RWA paper 2011 Infotech by Steve Rogers

Fault Tolerant Control of a Satellite Control System Simulation Using Reaction Wheel Assembly Actuators Steven C. Rogers MPL, Inc., Fairmont, WV 26554

A fault is something that changes the behavior of a system such that the system no longer satisfies its purpose or meets its requirements. It may be an internal event in the system, which stops the power supply, breaks an information link, or creates a leakage in a pipe. It may be a change in the environmental conditions that causes an ambient temperature increase that eventually stops or destroys a system. It may be a wrong control action given by the human operator that brings the system out of the required operation point, or it may be an error in the design of the system which remained undetected until the system comes into a certain operation point where this error reduces the performance considerably. An example of the above is satellite attitude control during an actuator failure. Four Reaction Wheel Assemblies (RWA) are occasionally used for attitude control of satellites. In the case that one of the RWA’s becomes damaged the satellite attitude may still be controlled in all three axes with reasonable reliability. The RWA’s are usually installed with each RWA axis offset from the principal axes by a specified angle, which still enables a torque control about any one of the satellite principal axes. The incapacity of any of the RWA’s can be compensated by the remaining suite of torque capabilities. A possible geometrical configuration of a control system may be based on four RWA’s each inclined to the Xb-Yb principal axis by an angle β. Because of the inclination each RWA applies a torque in the Zb principal axis as well. The RWA torque vector has four elements which have a matrix relationship to the three principal satellite axes. If one of the RWA torque contributions is degraded the other RWA’s must make up for the loss in effectiveness. In this paper several fault tolerant control schemes will be developed and applied to a satellite simulation with a control system based on four RWA’s. One of the RWA’s will be disabled and the control schemes will be compared based on their ability to accommodate the fault. Keywords: Adaptive Noise Control, ASE Filters, model-based adaptive filters, neural networks

Nomenclature FDI e s

= fault detection and isolation = error = signal of interest 1 American Institute of Aeronautics and Astronautics AIAA-2009-2068

Î˝1 y Î˝0 p q r d a b Îź c

= = = = = = = = = = =

disturbance corrupted measurement signal adaptive filter output noise source signal (strain gage equivalent) roll rate pitch rate yaw rate primary sensor signal filter coefficients of input filter coefficients of output adaptive gain for filter adaptive exogenous coefficients

I. Introduction

HE design of a fault tolerant control algorithm usually involves the incorporation or consideration of a plant model. The designer of a successful fault tolerant control algorithm must consider the criticality of the subsystem, legacy methods, types of faults, extent or impact of faults, risk analyses, and developmental resources. The flight control system must be robust against model uncertainties and external disturbances. The analytical fault-tolerant operation can be achieved either passively by the use of a control law designed to be insensitive to some known faults, or actively by an FDI (fault detection and isolation) mechanism, and the redesign of a new control law. The active methods are more realistic because all the faults that may affect the system cannot be known a priori. Figure 1. Typical Fault-Tolerant Control System. This paper shall be concerned with passive fault-tolerant control algorithms, that is, there is no redesign of the control law or a fault detection system. Passive algorithms include robust systems, adaptive components augmenting traditional SISO (single input single output) control algorithms, and multivariable robust control systems. Robust control designs, including H-infinity are designed to accommodate model uncertainties and structured disturbances. A model of a plant can never be perfect, and as such will always be an approximation of the true plant. Occasionally some characteristics of a plant will remain unmodelled. This occurs because their contribution is considered small, they are not well Figure 2. Actuator, Plant, and Sensor Faults. known, they may be difficult to model, or they change with varying flight conditions. A controller is said to exhibit good robust stability if it remains stable for all variations in plant behavior which are expected to occur. Example development and simulations of each category will be included in the paper. The control laws shown in this paper are pole placement, linear quadratic regulator, and H-infinity. They are implemented in single input single output (SISO) and multi-input multi-output (MIMO) form. The SISO forms are augmented with adaptive neural network components: adaptive linear combiner and adaptive radial basis functions.

