value problem. Future and past states of the body can be determined from these initial values by integrating the differential equations that describe the motion of the body. Cartesian coordinates are very general, easy to visualize and have excellent algebraic properties because Cartesian coordinates describe a orthogonal vector space. Cartesian coordinates have no specific advantages when used to describe an orbiting body. It does not contain direct information about the shape, size or orientation of the orbit in case of a two body problem. Cartesian coordinates only contain position and velocity information about a point and need integration to determine past and future position and velocities.

4.3.2

Kepler Elements

A Keplerian orbit is a conical section, six Kepler elements completely define the position and velocity of the body in three dimensional space. Kepler elements are a set of integration constants that have a physical interpretation for (instantaneous) Keplerian orbits. See Figure 3.3, Figure 3.4 and Section 3.1.4 for the definitions of the Kepler elements. e, the eccentricity a, the semimajor axis i, the inclination â&#x201E;Ś,the longitude of the ascending node

4.3. State Description

Chapter 4. Unified State Model

Figure 4.7: Position phase space of a Kepler orbit. Source: Altman [1] . ω, and the argument of periapsis τ , the time of pericenter passage; Some of the above mentioned Kepler elements can be interpreted as describing the motion (orbit) in position phase space (an ellipse) for the two body problem and others as angles for the orientation of the orbital plane. The method of variation of orbital elements (Section 3.4.2) and the method of Encke (Section 3.4.1) both use the fact that Kepler elements are constant for a two body problem. They use this constant orbit as reference orbit and only integrate (small) perturbations with respect to this reference orbit. This makes it possible to use larger integration stepsizes or a higher accuracy and precision. Position Phase Space Elements e, a can be interpreted as constants that define the shape and size of a Keplerian orbit in the position phase space, see Figure 4.7. Euler Angle Elements Ω, ω and i can be interpreted as variables that define the orientation of the orbit in three dimensional space. These three angles are now called Euler angles, their definition can be seen in Figures 3.4. The last element, τ , determines the position of the body in the orbit and is a function time.

4.3.3

Unified State Model Variables

The USM uses seven state variables to describe the state of the orbiting body. C, Rf 1 and Rf 2 are functions of the radial and angular momentum that determine the energy of the satellite and the size and position of the orbit in velocity phase space (velocity hodograph), see Figure 4.1. Rf 1 and Rf 2 ) are the components of R in the f1 and f2 direction. q4 , q1 , q2 , q3 are four Euler parameters defining a quaternion (see Section 4.1.2) that determines the orientation of the satellite fixed reference frame, Fb with respect to the inertial reference frame, Fi . [1] The USM variables differ from Cartesian coordinates because the USM uses the geometry of the velocity phase space and its orientation in inertial space to define the state of a satellite,

4.3. State Description

Chapter 4. Unified State Model

where as in Cartesian coordinates, the position and velocity vector determine the state. The USM variables are somewhat similar to Kepler elements because Kepler elements describe the geometry and orientation of the orbit in position phase space, and the USM describes the orientation and geometry of the orbit in velocity phase space. Velocity Phase Space Variables While the Kepler elements a and e define the shape and size of an ellipse in the position phase space (see Figure 4.7), the USM uses C and R to define the shape and size of a circle in the velocity phase space (see Figure 4.8). C and R are the magnitude of the velocity vector (v) components R and C, v=R+C (4.34) R always has the same direction, in the direction of the velocity at periapsis. C is always perpendicular to the radius vector. For an unperturbed two body problem, the magnitude of both R and C is constant. We can see from Figure 4.8 that the velocity phase space is a circle. We will show this mathematically and derive an expression for the radius, C, of this circle. For the acceleration of an orbiting body we have, µ d2 r = − 3r 2 dt r

(4.35)

˙ 2 pλ = mrve2 = mθr

(4.36)

1 m˙ m dθ = θ= r2 pλ pλ dt

(4.37)

we introduce the angular momentum,

which can be rewritten as,

substituting (4.37) into (4.35) yields, dv µm r dθ d2 r = =− 2 dt dt pλ r dt

(4.38)

for a small time interval ∆t, the magnitude of the velocity increment is given by [68] , |∆v| =

µm ∆θ pλ

(4.39)

the increment in speed (magnitude of the velocity) is proportional to the angle ∆θ. Furthermore, ∆v is always directed in the opposite direction of the radius vector, r, because the (gravitational) acceleration is always directed towards the center of the central body. A geometric interpretation is that the subsequent ∆v’s create a circle in velocity space, with radius µm pλ . We have thus derived the magnitude of C, C=

µm pλ

(4.40)

We can write C in terms of the velocity at periapsis and apoapsis using the fact that the orbital energy is constant, the orbital energy at periapsis and apoapsis are thus equal, ! 2 vp2 µ va µ m − =m − (4.41) 2 rp 2 ra

4.3. State Description

Chapter 4. Unified State Model

Figure 4.8: Top: Position phase space of a Keplerian orbit with velocity components C and R together with the radial and angular velocity components ve1 and ve2 . Bottem: The velocity phase space of the same Keplerian orbit, again with velocity components C and R as in the top figure.

4.3. State Description

Chapter 4. Unified State Model

we solve for µm, where we use that rp vp = ra va and mrp vp = pλ , µm =

(vp2 − va2 ) pλ 2(vp − va )

(4.42)

For which follows for C, C=

µm 1 = (vp + va ) pλ 2

(4.43)

We also note that C can be q written in terms of the velocity at periapsis, vp , and the circular velocity at periapsis, vcp = rµp , C=

vc2 µm µ = = p pλ rp vp vp

(4.44)

We will now derive an expression for the displacement of the centre of the velocity hodograph, R. From Figure 4.8 we can see that R is the vectorial sum of ve1 and ve2 − C, R = ve1 + (ve2 − C)

(4.45)

and for its magnitude we have, R=

2 + (v 2 ve1 e2 − C)

(4.46)

which can be written in terms of the radial and angular momentum, pr and pλ , pr = mve1 pλ = mrve2 as, s R=

p 2 r

pλ µm − mr pλ

2 (4.47)

and in terms of the speed at periapsis and apoapsis, R=

1 (vp − va ) 2

and, at last, in terms of C and the energy, E, p R = 2E + C 2

(4.48)

(4.49)

For a two body problem and no perturbation forces, C, Rf 1 and Rf 2 are constant. This means that they can be used in a method similar to the method of variation of orbital elements (Section 3.4.2) to efficiently compute the deviations from the reference orbit due to perturbing forces. The method of variation of USM variables will be presented in the next section, where the derivatives of the USM variables due to perturbing forces will be presented.

4.4. Variation of Unified State Variables

Chapter 4. Unified State Model

Quaternion Variables The USM uses a quaternion to keep track of the orientation of the orbital plane. Quaternions are chosen in the USM because at least four parameters are necessary for a singular free description of orientation. q4 , q1 , q2 , q3 are the four parameters in, q = q4 + q1 i + q2 j + q3 k and are called Euler parameters. To keep the number of degrees of freedom equal to three, the following constraint is introduced that completes the quaternion rotation description in the USM, q42 + q12 + q22 + q32 = 1 Quaternions have the following major advantages over Euler angles, see [1] [74] [75] [76] , • No singular conditions in rotation occur; the degrees of freedom are never lost. • All angular functions are algebraic; no trigonometrical functions.

4.4

Variation of Unified State Variables

The unified state variables C, Rf 1 and Rf 2 are constant for the unperturbed two body problem. If perturbing forces act on the body, the time rate of change of these state variables due to these force components in the e1 , e2 and e3 direction is described by the following first order differential equations, as derived by Altman [1] and corrected by Chodas [61] and Vittaldev [78]      C 0 −p 0 ae1 d  Rf 1  = cos λ −(1 + p) sin λ −γRf 2 /ve2  ae2  (4.50) dt Rf 2 sin λ (1 + p) cos λ −γRf 1 /ve2 ae3 where,

1 2q3 q4 sin λ = 2 cos λ (q3 + q42 ) q42 − q32

(4.51)

which becomes singular for q3 = q4 = 0. From (4.31) we can see that this corresponds with retrograde equatorial orbits (i = 180 degree). This means that the USM breaks down for retrograde equatorial orbits. This orbit can be avoided by using a negative orbital velocity and zero inclination. The body fixed velocity components are given by, ve1 0 cos λ sin λ Rf 1 = + (4.52) C − sin λ cos λ Rf 2 ve2 The parameter γ is used in (4.50) for compactness and is given by, γ=

q1 q3 − q2 q4 q32 + q42

(4.53)

This term also becomes singular for q3 = q4 = 0, which are polar orbits as explained above. The variable p is the ratio between C and the velocity perpendicular to the radius vector, p=

C ve2

(4.54)

4.5. Perturbing Forces in the Unified State Model

Chapter 4. Unified State Model

C and ve2 have the same direction. Only the magnitude of ve2 changes for a unperturbed two body problem. ve2 is largest at periapsis and smallest at apoapsis. C is given in terms of the velocity at periapsis and apoapsis by (4.44), so that we can write for the minimum and maximum values of p, vp + va 1 1−e 1 rp pmin = = +1 = +1 (4.55) 2vp 2 1+e 2 ra vp + va 1 1+e 1 ra pmax = = +1 = +1 (4.56) 2va 2 1−e 2 rp for elliptical orbits this means that p > 0 and for circular orbits p = 1. p is singular for ve2 = 0, which is the case when the body is not orbiting and moves in a straight lines to or away from the centre of gravitational attraction. This situation does generally not happen in reality and is of no concern for the simulation of satellite orbits. Finally, a is the acceleration vector with components in the e1 , e2 and e3 direction.   ae1 X ai (4.57) a = ae2  = ae3 The time rate of change of the quaternion variables is given by,      q1 0 ω3 0 ω1 q1  q2   1 −ω3 d  q 0 ω 0 2 1  =    −ω1 0 ω3  q3  dt q3  2  0 −ω1 0 −ω3 0 q4 q4

(4.58)

where ω are the body-fixed angular velocity components, ae3 ve2 ω2 = 0 ω1 =

ω3 =

2 Cve2

(4.59) (4.60) (4.61)

The angular velocity, ω2 , about the e2 axis is zero because there never is a velocity component out of the instantaneous orbit plane. It should be noted that the constraint on the quaternion, q12 + q22 + q32 + q42 = 1

(4.62)

still holds.

4.5

Perturbing Forces in the Unified State Model

The variational equations (4.50) and (4.58) are functions of the acceleration in the e1 , e2 and e3 direction. These perturbing force component can be derived for all perturbing forces given in Section 3.3. Only the atmospheric drag and J2 gravity perturbations are given in this section.

4.5. Perturbing Forces in the Unified State Model

Chapter 4. Unified State Model

Arbitrary Force An arbitrary force F on the body is included in the model as the acceleration in (4.50), a=

F m

(4.63)

where only the force components in the bode fixed reference frame (e1 , e2 , e3 ) are required. Transformation from inertial to body fixed reference frame can be done by the transformations given in Appendix A. This, however, means that the USM variables have to be converted to Cartesian variables to compute the force. This increases computation steps and may lead to overestimation in the in interval integration methods that will be discussed in Chapter 6 and are essential for guaranteed collision avoidance. Therefore a more direct expression for the acceleration components in the body-fixed reference frame were derived for atmospheric drag and disturbed gravity fields by Altman [1] . These will be presented in the next subsection, for Atmospheric Drag The drag is typically modelled by, 1 Fdrag = â&#x2C6;&#x2019; Ď CD Svv 2

(4.64)

where CD is the drag coefficient of the satellite that has to be determined from experiment, Ď the density of the atmosphere, S the frontal surface area of the satellite and v the velocity of the spacecraft with respect to the atmosphere. Altman used an atmospheric density model from Broglio [1] . This model from 1964 can not be found in the literature and is, according to Altman, based on the assumption that, â&#x2C6;&#x161;

Ď r

= constant

(4.65)

where Ď is the density, r the orbital radius and M the satelliteâ&#x20AC;&#x2122;s Mach number. This model, models the density as a function of the density at sea level. However, at the altitudes where satellites orbit (>100 km), the atmospheric density has little relation to the density at sea level and Broglioâ&#x20AC;&#x2122;s model is therefore not suitable for high altitudes [38] . Altman derived the body-fixed drag components in terms of the USM variables, where he used Broglioâ&#x20AC;&#x2122;s atmospheric model. The author was not able to reproduce and verify his derivation. His expressions are therefore not very accurate and will not be used in the collision avoidance system. His drag equations are given for completeness, ďŁŽ ďŁš ďŁŽ ďŁš Îś ae1 vr/e1 Ď C S Cv 0 D e2 ďŁ°ae2 ďŁť =â&#x2C6;&#x2019; vr ďŁ°vr/e2 ďŁť (4.66) 2 m Âľ vr/e3 ae3 drag where, ďŁŽ ďŁš ďŁŽ ďŁš ďŁŽ ďŁš ďŁŽ 11 13 vr/e1 ve1 0 q ďŁ° 12 13 ďŁ°vr/e2 ďŁť = ďŁ°ve2 ďŁť + rĎ&#x2030;e ďŁ°â&#x2C6;&#x2019; 33 ďŁť + [E] 0 (1 â&#x2C6;&#x2019; 213 )1/2 vr/e3 0 23 â&#x2C6;&#x2019;1 vr the relative velocity to the atmosphere, q 2 2 2 vr = vr/e1 + vr/e2 + vr/e3

ďŁš 12 cos Sz ďŁť 11 sin Sz 0

(4.67)

(4.68)

4.5. Perturbing Forces in the Unified State Model

Chapter 4. Unified State Model

r the distance to the origin of the inertial reference frame, follows from combining (4.43) and (4.36), Âľ r= (4.69) Cve2 Îś the square of the Mach number for the orbital velocity, â&#x2C6;&#x161; Îś= M

(4.70)

CD the drag coefficient of the orbiting body. Ď 0 the atmospheric density at sea level. S the cross section area of the orbital body. q the atmospheric wind velocity in the local horizon plane. Sz the direction of the wind velocity q in the local horizon plane, referred to the projection of the Z-axis on the plane. [E]0 the transformation matrix of body attitude rotation. And Ď&#x2030;e the planetary rate of rotation about its Z-axis. Better atmospheric density models for satellites are presented in Mentenbruck et al. [38] . Precomputed density tables can be used for local polynomial interpolation. This can replace Îś the Ď 20 CvÂľe2 density term in (4.71). The drag in terms of the USM variables then becomes, ďŁŽ ďŁš ďŁš ďŁŽ ae1 vr/e1 1 C S D ďŁ°ae2 ďŁť (4.71) =â&#x2C6;&#x2019; Ď vr ďŁ°vr/e2 ďŁť 2 m ae3 drag vr/e3 Gravity Field The acceleration due to the J2 gravity potential in the direction of e1 , e2 and e3 is given by [1] , ďŁŽ ďŁš ďŁŽ ďŁš 4 1 â&#x2C6;&#x2019; 3 213 ae1 3 Cv e2 ďŁ°ae2 ďŁť = â&#x2C6;&#x2019; ÂľRe2 J2 ďŁ° 2 13 23 ďŁť (4.72) 2 Âľ ae3 J2 2 13 33 where Re is the radius of the planet (Earth). J2 is defined in Section 3.3.2 and ii in Section 4.1.3.

Chapter 5

Conventional ODE Integration This chapter gives a brief overview of conventional ordinary differential equation (ODE) integration methods, to provide a comparison for the methods discussed in Chapter 6. Chapter 6 will discuss guaranteed (also called verified) ODE integration methods where the solution is guaranteed to be contained within a certain enclosure. In contrast with the methods in Chapter 6, the conventional methods in this chapter only yield approximate solutions. Furthermore, there is no way to be sure how accurate these approximation are, although approximate estimates are possible. Conventional methods are accurate (as can be seen from comparisons with experiments) enough for many ODEs, but for some ODEs and IVPs they are not. Moreover, if experiments verify the accuracy of a conventional method for a specific initial value problem, this does not guarantee accuracy for other initial values. This chapter summarizes the well known concepts and methods of ODEs and numerical integration. A more elaborate treatment of this subject can be found in the books by, for example, Stoer [79] , Butcher [80] , Hoffman [81] and Vuik [82] on which this chapter is based. The first Section, 5.1, of this chapter gives a brief introduction in the type of differential equations we will focus on in this report; ordinary differential equations and initial value problems. Section 5.2 gives an overview of the most common numerical integration methods for IVPs. Section 5.3 discusses the errors introduced by using these methods on a computer and the chapter ends with Section 5.4, a short discussion of the accuracy properties of conventional methods.

5.1

Introduction

Ordinary Differential Equations (ODE) are functions depending on a single independent [83] variable; only ordinary derivatives ( dx . dt ), appear in the differential equations An Initial Value Problem (IVP) is an ordinary differential equation together with a specified value, called the initial value (or initial condition), of the unknown function at a given point in the domain of the solution [83] . Trajectory propagation, starting from a certain given position and velocity, is an IVP. The initial position and velocity are the initial values. The equations of motion are the ODEs. Models in physics and engineering are often based on Newtonâ&#x20AC;&#x2122;s second law in which the force on a mass is a function of time, position and velocity. The initial state of the system, and thereby its future states, are determined by the initial values, in this case the position and velocity at the start of the integration, t = t0 . Such a model is an IVP of the following

5.1. Introduction

Chapter 5. Conventional ODE Integration

general form,

d2 r(t) = F(t, r(t)) dt2 r(t0 ) = r0 dr (t0 ) = v0 dt

(5.1)

where F is the force acting on a point with mass m, position r(t) and velocity v(t), and r0 and v0 are the initial position and velocity of the point mass. The second order differential equation can be written as a system of first order differential equations by combining the position and velocity vectors in a new six dimensional state vector x,   x y   z r  (5.2) x= =   v vx   vy  vz For which the derivative becomes, 

 vx    vy   vz  dx v  = = ax  = f(t, x(t)) a dt   ay  az

(5.3)

where a is the acceleration given by, F (5.4) m The resulting general system of first order ordinary differential equations, with initial value, can now be written as, a=

dx(t) = f(t, x(t)) dt x(t0 ) = x0

(5.5)

The system for Earth orbiting the Sun is given in Example 1. Integrating (5.5) yields, Z t x(t) = x0 + f(s, x(s))ds (5.6) t0

Differential equations of this form are generally difficult or impossible to solve analytically (exact). Numerical methods are developed to approximate the solution of differential equations using numbers, general algebraic operations (addition and multiplication) and finite steps, instead of symbols and infinite differential calculus operators. The finite and elementary algebraic nature of numerical methods make them exceptionally suitable for use on computers.

5.2. Numerical Methods

Chapter 5. Conventional ODE Integration

Example 1: System of first order ODEs for Earth orbiting the Sun.   x y   z re  xearth = x = =   ve vx   vy  vz 

vx vy vz



      dx   = f(x) =  msun  −G x dt r3    −G msun r3 y −G mrsun z 3 p r = x2 + y 2 + z 2

5.2

(5.7)

(5.8)

(5.9)

Numerical Methods

Many different numerical integration methods have been developed to integrate functions for which no closed form analytical solution exists. All methods can be divided into single or multi-step methods, and explicit or implicit methods. Single step methods use the current state to calculate the next, while multi-step methods use one or more previous steps to compute the next state. Explicit methods compute the next step from the current step, while implicit methods use the current and estimated next step to compute the next step.

5.2.1

Taylor Series

The Taylor series is a power series representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor series approximation of a function f (x), in the neighbourhood of a, is given by, f (x) =

∞ X f (n) (a) (x − a)n n! n=0

(5.10)

or written in finite terms, called a Taylor polynomial, f (x) = f (a) + f 0 (a)(x − a) +

f (n) (x(a)) f 00 (a) (x − a)2 + ... + (x − a)n + Rn (x) 2! n!

(5.11)

with Rn (x) the remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. We now substitute (5.11) into (5.6), which after a simple analytical integration results in the following Taylor series for the position x(t1 ), x(t0 + h) = x(t0 ) + hf (x(t0 )) + where

d(x(t)) dt

1 2 0 1 h f (x(t0 )) + ... + hn f (n−1) (x(t0 )) + Rn 2! n!

= f (x(t)), h = (t1 − t0 ) and Rn the remainder term.

(5.12)

5.2. Numerical Methods

Chapter 5. Conventional ODE Integration

The Taylor series converges to the true function value for some stepsize h and for most functions. The problem of applying Taylor series in dynamical models is obtaining the higher order derivatives of the state vector. Other numerical methods all try to approximate the Taylor polynomial by function evaluations of the first derivative of x(t), dx(t) dt = f(x(t)) (the force function), only. However, it is also possible to use recurrence relations to compute the exact values of all higher-order derivatives of x(t), at some time t, as long as f(x) is known. This so called ”Automatic Differentiation” is discussed in more detail in Section 6.6 and takes more computational time than methods that only use force-function evaluations (computation of the first order derivative of the state vector).

5.2.2

Euler Methods

The oldest and simplest numerical approximation method is Euler’s method. This method is expressed by the equation, xi+1 = xi + hf (xi ) (5.13) which is the Taylor series truncated after the second term. It is the ”linear” or ”first-order” approximation; it only uses the first derivative of x(t). A simple example is the approximation of the distance travelled, s(t + ∆t), at time, t + ∆t. For the distance and velocity at time t are s(t) and v(t), s(t + ∆t) = s(t) + v(t)∆t

(5.14)

When v(t) is constant, the higher order derivatives of s(t) are zero and the approximation becomes exact.

5.2.3

Runge-Kutta Methods

Runge-Kutta (RK) methods are methods for the numerical solution of the ordinary differential equations, dx = f (t, x(t)) (5.15) dt Runge and Kutta developed their numerical method by trying to create formulas that match the first n terms of the Taylor series (5.10), but without using derivatives of f (x). The RK Taylor series approximation, x(t + h) = x(t) + h

s X

bi gi

(5.16)

i=1

with,  gi = f t + ci h, x(t) + h

s X

 aij gj 

(5.17)

j=1

Euler’s method is in fact a Runge-Kutta method of first order with all coefficients in (5.17) zero except for b1 , which is 1. The most popular Runge-Kutta method is the classic fourth-order method, defined by, 1 x(t + h) = x(t) + (k1 + 2k2 + 2k3 + k4 ) 6

(5.18)

5.2. Numerical Methods

Chapter 5. Conventional ODE Integration

where, k1 = hf (t, x(t)) 1 k2 = hf t + h, x (t) + 2 1 k3 = hf t + h, x (t) + 2

(5.19) 1 k1 2

1 k2 2

(5.20)

k4 = hf (t + h, x(t) + k3 )

(5.21) (5.22)

The coefficients of a Runge-Kutta method, (5.17), can be conveniently organized in a so called ”Butcher tableau”, see Table 5.2.3. The Butcher tableau for the most common fourth order RK method is shown in Table 5.2.3. Table 5.1: General Butcher tableau of Runge-Kutta coefficients. [80] c

A bT

Table 5.2: Butcher tableau of classic fourth order Runge-Kutta method. [79] 0 1 2 1 2

1 2

0 0

1 6

1 3

1 6

RK methods up to at least order 12 are known. Although above order four, it is no longer possible to obtain order n with n function evaluations. For example, a 6th order method requires seven function evaluations and a 12th order 16 evaluations. This means that while the order does increase, the efficiency of the method decreases. Furthermore, there is no optimal set of coefficients for higher-order methods that is best (smallest error) for all differential equations. [84] Error estimates for numerical methods are useful for (adaptive) stepsize determination. ”Embedded” Runge-Kutta methods use the difference between two order approximations as an error estimate. The Dormand-Prince method is an example that uses six function evaluations to calculate the fourth and fifth order solutions, which can be used as error estimate for adaptive stepsize algorithms. The widely used Matlab ode45 solver is based on the Dormand-Prince method.

5.2.4

Multistep Methods

Euler and Runge-Kutta methods, discussed in the previous subsections, are one-step methods because they refer only to one previously calculated value to determine the next value. Multi-step methods use more than one previous value to determine the next value, they are used to try to increase the accuracy over one-step methods. A wide range of multi-step methods exists, the most commonly used are Adams-Bashforth, Adams-Moulton and Backwards Differentiation (BD) methods. Adams-Bashforth methods

5.3. Errors

Chapter 5. Conventional ODE Integration

are explicit, while Adams-Moulton and BD methods are implicit. Only one example of a forth-order explicit Adams-Bashforth method will be given here, h [55f (t, x(t)) − 59f (t − h, x(t − h)) 24 + 37f (t − 2h, x(t − 2h)) − 9f (t − 3h, x(t − 3h))]

x(t + h) = x(t) +

5.3

(5.23)

Errors

Numerical methods use finite numbers and finite steps to approximate continuous differential equations. This finite approximation introduces errors; the differences between the real and approximated solution. There are two kinds of errors, truncation and rounding errors. The error occurring in a single step, assuming no errors in previous steps, is called the local truncation error. The total accumulated error over the whole integration interval is called the global truncation error. Rounding errors are the result of the finite representation of numbers in computers. Truncation Errors For the following discussion about the truncation error, we assume that the computation is done using infinite digit numbers and therefore no rounding errors occur. Rounding errors are discussed separately in the next subsections. Truncation errors are the result of the finite number of steps used in numerical approximations of continuous (infinite small stepsize, infinite steps) differential equations. The truncation error is thus the difference between the real and approximated solution. The local truncation error is the truncation error resulting from a single step. The global truncation error is the summation of all local truncation errors over the integration interval. The global truncation error is thus the total difference between the real and approximate solution, at some time t, if infinite digit numbers would be used. It is not possible to determine the exact size of the truncation error, because the exact solution is not known for all practical applications of numerical methods. Nevertheless, mathematical analysis can give an estimate of the size and proportionality to the stepsize of the local and global truncation error, as we will show now. For Euler’s method, the local truncation error can be analysed by using a higher order truncated Taylor polynomial of x(t) about ti , x(ti + h) = x(ti ) + f (x(ti ))h +

f 0 (x(ti )) 2 h + Rn (x(ξ)) 2

(5.24)

where ξ is some point in the interval ti < ξ < ti + h. Subtracting Euler’s approximation (5.13) from (5.24) we find, x(ti + h) − xi+1 = (x(ti ) − xi ) + (f (x(ti )) − f (xi ))h +

f 0 (x(ti )) 2 h + Rn (x(ξ)) 2

(5.25)

For the local truncation error we assume no errors from previous steps, x(ti ) = xi , this gives the local truncation error, ei+1 as, ei+1 = x(ti+1 ) − xi+1 =

f 0 (x(ti )) 2 h + Rn (x(ξ)) 2

(5.26)

5.3. Errors

Chapter 5. Conventional ODE Integration

Rn (x(ξ)) can be expressed as the Lagrange Remainder term that encloses the remainder, see Section 6.2.1 for a more elaborate discussion and [85] , [86] , [87] and [88] for derivation and proof, Rn (x(ξ))

∈

f 00 (x(ξ)) 3 h 3! [ti , ti + h]

(5.27)

From (5.26) it follows that for small h, the dominant part of the local truncation error of the Euler method is proportional to the square of the stepsize, h2 , and the proportionality factor depends on f 0 (x). A uniform bound, valid on the time interval [ti , ti +h] is given by the Langrage Remainder term, h2 (5.28) |ei | ≤ M 2 where M is the maximum of |f 0 (x(ti ))| on the interval [ti , ti + h]. Since this is a worst case scenario, it may be a considerable overestimate of the actual local truncation error. The global truncation error is more important than the local error, but is also more difficult to analyse. A simple estimate, ignoring the propagation of previous errors, of the global truncation error can be made using the local truncation error. The error in n steps 2 is bounded by nM h2 , noting that n = (tend − t0 )/h, we find for the global truncation error, nM

h2 h = (tend − t0 )M 2 2

(5.29)

Although this is a very rough estimate, it can be shown that the global truncation error is indeed proportional to the first power of the stepsize, h1 . Since computers can only store floating point numbers in finite digits, a different error caused by rounding also occurs. This is the so-called rounding error. Rounding Errors Rounding errors are the result of the finite digit representation of real numbers in computers. Different applications require different precision and therefore different number of memory bits per stored number. When a new number is computed from two stored numbers, the result is rounded and stored in the computer’s memory. This rounding introduces a small error between the stored and real result of the computation. Rounding errors can accumulate and become large in the final result when many computation are involved. The number of bits that are used to store numbers can be defined by the user, although it should be kept in mind that a larger number of bits results in more memory and larger computation times.

Figure 5.1: Layout of 64bit double precision floating point number in computer memory. Image source: Wikipedia [89] .

5.4. Accuracy Properties

Chapter 5. Conventional ODE Integration

Most computer simulations use double precision, 64 bit, floating point numbers. 1 bit is used for the sign of the number, 11 bits for the exponent and 52 bits for the significant precision. This means that a double precision number contains approximately 16 significant decimal digits.

5.4

Accuracy Properties

The accuracy of numerical methods can be described by their convergence, order, stability and stiffness. This section gives a short overview of the concepts that determine the quality of a numerical method. The definitions vary and are not possible to describe exactly and quantitatively. Convergence, stability and stiffness are properties of an integration method in combination with a specific IVP, while order is only a property of the integration method.

5.4.1

Convergence and Order

A numerical method is convergent if the difference between the true solution and the numerical approximation goes to zero if the stepsize goes to zero. ||xi − x(ti )|| → 0 as i → ∞ (h → 0)

(5.30)

Higher order (> 3) Taylor Series and Runge-Kutta methods are convergent for almost all practical IVPs. Non-convergent methods are likely to generate increasingly meaningless results when smaller stepsizes, and more computational time, are used. Therefore, all numerical methods used in science and engineering should be convergent. The order, p, of a method is defined by the power of the stepsize to which the global truncation error is proportional. In general, the local truncation error is proportional to hp+1 and the global truncation error on a finite interval is bounded by a constant times hp . A smaller stepsize decreases the global truncation error but increases computational time and rounding errors.

5.4.2

Stability and Stiffness

A numerical method is stable, for a particular differential equation, if the difference between the real and numerical solution is bounded or goes to zero over time when small disturbances or errors are introduced. It means that a certain error in initial values or an error caused by truncation and rounding errors does not cause the numerical solution to deviate unbounded (the errors are magnified) from the numerical solution. Stability analysis is an important but complicated part of numerical analysis. The stability analysis of different numerical methods and differential equations can often be found in literature. [80] Numerical methods can only be stable for stable differential equations; a numerical method can not make an unstable ODE stable, but can make a stable ODE unstable. Stable differential equations are different equations in which the solutions stays ”close” to the reference solution (Lyapunov stability) or moves back to the reference solution (asymptotic stability) when there is a small disturbance or error in the initial conditions. Differential equations that require a very small (smaller than needed for accuracy) stepsize to become stable for some numerical methods are called stiff. An example of a stiff differential equation is given by Gear [90] , x0 = a(x − p(t)) + p0 (t) x(0) = v

(5.31) (5.32)

5.5. Random Error Propagation

Chapter 5. Conventional ODE Integration

where a is a constant. The analytical solution is, x(t) = (v − p(0)eAt ) + p(t)

(5.33)

For large negative values of a, the solutions converges very fast; the problem is super-stable. This super-stable property turns out to be a problem for numerical integration methods; (5.31) is stiff. Stiff differential equations require special attention and/or methods before the numerical solution can be assumed reasonably close to the real solution. Typical orbit integration problems are non-stiff. For numerical methods for stiff equations to reader is referred to the books by Gear [91] and Hairer and Wanner [92] .

5.5

Random Error Propagation

Current collision avoidance is based on estimates of the collision probability. This collision probability depends on errors in the knowledge of the state (for example, position and velocity) of the satellites at time t0 . These errors are estimated by propagating the (assumed) known error Probability Density Function (PDF) at time t0 to time t1 . [16] [35]

Figure 5.2: Two dimensional Wikipedia [93] .

Gaussian

(normal)

distribution.

Image

source:

The position error is typically modelled as a random vector with a three dimensional Gaussian (normal) PDF. This PDF can be defined in terms of the 3x3 position error covariance matrix, Cp [34] , −1 1 T 1 e−( 2 re Cp re ) f (re ) = p (5.34) 3 (2π) |Cp | where |Cp | is the determinant of Cp and re the position error. Figure 5.2 shows a two dimensional Gaussian PDF. The three dimensional PDF is similar, but with one extra dimension. Cp defines the (shape) spread of the normal distribution. A covariance matrix is a matrix of covariances between elements of a random state vector. The covariance is a measure of how much two state variables affect each other when they change and is defined as, Cov(xi , xj ) = E((xi − x ¯i )(xj − x ¯j ))

(5.35)

where E is the expected value operator and E(xi ) = x ¯i . The covariance Cov(xi , xi ) is called the variance of xi , V ar(xi ), which in turn is the square of the standard deviation σ. V ar(xi ) = σi2

(5.36)

5.5. Random Error Propagation

Chapter 5. Conventional ODE Integration

So for the covariance matrix we have,   V ar(x1 ) . Cov(x1 , xn )  . . . Cp =  Cov(xn , x1 ) . V ar(xn )

(5.37)

If the state of the system is given by position and velocity, the covariance matrix is a 6 × 6 matrix. Since the covariance matrix determines the n-dimensional error PDF, (5.34), we can use the covariance matrix to generate an ellipsoid of constant probability around a reference position. Two common methods to propagate error PDFs are Monte Carlo and covariance matrix propagation. Both methods are biased (approximations) for non-linear systems because they use a truncated series expansion and a finite number of integrations. This makes the methods non-guaranteed (or non-verified).

5.5.1

Monte Carlo

Monte Carlo methods are a class of computational algorithms that use a finite number of random or pseudo-random simulation parameters to approximate a solution. Monte Carlo methods have a wide range of applications including sensitivity analysis of dynamical systems, optimization and particle modelling. [94] Monte Carlo error propagation works by picking pseudo-random (based on the PDF of the initial state) initial values from a specified region around the reference position. It then integrates the orbits with these initial conditions using numerical integration. The resulting state, after integration over a certain time, can be used in estimation methods to estimate the PDF of the simulated state by assuming a Gaussian distribution and estimating the standard deviation (or variance) from the simulation results. This results in a continuous PDF from a finite number of simulations. The advantages of Monte Carlo error propagation is that it is highly general and easy to apply. The continuous PDF allows for an easy way to estimate collision probability. Its generality makes it possible to use accurate high-order numerical methods. And it is easy to apply, because existing orbit integration models can be used. [95] However, Monte Carlo methods are computationally expensive and uncertain. It is a type of brute force method that relies on computational power instead of a analytical solution. The uncertainty comes from the truncation error in numerical integration (see section 5.3) and estimation of the PDF from a finite number of simulations.

5.5.2

Covariance Matrix Propagation

Covariance matrix propagation methods uses the first order state transition matrix of a system to propagate a covariance matrix, 5.37, from time t0 to t1 . Since the n × n covariance matrix defines the n-dimensional PDF, this is a way to propagate the error PDF of a satellite. When the error PDFs of two satellites at time t1 are known, the resulting collision probability can be determined. A overview of the concept is shown in figure 5.3. [35] [97] [16] [98] The state transition matrix describes the first order relation between the change of initial state and the change in final state. The state transition matrix Φ(t, t0 ) for the orbit state vector x = [r, v]T is given by [38] , Φ(t0 , t) =

∂x(t) ∂x0 (t0 )

(5.38)

5.5. Random Error Propagation

Chapter 5. Conventional ODE Integration

Figure 5.3: Analytic and numerical error covariance matrix propagation with exaggerated error ellipsoids at times t0 and t. Image source: Der [96] .

Figure 5.4: Two dimensional mapping onto the B-plane of three dimensional combined position uncertainty of two objects at the time of closest approach. Image source: Klinkrad [35] . The state transition matrix can be determined in a simplified analytical way or by using numerical methods or a combination of both. Different methods for determining the state transition matrix and a comparison between the different methods can be found in [96] . Numerical methods rely on finite difference methods or automatic differentiation (see Section 6.6).

5.5. Random Error Propagation

Chapter 5. Conventional ODE Integration

The state transition matrix can be used to propagate the covariance matrix from time t0 to t1 using the following relation [95] [35] , Cp (t1 ) = Φ(t, t0 )Cp (t0 )Φ(t, t0 )T

(5.39)

A major advantage of covariance matrix propagation is its computational efficiency. This efficiency is the result of the linearisation of the error propagation, which is also the major disadvantage of the method. This linearisation means that possible large truncation errors are ignored, which makes the result an approximation. Several authors have developed methods to determine the probability collision using covariance matrix propagation. They all use the following assumptions [35] , • During the conjunction, the target and risk object move along straight lines at constant velocities. • The uncertainties in the velocities can be neglected. • The target and risk object position uncertainties are not correlated. • The position error covariances during the encounter are constant, corresponding to those at the time of closest approach. • The position uncertainties can be combined in a common three-dimensional Gaussian distribution. Because of these assumptions and the first order approximation of the method, covariance propagation is not suitable for guaranteed integration and thus also not for guaranteed collision avoidance.

Chapter 6

Interval ODE Integration Finding numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs) is one of the fundamental problems in scientific computing. A wide range of conventional integration methods is available to approximate solutions (see Chapter 5). However, these methods yield only an approximate solution and do not take any rounding or truncation errors into account. Error estimates are possible with some conventional methods, but these are non-guaranteed estimates mainly used for stepsize control. [99] Moreover, most can only solve for single point initial values instead of a set of initial values; the solution is a single point in n-dimensional space and not a bounded set of points. For a set of initial values, solutions can be obtained by variational equations or Monte Carlo methods, but remain non guaranteed estimates. Verified (guaranteed) integration methods are part of a group of integration methods that guarantees solutions to differential equations to be within a certain solution set. The methods bound rounding errors, truncation errors and can often handle an interval initial value and parameter set, instead of a single point. This guarantees, by mathematical proof, a solution to be within certain bounds for uncertain initial values and parameters. This guarantee is particularly interesting for critical dynamical systems with high safety requirements, but can also be used for global optimization or simply to check the results of conventional methods. Verified integration has one important disadvantage, it often overestimates the solution bounds. This overestimation is mainly the result of the dependency problem and the wrapping effect, which will be explained later. Most research focusses on reducing this overestimation. Verified integration is also computationally more expensive than conventional integration. The first idea of verified integration came from Moore (1968) [100] when he worked on interval analysis, a method to bound functions for a range (interval) of variables. His method is based on bounding truncated Taylor series integration methods. Development of his method was continued by Eijgenraam [101] , Lohner [102] , Corliss [103] [104] , Rihm [105] [106] and others. These methods, based on interval analysis and Taylor series, are called Interval Taylor Series (ITS) methods. A second group of verified integration methods was developed by Makino and Berz, initially to analyse the stability of particles in particle accelerators [107] [108] [109] . Their method is also based on truncated Taylor series integration, but uses Taylor models in combination with interval analysis to bound solutions. These methods are called Taylor Model (TM) methods. They symbolically propagate initial value sets and use interval analysis to bound rounding and truncation errors.

6.1. Interval Analysis

Chapter 6. Interval ODE Integration

A third group of verified methods was developed by Marciniak [110] and others. These methods modify conventional Runge-Kutta methods to yield guaranteed solutions. This method is more complex than ITS and TM methods, because bounding the truncation error of Runge-Kutta methods is more complex. It has the advantage that conventional Runge-Kutta methods can be used. This chapter focusses on the verified integration of IVPs of ODEs, which we will call Interval ODE Integration. It should give all information about interval ODE integration methods to apply them to spacecraft trajectory propagation with the objective to determine for two satellites whether A: the satellites are 100% guaranteed not to collide or B: an orbit maneuver is required to avoid collision. Since all integration methods use interval analysis, a short introduction to interval analysis is given in Section 6.1. Section 6.2 gives an overview of the Interval Taylor Series methods, Section 6.3 of the Taylor Model methods and Section 6.4 of the Interval Runge-Kutta methods. The bounding of rounding errors is discussed in Section 6.5. A useful differentiation method for Taylor series integration is given in Section 6.6. Software implementations of the methods described in this chapter are shown and compared in Section 6.7. The chapter ends with a few applications of interval (verified) ODE integration in Section 6.8.