II. Control Laws Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. Root locus uses compensator gains to move closed-loop poles to achieve design specifications for SISO systems. You can, however, use state-space techniques to assign closed-loop poles. This design technique is known as pole placement, which differs from root locus in the following ways: 2 American Institute of Aeronautics and Astronautics AIAA-2009-2068

• •

Using pole placement techniques, you can design dynamic compensators. Pole placement techniques are applicable to MIMO systems.

Pole placement requires a state-space model of the system. In continuous time, such models are of the form x = Ax + Bu , y = Cx + Du , where u is the vector of control inputs, x is the state vector, and y is the vector of measurements. Under state feedback u = −Kx , the closed-loop dynamics are given by x = ( A − BK ) x and the closed-loop poles are the eigenvalues of A-BK. A gain matrix K can be computed that assigns these poles to any desired locations in the complex plane (provided that (A, B) is controllable). The state-feedback law can’t be implemented unless the full state x is measured. However, you can construct a state estimate such that the law retains similar pole assignment and closed-loop properties. You can achieve this by designing a state estimator (or observer) of the form x = Ax + Bu + L( y − Cx − Du ) . The estimator poles are the eigenvalues of A-LC, which can be arbitrarily assigned by proper selection of the estimator gain matrix L, provided that (C, A) is observable. Generally, the estimator dynamics should be significantly faster than the controller dynamics (eigenvalues of A-BK). The Linear Quadratic Regulator (lqr) is a commonly used type of state space control algorithm. It is frequently used for comparison purposes as it is a very useful means to obtain a state estimator as well as a state feedback control. It does require a linearized dynamic model of the system to be controlled, or a nonlinear model that is convenient to linearize. For a linear model expressed as: x = Ax + Bu , x ∈ℜn (1) We can propose a performance index: ∞ 1 J = ∫ x T Qx + u T Rudt (2) 20 In these equations x is the bounded state vector, u is the control input vector, A is the control matrix, B is the input matrix, Q is the state performance matrix, and R is the input performance matrix. There are no terminal constraints on the control system. The above may be solved by use of the maximum principle as below.

H = x T Qx + u T Ru + λT ( Ax + Bu ) T

 ∂H  x =   = Ax + Bu  ∂λ  T

 ∂H  T − λ =   = Qx + A λ  ∂λ  ∂H 0= = Ru + λT B ⇒ u = −R −1 B T λ ∂u By using the terms of eqn 3 we may arrive at the classical lqr solution as:

(3)

λ = Px ⇒λ = P x + Px

λ = P x + P ( Ax − BR −1 B T P ) x = −Qx + AT Px − P = PA + AT P − PBR −1 B T P + Q (4)

u = −R −1 BPx

 is set to 0, then we have The 3rd line in the equation above is called the Riccati ordinary differential equation. If P the algebraic Riccati equation which may be solved by numerous software packages. With minor modifications the same approach can produce a state estimator. A direct H∞ solution is given by minimizing γ in the equation below. 3 American Institute of Aeronautics and Astronautics AIAA-2009-2068

γ = inf K

K  −1 ~ −1 ( )  I  I −GK M  

∞

(5) Note that in this approach the classical γ -iteration is avoided. Given a minimal realization [A, B, C, 0] of a controllable and observable plant the control algebraic riccati equation (CARE) and filtering algebraic riccati equation (FARE) are given. A* X + XA − XBB * X + C * C = 0 AZ + ZA* − ZC *CZ + BB * = 0

The optimal γ is given by:

(6)

γ min = (1 + λmax ( XZ ) ) 1 2

(7)

The controller is then:

 A + HC + γ 2 BB * XW * − 1 − H  1 K=  2 * * −1 γ B XW1 0   W1 = I + XZ − γ 2 I

(8)

H = − ZC * III. Adaptive Components The adaptive linear combiner is the basic building block for adaptive filters. The linear combiner input sequence is not necessarily temporal delayed samples of one single input, and it is therefore a generalized form of the transversal structure. The adaptive linear combiner filter structure is shown in Figure 2. The input vector in the case of the linear combiner consists of temporal samples of several signals that might be Figure 3. Adaptive Linear Combiner coming from an array of sensors, and is expressed as

x ( n ) = [ x 0 ( n)

x1 (n)  x N −1 ( n)]