6.1

Interval Analysis

Interval Analysis (also called interval arithmetic or interval computing) was developed by Moore in the early 1960’s [100] . It can be seen as an extension of real numbers, an interval, that is represented by a pair of real numbers. This new interval contains all real numbers between the two; an interval represents a range of numbers. This section gives a brief overview of the most important rules of interval analysis. It is based on the works by Moore, [111] and [112] , to which the reader is referred for more information about interval analysis. In mathematical terms, an interval is a closed bounded set of real numbers, [a, b] = {x : a ≤ x ≤ b}

(6.1)

We denote intervals by [x], and the endpoints of the interval by x and x ¯. Thus [x] = [x, x ¯]. ¯ ¯ An n-dimensional interval vector is written as a bold letter, [x], which is a vector that contains the intervals x1 , x2 , .., xn ,   [x1 ]  [x2 ]     [x] =  (6.2)  .   .  [xn ] In this chapter, however, we do not use the bold notation and assume that all intervals, [x] can also be interval vectors. This because it clutters the sometimes long equations and is consistent with the notation in interval literature. Two intervals are equal if their endpoints are equal, that is, [x] = [y]

if x = y and x ¯ = y¯ ¯ ¯ The intersection of two intervals is empty if either x > y¯ or y > x ¯ and else, ¯ ¯ [x] ∩ [y] = [max(x, y), min(¯ x, y¯)] ¯ ¯

(6.3)

(6.4)

6.1. Interval Analysis

Chapter 6. Interval ODE Integration

If two intervals have non-empty intersections, their union is again an interval, [x] ∪ [y] = [min(x, y), max(¯ x, y¯)] ¯ ¯ The set inclusion of two intervals is given by, [x] ⊆ [y]

if and only if y ≤ x and x ¯ ≤ y¯ ¯ ¯ The width of an interval is defined by,

(6.5)

(6.6)

w([x]) = x ¯−x ¯

(6.7)

(x + x ¯) m([x]) = ¯ 2

(6.8)

|[x]| = max(|x|, |¯ x|) ¯

(6.9)

The midpoint as,

And the absolute value as,

6.1.1

Interval Arithmetic

Interval arithmetic describes the rules of elementary operations, like addition and multiplication, for intervals. The rules for addition and subtraction are, [x, x ¯] + [y, y¯] = [x + y, x ¯ + y¯] ¯ ¯ ¯ ¯ [x, x ¯] − [y, y¯] = [x − y¯, x ¯ − y] ¯ ¯ ¯ ¯ The rules for multiplication are,

(6.10) (6.11)

[x] · [y] = [min(xy, x¯ y, x ¯y, x ¯y¯), max(xy, x¯ y, x ¯y, x ¯y¯)] (6.12) ¯¯ ¯ ¯¯ ¯ ¯ ¯ By testing the signs of the endpoints of [x] and [y], this can be reduced to nine special cases (see Moore [112] for details). The multiplicative inverse (reciprocal) is defined as, 1 1 1 = , (6.13) [x] x ¯ x ¯ if [x] is an interval not containing the number 0. If [x] contains 0, the set is unbounded and cannot be represented as an interval whose endpoints are real numbers; both bounds go to infinity. This loses valuable information about the division that cannot be contained in a single interval. However, splitting the interval in two parts, [−∞, 1/y1 ] and [1/y2 , ∞], and using two intervals can contain the information. This has been implemented in several interval software tools discussed in Section 6.7.1. The division of two intervals is defined by, 1 [x] = [x] · [y] [y] and can therefore be computed using relation (6.12) and (6.13).

(6.14)

6.1. Interval Analysis

6.1.2

Chapter 6. Interval ODE Integration

Interval Algebraic Properties

Interval arithmetic has algebraic properties similar to real numbers. A short overview of the most important properties is presented in this section. Addition and multiplication are associative, [x] + ([y] + [z]) = ([x] + [y]) + [z]

(6.15)

[x]([y][z]) = ([x][y])[z]

(6.16)

[x] + [y] = [y] + [x]

(6.17)

[x][y] = [y][x]

(6.18)

and also commutative,

although not always distributive, for example, [1, 2] · (1 − 1) = [0, 0] = 0

(6.19)

[1, 2] · 1 − [1, 2] · 1 = [−1, 1] 6= 0

(6.20)

This means that interval functions are not uniquely defined; the resulting bounds depend on the form of the function. Some particular useful cases were distributivity does hold are, x([y] + [z]) = x[y] + x[z] [x]([y] + [z]) = [x][y] + [x][y]

(6.21) [y][z] > 0

(6.22)

However, the following algebraic property always holds, [x]([y] + [z]) ⊆ [x][y] + [x][z]

(6.23)

This property is called subdistributivity and implies that if we rearrange an interval expression, we may obtain tighter bounds (smaller interval) on the range of the expression or function [113] .

6.1.3

Inclusion Monotonic Interval Extensions of Functions

An interval extension of function f is an interval valued function F of n interval variables [x]1 , ..., [x]n , with the property, F (x1 , ..., xn ) = f (x1 , ..., xn )

(6.24)

So an interval extension of f is an interval valued function which has real values when the arguments are all real and coincides with f . F is an inclusion monotonic interval extension of f when, f ([x]1 , ..., [x]n ) ⊆ F ([x]1 , ..., [x]n )

(6.25)

This means that an inclusion monotonic interval extension F gives bounds on the function when the function variables are within the intervals [x]1 , ..., [x]n . All rational functions (the function can be expressed as a ratio of two polynomials) have natural inclusion monotonic interval extensions; we only have to replace the real variables with interval variables and real arithmetic operations with interval arithmetic operations.

6.1. Interval Analysis

Chapter 6. Interval ODE Integration

For example (from Moore [112] ), consider the polynomial, 1 p(x) = 1 − 5x + x3 3

(6.26)

and suppose we want the range of all values of p(x) when x is a number in the range [2, 3], x ⊆ [2, 3]. Then a natural interval extension of p is, 1 P ([x]) = 1 − 5[x] + [x] · [x] · [x] 3

(6.27)

1 34 P ([2, 3]) = 1 − 1 − 5[2, 3] + [8, 27] = [− , 0] 3 3

(6.28)

Computing P([2,3]),

This result means that we have found the range of values of p(x) when x is in [2, 3]. p(x) is contained in the interval [− 34 3 , 0] We can construct inclusion monotonic interval extensions of all commonly used nonrational functions. Software tools implementing interval extensions for all common functions are discussed in Section 6.7.1.

6.1.4

The Dependency Problem

Interval analysis cannot detect multiple occurrences of the same variable. This leads to overestimation in some cases, as was already noted by Moore. For example, x − x = 0 for all x ∈ [1, 2], but if we evaluate [1, 2] − [1, 2] in interval arithmetic, the result is [−1, 1]. The true solution [0, 0] is contained in the result, but contains a large overestimation. This type of overestimation ”problem” is called the dependency problem. Writing an interval function in a different form can change the enclosure. It may be useful to rewrite the function such that the overestimation is reduced, but removing the dependency is not always possible. It can be the cause of significant overestimation and should always be kept in mind when using interval arithmetic. [112] Apart from rewriting the interval function, the dependency problem can be reduced by the subdivision strategy described in Section 6.2.4. This method works by dividing the interval variables in two or more subintervals and combining the result of subinterval computations. The longer computation time is the only disadvantage of this strategy. Computation time grows roughly linear with the number of n-dimensional subintervals. [114]

6.1.5

Interval Intersections

If the possible positions of two satellites are represented by two intervals, the satellites may collide if these intervals intersect. We can thus rule out collision if the intervals do not intersect. The problem of collision determination is thus reduced to determining whether two intervals intersect. This is easy if we assume the intervals to be in the same coordinate system. As an example we take the two 2-dimensional interval vectors [x] and [y], [x1 ] [y1 ] [x] = [y] = (6.29) [x2 ] [y2 ]

6.1. Interval Analysis

Chapter 6. Interval ODE Integration

The intervals [x] and [y] intersect if, and only if, one of the corners of [x] is within [y], y1 ≤ x1 ≤ y¯1 and y2 ≤ x2 ≤ y¯2 or ¯ ¯ ¯ ¯ y1 ≤ x ¯1 ≤ y¯1 and y2 ≤ x ¯2 ≤ y¯2 or ¯ ¯ y1 ≤ x1 ≤ y¯1 and y2 ≤ x ¯2 ≤ y¯2 or ¯ ¯ ¯ y1 ≤ x ¯1 ≤ y¯1 and y2 ≤ x2 ≤ y¯2 ¯ ¯ ¯ For three dimensional interval vectors, we can determine in a similar way whether the 8 corners of vector one are within the box represented by vector two.

[y]

[x]

Figure 6.1: Intersection of two intervals [x] and [y].

6.2. Interval Taylor Series Methods

6.2

Chapter 6. Interval ODE Integration

Interval Taylor Series Methods

We consider the following ordinary differential equation and initial value, x0 (t)

= f (x(t))

x(t0 )

= x0

(6.30)

Which can be written in integral form as, Z x(t)

f (x(τ ))dτ

x(t0 ) +

(6.31)

x(t0 )

We can locally approximate f (x(t)) and use it in the integral form (6.31) if we can find a Taylor polynomial of f (x(t)) about t0 . After integrating (6.31), we obtain a new polynomial which is the Taylor series approximation of x(t) about t0 , x(t0 + h) = x(t0 ) + hf (x(t0 )) +

1 2 0 1 h f (x(t0 )) + ... + hn f (n−1) (x(t0 )) + Rn (t) 2! n!

(6.32)

A polynomial can be evaluated on a computer because it involves only elementary algebraic operations, in contrast with the integral operator. See Chapter 5 for more details on conventional numerical integration using Taylor series. Interval Taylor Series (ITS) methods are a type of interval integration methods that use Taylor series to construct guaranteed bounds, in interval form, on the solutions of ODEs. It can handle interval initial values [x0 ], such that x(t0 ) ⊆ [x0 ] and interval parameters (constants) in the function f (x(t)) as [cn ], such that cn ⊆ [cn ]. ITS methods create an inclusion monotonic interval extension of the Taylor polynomial of f (x(t)), to create guaranteed bounds on this polynomial, for interval valued variables and parameters. Because the polynomial can only be evaluated to a finite number of terms, the series has to be truncated at some term which introduces an error with respect to its true value. We call this error the remainder term. This remainder term also has to be bounded for a guaranteed solution of the ODE. Moore was the first to publish the idea of interval integration using interval bounded Taylor series in 1966 [111] . Kr¨ uckeberg [115] , Eijgenraam [101] , Lohner [116] and others used Moore’s ideas and improved his method to solve initial value problems of ODE. Lohner was the first to implement his ODE interval integration method in the computer software called AWA (AnfangsWertAufgabe) in 1989 [102] . Although many mathematicians are interested in the concept of guaranteed integration over a range of initial values [80] , application of the method is limited to simple example or benchmark problems such as the mathematical pendulum, double pendulum and Lorenz equations. Mainly because the method is computational more expensive and more complex to implement than conventional integration methods.

6.2.1

Bounding Taylor Series

To create guaranteed bounds on the solutions of IVPs using Taylor series integration, one has to bound the Taylor polynomial of a single integration step, including the local truncation error (the remainder, Rn in 6.32). The next step can then use these bounds as initial interval values for the next integration step.

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

The global truncation error is thereby propagated, throughout the complete integration, in the initial values of every step. The same is true for the interval initial values and parameters. In case of the Taylor series method, the local truncation error is the remainder in the Taylor polynomial of x(t) about some point (see Section 5.2.1). To bound the real solution x(t) for the IVP with interval initial values, we have to find bounds on this polynomial and its remainder term, for interval initial values and parameters. We also have to bound rounding errors, because the method is used on a computer that uses finite precision numbers. The remainder and rounding error are both the result of how digital computers work. The remainder term is the result of the finite evaluation of Taylor series and the rounding error the result of finite precision numbers used in computers. Interval Variables and Constants The Taylor series expansion of x(t) contains the model’s variables and constants. We want to use interval valued variables and constants and still guarantee bounds. This can be done by using the inclusion monotonic interval extension (see Section 6.1.3) of the Taylor series expansion of x(t). This simply means evaluating the polynomial using interval arithmetic and interval variables and constants. The resulting interval bounds the truncated polynomial for the interval variables and constants. However, to bound the infinite Taylor series, we also have to bound the remainder term. Remainder Term We take a look at the Taylor polynomial of x(t) in ODE (6.30) about t0 , x(t0 + h) = x(t0 ) + hf (x(t0 )) +

1 1 2 0 h f (x(t0 )) + ... + hn f (n−1) (x(t0 )) + Rn (t) 2! n!

(6.33)

where Rn (t) is the remainder term which represents all the higher order terms, up to infinity, that will not be computed. This remainder term is the local truncation error of the integration method. This subsection shows a method to bound the remainder term. The exact value of the remainder term can only be found by computing infinite Taylor terms, which is not possible. Luckily, there are methods that bound the remainder term of Taylor series; the remainder term is guaranteed to be within these bounds, although these bounds may overestimate the remainder. Conventional methods sometimes estimate the remainder for stepsize control, but never try to bound this error. The remainder terms can be enclosed by several expressions, the most common is the Lagrange form of the remainder term. For the general Taylor series of g(x) about a point a, g(x) = g(a) + g 0 (a)(x − a) +

g 00 (a) g (n) (x(a)) (x − a)2 + ... + (x − a)n + Rn (x) 2! n!

(6.34)

the Langrange remainder is given by, Rn (x)

∈

g n+1 (ξ) n+1 h (n + 1)! [a, x]

(6.35)

h = x−a However, this form is not directly applicable to the autonomous ODE (6.30) because x(t) is the dependent and t the independent variable. Also note that in the Taylor series

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

expansion of the solution x(t), we have x(t) ˙ = f (x) so that we can replace g n+1 (ξ) in (6.35) n with f (x(ξ)), which yields the following remainder term of (6.33) Rn (t)

∈

f n (x(ξ)) n+1 h (n + 1)! [ti , ti+1 ]

(6.36)

h = ti+1 − ti In (6.34) we can evaluate the remainder term as long as the time interval and function g(x) are known. This is, however, not possible for (6.36) because we first need to enclose the solution x(t) on the interval t ∈ [ti , ti+1 ], which is what we are already trying to do by bounding the remainder term Rn (t); we need bounds on x(t) to create bounds on x(t). Therefore, we first need a different way to enclose the solution x(t) on the time interval, before we can bound the remainder term. This ”a priori” enclosure does not need to be tight because the new enclosure of x(t) gained from using the a priori enclosure in the remainder term, can be used again in evaluating the remainder term. This operation can be performed iteratively until the bounds do not decrease any more, which occurs for many ODEs after one iteration [105] . Example (Taylor Remainder Bounds) We want to find bounds on the Taylor remainder for the third order Taylor approximation of x(t) = et about x = 0. The third order Taylor polynomial plus remainder term is, t2 t3 t4 x(t) = 1 + t + + + eξ ξ ∈ [0, t] (6.37) 2! 3! 4! In this case we can simply use ξ = t in eξ as upper bound of the remainder error and ξ = 0 as lower bound. This is only possible because et is monotonically increasing; the largest value of t gives the largest value of et . If we have a function that is not monotonically increasing, we have to find its inclusion monotonic interval extension and use interval arithmetic to generate bounds on eξ . Figure 6.2 shows the graph of the Taylor approximation and real function. It also shows the graph of the difference between the approximation and real function (the value of the remainder term) and the bounds on the remainder term computed by t4 ξ 4! e with ξ ∈ [0, t].

6.2.2

A Priori Enclosure (Verifying Existence and Uniqueness)

An a priori rough enclosure of the solution is required to bound the remainder term of the Taylor series in (6.36). The a priori bound is found by validating existence and uniqueness of a solution within a certain box, [ˆ x], on a certain time interval h = (t1 − t0 ). Validating the existence and uniqueness is usually done using the Picard-Lindel¨of theorem and Picard iterator, which are results of the Banach fixed-point theorem [117] . These theorems are modified for the interval case and require an initial guess for the enclosure, [˜ x], and stepsize, h, which can be checked for existence and uniqueness using the Picard iterator in interval arithmetic. An overview of the important theorems used to prove the a priori enclosure method is presented below. The full proofs are rather complicated and long, they can be found in [104] , [117] and [118] .

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

4.5 4

x(t)

3.5

Taylor series approximation real value

3 2.5 2 1.5 1 0

0.5

1.5

1 0.8

local truncation error lower error bound

x(t)

0.6

upper error bound

0.4 0.2 0 0

0.5 t

Figure 6.2: Top: Plot of x(t) = et and its 3th order Taylor approximation about x = 0. Bottom: Plot of the difference between the real and approximated solution and the determined error bounds. Theorem 6.2.1. (Picard-Lindel¨ of Theorem) [119] Consider the initial value problem x0 (t) = f (t, x(t)),

x(t0 ) = x0 ,

t ∈ [t0 − α, t0 + α].

(6.38)

Suppose f is Lipschitz continuous in x and continuous in t. Then, for some value ε > 0, there exists a unique solution x(t) to the initial value problem within the range [t0 − ε, t0 + ε]. Theorem 6.2.2. (Picard Iteration) [120] The solution to the IVP in (6.38) is found by constructing recursively a sequence xn (t)∞ n=1 of functions, x(t0 ) = x0 Z t xn+1 (t) = x0 + f (xn (s))ds (6.39) t0

Then the solution, x(t), to (6.38) is given by the limit, x(t) = lim xn (t) n→∞

The interval version of Picard iteration becomes,

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

Theorem 6.2.3. (Picard Interval Iteration) [106] [105] Let [x0 ] be an inclusion of x0 , let [x](t) be an interval vector valued function with continuous bounds, and let [x](t) ⊆ D for t ∈ [t0 , t1 ]. If the inclusion Z

f ([˜ x](τ ))dτ ⊆ [˜ x](t)

[ˆ x](t) := [x0 ] +

(6.40)

holds for t ∈ [t0 , t1 ], then a unique solution x∗ exists on this interval and x∗ (t) ∈ [ˆ x](t)

for

t ∈ [t0 , t1 ]

One can check the validity of a constant inclusion without integrating by, Corollary 6.2.4. (Linear Enclosure) [116] If the inclusion [ˆ x](t) := [x0 ] + (t − t0 )f ([˜ x]) ⊆ [˜ x]

(6.41)

holds for all t ∈ [t0 , t1 ] with x0 ∈ [x0 ] ⊆ [˜ x] ⊆ D, then a unique solution x∗ exists on [t0 , t1 ] and x∗ (t) ∈ [ˆ x] for t ∈ [t0 , t1 ]. Verifying whether condition (6.41) is true for all t ∈ [t0 , t1 ] can be done using interval analysis. We simply replace (t−t0 ) with the interval [0, h] so that we have a constant interval enclosure, Corollary 6.2.5. (Constant Enclosure) [116] [104] If the inclusion [ˆ x] := [x0 ] + [0, h]f ([˜ x]) ⊆ [˜ x]

(6.42) ∗

holds with x0 ∈ [x0 ] ⊆ [˜ x] ⊆ D and h := (t1 − t0 ), then a unique solution x exists on [t0 , t1 ] and x∗ (t) ∈ [ˆ x] for t ∈ [t0 , t1 ]. The resulting constant enclosure, [ˆ x](t), encloses x(t) on the time interval [t0 , t0 + h]. With the use of (6.42) we can now create the following computer algorithm to compute an a priori enclosure. This algorithm is used in the AWA software by Lohner [116] . Algorithm 1 Constant a priori enclosure [105] 1: set [˜ x] := [x0 ], 2:

produce an inflation [x] of [˜ x], i.e. an enlarged interval [x] containing [˜ x] in its interior

compute [˜ x] := [x0 ] + [0, h]f ([t0 , t1 ], [x]) ⊆ [x]

if [˜ x] ⊆ [x] go to 5. Otherwise, there are two options, a: decrease h and go to 1. or b: go to 2.

set [ˆ x] := [˜ x]

With a constant or linear a priori enclosure, [ˆ x], we can write the remainder term in (6.36) as, f n ([ˆ x]) n+1 [Rn (t)] = h (6.43) (n + 1)!

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

The main disadvantage of the constant enclosure method is the allowable stepsize h for which the validation can be done using Algorithm 1. This stepsize is limited to a step appropriate for Euler’s method, no matter how high the order of the method to tighten the enclosure after an a priori enclosure is found. These enclosure tightening methods use high order Taylor polynomial and could use large timesteps if not limited by the stepsize restriction from the a priori enclosure computation. [105] [104] Example (A Priori Enclosure) We want to find an a priori enclosure of the following interval initial value problem, x0 = x2

x(0) ∈ [1, 1.1]

(6.44)

We enlarge the initial enclosure and create x ˜ ∈ [0.9, 1.2], we then use the constant enclosure method (6.42) to check for which h the enclosure is valid, [ˆ x] = [1, 1.1] + [0, h]([0.9, 1.2]2 ) ⊆ [0.9, 1.2] For h = 0.2, x ˆ = [1, 1.39] * [0.9, 1.2]. For h = 0.1, x ˆ = [1, 1.24] * [0.9, 1.2]. For h = 0.05, x ˆ = [1, 1.17] ⊆ [0.9, 1.2]. This guarantees that the solution of the IVP is contained within x ˆ ∈ [1, 1.17] for h ∈ [0, 0.05]. The linear enclosure method (6.41) yields for h = 0.05, [ˆ x] = [1, 1.1] + 0.05([0.9, 1.2]2 ) = [1.04, 1.17] ⊆ [0.9, 1.2] Thus a guaranteed enclosure at t = 0.05 is x ˆ ∈ [1.04, 1.17]. Figure 6.3 and 6.4 show the constant enclosure as function of h for the IVP with two sets of initial guesses [˜ x]. The constant enclosure, [ˆ x], is valid as long as it is enclosed by [˜ x], the straight purple lines. As one can see from these figures, the largest valid h become larger when [˜ x] becomes wider. Higher Order Enclosure (HEO) can increase the stepsize, but is more complicated to implement and more difficult to validate the inclusion over the whole interval t ∈ [t0 , t1 ], Corollary 6.2.6. (High Order Enclosure) [104] [117] Let x0 ∈ [x0 ] ⊆ D. Let p X (t − t0 )n (t − t0 )p+1 f ([x0 ])n−1 + f ([˜ x])p ⊆ [˜ x] [ˆ x](t) := [x0 ] + n! (p + 1)! n=1

(6.45)

for t ∈ [t0 , t1 ]. Then x(t) ∈ [ˆ x](t)

for

t ∈ [t0 , t1 ]

Finding the largest t1 such that (6.45) holds for all t ∈ [t0 , t1 ] is not trivial and computational expensive; it requires rigorous lower bounds for the positive real roots of 2n algebraic equations [117] . However, if p X [0, hn ] [0, hp+1 ] [ˆ x](t) := [x0 ] + f ([x0 ])n−1 + f ([˜ x])p ⊆ [˜ x] n! (p + 1)! n=1

(6.46)

then (6.45) holds for all t ∈ [t0 , t1 ]. Verifying (6.46) is not difficult and can verify that (6.45) holds for all t ∈ [t0 , t1 ]. Details about the high order enclosure methods and how to find apropriate [˜ x] can be found in papers by Corliss (1996) [ 104 ] and Nedialkov (1999) [ 117 ] .

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

1.5 constant enclosure 1.4

constant enclosure initial guess initial guess

1.3

true max true min

x(t)

1.2

1.1

0.9

0.8 0

0.02

0.04

0.06

0.08

0.1 t

0.12

0.14

0.16

0.18

0.2

Figure 6.3: A priori enclosures for the constant interval enclosure method for the differential equation x0 (t) = x2 (t) with initial value x(0) ∈ [1, 1.1] and x ˜ ∈ [0.9, 1.1]. A constant enclosure is guaranteed if its width is smaller than the with of the initial guess. In this case the constant enclosure is true for a time interval [t0 , t1 = 0.7].

6.2.3

Interval Integration using Bounded Taylor Series

Taylor series approximate functions and can be integrated analytically due to their elementary algebraic structure, resulting in a new Taylor series. The previous subsections showed how we can create guaranteed bounds on infinite Taylor series. We can apply this guaranteed enclosure method to the Taylor series of x(t) in (6.30), about a point t, and produce guaranteed bounds on the solution x(t). If we use interval arithmetic, we also include interval initial value [x0 ] and parameters [c]. Because we work with intervals and interval arithmetic, not all properties of normal arithmetic apply; different ways of evaluating a function may yield different results (intervals of different width). Moreover, the bounding of Taylor series is not exact. That is, the bounded solution set contains the solutions, but also empty points. These empty points or ”non-solutions” are carried through in the next integration step as initial values. This means that more and more non-solutions are picked up every integration step; the interval bounds on the solution set become a large overestimation of the real solution set. This subsection presents the two basic ways of evaluating the Taylor series of x(t); Moore’s Direct method and the Mean Value method. The last subsection goes into details about reducing the wrapping effect, the most prominent cause of overestimation.

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

1.5 1.4 1.3 1.2

x(t)

1.1 1 constant enclosure 0.9

constant enclosure initial guess

0.8

initial guess true max

0.7

true min 0.02

0.04

0.06

0.08

0.1 t

0.12

0.14

0.16

0.18

0.2

Figure 6.4: A priori enclosures for the constant interval enclosure method for the differential equation x0 (t) = x2 (t) with initial value x(0) ∈ [1, 1.1] and x ˜ ∈ [0.7, 1.4]. A constant enclosure is guaranteed if its width is smaller than the with of the initial guess (The red lines have to stay within the purple lines). In this case the constant enclosure is true for a time interval [t0 , t = 0.15]. Moore’s Direct Method Section 6.2.1 explained how Taylor series are bounded by using the inclusion monotonic function extension of f , an a priori enclosure in the Lagrange remainder term of Taylor series and bounds on the rounding error. In short, Moore’s direct method uses these bounded Taylor series to bound the solution of ODEs with interval initial values and parameters. The method generates the interval enclosure, p X 1 n n−1 [x1 ] = [x0 ] + h f ([x0 ]) + [Rn ] (6.47) n! n=1 with, [Rn ] =

hp+1 p f ([ˆ x]) (p + 1)!

(6.48)

where x(t) ∈ [ˆ x]

for t ∈ [t0 , t1 ].

and x(t1 ) ∈ [x1 ]

(6.49)

The interval Taylor coefficients in (6.47) can be computed using automatic differentiation (see Section 6.6). An a priori enclosure, [ˆ x] for t ∈ [t0 , t1 ], follows from a method in Section 6.2.2. This a priori enclosure can be used to compute the remainder term, [Rn ].

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

The result is an enclosure of the solution [x1 ] at time t1 = t0 + h. This procedure can be repeated for next steps in the integration where the previous enclosed solution becomes the initial values of the next step. If we have computed an enclosure [x1 ] of the solution [x(t1 )], we can improve the constant a priori enclosure on [t0 , t1 ] a posteriori by, [ˆ xnew ] := [ˆ x] ∩ ([x0 ] + [0, h]f ([ˆ x])) ∩ (([x1 ] − [0, h]f ([ˆ x]))

(6.50)

Using this new interval vector, we can improve the Taylor remainder term. This procedure can be iterated, but does generally not improve after a single iteration. [105] Moore’s direct method evaluates (6.47) ”directly”; all intervals are inserted into (6.47) and computed using interval arithmetic. This method is width increasing (the enclosures become larger every step), w([x1 ]) = w([x0 ]) +

p X 1 n h w(f n−1 ([x0 ])) + w([Rn ]) ≥ w([x0 ]) n! n=1

(6.51)

since in interval arithmetic w([x] ± [y]) = w([x]) + w([y]). Alefeld and Herzberger showed in 1983 [121] that the mean value form of (6.47) often yields tighter enclosures than Moore’s direct method. For wide intervals the form may actually give wider bounding intervals than the natural form [112] . Mean Value Method The following mean value theorem can modify (6.47) such that sharper and width decreasing bounds can be created, Theorem 6.2.7. (Mean Value Theorem) Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that, f (b) = f (a) + f 0 (c)(b − a)

(6.52)

The interval mean value form follows directly from the mean value theorem [112] [122] , Theorem 6.2.8. (Interval Mean Value Theorem) Let f : [b] → R be continuous differentiable on the interval [b]. Then for any y, a ∈ [b], f (y) ∈ fM ([b], a) = f (a) + f 0 ([b])([b] − a)

(6.53)

Using this interval mean value form, we can write (6.47) as,

x(t1 ) ⊆ [x1 ] = x ˜0 +

p X hn (n−1) h(p+1) p f (˜ x0 ) + f ([ˆ x]) n! (p + 1)! n=1

(6.54) + I+

p X

hn J(f (n−1) ; [x0 ]) ([x0 ] − x ˜0 ) n! n=1

where [ˆ x] is the a priori enclosure of x(t) for all t ∈ [t0 , t1 ] and J(f (n−1) ; [x0 ]) the Jacobian (n−1) of f evaluated in interval arithmetic at [x0 ],

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

∂f1  ∂x1  . J =  ..  ∂f m ∂x1 

··· ..

···

 ∂f1 ∂xn  ..   .  ∂fm  ∂xn

(6.55)

where fi and xi are the i-th element of the m-dimensional vector function f (x(t)) and ndimensional state x. The Jacobian can be computed using Automatic Differentiation (see Section 6.6) [123] [117] . We choose x ˜0 to be the midpoint of [x0 ], m([x0 ]). We introduce the convenient notation used by Nedialkov [117] , [v1 ] = m([x0 ]) +

p X hn (n−1) h(p+1) p f (m([x0 ])) + f ([ˆ x]) n! (p + 1)! n=1

p X hn [S0 ] = I + J(f (n−1) ; [x0 ]) n! n=1

and

(6.56) (6.57)

So that we can write (6.54) as, [x1 ] = [v1 ] + [S0 ]([x0 ], m([x0 ]))

(6.58)

[x1 ] can now be smaller than [x0 ] if the remainder term Rn is small enough and the diagonal elements of J are negative [105] [117] . We can again take subsequent integration steps by using the enclosed solution of the previous step as the interval initial value in the next integration step. Algorithm 2 describes the mean value interval method. Algorithm 2 Direct Mean Value Interval Method [105] Input: h, [x0 ], [ˆ x], m([x0 ]) Pp hn (n−1) hp+1 p 1: [v1 ] = m([x0 ]) + (m([x0 ])) + (p+1)! f ([ˆ x]) n=1 n! f Pp hn (n−1) 2: [S0 ] = I + ; [x0 ]) n=1 n! J(f 3:

[x1 ] = [v1 ] + [S0 ]([x0 ], m([x0 ]))

m([x1 ]) = m([v1 ])

Output: [x1 ], m([x1 ])

6.2.4

Reducing the Wrapping Effect

One dimensional intervals can be seen as lines parallel to the coordinate axis. A two dimensional interval vector is a rectangle and a three dimensional vector is a box with sides parallel to the (perpendicular) coordinate axis. Figure 6.5 gives a graphical representation of the following one, two and three dimensional interval vectors in a Cartesian coordinate

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

system,   [x] intvec3 = [y] [z]

[x] intvec1 = [x] intvec2 = [y] [x] = [3, 7]

[y] = [3, 7]

(6.59)

[z] = [3, 7]

(6.60)

4 2 10

0 0

6 x

0 0

5 2

8 x

10 0

Figure 6.5: Graphical representation of a one, two and three dimensional interval vectors. The solution set of an ODE with interval initial values is generally not a rectangle (or box etc.), which means that interval enclosures of solutions contain points that are not part of the solution set. These extra points are used as initial values in the next integration step and introduce a larger solution set than the true solution set. This already larger solutions set has to be bounded again in a rectangle which contains new empty points. This causes an increasing overestimation of the solution set and is called the ”wrapping effect”. [112] The wrapping effect can be illustrated by Moore’s example [111] , we take the following system of differential equations, x01 = x2 x02 = −x1 The solution, with initial value x0 is given by x(t) = A(t)x0 , where, cos t sin t A(t) = − sin t cos t

(6.61)

(6.62)

With interval initial value x0 ∈ [x0 ], the interval vector [x0 ] can be viewed as a rectangle in the (x1 , x2 ) plane. At t1 , [x0 ] is mapped by A(t1 ) into a rectangle of the same size (see Figure 6.6). If we want to enclose the rotated rectangle in an interval vector, we have to wrap it in another rectangle with sides parallel to the x1 and x2 axis (a new interval vector, [x1 ]). This interval is lager. In the next integration step, the larger rectangle is rotated again and has to be enclosed in an even larger rectangle (a rotated box only fits in a larger box). This means that the enclosing rectangle (interval vector) becomes larger and larger, while the true solution set {A(t)x0 |x0 ∈ [x0 ], t > t0 } remains a rectangle of the same size. Therefore, the interval vector enclosing the solution at some time t contains extra points not in the solution resulting from [x0 ]. This phenomenon happens for all flows not parallel to

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

the coordinate system. The wrapping effect is the major problem in interval integration and causes bounds to blow up and make solutions useless. For the above example the solution bounds are inflated by a factor e2π ≈ 535 for a stepsize going to zero. Moore [112] already suggested a few ways to reduce the wrapping effect and many others developed new wrapping reduction methods after him. Although the wrapping effect is inherent to interval enclosures, there are a few effective ways to reduce it. The most common are subdivision, the parallelepiped and QR-factorization method; these three are presented in the next sections. Nevertheless, reducing the wrapping effect for interval methods is always limited by the fact that the interval enclosure sets are convex. A set is convex if for every pair of points within the set, every point on the straight line segment that joins them is also within the set. For non-linear ODEs the solution is a non-convex set that cannot be contained in a convex set without overestimation; ”any interval wrap must be at least as large as the convex hull of the solution” [99] . No method is best for all IVPs and it is difficult to tell without trying which method will be best at reducing the wrapping effect for a new problem.

Figure 6.6: Wrapping of a rectangle specified by the interval vector ([−1, 1], [10, 11])T . Source: Nedialkov [123]

Subdivision Subdivision is the division of intervals in subintervals and evaluating each subinterval separately. The wrapping and dependency problem are reduced by subdivision because many small intervals can more accurately describe a solution set than a single large interval. That is, separate intervals can more accurately describe a non-convex set than one interval. There are two types of subdivison strategies [114] . The first divides the initial interval in

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

smaller subintervals at the beginning of the method and work with these subintervals to the end. The second strategy divides the initial interval in subintervals just before the evaluation of the variational part, (6.55), which produces the wrapping, then recombining the subintervals in one interval after the evaluation. This strategy has the lowest impact on computational time. A problem with subdivision is its linear increase of computational time with the number of subintervals. The good thing about this problem is that it is easily parallelized and can therefore run on multiple CPUs at the same time. This means that with enough CPUs, results do not take longer compared to using a single interval. Coordinate Transformation (Lohner’s Method) Moore was the first to notice the wrapping problem in 1965 [124] . He also mentioned a way to reduce the wrapping effect by performing a local coordinate transformation in the direction of the flow. Instead of using a box enclosure, [xj ], defined by the unit vectors of the original coordinate system, we can also represent the solution as a single point with a locally transformed interval box around it [123] , [xj ] = m([xj ]) + Aj [rj ] (6.63) where Aj is transformation matrix and [rj ] an interval vector. Lohner [117] developed a method to determine rj and m([xj ]) in (6.63) for the ITS method when Aj is given. Aj is, for example, determined by the parallelepiped or QR-factorization method discussed in the following subsections. He introduced the following notation based on Algorithm 2, hp+1 p f ([ˆ x]) (p + 1)! = m([zj+1 ])

[zj+1 ] = sj+1

m([xj+1 ]) = m([xj ]) + [Sj ] = I +

p X

hn (n−1) f (m([xj ])) + sj+1 n! n=1

p X hn J(f (n−1) ; [xj ]) n! n=1

(6.64) (6.65) (6.66) (6.67)

A0 = I

(6.68)

[r0 ] = [x0 ] − m([x0 ])

(6.69)

From (6.54) and the above notations, we obtain, x(t1 ) ∈ [x1 ] = m([x1 ]) + ([S0 ]A0 )[r0 ] + [z1 ] − s1 [x1 ] = m([x1 ]) + A1 [r1 ] −1 [r1 ] = (A−1 1 ([S0 ]A0 ))[r0 ] + A1 ([z1 ] − s1 )

where A1 is non-singular. The next integration step now becomes, x(t2 ) ∈ [x2 ] = m([x2 ]) + ([S1 ]A1 )[r1 ] + [z2 ] − s2 [x2 ] = m([x2 ]) + A2 [r2 ] −1 [r2 ] = (A−1 2 ([S1 ]A1 ))[r1 ] + A2 ([z2 ] − s2 )

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

Continuing this process results in Lohner’s Method, see Algorithm 3. Algorithm 3 Lohner’s Method [117] Input: h, [x0 ], [ˆ x], m([x0 ]), A0 , [r0 ] hp+1 p x]) (p+1)! f ([ˆ

[z1 ] =

s1 = m([z1 ])

Pp n m([x1 ]) = m([x0 ]) + n=1 hn! f (n−1) (m([x0 ])) + s1 Pp hn (n−1) ; [x0 ]) 4: [S0 ] = I + n=1 n! J(f

Choose A1 by the parallelepiped or QR method, see below.

[x1 ] = m([x1 ]) + ([S0 ]A0 )[r0 ] + [z1 ] − s1

−1 [r1 ] = (A−1 1 ([S0 ]A1 ))[r0 ] + A1 ([z1 ] − s1 )

Output: [x1 ], m([x1 ]), A1 , [r1 ] Two common ways to select Aj are the parallelepiped and QR-factorization method. The parallelepiped method uses a local transformation to a non-orthogonal coordinate system; Aj [rj ] is a parallelepiped. The QR-factorization method uses a local rotation transformation to a different orthogonal coordinate system; Aj [rj ] is a rotated box. Although the parallelepiped method often gives tighter enclosures, Ai can become singular which makes verified inversion of Aj impossible. [125] [99] More information about the reducing the wrapping effect can be found in [117] [125] [114] [126] [99] [123] [105] and [127] . Parallelepiped We obtain the parallelepiped method from Lohner’s method (Algorithm 3) if [117] , Aj+1 = m([Sj ]Aj )

(6.70)

where [Sj ] is defined by (6.67). ”In the parallelepiped method, the flow of the ODE at intermediate time steps is enclosed by parallelepipeds instead of rectangular boxes. This choice is motivated by the shape of the flow of a linear ODE with interval initial values, which is a parallelepiped at any time.” [99] The only source of overestimation for a linear flow and the parallelepiped method, are the truncation and rounding errors. This means that this method is as tight as possible for linear ODEs. For non-linear ODE’s, like orbital equations of motion, the wrapping reduction depends on the specific ODE. The problem with the parallelepiped method is the transformation matrix Aj which becomes singular after some time. The results is impossible matrix inverses. This breakdown of the parallelepiped method is a rule rather than an exception for most ODEs. A verified way to compute the inverse of Aj can be found in [114] . QR-factorization To solve the problem of inverting a singular matrix in the parallelepiped method, Lohner stabilized the iteration by orthogonalization of the matrices, so that the algebraic problem of inverting the matrices is reduced to taking the transpose. [114] [99] [128] [105]

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

Let A˜j+1 ∈ [Sj ]Aj and let Aˆj+1 = A˜j+1 Pj+1 , where Pj+1 is a permutation matrix. A permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0’s elsewhere. Each such matrix represents a specific permutation of m elements and, when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix; Pj+1 rearranges the columns of A˜j+1 We perform the QR-factorization Aˆj+1 = Qj+1 Rj+1 , where Qj+1 is an orthogonal matrix and Rj+1 an upper triangular matrix. If Aj+1 in Algorithm 3 is chosen to be Qj+1 , we have the QR-factorization method for reducing the wrapping effect. QR factorization (also called a QR decomposition) decomposes a matrix A in a upper triangular matrix, R and a orthogonal matrix, Q. There are several algorithms to compute Q and R, such as the Gram-Schmidt algorithm, Householder transformations, or Givens rotations. Here we only present the modified Gram-Schmidt algorithm [129] , A = a1 A2 Q = q1 Q2 r R12 R = 11 0 R22 And thus, a1

= q1

r11 Q2 0

R12 = q1 r11 R22

q1 R12 + Q2 R22

(6.71)

where, r11 = ||a1 || 1 a1 q1 = r11 R12 = q1T A2 Q2 R22 = A2 − q1 R12 The matrix Qj+1 introduces an orthogonal coordinate system, where the axis corresponding to the first column of Qj+1 is parallel to the first column of Aˆj+1 . We expect a tighter enclosure if the first column of Qj+1 is parallel to the longest edge of the parallelepiped A˜j+1 [rj + 1] than to a shorter edge. Pj+1 should therefore be chosen such that the first column of A˜j+1 corresponds to the longest edge, then the second longest and so on. An eigenvalue type stability analysis shows that the QR method provides good stability for interval methods. [117] [128] [130] Figure 6.7 shows the resulting enclosures after QR-factorization. One along the shortest axis and one along the longest axis of the parallelepiped.