. The

adaptive linear combiner can be split into two main parts, the filter part and the update part. The function of the former is to calculate the filter output y(n), while the function of the latter is to adjust the set of N filter coefficients wi , i = 0,1,  , N −1 (tap weights) so that the output y(n) becomes as close as possible to a desired signal d(n). The filter part processes the set of input signals at each time index n to produces a single output sample y(n) (assuming sample per sample implementation). The filter output at time index n is calculated as a linear combination of the input signals sampled at that time instance

(9) Expressing

the

w( n) = [ w0 ( n)

set

filter

coefficients

w1 (n)  wN −1 ( n)]

time

index

vector

notations

such

that

, where (.) is the vector transpose operator, eq (9) can be written

as y ( n) = w( n) x ( n) = x( n) w( n) . The most commonly used learning algorithm is called LMS (Least Mean Square), uses an instantaneous gradient and is simple and effective. The algorithm is described by the simple recursive formula: T

4 American Institute of Aeronautics and Astronautics AIAA-2009-2068

Wk +1 = Wk + 2 µek X k ek = d k − X kT Wk (10) The present weight vector is Wk, the next weight vector is Wk+1, the present error is ek, the present desired response is dk, and the constant µ is a design parameter that determines stability and rate of convergence.

A radial basis function network4 as shown in Figure 4 is an artificial neural network that uses radial basis functions as activation functions. It is a linear combination of radial basis functions. They are used in Figure 4. Architecture of a Radial function approximation, time series prediction, and control. An input Basis Function Network. vector x is used as input to all radial basis functions, each with different parameters. The output of the network is a linear combination of the outputs from radial basis functions. Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The output

of the network is thus

, where N is

the number of neurons in the hidden layer, is the center vector for neuron i, and ai are the weights of the linear output neuron. In the basic form all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance and the basis function is taken to be Gaussian. RBF networks are universal approximators on a compact subset of . This means that a RBF network with enough hidden neurons can approximate any continuous function with arbitrary precision. The weights optimizes the fit between and the data.

or parameters are determined in a manner that

An adaptive radial basis function network4 may be constructed using a Gaussian basis function

 − u − tk

ϕ ( u; t k ) = exp  

2σ

2 k

 , σ > 0  k 

(11) The adaptive equations are as follows: K

y (n) = ∑ wk (n)ϕ( u (n); t k (n) ) k =1

e( n) = d ( n) − y ( n) wk (n + 1) = wk (n) + µw e( n)ϕ( u (n); t k ( n) ) t k (n + 1) = t k (n) + 2µt e(n) wk (n)ϕ( u (n); t k (n) )

u ( n) − t k ( n) σ k2 (n)

σ (n + 1) = σ (n) + µσ e(n) wk ( n)ϕ( u (n); t k (n) ) 2 k

2 k

u ( n) − t k ( n)

σ k2 (n)

(12) Note that σ is standard deviation which is always positive, µw , µt , µσ in general are different positive values, u(n) is the input vector at sample point n, t is the center width of the kth radial basis function, w is the kth weight in the output function.

5 American Institute of Aeronautics and Astronautics AIAA-2009-2068

IV. syscon roll control syscon

3 Mom 3

SISO_Mom

pitch control 1 err

syscon

Integration of Adaptive Components with SISO Control Laws

Rerr 1 Perr 2 Yerr 4

5 5 Delays 5 5 Delays 5 5 Delays

5 5 5 5 5 5

30 30

MATLAB 3 Function LC 3 MATLAB 3 1 3 Function Maug Manual Switch rbf

Rt The control 5 5 3 Delays laws are all Pt 5 proportional Rerr 5 5 Delays Perr 3 Rt derivative (PD) in 3 Maug Y err Yt Maug 3 5 Adap_Mom Pt 5 design, however, 6 Delays Yt LinComb the adaptive components only operate on error Figure 6. SISO Adaptive Components Figure 5. SISO Control Integration (deviation from desired set point). As mentioned above the adaptive components were added to the SISO control designs and the adaptive components are SISO as well. The conventional control laws are applied to the roll, pitch, and yaw signals independently and the adaptive components are designed for roll, pitch, and yaw. Figure 11 shows the architecture for the integration. The top three simulink blocks are for the roll, pitch, and yaw channel control laws. The bottom simulink block contains the adaptive SISO components. The outputs of the top control laws and the adaptive components are added to together to obtain the total control input. Figure 12 shows the inputs to the adaptive components. The inputs for both the adaptive linear combiner and the adaptive radial basis function are the same. They consist of set point deviation error and previous moment inputs. For each control channel there are two inputs, which are fed to a five tapped delay line. yaw control