100

6.2. Interval Taylor Series Methods

Chapter 6. Interval ODE Integration

Figure 6.7: Visualization of inclusion sets using Lohner’s QR-factorization method. (a) The set {r|r ∈ [r]}. (b) {Ar|r ∈ [r]} enclosed by A[r]. (c) {Ar|r ∈ [r]} enclosed in the coordinate system induced by Q. (d) {(Q−1 A)r|r ∈ [r]} enclosed by ˆ (f) Q−1 A[r]. (e) {Ar|r ∈ [r]} enclosed in the coordinate system induced by Q. −1 −1 [117] ˆ ˆ {(Q A)r|r ∈ [r]} enclosed by (Q A)[r]. Source: Nedialkov

101

6.3. Taylor Model Methods

6.3

Chapter 6. Interval ODE Integration

Taylor Model Methods

Taylor Model (TM) methods use symbolic propagation of a (symbolic) initial value set to relate initial values to the final solution. It uses an interval remainder term to bound truncation and rounding errors. While Taylor Models use interval analysis for the truncation and rounding errors, it does not use interval arithmetic to propagate the initial value set in intermediate interval boxes. Taylor Models are developed by Makino and Berz, initially to analyse the stability of particles in particle accelerators [107] [108] . TM are a combination of Differential Algebras [131] and Interval Arithmetic. Taylor Models are very general and can be used to analyse a wide range of dynamical systems. In this chapter we only discuss the application of TM methods to verified ODE integration. Makino and Berz created the software tool COSY Infinity [109] that implements their method. The main difference between ITS and TM methods is their way of determining the dependency between initial values and the solution set. Taylor Model methods also use Taylor series expansion for the numerical integration of the differential equations, but express the interval initial values in symbols. The solution set, after one integration step, is thus a function of symbolic initial values, a and b, {x(t1 , a, b) â&#x160;&#x2020; T M

| a â&#x2C6;&#x2C6; [amin , amax ],

b â&#x2C6;&#x2C6; [bmin , bmax ]}

(6.72)

This symbolic solution set is used as initial values for the next integration step until the final integration time is reached. The example in Section 6.3.4 will make this more clear. This in contrast with ITS methods that describe the bounds on the initial values, intermediate solutions and final solution in numerical intervals instead of symbolic Taylor Models. Taylor Model methods, however, do also use interval arithmetic to bound truncation and rounding errors, although they use slightly different ways to bound the Taylor series truncation error and to reduce the wrapping effect. The major advantage of TM over ITS methods is the avoidance of large part of the wrapping effect due to the symbolic propagation of the initial value set. Since interval arithmetic is only used for the remainder term, the wrapping effect occurs in the often small interval remainder term only. This results in sharper bounds. The disadvantages are its computational cost and the representation of the solutions set. Taylor Models are up to 50 times slower than ITS methods [117] . Furthermore, the TM representation of the solutions set is more complex than an interval representation. This generates sharper non-convex sets, but makes it harder to determine whether an element (point) is contained in the set. Although a single point can be easily checked using root finders, the intersection between two Taylor Model sets is much more difficult to compute. As far as the author knows, no method can guarantee that two Taylor Model sets do or do not intersect without wrapping the TM in a convex enclosure. This is a major disadvantage when using Taylor Models in guaranteed collision avoidance. However, the final Taylor Model solution set can be wrapped in an interval by using interval arithmetic. This adaptation does increase overestimation, but makes it easier to check for intersecting solutions. We will call this method the Hybrid Taylor-Interval method and is explained in Section 6.3.5. The next subsections present the general TM algorithm description, the methods to reduce the wrapping effect and an example to clarify the method. More information about Taylor Model methods can be found in [132] [99] [133] [134] [135] [136] [137] [138] [139] [114] and [140] .

102

6.3. Taylor Model Methods

6.3.1

Chapter 6. Interval ODE Integration

Taylor Models

In contrast with ITS methods, the basic data type for TM is not an interval but a ”Taylor Model”. A Taylor Model consists of a polynomial, P , and interval, [R], part, T M := P + [R]

(6.73)

Also denoted as, T M = (P, [R]). The polynomial part is the Taylor series expansion of finite order n and the interval part contains truncation and rounding errors and depends on the algorithm used to compute it. Any function that can be represented on a computer and is continuously differentiable can be modelled (bounded) by a Taylor Model. Just like interval arithmetic describes mathematical operations for intervals, Taylor Model arithmetic (part of Remainder-enhanced Differential Algebra developed by Berz and Makino [108] ) defines operations for Taylor Models. Taylor Model arithmetic can be used to compute the Taylor models of complex functions by combinations of simple Taylor models. Addition and subtraction, for example, can be obtained by, T Mg ± T Mf = (Pg + Pf , [Rg ] ± [Rf ]) Intrinsic functions, derivative and anti-derivative are also defined within Remainder-enhanced Differential Algebra and can be found in [108] and [131] .

6.3.2

Verified ODE Integration

Makino and Berz developed TM and TM arithmetic for a wide variety of applications including the verified solution of explicit and implicit ODEs, but also the more general Differential Algebraic Equations (DEA) [136] , dx , x, y, t = 0 f dt This generality, however, makes it difficult for engineers and scientist outside the field of Taylor models to apply the method to a specific problem. As Neher notes ”the sometimes cursory description of technical details of Taylor model arithmetic, may be obvious to the experts of Taylor models, but are less trivial to others.” [99] . This section provides a discussion of its application to the ODE, d x(t) = f (x(t), t) dt

(6.74)

The algorithm to compute the remainder term of the truncated Taylor series for the Taylor model method is similar to the one for the Interval Taylor method. It uses Schrauder’s and Banach’s fixed point theorem to prove existence and uniqueness of a solution. We introduce the Banach operator A(), Z t A(g)(t) = x0 + f (g(s), s)ds (6.75) t0

So a general function g is transformed into a new function via the insertion into f and subsequent integration. The problem of finding a solution to the differential equations is now reduced to a fixedpoint problem, x = A(x) (6.76)

103

6.3. Taylor Model Methods

Chapter 6. Interval ODE Integration

Schauder’s fixed point theorem can now be used to obtain a Taylor Model for the solution describing the dependency on the initial conditions, Theorem 6.3.1. (Schauder’s fixed point theorem) [141] Let A be a continuous operator on the Banach space X. Let M ⊂ X be compact and convex, and let A(M ) ⊂ M . Then A has a fixed point in M , i.e. there is an x ∈ M such that A(x) = x. One can find such an M in 3 steps [141] , 1. Determine a family Y of subsets of X from which to draw candidates for the set M . The sets in Y have to be compact and convex and they should be contained in a Taylor Model. 2. Construct an initial set M0 ∈ Y that satisfies the inclusion property A(M0 ) ⊂ M0 . All requirements of the fixed point theorem are satisfied and the existence of a solution in M0 is verified. M0 is a Taylor Model inclusion of a solution of the ODE. 3. Iteratively reduce the set M0 to obtain a bound that is as sharp as possible; Mi = A(Mi−1 ). The problem now boils down to finding a Taylor Model P + [R] such that, A(P + [R]) ⊂ P + [R]

(6.77)

where we want the remainder interval, [R], to be as small as possible. This means finding a Taylor Model with small [R] that still contains the solution. This lead Makino and Berz to try sets M ∗ of the form, M ∗ = MPn (x,t)+[R∗ ] (6.78) where Pn (x, t) is the n-th order Taylor expansion in time and initial values of the solution, and [R∗ ] an interval remainder term. Pn (x, t) is the approximation of the solution on the time interval [t0 , t]. This quantity (0) can be obtained by iteration of (6.75). Set Pn (x, t) = I, I is the identity function, then iterate, Pn(k+1) = A(Pn(k) ) (6.79) Next we try to find an [R] such that, A(Pn (x, t)) + [R∗ ]) ⊂ Pn (x, t)) + [R∗ ]

(6.80)

which is the inclusion property for Schauder’s theorem. Once such an inclusion has been determined, a solution of the ODE is contained in the Taylor Model: Pn (x, t)) + [R∗ ]. [141] [142] [99] With the initial values sets a ∈ [amin , amax ] and b ∈ [bmin , bmax ], the solution set, in terms of a and b can be written as, {x(te , a, b) ⊆ T M := P (te , a, b) + [R](te ) | a ∈ [amin , amax ],

6.3.3

b ∈ [bmin , bmax ]} (6.81)

Reducing the Wrapping Effect

The wrapping effect (see Section 6.2.4) is already greatly reduced by enclosing only the remainder term by an interval [R]. However, this interval can still grow out of bounds for long integration times. Berz and Makino introduced two methods to further reduce the wrapping effect of the interval remainder term; Shrink Wrapping and Preconditioning [99] .

104

6.3. Taylor Model Methods

Chapter 6. Interval ODE Integration

Shrink Wrapping Shrink Wrapping absorbs (part) of the interval remainder term in the symbolic polynomial part of the Taylor Model, P , by modifying the polynomial’s coefficients. The polynomial and interval part of the TM are wrapped in a new, but pure polynomial, TM. This TM has an interval part of zero now. This new set may be larger than the original set, but is less prone to wrapping in successive integration steps and to the dependency problem. Shrink Wrapping is performed when the interval part of the TM becomes larger than a specified value. It resembles the parallelepiped method. By multiplying the non-constant coefficients of the Taylor polynomial, for linear autonomous ODEs, the interval term is absorbed as in the parallelepiped method. Shrink Wrapping uses the same linear map as the parallelepiped method and has the same problems as the parallelepiped method discussed in Section 6.2.4. Preconditioning stabilizes the integration process that became unstable due to the Shrink Wrapping. [99] [143] [144] [132] [145] Preconditioning Preconditioning aims at maintaining a small condition number for the shrink wrapping map. It thereby stabilizes the integration process, like the QR factorization method does for the ITS method. Preconditioning also uses a factorization into two Taylor Models; only one Taylor Model is used for the continuation of the integration, just as in the QR method. It is a coordinate transformation that is more suitable for continuing the integration, just like the QR and parallelepiped method,. For preconditioning Taylor models, a large variety of well-conditioned transformations is possible. The optimal choice is still an open question for future research. Both shrink wrapping and preconditioning are implement in the TM software made by Berz and Makino, COSY Infinity. See Section 6.7.3 for more information about this software. [144] [99] [132] [145]

6.3.4

Example

Example (Propagation of Symbolic Initial Values) Consider the first order interval initial value problem, x˙ = y 2 , y˙ = x,

x(0) = −1 + b y(0) = 1 + a

a, b ∈ [−0.05, 0.05]

(6.82) (6.83) (6.84)

where the interval initial values are given symbolically; a and b are intervals around the reference initial values. This representation is equal to interval initial values x(0) ∈ [−1.05, −0.95] and y(0) ∈ [0.95, 1.05]. The third order Taylor series expansion of x(t) and y(t) about t = 0, 2 2 h3 a h ah3 bh3 1 h2 2 3 + ah − + − + + abh + [Rx ] x(t0 + h, a, b) = 1 + a − h + bh + 2 3 2 3 3 3 3 2h y(t0 + h, a, b) = −1 + b + h + 2ah + a2 h − h2 − ah2 + bh2 + 3 3 2 3 3 3 2bh b h a h − + + [Ry ] + abh2 + ah3 + a2 h3 + 3 3 3

105

6.3. Taylor Model Methods

Chapter 6. Interval ODE Integration

The terms between the brackets have a order higher than three because a and b are very small compared to the midpoint of the initial values, 1 and -1. These terms are taken into the remainder interval [R]. We now integrate one step from t0 = 0 to t1 = 0.1, x(0.1, a, b) = −0.91 + 0.191a + 0.101a2 + 1.01b + [Rx ] y(0.1, a, b) = 0.9 + 1.01a + 0.1b + [Ry ] We again expand x(t) and y(t), but now about t1 = 0.1, using x(0.1, a, b) and y(0.1, a, b) as calculated above, x(t1 + h, a, b) = −0.91 + 0.191a + 0.101a2 + 1.01b + 818h + 1.83ah + 0.181bh − 0.823h2 + 1.02a2 h + 0.202abh + 0.01b2 h − 0.747ah2 + 0.823bh2 + 0.522h3 + [Rx ] y(t1 + h, a, b) = 0.9 + 1.01a + 0.1b0.909h + 0.19ah + 1.01bh + 0.409h2 + 0.1a2 h + 0.914ah2 + 0.0905bh2 0.274h3 + [Ry ] And again integrate one step, now from t1 = 0.1 to t2 = 0.2, x(0.2, a, b) = 0.835 + 0.365a + 1.04b + 0.202a2 + 0.0202ab + 0.001b2 + [Rx ] y(0.2, a, b) = 0.818 + 1.04a + 0.202b + 0.01a2 + [Ry ] This process is repeated until the required integration time is reached. The solution of the interval IVP at time t = 0.2 (6.82) is contained in the set, {(x(0.2, a, b), y(0.2, a, b)|a, b ∈ [−0.05, 0.05]} This solution set without remainder, and thus not guaranteed, is shown in Figure 6.8. Although the set looks like a parallelogram, it has slightly curved edges and can therefore not be represented by an interval vector. Only the initial values are a square around a reference point that can be represented by an interval vector, the sets become multivariate polynomials after a single integration step. This can be clearly seen in the solutions sets plotted in Figure 6.9, which shows solutions sets at time intervals of 0.2.

106

6.3. Taylor Model Methods

Chapter 6. Interval ODE Integration

Figure 6.8: Graphical representation of the solution set of (6.82) at time t = 0.2 without remainder (non-verified). The edges are slightly curved.

Figure 6.9: Graphical representation of flow of (6.82). Source: Neher [132] .

107

6.3. Taylor Model Methods

6.3.5

Chapter 6. Interval ODE Integration

Interval Evaluation of Taylor Model Solutions

Interval evaluation of Taylor Models encloses (or wraps) the TM in an interval. This may be useful when we need intervals to work with. To easily check whether two trajectory enclosures intersect, for example. To enclose a TM in an interval, we simply have to evaluate the TM using interval arithmetic. If we have the following TM, without remainder term, from Section 6.3.4, {(x(0.2, a, b), y(0.2, a, b)|a, b ∈ [−0.05, 0.05]} where, x(0.2, a, b) = 0.835 + 0.365a + 1.04b + 0.202a2 + 0.0202ab + 0.001b2 y(0.2, a, b) = 0.818 + 1.04a + 0.202b + 0.01a2 We can wrap this TM by evaluating x(0.2, a, b) and y(0.2, a, b) using interval arithmetic where a and b are the intervals [a] = [b] = [−0.05, 0.05], x(0.2, [a], [b]) = 0.835 + 0.365[−0.05, 0.05] + 1.04[−0.05, 0.05] + 0.202[−0.05, 0.05]2 + 0.0202[−0.05, 0.05][−0.05, 0.05] + 0.001[−0.05, 0.05]2 = [−0.905, −0.764] y(0.2, [a], [b]) = 0.818 + 1.04[−0.05, 0.05] + 0.202[−0.05, 0.05] + 0.01[−0.05, 0.05]2 = [0.76, 0.88] So that we obtain an interval vector [x] that encloses the TM, [0.76, 0.88] [x] = [−0.91, −0.77]

(6.85)

Which can be seen from Figure (6.8) to enclose the Taylor Model for the example at t = 0.2.

108

6.4. Interval Runge-Kutta Methods

6.4

Chapter 6. Interval ODE Integration

Interval Runge-Kutta Methods

Runge-Kutta (RK) methods are the most popular conventional integration methods (see Section 5.2.3). They are, however, mainly used for their high efficiency and low complexity and not because the error can be bounded easily, which is a requirement for interval integration. The conventional application of Runge-Kutta methods rarely focusses on, or even discusses, the quality of the solution (the size of the error) and often simply assumes that the solution is the true solution [146] . Moreover, the only regularly used error analysis is a local truncation error estimation by taking the difference between a n-th and (n + 1)-th RK method. However, for many safety critical systems and optimization problems, this estimation is not enough and guaranteed solution sets are required. Finding guaranteed bounds for the truncation errors in RK methods is not easy and prone to errors due to its complexity. Only a few papers provide a proof for the bounds of RK methods and only a single application of these verified error bounds was found in literature [146] . Almost all interval integration methods use Taylor series instead of RK because the remainder of Taylor series is easier to bound than RK methods. Other advantages or disadvantages of ITS Taylor series over RK methods in interval integration may also exist, but no good comparison between RK and Taylor interval integration methods was found in literature. A guaranteed local truncation error bound was first given by Bieberbach in 1951 [147] , his method was improved by Lotkin in the same year [148] , Carr provided a way to bound the global error by propagating local truncation errors in 1958 [149] . A more recent method with computer implementation is given by Bouissou and Martel in 2006 [146] . Other work on interval RK methods have been published by Marciniak, Szyszka and Gajda between 1999 and 2008 [110] [150] [151] .They bounded the solution of the chaotic Lorenz equations using their Guaranteed Runge-Kutta Library (GRKlib) and compared it with different Taylor series methods. Although their method showed promising results, little information about their method and no follow-up studies were found. This makes the error bounding of RK methods interesting for further mathematical development, but Taylor series methods are still preferred for applied interval integration in real world problems because they are more developed and verified. The following, and last, part of this subsection gives an overview of Bouissou and Martels method for bounding the errors in a RK integration. We consider the system of ODEs with initial values, x(t) Ë&#x2122; = f (x(t))

x(t0 ) â&#x2C6;&#x2C6; [x0 ]

(6.86)

where [x0 ] is an interval initial value. We now use the following notation for the values of xn to distinguish between different errors, xvn is xvn+1 xn is xn+1 xrn+1 xâ&#x2C6;&#x2014;n+1

the real value at step n. is the real value at step n + 1. the numerically integrated value at step n. is the numerically integrated value at step n + 1. is the numerically integrated value at step n + 1 without rounding errors. is the numerically integrated step n + 1 if we had started from xvn , no rounding errors.

Figure 6.10 gives a graphical representation of the different definitions and corresponding

109

6.4. Interval Runge-Kutta Methods

Chapter 6. Interval ODE Integration

Figure 6.10: Three kinds of errors. Source: Bouissou, 2006 [146] . introduced errors. The errors can now be described as the differences between the above variables, v â&#x2C6;&#x2014; Ď&#x2021;n+1 = yn+1 â&#x2C6;&#x2019; yn+1 , the propagated error. â&#x2C6;&#x2014; r Îˇn+1 = yn+1 â&#x2C6;&#x2019; yn+1 the local truncation error. r en+1 = yn+1 â&#x2C6;&#x2019; yn+1 , the rounding error. v n+1 = yn+1 â&#x2C6;&#x2019; yn+1 , the total error.

The bounds on these errors are intervals and will be denoted by [Ď&#x2021;n+1 ], [Îˇn+1 ], [en+1 ] and [ n+1 ]. These errors are given by the following relations, Îˇn+1 â&#x2C6;&#x2C6;

d4 f d 5 Ď&#x2020;n v (x([t , x ]; t , x )) â&#x2C6;&#x2019; ([tn , xt+1 ]) n t+1 n n dt4 dt5

(6.87)

Ď&#x2021;n+1 â&#x2C6;&#x2C6; J(Ď&#x2C6;n , ynv + [ n ]) Âˇ [ n ]

(6.88)

[ n+1 ] = [Îˇn+1 ] + [Ď&#x2021;n+1 ] + [en+1 ]

(6.89)

â&#x2C6;&#x2014; r where Ď&#x2020;n (tn+1 ) = yn+1 , Ď&#x2C6;n (yn ) = yn+1 and J the jacobian matrix of Ď&#x2C6;n evaluated at v yn + [ n ]. To reduce the wrapping effect, Bouissou and Martels used Lohnerâ&#x20AC;&#x2122;s QR factorization (see Section 6.2.4), a method that is also frequently used in ITS methods. The main difference between the GRKlib method and ITS methods is how the local truncation error is bounded. The method also differs from basic interval integration methods in that it only uses interval arithmetic for the errors and uses conventional floating point numbers for the integration.

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6.5

Chapter 6. Interval ODE Integration

Bounding Rounding Errors

The main point of verified ODE integration is to provide guaranteed bounds on ODE solutions. The previous sections show in detail how it can be mathematically proven (guaranteed) that a solution exists and is unique within the bounds provided by different verified ODE integration methods. Because these methods are implemented on a finite precision computer, it should also take rounding errors into account; rounding errors should also be bounded. Bounding rounding errors is trivial and easily implemented, but should not be forgotten as it makes the methods discussed in previous sections non-guaranteed; the proof fails when rounding errors are not bounded. The interval [x] = [x, x ¯] may not be representable on a finite precision computer if x and ¯ ¯ x ¯ are not finite numbers. On a computer x and x ¯ must be rounded and the default is usually ¯ rounding to the nearest representable number. A rounded interval in this way, [xr ], may not bound the original interval [x]. In order that [xr ] ∈ [x], x must be rounded downward and ¯ x ¯ must be rounded upward, which is called outward rounding. The IEEE 754 standard for floating point arithmetic [152] has four rounding modes, nearest, round down, round up and round towards zero, thereby making interval arithmetic possible on essentially all current computers. [153] The interval software tools discussed in Section 6.7.3 all have outward rounding implemented. The overestimation and increase of interval widths is very small and insignificant to the wrapping effect and dependency problem that cause overestimation in interval ODE integration, as long as double precision (64bit) floating point numbers are used.

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6.6

Chapter 6. Interval ODE Integration

Automatic Interval Differentiation

”Automatic Differentiation (AD) is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as a computer program.” [154] Every computer program uses basic elementary arithmetic operations such as addition and multiplication. By dividing a complex function in small parts and applying the chain rule of derivative calculus recursively to these operations, we can automatically compute derivatives of arbitrary order, accurate to working precision (only rounding errors). When we combine AD with interval arithmetic, rounding errors and interval variables can be used to bound the derivatives. A good introduction was published by Rall and Corliss in 1996 [155] . Conventional differentiation methods are symbolic differentiation and finite difference numerical differentiation. Symbolic differentiation is slow and requires large amounts of memory for high order derivatives. Numerical methods are only approximations and contain truncation errors. AD is a fast and precise method to calculate total and partial derivatives of complicated functions at certain points. The use of derivatives in scientific computing is limited due to the misunderstanding that precise derivatives are hard to obtain. Numerical differentiation is used, but is not precise and can lead to inaccurate results. AD can be used to obtain high order Taylor series and can therefore replace Runge Kutta and other methods that avoid high order derivatives at the cost of accuracy. [156] The first two subsections give a short summary of conventional differentiation methods; finite difference numerical methods and symbolic methods. The last subsection discusses how to obtain high order Taylor coefficients using AD. AD software tools are discussed in Section 6.6.

6.6.1

Finite Difference Numerical Differentiation

This section is a based on the sections about numerical differentiation in the books on numerical calculus by Stoer [79] , Butcher [80] , Hoffman [81] and Vuik [82] . The simplest method to approximate the first derivative of a function f (x) is to use Newton’s finite difference approximation, f 0 (x) ≈

f (x + h) − f (x) h

(6.90)

As h approaches zero, the approximation becomes closer and closer to the tangent line to the function f (x) at point x. Since we cannot let h go to zero on computers, we always have an approximation of the true derivative. Even if we make h very small on computers, we introduce large rounding errors so we cannot use extremely small h. Higher order methods for the first and higher derivatives exist. These require more function evaluations, but are generally more accurate. An example of a second order method is, 3f (x) + 4f (x + h) − f (x + 2h) f 0 (x) ≈ (6.91) 2h and an example for the second derivative, f 00 (x) ≈

f (x + h) − 2f (x) − f (x − h) h2

(6.92)

The advantage of finite difference approximations is that it only requires function evaluations of f (x) to approximate its derivatives. The disadvantage is that it yields approximations only.

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6.6. Automatic Interval Differentiation

6.6.2

Chapter 6. Interval ODE Integration

Symbolic Differentiation

Symbolic Differentiation (SD) uses the same differentiation techniques as AD, but it never uses numerical values and always leaves the derivatives in symbolic form. Symbolic differentiation has some advantages and disadvantages over automatic differentiation. The advantage of SD over AD is that SD is exact (no rounding errors), general (not evaluated at a certain value of the function variables) and symbolic (more options for qualitative analysis). The major disadvantages are the computation time and memory requirements for higher order derivatives. Derivatives are expressed in lower order derivatives, which makes most derivatives exponentially increase in length/complexity with higher order. This does not happen with AD because lower order derivatives are evaluated for given function variables and only a single floating point value (or interval value) represents the lower order derivative. Maple and Mathematica are mathematical software environments that can do symbolic (analytical) differentiation. The symbolic Maple engine is also available as a MATLAB toolbox and can thus also be used in MATLAB. Example (Symbolic Differentiation with Mathematica) f = ex sin(x) df = ex sin(x) sin(x) + xex sin(x) cos(x) dx d2 f = x2 ex sin(x) cos2 (x) + ex sin(x) sin2 (x) − xex sin(x) sin(x) dx2 + 2ex sin(x) cos(x) + 2xex sin(x) sin(x) cos(x) d3 f = x3 ex sin(x) cos3 (x) + 3x2 ex sin(x) sin(x) cos2 (x) − 3x2 ex sin(x) sin(x) cos(x) dx3 + ex sin(x) sin3 (x) − 3xex sin(x) sin2 (x) − 3ex sin(x) sin(x) + 6xex sin(x) cos2 (x) + 3xex sin(x) sin2 (x) cos(x) − xex sin(x) cos(x) + 6ex sin(x) sin(x) cos(x)

6.6.3

Interval Taylor Coefficients using Automatic Differentiation

Interval integration, in contrast with conventional integration, requires that the error term of the integration method can be bounded. Taylor series integration is rarely used in conventional integration because it requires explicit derivatives up to high order, instead of only function evaluations, to approximate the derivatives. However, the error term in Taylor series can be easily bounded and are therefore useful in interval integration. This, however, does require the determination of the derivatives of the (force) function, f (x(t)) in (6.30), up to high order, which is often stated as computational intensive, complex and prone to errors [112] . And is the reason why Runge-Kutta like integration methods are used in conventional integration. According to R. Moore [112] , however, the general notion about its computational intensity is not correct and general algorithms can be developed to find numeric, but exact, Taylor

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coefficients of a wide range of functions. Such algorithms for the automatic determination of the Taylor series coefficients (function derivatives) in a recursive way are discussed in this section. Today this method called automatic differentiation. We consider the system of first order differential equations, d x(t) = f (x(t)) dt

(6.93)

for which the Taylor polynomial, f (x(t)) = f (x(ti )) + f 0 (x(ti ))h +

f 00 (x(ti )) 2 f (n) (x(ti )) n h + ... + h + Rn 2! n!

(6.94)

approximates f (x(t)) in the neighbourhood of t = ti . Here, h = (t−ti ) and Rn the remainder term. We now define Taylor coefficients for f (x(t)) as, cfn (ti ) =

f (n) (ti ) n!

n≥0

(6.95)

Such that (6.94) can be written as, f (ti + h) = cf0 (ti ) + cf1 (ti )h + cf2 (ti )h2 ... + cfn (ti )hn + Rn

(6.96)

Or in short, f (ti + h) =

N −1 X

cfn (ti )hn + Rn

(6.97)

n=0

To explicitly solve (6.93) for x(t), we substitute (6.94) in (6.93) and integrate over time from ti to t which yields, x(ti + h) = x(ti ) + hf (x(ti )) +

1 2 0 1 h f (x(ti )) + ... + hn f (n−1) (x(ti )) + Rn 2! n!

(6.98)

We can now define Taylor coefficients for x(t) as, cxn (ti ) =

x(n) (ti ) n!

n≥0

(6.99)

such that (6.98) can be written as, x(ti + h) =

N −1 X

cxn (ti )hn + RN

(6.100)

n=0

From (6.99) we find that, x0 (ti ) = cx1 (ti )

(6.101)

and, cxn =

1 c(cx1 )n−1 n

(6.102)

where, (n−1)

c(cx1 )n−1 =

d(n−1) cx1 (ti ) (ti ) 1 cx = 1 (n−1) (n − 1)! (n − 1)! dt

(6.103)

x0 (t) = cx1 (t) = f (x(t))

(6.104)

now we use from (6.93) that,

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so that we can use, 1 cfn−1 (t) n≥1 (6.105) n to compute cxn (t), the Taylor coefficients for x(t), if we can recursively analyse the derivatives of f (x(t)) in cfn . The computation of the Taylor coefficients for cfn is possible for most functions by splitting the function f (x(t)) in elementary operations. Let u and v be analytical functions of t in the neighbourhood of t = ti . The following arithmetic operations of the Taylor coefficients of u and v hold, c(u + v)n = cun + cvn (6.106) cxn =

c(u − v)n = cun − cvn c(uv)n =

n X

cuj cvn−j

(6.107) (6.108)

j=0

  n X 1 cun − cuj c(u/v)n−j  c(u/v)n = v j=1

(6.109)

And the following elementary functions, n−1 1X n(a + 1) a c(u )n = a− cun−j c(u )j u j=0 k

(6.110)

n−1 1X (j + 1)c(cos u)n−1−j cuj+1 n j=0

(6.111)

c(sin u)n =

c(cos u)n = −

n−1 1X (j + 1)c(sin u)n−1−j cuj+1 n j=0

(6.112)

The above relations make it possible to compute an arbitrary number of Taylor coefficients which make very precise approximations or long integration steps possible.

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Example (Automatic Differentiation) We use the following simple first order ODE to give an example of the recursive calculation of the Taylor coefficients of x(t), dx = sin(t) dt

x(0) = 0

The analytical solution is given by, Z t x(t) = x(0) + sin(t)dt = x(0) + 1 − cos(t)

(6.113)

(6.114)

Moreover, the symbolic computation of the Taylor coefficients of x(t) is also simple in this case and does not increase in complexity for higher order derivatives. This makes it easy to check the recursive method. The recursive (numeric) computation of cxn (ti ) is done by using (6.111) and (6.112) to derive the Taylor coefficients for sin(t), c(sin(ti ))n , and (6.105) to compute the Taylor coefficients for x(t), cxn . Table 6.1 shows the symbolic Taylor coefficients, their values for t = ti and the recursively computed values. The resulting approximation of x(t) about t = 0 and the deviation from the real solution, for a Taylor polynomial of order 100, is shown in Figure 6.11. This figure shows that, for this periodic example, three periods can be approximated in a single integration step before the error grows larger than 2 × 10−9 . This in contrast with a typical fourth order integration method that can approximate only 1/400 of a period before the error becomes larger than 2 × 10−9 .

Table 6.1: Analytical and AD derived Taylor coefficients for the solution of the differential equation x0 (t) = sin(t) about t = ti . n 0 1 2 3 4 5 6 7 8 9 10

cxn (ti ) x(ti ) sin(ti ) 1 2!

cos(ti ) 1 − 3! sin(ti ) 1 − 4! cos(ti ) 1 sin(ti ) 5! 1 cos(t i) 6! 1 − 7! sin(ti ) 1 − 8! cos(ti ) 1 sin(t i) 9! 1 cos(t i) 10!

cxn (0) analytic 0 0 1 2!

0 1 - 4! 0 1 6!

0 1 - 8! 0 1 10!

cxn (0) recursive 0 0 0.500000000000000 0 -0.041666666666667 0 0.001388888888889 0 -0.000024801587302 0 0.000000275573192

If we can use the procedure described above to compute the floating point Taylor coefficients of f (x(t)), we can easily find the interval Taylor coefficients since only elementary algebraic operations are used. We therefore simply perform the computations in interval

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x(t)

1.5 1 0.5 0 0

10 t

−9

x 10 1.5

x(t)

1 0.5 0 −0.5 −1 −1.5 0

Figure 6.11: Top: The solution to IVP 6.113. Bottom: The difference between the true solution and the Taylor series approximation of order 100 about t = 0.

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arithmetic with F ([x](t)) instead of f (x(t)). This completes the procedure of the computation of the interval Taylor coefficients of f (x(t)).

6.7

Software

This chapter presents an overview of existing software that can be used for interval ODE integration. Chapter 6.7.1 presents interval arithmetic software, Section 6.7.2 automatic differentiation tools and Section 6.7.3 interval integration software. Some of these tools will be used for a collision avoidance system.

6.7.1

Interval Arithmetic

Interval arithmetic software tools provide a way to do interval computations (see Section 6.1.1) instead of fixed point arithmetic. The available tools differ in their language support, speed, rounding methods and floating point precision. This section gives an overview of available interval arithmetic tools and gives a comparison of the tools for use in interval ODE integration. The University of Texas at El Paso has an on-line catalog of interval software that provides a good overview of all available interval related software [157] . Most interval arithmetic software is written in C, C++ or MATLAB. The following subsections give the most commonly used C++ and MATLAB tools. C/C++ The most commonly used C++ (or C++ frontend) interval tools are PROFIL/BIAS [158] , CGAL [159] , Filib++ [160] , MPFI [161] and BOOST [162] . They are all free for non-commercial use. Sun also has a fast and accurate interval library, SUN Forte C++, but is commercial and non free and will therefore not be discussed. [163] PROFIL/BIAS and CGAL only handle double precision floating point numbers, while Filib++, MPFI and BOOST can also do long double precision numbers. Other differences between the methods are in the methods they use to round, handle infinite values (extended interval arithmetic) and the ability to use integers instead of floating point numbers. [164] BOOST is made to emulate all the above mentioned tools without sacrificing the precision, validity and speed of the computations [163] . BOOST is therefore the most flexible tools, but is relative new and may be more difficult to implement than one of the other methods due to the lack of examples and complete documentation. PROFIL/BIAS is very fast in basic interval operations because it uses hardware efficiently. The mathematical functions are slow compared to other tools because it does not use fast look-up algorithms. However, these look-up tables are less accurate [165] . When more than basic operations are needed, Filib++ is fastest and most accurate according to a comparison by Zilinskas [165] . PROFIL is three to seven times faster than Filib++ in basic operations, but Filib++ is three to thirty times faster for other functions. MATLAB Interval tools for MATLAB are INTLAB [166] and b4m [167] . INTLAB is written in MATLAB and the most commonly used tool. b4m (BIAS for MATLAB) is based on BIAS mentioned in the previous subsection. A comparison between the two tools seems not available. Both will be sufficient for most applications although they may greatly differ in speed.

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INTLAB is the most updated, used and active tool and is therefore preferred over b4m. Although, users who have worked with BIAS may prefer b4m because it is the MATLAB version of BIAS.

6.7.2

Automatic Differentiation

Section 6.6 gave a short introduction to Automatic Differentiation (AD). AD is used in interval integration to compute interval Taylor coefficients. This section discusses software tools that can be used for AD, and more specific, computing interval Taylor coefficients. A good overview of different AD software tools for different purposes and programming languages can be found on-line at www.autodiff.org. The next two subsections give a brief overview of available AD software tools for C++ and MATLAB. The C++ tools are also compatible with Fortran and other languages that interface with C. Most tools are written for C++/Fortran and MATLAB; the most common scientific computing languages. Python tools are also available. For computation of interval Taylor series, for use in interval ODE integration, FADBAD++ is probably the best C++ tool and INTLAB the best MATLAB tool. C/C++ There are numerous AD tools written in the C++ language, the most popular are ADIC [168] , ADOL-C [169] , FADBAD++ (FADBAD/TADIFF) [156] and TAPENADE [170] . All can compute higher derivatives of vector functions written in C or C++. The resulting routines can be called from any other language that can be linked with C, like Fortran and C++. ADOL-C seems to be the most popular in scientific literature with at least 50 publications related to ADOL-C. All are open source and free for non-commercial purposes. Only FADBAD++ is known to be able to do AD with respect to interval variables in interval arithmetic using the C++ interval tool PROFIL/BIAS [171] . Other AD tools may be modified to use an C++ interval arithmetic package, but this seems unnecessary since FADBAD++ can already do that. The only reason to try other AD tools, for example the most popular ADOL-C, is to try to increase computational speed. MATLAB TomSym [172] and INTLAB [166] are MATLAB toolboxes that can do AD of high order derivatives. INTLAB is an interval arithmetic toolbox that has an integrated AD tool. This means that INTLAB can compute derivatives with respect to interval variables (using interval arithmetic). TomSym is not able to do this and INTLAB will therefore be the obvious and probably only MATLAB toolbox for AD with respect to interval variables.

6.7.3

Interval ODE Integration

All major interval ODE integration software tools are mentioned an summarised below. The overview includes ITS, TM and RK interval integration methods. AWA AWA is the first interval ODE integration software made by Lohner in 1988 [102] , written in PASCAL-XCS. AWA is the most used ITS method software available. It has been updated with new methods to reduce the wrapping effect and stepsize control a few times since its creation [132] .

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AWA uses a constant a priori enclosure, interval Taylor series and Lohner’s QR factorization to create an interval enclosure of ODE solutions. The constant a priori enclosure restricts the stepsize of AWA to Euler like stepsize. [173] AWA is used to benchmark new ODE integration tools and is verified by many scientist. It is free for commercial use, although currently not available for download on the original AWA website. Although updated a few times, the last known update was in 1994 and the lack of a website suggest that AWA is not improved and used anymore. AWA can be used as a good benchmark for other methods and is sufficient for some problems, but does not use the latest methods to reduce the wrapping effect and is not able to use larger stepsize. Other tools such as VNODE/VNODE-LP, VSPODE and COSY Infinity are newer tools that produce tighter bounds for most problems and are more actively developed and uesd. ADIODES ADIODES is created by Stauning in 1997, written in C++. ”It is used to prove existence and uniqueness of periodic solutions to specific ordinary differential equations occurring in dynamical systems theory. These proofs of existence and uniqueness are difficult or impossible to obtain using other known methods.” [174] ADIODES uses a constant a priori enclosure, interval Taylor series and a rotating rectangular enclosure to create guaranteed bounds; it implements an ITS method. It is implemented in C++ and uses PROFIL/BIAS for interval arithmetic and FADBAD++ for automatic differentiation. ADIODES has no known published applications besides the examples given by Stauning. Other tools are more popular and developed. This makes the validity of ADIODES uncertain and not a preferred choice for use in real applications. VNODE/VNODE-LP VNODE [175] and its updated version VNODE-LP [176] are made by Nedialkov and written in C++. VNODE was created in 2001 and its update VNODE-LP in 2006. [177] VNODE-LP is the update of VNODE completely written in literate programming so that its correctness can be verified by a human expert, similar to how mathematical proofs are verified [123] . VNODE uses the high order a priori enclosure method which makes larger stepsizes possible, compared to the constant enclosure used in AWA. It uses an interval HermiteObreschkoff method that is a generalization of interval Taylor series and is more efficient and produces tighter bounds. Moreover, it also provides the option for an interval Taylor series method. Lohner’s QR-factorization is used to reduce the wrapping effect. [177] The VNODE tool has been applied to multi-body simulation, surface intersection, eigenvalues, state estimation, shadowing and theoretical computer science [123] . The use of high order a priori enclosure and the interval Hermite-Obreschkoff results in larger stepsizes and tighter bounds for most problems, compared to AWA. COSY Infinity COSY Infinity is a Fortran-based code originally developed by Berz and Makino to study and design beam physics systems [109] . The method uses higher order Taylor polynomials with respect to time and initial values. It is not a true interval ODE integration method because it does not return an interval bound. However, the bounds it produces are guaranteed and the method is part of verified integration. The wrapping effect is reduced by establishing functional dependency between initial and final values. [177]

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This method differs from interval Taylor methods like AWA and VNODE in that it does not use interval arithmetic to bound solutions. The resulting bounds are thus not intervals, but a set of Taylor series. It only uses interval arithmetic to bound the truncation error. This has the advantage that the wrapping effect is greatly reduced and non-convex bounds are possible that are much better to bound the real solution set. VSPODE VSPODE was created by Lin and Stadtherr in 2005 and is written in C++. It was written to deal with interval parameters instead of the initial initial values most methods focus on. Interval parameters can be treated the same as initial values, but this leads to the wrapping effect or a significant increase in computational expense that grows exponentially with the number of interval parameters. [178] VSPODE is a hybrid method that uses a combination of ITS and TM methods. VSPODE uses the a priori enclosure method from interval Taylor series, but uses Taylor models with symbolic propagation of initial values and parameters to tighten the enclosure. The set of Taylor models is bounded by an interval and used as initial interval in the next integration step. This is different from COSY Infinity, where the initial values are symbolically propagated through every integration step and the solution set always remains in Taylor model form with only the remainder term in interval form. VSPODE uses QR-factorization to reduce the wrapping effect. VSPODE has been tested against VNODE and produced tighter enclosures for most non-linear test cases. [179] GRKLib GRKLib is a guaranteed Runge Kutta library made by Bouissou and Martel in 2006 [146] . It uses the conventional Runge Kutta integration method (see Section 5.2.3) instead of Taylor series, which are used in most verified integration methods. No details about the methods and no applications can be found in the literature. The method used for bounding the truncation error and the implementation should be independently verified before this method can be applied to real engineering problems. The advantage of this method is that it uses a conventional integration method that does not need high order derivatives and is thus simpler and faster. A disadvantage is the fixed order of the current implementation to four, although higher order Runge Kutta methods can probably be implemented in the library. ValEncIA-IVP ValEncIA-IVP [180] is a guaranteed ODE solver for IVPs based on a novel method not described in this chapter. It is made by Rauh and Auer in 2005 and written in C++. Only a few demonstration examples are given by the authors and no other applications are known. The method ValEncIA-IVP implements, is based on a conventional fixed point numerical integration and thereafter a Picard iteration (see Section 6.2.2) to obtain an interval remainder in which existence and uniqueness is guaranteed [181] [182] [183] . It uses the mean value form and monotonicity tests to reduce the wrapping effect. In the double pendulum example given by the authors, it produces tighter enclosures than VNODE and comparable to COSY Infinity. The disadvantage of ValEncIA-IVP is the current lack of verification of the method, applications and documentation. If this tool is developed further, it may proof to be a fast and tight enclosure method.