3 3

MomTot

V. Simulation The simulation is based on a generic satellite model. The satellite rigid body dynamics is modeled as a simple set

 0  = −ΩJw + M , Ω =  w3 of nonlinear equations Jw  − w2

− w3 0 w1

w2  − w1  , where w is the angular rate vector [roll, 0 

pitch, yaw], J is the mass moment of inertia matrix, and M is the moment vector. For this application the matrix 0  1269 −106  0  moment of inertia J = −106 1272  . The moment vector is given in the three axes. Note that for  0  0 1524  redundancy most satellites are designed with four reaction wheel assemblies (RWA’s). Each is placed on an axis, but tilted at an angle β in order to influence more than one axis. Following the example in ref 5 we have the equations:

T  T1  Tˆcx  Tcx / cβ  1 0 − 1 0  1  T  T2  ˆ        2 Tcy  = Tcy / cβ  = 0 1 0 − 1 T  = [ Aw ] T  = [ Aw ]T 3 Tˆ  T / sβ   1 1 1 1   3       T  cz   cz  4 T4  (13) In the above cβ = cos(β), sβ = sin(β), and β is the RWA inclination angle to the appropriate axis. The matrix Aw in equation 13 is not square and is not easily inverted, therefore another equation is needed. Since the summation of moments is zero, we can use the following 0 = T1 − T2 + T3 − T4 . We can now rewrite equation 13 as:

6 American Institute of Aeronautics and Astronautics AIAA-2009-2068

Tˆcx  1 ˆ   Tcy  = 0 Tˆcz  1     0  1

−1

1 1 −1

0 1 1

0  T1  − 1 T2  1  T3    − 1 T4 

(14)

0 T1  1 T  0 1  2= 1  T3  2 − 1 0    T4   0 −1

0.5 0.5 0.5 0.5

0.5 Tˆcx    − 0.5Tˆcy  0.5 Tˆcz    − 0.5 0 

(15) With the above we have the transformation between the three axes command control torques and the four wheels’ command control torques. To achieve a zero-reaction failure for a single RWA a column of the matrix in equation 14 may be zeroed out. The above sequence of equations was implemented in matlab and simulink.

7 American Institute of Aeronautics and Astronautics AIAA-2009-2068

VI. Results The nine different strategies as explained above were tested on the above satellite model. The plots are shown below. The first comparison (Figures 7-12) is with closed-loop frequency response. The singular values for the three axes are plotted on top and the classical stability metrics including gain margin (GM) and phase margin (PM) are in the table. The pole placement and lqr approaches appear to be the best for both SISO (single input single output) and p oleplace T rackingerror, S VD , roll off 10

Hinf SISOTracking error, SVD , roll off 40

-1 0 Singular Values (dB)

S ingular V alues (dB )

-2 0 -3 0 -4 0 -5 0

-20

-40

-6 0 -60

-7 0 -8 0 -4

-2

-80 -5 10

F reque ncy (rad /se c)

-4

-3

-2

-1

Frequency (rad/sec)

Figure 8. SISO H-infinity Sensitivity Matrix Figure 7. SISO Matrix

Pole

Placement

Sensitivity

MV poleplaceM VTrackingerror, SVD , and roll off 10

lqr S IS O T rackin gerror, S VD , an droll off

Singular Values (dB)

S ingular V alues (dB )

-1 0

-2 0

-3 0

-10 -20 -30 -40 -50 -60

-4 0

-70 -5 0

-80

-4

-2

Frequency (rad/sec)

-6 0 -5 10

-4

1 0

-3

-2

1 0

-1

F reque ncy (rad/sec)

Figure 10. Multivariable Sensitivity Matrix

Figure 9. SISO LQR Sensitivity Matrix

Pole

Placement

lqr M VT rackingerror, S VD , androll off

H inf M VT rackingerror, S VD , androll off

S ingular V alues (dB )

0 -10 -20 -30 -40

-20

-40

-60

-50

-80

-60 -70

-4

-2

-100

-4

F requency (rad/sec)

-2

F requency (rad/sec)

Figure 11. Matrix

Multivariable

H-infinity

Sensitivity

Figure 12.