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Comparison VNODE-LP is the most modern and currently most used ITS software. It is open source and free for non-commercial usage. AWA is older and not up to date with the newest wrapping reduction techniques. VNODE-LP is made to improve on AWA and does so for most problems. COSY Infinity is a mature implementation of the TM method with over 2000 registered users. VSPODE also uses the TM methods, but in combination with ITS methods. It is the only known hybrid method and has shown promising results in a few benchmark problems. For the collision avoidance system, the VNODE-LP, COSY Infinity and VSPODE software will be applied and evaluated for the ITS, TM and hybrid method, for the reasons given above. Table 6.2 provides an overview of all interval integration methods. Table 6.2: Comparison of interval (verified) integration methods. Name AWA ADIODES VNODE VNODE-LP VSPODE COSY Infinity

Integrator Taylor series Taylor series Taylor series Taylor series Taylor series Taylor series

Method Interval Interval Interval Interval Hybrid Taylor

Year 1988 1997 2001 2006 2005 1997

GRKlib ValEncIA-IVP

Runge Kutta Runge Kutta

Interval Interval Jacobian

2006 2005

122

Language Pascal-XSC C++ C++ C++ C++ Fortran, C++ IF C++ C++

Reference [102] [174] [175] [176] [178] [109]

[146] [180]

6.8. Applications

6.8

Chapter 6. Interval ODE Integration

Applications

Interval ODE integration can be used to analyse or optimize all dynamical systems that are modelled by ordinary differential equations. Critical systems with a high safety requirement can use interval integration to guarantee that the system will operate within certain limits (no collision, max deflection) for given uncertainties within the system. It can also be used for sensitivity analyses, where it can replace Monte Carlo simulations and variational approximations. Or just verify the solutions obtained from conventional integration methods. Guaranteed global optimization of dynamical systems is possible with the combination of interval integration and interval analysis. It is, so far, the only way to guarantee a global optimum, although it is still extremely computational expensive for realistic dynamical systems. Every numerical integration would benefit from the use of interval integration, if only to bound rounding and truncation errors. The disadvantage is the large computational time and wrapping effect that make the bounds useless after long integration times. Space applications of interval integration are presented in the first part of this section. The second part presents some other applications of interval integration.

6.8.1

Space

Space related applications of interval integration include the detection of asteroid collisions, low-thrust missions and trajectory optimization. Asteroid Collision Berz and Makino were the first to apply an interval integration method to orbit integration in 2001 [142] . Hoefkens did the same in 2001 [136] and 2003 [143] , but compared two interval integration methods. Alessi et al. used interval integration to integrate the orbit of asteroid Apophis in 2007 [114] Berz and Makino used their software COSY Infinity that implements the Interval Taylor Model method to integrate the orbit of near Earth asteroid ”1997 XF11” with interval initial values. They used an uncertainty of 10−6 AU in the position and velocity of the asteroid and were able to integrate 3.5 years with a resulting solution set with a size in the order of 10−5 AU. They used a dynamical model that includes all perturbation forces due to the planets, the Moon and a relativistic correction. Not much detail about the models, coordinate sets and implementation used, are given in their paper. Hoefkens also applied the TM method for the verified integration of asteroid 1997 XF11 with interval initial values and physical constants. He notes about the problem, after private communication with Moore, that ”the motion of asteroids in the solar system poses several challenges to verified integration methods, mostly due to the large uncertainties in the initial values, measured physical constants and strong non-linearity in the mathematical models. Therefore, verified tools have not found widespread application for these problems. However, due to the potential seriousness of the outcome of the integration, they are important applications and test cases of verified [integration] methods.” [136] . Hoefkens model includes all planets and a relativistic correction. He uses perturbation methods (Section 3.4) and Kepler elements because the orbital elements are almost linear functions with time, which probably increases the accuracy of the enclosure. He compares the implementation of the Interval Taylor Series method in AWA with the implementation of the Taylor Model method in COSY Infinity. Figure 6.12 shows the position enclosure size

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versus time for AWA and Figure 6.13 and 6.14 the position and velocity enclosure for COSY Infinity. The integrations were started wit a uncertainty of 10−7 AU in position uncertainty. For COSY, the position uncertainty rapidly goes to 10−5 AU and is in the order of 10−4 AU after 10 years of integration. AWA performs worse, although it is up to 50 times faster, and it generates useless bounds after 7 years of integration. Zazzera et al. [128] compared AWA and VNODE for a great number of different Earth orbits in 2004. AWA performs best for interval initial position and velocity, while VNODE is better for point initial conditions or interval initial position or velocity. They recommend investigating Taylor Model methods, which may even be used without interval remainder for missions where guaranteed bounding is not necessary.

Figure 6.12: Diameter (in AU) of the position enclosure versus time (in years) for asteroid 1997 XF11 during a 10 year integration interval using AWA. Source: Hoefkens [143] . Alessi et al. (2007) [114] integrated the orbit of asteroid Apophis with an uncertainty in the initial values of 10−6 AU using the ITS method. They compared the parallelepiped and QR-decomposition methods to reduce the wrapping effect. With a stepsize of 0.625 days, the QR method produced the best results and was able to integrate for 749 days, compared to 649 days of the parallelepiped method. It is unclear what makes the integration stop, but it is probably because the solutions enclosure becomes too large and the algorithm breaks down. The QR method produced errors with the inversion of matrices, while this did not happen in the parallelepiped method. They conclude their research with important difficulties, requirements for useful orbit integration and recommendations for future research. The difficulties are the small stepsizes due to the determination of the rough enclosure, the dependency problem and the wrapping effect. Small stepsizes cause large computation times. The dependency problem is best reduced by subdivision (see Section 6.2.4) and the wrapping effect by using subdivision, parallelepiped or QR method. The successful interval integration requires small initial value intervals, moderate integration times and small wrapping effect. The authors do not give quantitative information about these requirements. They do advice to research Taylor Model methods, because this method is generally better in dealing with the wrapping effect.

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Figure 6.13: Diameter (in AU) of the position enclosure versus time (in years) for asteroid 1997 XF11 during a 10 year integration interval using Taylor models. Source: Hoefkens [143] .

Figure 6.14: Diameter (in AU/TU , TU = year 2Ď&#x20AC; ) of the velocity enclosure versus time (in years) for asteroid 1997 XF11 during a 10 year integration interval using Taylor models. Source: Hoefkens [143] . Low-thrust Mission Besides interval integration of an asteroid orbit, Alessi et al. [114] also analysed the use of interval integration for low-thrust missions to escape from Earth. Again, they applied

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Figure 6.15: In red, the part of trajectory integrated with a first order QRLohner method, about 43 days. In blue, with a first order parallelepiped one, about 9 days. In black, the Earth. An uncertainty in space of 1012 AU and null uncertainty in the thrust. Unit AU. Source: Alessi et al. [114] . the Interval Taylor Series method with the parallelepiped and QR method to reduce the wrapping effect. They used an uncertainty in the initial position, 10â&#x2C6;&#x2019;12 AU, and the thrust, 10â&#x2C6;&#x2019;9 AU/day2 . With the QR-decomposition method the maximum integration time was 43 days compared to 9 days for the parallelepiped method. Figure 6.15 shows the interval trajectories of the low thrust spacecraft for different wrapping reduction techniques. Trajectory Optimization One of the most useful applications of interval analysis is in global optimization, where the search space is divided in intervals. Computing the cost function using interval arithmetic will yield intervals containing the cost related to a certain part of the search space. Removing the intervals that certainly do not contain the optimal solution and dividing the remaining search space in smaller intervals yields, after recursion, a small interval which contains the global optimum. Interval optimization is the only generally applicable method that guarantees a global optimum. [184] In case of global spacecraft trajectory optimization, the cost function depends, among others, on the (integrated) trajectory of the spacecraft. This means that the cost function contains an integral, which in turn means that interval integration has to be used if we want to use interval analysis for the global optimization. The search space is, for example, the possible initial values. These can be divided into multiple intervals and integrated using an interval integration method to yield an interval valued cost related to the interval initial values. Although the idea of global trajectory optimization was already mentioned by Moore in 1968 [112] [184] , there is almost no literature on the subject. Chu [185] [186] applied interval analysis to global optimization of a re-entry trajectory, her efforts were continued by Filipe [187] who noted a significant flaw in Chuâ&#x20AC;&#x2122;s method. Although Chu used interval analysis to evaluate most of the cost function, she did not use interval integration for the integrated part. Bounding the solution set for interval initial values was done using normal numerical

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integration of the upper and lower bounds of the initial values [187] . This method is only correct for monotonically increasing or decreasing integrals, which is generally not the case for the non-linear differential equations of spacecraft trajectories. That said, she notes that, even without using interval integration, the optimization for a re-entry trajectory is time consuming and impossible to do globally with modern computers. Moreover, the wrapping and dependency problem resulted in many feasible solutions that are difficult to distinguish. Filipe tried to include interval integration in his method to correct and improve Chuâ&#x20AC;&#x2122;s method. He used the VNODE-LP (see Section 6.7.3) tool for the interval integration, although he identified VSPODE as the most promising interval ODE solver, but did not use it due to his collaboration with ESA who preferred VNODE-LP for unspecified reasons. Filipeâ&#x20AC;&#x2122;s results show that global interval optimization is possible, but currently not practical due to the enormous computation time needed for global optimization of several parameters. The wrapping problems causes large overestimates which can only be reduced by choosing interval initial values of small width. This means that the search space is divided in a huge amount of intervals and a corresponding number of simulations. The results of both Chu and Filipe motivated the author to research the most basic and largest problem with global interval trajectory optimization; the interval integration part with interval initial values and parameters. Before a complex problem like re-entry optimization can be solved using interval integration, the interval integration itself has to be improved and researched to find the best methods that can handle the largest range of initial values with the smallest overestimation. This literature research and the planned scientific computing research are hoped to yield more information about the different methods that can be used for global trajectory optimization and to make a small step towards feasible global trajectory optimization using interval analysis.

6.8.2

Other

Besides space applications, interval ODE integration has many possible applications in all engineering and science fields where analysis, optimization or verification of a dynamical systems, modelled by ODEs, is required. This section gives a short overview of its applications to particle accelerators, chemical engineering and general optimization of dynamical systems. Particle Accelerators The Taylor Model method and implementation in the COSY Infinity package were originally invented by Berz in 1994 [107] , to analyse the stability of particles in particle accelerators like the Large Hadron Collider (LHC) and Tevatron. Storage rings within these accelerators store (moving) particles that are later injected into the main accelerator ring where they collide and yield data for particle research. The particles in the storage rings may collide with the wall, which effectively makes the use of that particle impossible. Berz method finds six approximate invariants of motion and give guaranteed bounds on the lifetime of a particle. The method relies on bounding of a deviation function which describes fluctuations of the approximate invariants. This method is comparable to the method of variation of parameters (see Sections 3.4.2), although the stability analysis differs from any method mentioned in this report. By also bounding rounding and truncation errors with interval arithmetic, 1 million stable turns in the PSR II particle storage ring can be guaranteed. However, this requires

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splitting of intermediate intervals in smaller intervals because otherwise the bounds blow up too fast. [188] [189] [107] Chemcial Engineering Lin and Stadtherr are scientists at the chemical and biomolecular engineering department of the university of Notre Dame. In [178] they published their software implementation of interval integration called VSPODE along with a few example problems in chemical engineering. One example is the a microbial growth process in a bio-reactor. The concentrations of the biomass and substrate and their change are modelled by a set of two first order differential equations. Using VSPODE, they show that both concentrations can be bounded. Optimization As is already explained above, interval analysis is the only general method to find a guaranteed global optimum. In the optimization of dynamical problems, the optimization often also needs guaranteed interval integration. Interval integration is not yet used in realistic optimization where interval integration is needed because the wrapping effect becomes too large for realistic interval initial values. With the current wrapping methods, a non-linear problem has to be divided in a very large number of sub-intervals. All these sub-intervals have to be integrated using an interval integration method. This takes currently too much computational time to be practical. Two types of improvements can make global optimization of dynamical systems feasible; faster computers and better methods to reduce the wrapping effect. Both improvements are possible and required for global interval optimization. Future research will have to show how much improvement in both methods and computational speed are required.

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Chapter 7

Thesis Assignment This literature survey is a preparatory work for the MSc. Thesis at the Aerospace Engineering faculty of Delft University of Technology. This chapter defines the research question, intermediate objectives and the general methodology to answer and achieve these.

7.1

Research Question

All topics discussed in this literature survey are related to the problem of space collision avoidance using interval analysis (integration). All necessary information to investigate whether such an Interval Collision Avoidance System (ICAS) is feasible, is gathered and organised in this survey. ICAS will use the interval ODE integration methods from chapter 6 to guarantee, by mathematical proof under certain assumptions, for two orbiting bodies, â&#x20AC;˘ A: The absence of possibility of collision without intervention, during a specified time or, â&#x20AC;˘ B: The absence of possibility of collision after a certain orbital maneuver, during a specified time. A is the case when the guaranteed bounds on the trajectory of both bodies do not intersect. When they do intersect, case B, collisions may occur and an orbital maneuver has to be performed to rule out collisions again. Researching the feasibility of such a system is the objective of the thesis. The thesis research question is therefore defined as: Is it feasible to use interval analysis to rule out collisions in space? This question is general and cannot be answered by a simple yes or no. It merely defines the application and focus of research of interval trajectory integration and the Unified State Model (USM) (see Chapter 4). A few relevant and important sub-objectives related to the USM, interval trajectory integration and collision avoidance will be treated while trying to answer the main question. These intermediate objectives are discussed in the following section.

7.2

Objectives

The main objective of this thesis is to answer the above mentioned research question. In order to do so, an ICAS will be developed.

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However, there are also many unknowns about the application of interval integration and the USM. Important new information about these methods will directly result from the application of these methods to space collision avoidance. The USM has not yet been applied to orbit simulation and comparisons between other methods like the variation of orbital elements and Cartesian variables do not exist. For interval integration, there is not yet a complete and useful overview of the quality of different methods for guaranteed trajectory integration. The author was introduced to interval analysis by the thesis work of Chu [186] and Fil[187] ipe who tried to use interval analysis for optimization of dynamical systems (re-entry). Although both made progress in global optimization using interval analysis, Chu did not use interval integration and Filipe used interval integration but could not reduce the overestimation enough for practical applications to re-entry problems (see Section 6.8). Performance and requirements for global trajectory optimization using interval analysis will be investigated and documented to support further research on this topic. We can put the intermediate objectives in three categories; the USM, interval trajectory integration and space collision avoidance: Unified State Model: • Research the computational efficiency of the USM compared to Kepler and Cartesian models. Which model is most efficient for a certain accuracy? • Research the tightness and efficiency of the USM in interval trajectory integration. Does the USM yield smaller enclosures than the method of Variation of Orbital Elements or Cartesian variables? Interval Trajectory Integration: • Research size and growth of solution enclosures for Interval Taylor Series and Taylor Model methods, applied to trajectory simulation. Which method yields the smallest solution enclosure for orbit integration? • Research the relation between the interval initial-value enclosure size and the solution enclosure size. This is important information to determine global trajectory optimization feasibility. Is global trajectory optimization possible using interval analysis? • Research the enclosure sizes generated by different dynamical models (USM, variation of orbital elements and Cartesian). Which dynamical orbit model produces the smallest solution enclosures for a certain computational effort? Space Collision Avoidance: • Research the requirements on the uncertainty in position and velocity, from orbit determination, for collision avoidance (Earth satellites and asteroids). How precise do we need to know the position and velocity of space objects for interval collision avoidance? • Research the required computational power for a full scale Space Collision Avoidance system with current measurement uncertainties. Is a global ICAS feasible with current computers and measurements?

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â&#x20AC;˘ Research the number of collision avoidance manoeuvres with current measurement uncertainty. How often will satellites need to change orbit because ICAS cannot rule out collisions? Is this worth the risk of collision?

7.3

Computer Implementation

The models and methods described in this literature survey will be implemented, during the thesis research, in order to fulfil all objectives stated in the previous section and answer the research question. Equations of motion, coordinate transformations, reference frames, perturbing forces, integration methods, interval methods and collision detection algorithms will all be integrated in a single, but modular, ICAS that forms a prototype for an interval collision avoidance system. C++ will be the programming language of choice for ICAS, Matlab will be used for data analysis and visualization. There are two reasons for choosing C++. First, most interval integration, automatic differentiation and interval analysis software tools are written in C/C++. Four of these tools will be used in ICAS; FADBAD++ PROFIL/BIAS, VNODELP, VSPODE and COSY INFINITY. The use and implementation of these open source C++ tools in ICAS is the main reason for choosing C++. Second, C++ performs generally faster than Matlab and Java for ODE integration.

7.3.1

Software Tools and Libraries

ICAS will use the following tools and libraries. The standard BLAS and LAPACK routines for fast linear algebra computations. PROFIL/BIAS for interval analysis. FADBAD++ for automatic differentiation and TADIFF for automatic differentiation of Taylor coefficients. VNODE-LP, VSPODE and COSY Infinity are used for interval ODE integration and wrapping reduction. All the above packages have a C++ interface and are free for non-commercial use, except for VSPODE for which no information about licence, availability and installation is available. Personal communication with the author is the only way to obtain VSPODE. The Matlab package INTLAB is an interval library that will be used to verify interval results.

7.3.2

Software Requirements

The main requirement of the program is to detect collisions between satellites. It therefore has to produce an interval trajectory enclosure, check for intersection of two intervals and validate its methods by comparison with conventional integration method in combination with Monte Carlo simulations. A dynamical orbit model, an interval integration method, satellite initial data and transformations are required to produce an interval trajectory enclosure. All primary software requirements are shown in a requirements tree in Figure 7.1.

7.3.3

Units

Figure 7.2 gives an overview of the different units in the proposed program. The units are put in three different groups; model, viewer and controller. These function as independent parts within the program. The model contains the groups of equations of motion, transformations, integrators, detection methods, etc; the viewer the user interface, data processing

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Detect Possible Collisions

Detect Interval Intersection

Interval Intersection Method

Produce Interval Position Enclosure

Integrate Interval Initial Values

Dynamical Model

Perturbing Forces

Validating Enclosure

Equations of Motion

Interval Taylor Series

Transform Coordinate Sets

Transform Reference Frames

Initial Data

Dynamical Model

Conventional Integration

Taylor Model Hybrid

Taylor Series

RungeKutta

Monte Carlo

Figure 7.1: Software Requirements Discovery Tree.

Model

Controller Coordinate Transformations Cartesian, Spherical, Kepler, USM

Reference Frame Transformation Earth Centered Inertial International Celestial

Equations of Motion Unified State Model Cartesian Vectors Gauss's Planetary Equations

Conventional Integrators Runge Kutta Taylor Series (Automatic Differentiation)

Interval Integrators Interval Taylor Series Taylor Models

Collision Detection Interval Intersection Detection Closest Distance between Orbits

Perturbing Forces J2 gravity term Major Planets and the Moon Aerodynamic drag

Monte Carlo

Viewer Solar System Data

Position, mass, velocity Planets, Sun and SS Barycenter

Earth Data Position, mass, velocity Atmospheric density model J2 gravity constant

Satellite Data Test scenarios for various Earth orbiting satellites Position, velocity, mass, size, surface area, drag coefficient,

Select Dynamical Model Unified State Model Cartesian Vectors Gauss's Planetary Equations

Command Line Interface 3D Cartesian Trajectory Plot 2D “varx vs vary” Plots Cartesian, Spherical, Kepler, Unified State Variables and Time.

Interval Enclosure Visualization 3D interval plotter Enclosure vs time for all variables

Interval Intersection Plot Distance Between Two Objects

Select Integrator Conventional: RK, Taylor Interval: VNODELP, VSPODE, COSY

Select Output Data Coordinate set Intervals

Select Reference Frame

Figure 7.2: Preliminary software units of the Interval Collision Avoidance System.

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and visualization; and the controller is the interface between the model and viewer and contains initial data and event handlers. The model is the most important and complex part of the program. Its units will be described in the next paragraphs. Transformations Transformations between different coordinate sets are necessary for data analysis, to transform initial conditions to the variables of the model and to transform intervals to a single coordinate set for interval intersection detection. The coordinate sets used are Cartesian coordinates, Kepler elements and Unified State variables. The transformation unit can transform between these sets, in interval analysis. Transformation between reference frames is generally not required because all simulations occur in an inertial Celestial or inertial Earth centred reference frame. A transformation between these two references frames is only needed for collision detection between asteroids and Earth satellites and for comparison with measurement data and to use measurement data as input. The absence of non-inertial reference frames and the lack of reference frame transformations in general make this part of the transformation unit relatively simple. Equations of Motion Different orbital models (coordinate sets) use different equations of motion. The equations of motion unit returns the variational (or â&#x20AC;?force functionâ&#x20AC;?) values for for a certain state. The models used are the USM, Cartesian and Gauss form of Lagrangeâ&#x20AC;&#x2122;s planetary equations (Variation of Kepler elements). All equations of motion have to be evaluated using interval analysis. The resulting force is therefore also an interval. Perturbation Forces The perturbation forces unit provides the perturbation force due to atmospheric drag and J2 gravity term of the central body. It provides perturbation force components in the directions required for the Cartesian model, Unified State Model and Gauss form of the Planetary equations (Kepler Elements). The unit has the state variables as input and returns the corresponding force components. All function can be evaluated in interval arithmetic. An exponential atmospheric density model simulates the varying atmospheric density with altitude. Integrators Integrators take initial values, stepsize and integration time as input and return the integrated state. An integrator is composed of an equation of motion unit, perturbation unit and an integrator type. Interval Wrapper The interval wrapper is a unit that wraps a Taylor Model in an interval. The wrapper has a Taylor model as input and evaluates the Taylor model in interval arithmetic, using the PROFIL/BIAS library, which yields an interval that encloses the Taylor Model. It uses the transformation unit to transform between coordinate sets and wrap an interval expressed in one coordinate set in an interval in a second coordinate set.

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Change Display

Controller

Viewer

User Input Change Model State

Status/Data request

Model

Status/Data update

Figure 7.3: Interfaces between the model, viewer and controller components. Collision Detection The collision detection unit wants two interval trajectories as input and returns whether, when and where these trajectories intersect. This unit uses the transformation and interval wrapper unit to compare intervals in different coordinate sets. Monte Carlo The Monte Carlo unit generates a large number of initial values for conventional integration methods. It pseudo-randomly (based on a probability density function) samples these initial values from a given interval. The density functions used are a Gaussian(normal) and uniform distribution. In case of a Gaussian distribution within the interval, for example, with the mean in the centre of the interval, more samples will be drawn around the centre than at the edges of the interval; the probability density is approximated by the number of samples per unit space.

7.3.4

Interfaces

The model, viewer and controller components of ICAS all operate independently and can only access and change each other via interfaces between the components. The viewer is dumb and only submits events to the controller. The controller decides what to do with viewer event and gives possible commands to the model. The model follows the controller instructions, keeps track of status changes and updates the viewer. The viewer is an observer of the model and receives an update from the model when the model’s status changes and when new simulation data are available. The model can also send status and simulation data on the viewer’s request. The controller and viewer cannot change any data or functions in the model. The model itself is internally governed by ICAS model, which is composed of different units. ICASmodel can instantiate a simulation. This simulation requires simulations options and returns the

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simulation data to the ICAS model. The internal data and functions of the simulation object are not accessible from outside. The user can give commands to ICAS via the viewer component in the form of a command line interface (or GUI). Simulation settings (integrator type, stepsize, time, initial values, number of integrations, coordinate set, etc) can be set and run. The viewer saves the settings and sends the commands to the controller which decides on an action and actives the model. Simulation data is saved in a standard data file which can be imported in Matlab for visualization and analysis.

7.3.5

Unit Tests

Unit testing is performed to verify the correct implementation of the individual units. The following unit tests will be performed. Transformations Coordinate transformations can be can be verified by comparing the results to manual computations. The transformation from Cartesian coordinates to USM variables and back can also be verified by comparison with Vittaldevâ&#x20AC;&#x2122;s [78] implementation. Transformations between reference frames will have to be verified by comparison to manual calculations and visual inspection of the code. Data of, for example, the Moon can also be found on NASAâ&#x20AC;&#x2122;s Horizon website for different reference frames. The transformations unit should be able to reproduce these data. Equations of Motion and Perturbation Forces Both the perturbation forces and equations of motions have to be verified by comparison with manual calculations, average values for different perturbing sources from literature and compared with other implementations. The USM implementation of Vittaldev [78] can, for example, be used to compare and verify the equations of motion unit for the USM. Integrators Integrator units can be verified by integrating differential equations with known analytical solutions and comparison with existing integrators like the Matlab ode45 integrator. Interval integrators can be verified by comparison with each other and with Monte Carlo simulations. The results from the Monte Carlo simulations should all be contained in the interval enclosure from the interval integration methods. Collision Detection Collision detection boils down to interval intersection detection. Interval intersection methods can be verified by visualization of the intervals and visual inspection. Manual calculations can also easily verify the interval intersection unit. Interval Wrapper The interval wrapper needs to completely wrap a rotated interval in a new interval. This unit can be verified by comparing it to manual calculations, visualization and interval intersection methods. The interval intersection method can verify that the first interval is completely wrapped in the second.

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Chapter 7. Thesis Assignment

Simulator

ICASmodel

-initialValues -equationsOfMotion -integrator -pertForces -options +integrate() +updateStatus()

-settings

PROFIL/ BIAS

+newIntegration() +updateViewer() +checkIntersection() +transformCoordinates() +transformReference()

FADBAD++

TaylorInt

Rk4Int

Rk8Int

-order +integrate()

+integrate()

Tranformations -cart2usm -cart2kepler -kepler2cart -usm2cart -celestial2earthfix +transformCoord() +transformFrame()

CollisionDetection MonteCarlo -gausPDF -uniformPDF +monteCarloSim()

-orbitDateOne -orbitDataTwo +checkIntersection() +updateStatus()

ConvInt

Integrator

VnodeInt VspodeInt +createTemplate()CosyInt +createTemplate()

+integrate()

EquationsOfMotion

+createTemplate()

+returnValues()

VNODE-LP PerturbationForce VSPODE

COSY

-parameters -densityModel +returnValues() +setForces()

USM CartPert +usmFunction() Kepler +cartFunction()

USMPert

+gaussFunction()

CartPert +usmPert() KeplerPert +cartPert() +keplerPert()

Figure 7.4: Class diagram of the ICAS model component. The tightness of this wrapping is analysed by visual inspection of the interval around the Taylor Model. Taylor Models can be easily visualized with Mathematica.

7.3.6

Integration

The most important and complex component of ICAS is the model component, ICASmodel. ICASmodel governs simulators, collision detection and transformations. It is continuously running and waiting for instructions from the controller. It can instantiate single simulators that simulate the orbit of a single satellite. It keeps track of all simulations and their results. The simulator is composed of equations of motion, perturbation forces and an integrator. The type of equations of motion, perturbation forces and integrator can all be chosen by the ICASmodel when initializing a single simulator. Multiple single simulators can be run at the same time or queued. Once an integrator is finished, it notifies ICASmodel and returns the integrated trajectory. ICASmodel can, next to simulating, also transform data sets into different coordinate sets and reference frames. It can check for collisions with its interval intersection detection unit. ICASmodel reports an update to the viewer when the systemâ&#x20AC;&#x2122;s state changes and provides a status update when the viewer requests one. It can only receive instructions from the controller. The class diagram of the model component is shown in Figures 7.4.

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7.3.7

Chapter 7. Thesis Assignment

Verification

Verification of ICAS is necessary to guarantee the program works as intended. Possible errors within the program have to be revealed by verification. Individual units are already verified so any errors in the integrated program are the result of erroneous integration or incompatible interfaces. Verification of the equations of motion, perturbing forces, transformations and conventional integrators is mainly done by comparing the ICAS simulation results with other commonly used trajectory integration models, analytical solution, integration constants and when possible with real satellite or asteroid data. The different integrators of ICAS will also be compared to easy other. The results will be compared to simulation data from NASAâ&#x20AC;&#x2122;s Horizons and other sources. These sources provide thoroughly verified data which can be compared to ICAS. Possible trajectories to compare are the planets and their moons. NASAâ&#x20AC;&#x2122;s Horizons does not provide active satellite ephemeris. The simulated trajectories will also be compared to real satellite data to see how well ICAS models reality. Space-Track.org [190] provides two line elements (TLE) from the US Space Surveillance Network. These TLE accuracy is undisclosed, but is thought to be in the order of kilometers. More accurate special perturbations data is classified and maintained by the US Air Force. The collision between Iridium 33 and Cosmos 2251 can be simulated, but only with the inaccurate TLE data. More precise For some orbits and assumptions, analytical (exact) solutions do exist. These orbits and assumptions (no perturbation forces in two body problem) can be used in the ICAS simulation which then should yield the analytical solution. Last, for the unperturbed n-body problem there are constants of motion, like energy and momentum (see Section 3.1.3). These quantities do not change in our physical models. This means that these quantities should also not change in ICAS, for the same conditions. The interval integration results can be verified by comparison with existing verified interval integration software tools and by Monte Carlo methods (see Section 5.5.1). Monte Carlo methods use a large number of integrations to analyse a system. In case of interval integration, all Monte Carlo solutions starting from within the interval initial values should be enclosed by the interval integration methods. If only one point falls outside this enclosure, there have to be errors in the interval integration method.

7.3.8

Parallel Computing

Parallel computing uses many computational independent simultaneous computations, instead of serial computations, to increase the computational speed. A single CPU can only do serial computations, but a network of CPUs can do parallel computing by all doing individual serial computations and communicate the results. Although a single trajectory simulation is not suitable for parallel computing, subdivision (see Section 6.2.4) and optimization require many independent parallel integrations. For very large numbers of parallel integrations, relative cheap Graphics Processing Units (GPUs) can be used for parallel computing (a single GPU has tens to hundreds of cores) may greatly reduce computational time. Rewriting ICAS for use on NVIDIA GPUs using CUDA (a parallel computing architecture developed by NVIDIA) will be looked into when time allows. Modern CPUs have two to eight cores. This means that they are suitable for parallel computing. They actually only use one core if the software is not specifically programmed

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Chapter 7. Thesis Assignment

to use more than one core. ICAS will be able to run simulations on multiple cores to use the full potential of modern CPUs.

7.4

Simulation

Implementing and verifying the computer implementation is not sufficient to achieve the above mentioned objectives and answer the research question; simulations of realistic situations are necessary. The following simulations are planned, all done using Cartesian variables, variation of orbital elements and the USM; conventional and interval integration (multiple methods) and with different wrapping reduction methods: • Earth satellites: LEO, MEO, GEO and HEO. • Near-Earth Asteroids. • Collision scenarios. Conventional simulation backwards in time from collision situation to pre-collision situation. The pre-collision situation is then integrated using interval integration with collision detection. For these orbits and mentioned orbit models and integration methods, we want to determine, • The size of solution enclosure (interval) for interval integration methods. • The computational speed. • The integration time after which interval integration methods break down; the wrapping effect becomes extremely large.

7.5

Tasks and Schedule

The MSc. thesis is worth 42 ECTS. 1 ECTS is equal to 28 hours, which means that the thesis should take about 33 weeks of 36 hours per week. Figure 7.5 shows an overview of all tasks together with a time estimate per task. The blue tasks indicate lower priority than the black tasks. The computer implementation tasks correspond with the units described in Section 7.3.3. Integration tasks in Section 7.3.6, verification tasks in Section 7.3.7 and simulation tasks in Section 7.4. The document related tasks include writing of introduction, conclusion, recommendations and summary, and proof reading and rewriting the thesis.

138

7.5. Tasks and Schedule

Chapter 7. Thesis Assignment Oct 2009

Task Name

Nov 2009

4/10 11/10 18/10 25/10 1/11

Computer Implementation Design

10d

Coordinate Transformations

Cartesian, Spherical

Kepler Elements

Unified State Variables

Reference Frames

Equations of Motion Cartesian Vector

Gauss’s Planetary Equations

Unified State Model

Perturbing Forces

Conventional Integration Runge-Kutta (order 4-6)

2.5d

Taylor series expansion using AD

2.5d

FADBAD AD implementation Interval Analysis

31/1

7/2

14/2

Mar 2010 21/2

5d 2d

Getting familiar with PROFIL/BIAS and INTLAB

Interval Automatic Differentiation Interval Integration

1d 1d 42.5d

VNODE-LP implementation

COSY INFINITY implementation

VSPODE implementation

A priori enclosure

Moore’s method

Mean value method

Subdivision

Parallelepiped method

10d

Monte Carlo

Collision Detection

5d 2.5d

Interval Intersection Detection

Closest distance between objects

Integration & Verification

10d 5d 5d

15d

Unit tests

Unit integration

Integration tests

Simulation

37d

USM

Compare vs Cartesian

Compare vs Gauss’s planetary equations (Kepler) Interval Integration

2d 3d 15d

Compare Taylor Series and Taylor Model methods

Compare VNODE-LP, VSPODE and COSY

Compare USM, Kepler and Cartesian

Compare wrapping reduction methods

Interval sizes for optimization

Feb 2010 24/1

17/1

10d

10/1

Jan 2010 3/1

15d

8/11 15/11 22/11 29/11 6/12 13/12 20/12 27/12

135.5d

Dec 2009

Duration

Collision Avoidance

3d 17d

Multiple Earth Satellites on near collision course

10d

Near Earth Asteroid

Uncertainty Requirements

Interpretation and Visualization

15d

3D Cartesian trajectory plots

Variable x vs variable y plot

Interval Enclosure Visualization

Interval Intersection plot

Distance between two objects

57 58

Report Related

30d

Figure 7.5: Thesis schedule. Blue tasks have lower priority. 139

28/2

7/3

14/3

21/3

Appendix A - Transformations Cartesian Coordinates to Kepler Elements The transformation from Cartesian position and velocity, x, y, z, x, ˙ y, ˙ z, ˙ to Kepler elements, a, e, i, ω, Ω, M , is given by [50] , p (1) r = x2 + y 2 + z 2 p (2) V = x˙ 2 + y˙ 2 + z˙ 2 r a= (3) 2 2 − rVµ e cos E = 1 −

r a

(4)

1 (xz˙ + y y˙ + z z) ˙ µa r θ 1+e E tan = tan 2 1−e 2 s a3 τ =t− (E − e sin E) µ p H = µa(1 − e2 )

e sin E =

i = arccos

xy˙ − y x˙ H

(5)

(6)

(7) (8)

(9)

sin Ω =

y z˙ − z y˙ H sin i

(10)

cos Ω =

xz˙ − z x˙ H sin i

(11)

cos(ω + θ) =

x y cos Ω + sin Ω r r

M = E − e sin E

140

(12) (13)

7.5. Tasks and Schedule

Chapter 7. Thesis Assignment

Kepler Elements to Cartesian Coordinates The transformation from Kepler elements, a, e, i, ω, Ω, M , to Cartesian position and velocity, x, y, z, x, ˙ y, ˙ z. ˙ First get the eccentric anomaly E from the mean anomaly M and eccentricity e, by iteratively solving Keplers equation [191] . This is commonly done by the simple and effective Newton-Raphson root finder, f (x0 ) x1 = x0 − 0 (14) f (x0 ) For this problem we use for f , f (Ei ) = Ei − e sin Ei − M 0

f (Ei ) = 1 − e cos Ei

(15) (16)

So that the following iteration converges to the true value of E, Ei+1 = Ei −

Ei − e sin Ei − M 1 − e cos Ei

(17)

where E0 = M . Next, we get the position and the velocity in the q-frame, which has its z-axis perpendicular to the orbital plane and its x-axis pointing to the perigee,   a(cos E − e) √ (18) q = a 1 − e2 sin E  0   √ − sin E na  1 − e2 cos E  q˙ = (19) 1 − e cos E 0 where n is the mean motion, r

µ (20) a3 The transformation from inertial frame to the q-frame is performed by the rotation sequence R3 (ω)R1 (i)R3 (Ω). So, vice versa, the inertial position and velocity are obtained by the reverse transformations, n=

r = R3 (−Ω)R1 (−i)R3 (−ω)q r˙ = R3 (−Ω)R1 (−i)R3 (−ω)q˙

(21) (22) (23)

where,   cos Ω − sin Ω 0 R3 (−Ω) =  sin Ω cos Ω 0 0 0 1   1 0 0 R1 (−i) = 0 cos i − sin i 0 sin i cos i   cos ω − sin ω 0 cos ω 0 R3 (−ω) =  sin ω 0 0 1

141

(24)

(25)

(26)

7.5. Tasks and Schedule

Chapter 7. Thesis Assignment

Cartesian Coordinates to USM variables The transformation from Cartesian position and velocity, x, y, z, x, ˙ y, ˙ z, ˙ to USM variables, C, Rf 1 , Rf 2 , q1 , q2 , q3 , q4 , are given by Altman [1] , µ H

(27)

Rf 1 = −R sin(λ − φ)

(28)

Rf 2 = R cos(λ − φ)

(29)

where, H=

p (y z˙ − z y) ˙ 2 + (z x˙ − xz) ˙ 2 + (xy˙ − y x) ˙ 2

(30)

f = (y z˙ − z y)/H ˙

(31)

g = (z x˙ − xz)/H ˙

(32)

h = (xy˙ − y x)/H ˙

(33)

(1 − h) 2

(34)

(1 + h) 2

(35)

(1 − cos Ω) 2

(36)

(1 + cos Ω) 2

(37)

sin i/2 = + r cos i/2 = + r sin Ω/2 = ± r cos Ω/2 = ±

1 sin Ω = f p 2 f + g2 cos Ω = −g p r sin u/2 = ± r cos u/2 = ±

u = sign(z) arccos

1 f2

(39)

+ g2

(1 − cos u) 2

(40)

(1 + cos u) 2

(41)

(−gX + f Y )(f 2 + g 2 ) p x2 + y 2 + z 2

sin λ = sin(Ω + u)

142

(38)

! (42) (43)

7.5. Tasks and Schedule

Chapter 7. Thesis Assignment

cos λ = cos(Ω + u) sin φ =

xx˙ + y y˙ + z z˙ p x2 + y 2 + z 2

(44) (45)

cos φ =

µ(x2 + y 2 + z 2 )− 2 − C 2 CR

(46)

2µ R = + (x˙ 2 + y˙ 2 + z˙ 2 ) − p + C2 x2 + y 2 + z 2 Quaternion in terms of the Kepler elements (Ω, i, ω, θ),     sin 2i cos Ω−u q1 2 q2   sin i sin Ω−u  2 2   = q3   cos i sin Ω+u  2 2 q4 cos 2i cos Ω+u 2

(47)

(48)

USM variables to Cartesian Coordinates The transformation from USM variables, C, Rf 1 , Rf 2 , q1 , q2 , q3 , q4 , to Cartesian position and velocity, x, y, z, x, ˙ y, ˙ z, ˙ are given by Altman [1] ,     X r  Y  = [E]T0 0 (49) Z 0     ve1 X d   Y = [E]T0 ve2  (50) dt 0 Z r=

µ Cve2

ve1 0 cos λ sin λ Rf 1 = + ve2 C − sin λ cos λ Rf 2 1 sin λ 2q03 q04 = 2 2 ) q2 − q2 cos λ (q03 + q04 04 03

143

(51) (52) (53)

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