Multivariable LQR Sensitivity Matrix

8 American Institute of Aeronautics and Astronautics AIAA-2009-2068

(multivariable). The MV approach appears also to be less sensitive to disturbance as noted by the smaller peak as the singular values begin to roll off. A 300 second trajectory was simulated for each of the seven strategies. Across the title of each plot is listed the control method, whether RWA 1 is failed or not, and poleplace, Satellite RWA No. 1 is 100% RPY 2.30 1.87 x 10error = 1.34 0.1 1 the roll-pitch-yaw error vector. Figures 13-18 are the SISO without adaptive components added. The pole 0 0 placement appears to be the best in this case. -0.1 -1 roll (rad)

-3

100

200

300

100

200

300

100

200

300

100 200 time (sec)

300

pitch (rad)

-3

0.2

-0.2

-5

100

200

300

x 10

yaw (rad)

-3

0.1

-0.1

Figure 13.

100 200 time (sec)

300

-5

x 10

SISO Pole Placement without Failure

9 American Institute of Aeronautics and Astronautics AIAA-2009-2068

According to the summary table below, multivariable control appears better than a SISO approach. In the case of H-Infinity the performance was remarkably better, with or poleplace, Satellite RWA No. 1 is 0% RPY 2.30 10.78 x 10error = 4.61 0.1 5 without failure. A multivariable adaptive component was not developed. However, an adaptive component improved SISO 0 0 control performance, with or without failure. The radial basis -0.1 -5 0 100 200 300 0 100 200 300 function adaptive component generally produces a better x 10 improvement than the linear combiner adaptive component. 0.2 5 The easiest methods to develop were the pole placement and 0 0 the linear combiner adaptive component. It should be noted that -0.2 -5 0 100 200 300 0 100 200 300 the adaptive components are independent add-ons, that is, they are not integrated into the other control laws. They may be 0.2 0.01 easily integrated with most legacy control laws. roll (rad)

-3

yaw (rad)

0.02

-0.2

-0.02

100

200

300

0.1

0.01

-0.1

-0.01

Figure 17.

100 200 time (sec)

300

100

200

300

0.2

100 200 time (sec)

SISO LQR without Failure

300

100 200 time (sec)

300

0.01

-0.2

-0.01

100

200

300

0.2

roll (rad) pitch (rad)

0.2

100 200 time (sec)

Figure 14.ErrorSISOPitch PoleError PlacementYaw withError Failure Roll 1.34 2.3 1.87 4.61 2.3 10.78 1.33 2.23 1.83 Hinf, Satellite RWA No. 1 is 0% RPY error = 5.07759 4.33 2.23 8.37 2.84162 40.6475 0.1 0.01 1.17 2.23 1.8 0 0 4.04 2.24 9.57 -0.1 -0.01 0 100 200 0 100 200 300 1.97 2.91 300 2.57 5.08 2.84 40.65 0.2 0.01 1.97 2.822 2.23 0 0 5.25 2.71 9.79 -0.2 -0.01 0 100 200 0 100 200 300 1.77 2.79 300 2.19 4.83 2.71 11.45 0.2 0.05 2.62 3.96 3.5 0 0 5.94 3.55 47.65 -0.2 2.47 4.08 300 -0.05 0 3.37 0 100 200 100 200 300 time (sec) time (sec) 7.68 4.09 13.84 1.99 4.11 3.3 Figure 16. SISO H-infinity with Failure 6.84 4.12 16.49 1.31 2.01 1.58 4.32 2.02 9.17 lqr, Satellite RWA No. 1 is 0% RPY error = 5.94 3.55 47.65 0.05 0.05 0.05 0.1 0.01 0.06 0.06 0.17 0 0 4.12 6.57 5.27 -0.1 -0.01 0 100 200 0 100 200 300 13.33 6.6 300 29.86

yaw (rad)

pitch (rad)

roll (rad)

-3

pitch (rad)

roll (rad)

-3

0 -0.01

yaw (rad)

SISO pole placement, no Failure, no adaptive component SISO pole placement, RWA 1 Failure, no adaptive component SISO pole placement, no Failure, Linear Combiner Hinf, Satellite RWA No. 1 is 100% RPY xerror = 1.9703 2.9081 2.5367 10Linear SISO pole0.1placement, RWA 1 Failure, Combiner 5 SISO pole 0placement, no Failure, Radial Basis Function 0 SISO pole-0.1placement, RWA 1 Failure, Radial Basis Function -5 0 100 200 300 0 100 200 300 SISO H-Infinity, no Failure, no adaptive component SISO H-Infinity, RWA 1 Failure, 0.01 no adaptive component 0.2 SISO H-Infinity, no Failure, Linear Combiner 0 0 SISO H-Infinity, RWA 1 Failure, Linear Combiner -0.2 -0.01 0 100 200 300 0 100 200 300 SISO H-Infinity, no Failure, Radial Basis Function SISO H-Infinity, RWA 1 Failure, Radial Basis Function 0.1 0.01 SISO LQR, no Failure, no adaptive component 0 0 SISO LQR, RWA 1 Failure, no adaptive component -0.1 -0.01 SISO LQR, Linear300Combiner 0no Failure, 100 200 0 100 200 300 (sec) time (sec) SISO LQR, RWAtime 1 Failure, Linear Combiner SISO LQR, no Failure, Radial Basis Function Figure 15. SISO H-infinity without Failure SISO LQR, RWA 1 Failure, Radial Basis Function MV pole placement, no Failure, no adaptive component MV pole placement, RWA 1 Failure, no adaptive component lqr, Satellite No. 1 is 100% RPY = 2.62 3.96 3.50 x 10error MV H-Infinity, noRWA Failure, no adaptive component 0.1 5 MV H-Infinity, RWA 1 Failure, no adaptive component 0 0 MV LQR, no Failure, no adaptive component -0.1 -5 0 100 200 300adaptive 0 100 200 300 MV LQR, RWA 1 Failure, no component

0 -0.2

roll (rad)

Summary Table of Tracking Error Performance Characteristics

yaw (rad)

pitch (rad)

-3

100

200

300

100 200 time (sec)

300

0.1

-0.2

-0.1

Figure 18.

100 200 time (sec)

300

SISO LQR with Failure

10 American Institute of Aeronautics and Astronautics AIAA-2009-2068

VII. Conclusion Control law principles were discussed in the context of failure of a satellite reaction wheel control assembly. Twelve different combinations of control laws were compared for control of a generic satellite attitude control system using four reaction wheel assemblies (RWA). The combinations are listed in the following table:

SISO Pole placement SISO H-Inf SISO LQR MV Pole placement MV H-Inf MV LQR

No extra X X X X X X

LC X X X

RBF X X X

SISO means single-input single-output, No extra means that it operated by itself, LC is an adaptive linear combiner component, RBF is an adaptive radial basis function component, H-Inf is H-Infinity controller, LQR is a linear quadratic regulator, and MV is a multi-variable control law. A control law was applied to all three roll-pitch-yaw channels of the satellite attitude control system. In all cases the SISO control performance was improved by the addition of an adaptive component, with the RBF performance slightly better. The MV approaches for pole placement and H-Infinity were showed improvement, but not so for the LQR. Overall, the MV H-Infinity method showed dramatic improvement. Future work should include additional aerospace model applications, addressing the stability issues for adaptive components, additional neural network structures, and testing on more realistic maneuvers (sensor calibration, orbital transitions, and transitioning to different sensors).

References 1

Glover, K & McFarlane, DC, ‘Robust Stabilization of Normalized Co prime Factor Plant Descriptions with H∞-Bounded Uncertainty.’ IEEE Transaction on Automatic Control, 34(8):821-830, Aug., 1989. 2 McFarlane, DC, ‘Robust Controller Design Using Normalized Co prime Factor Plant Descriptions,’ PhD thesis, University of Cambridge, 1988. 3 Jun Tan, F.N. Cornett, "A fast learning algorithm for adaptive linear combiner," ssst, pp.399, 29th Southeastern Symposium on System Theory (SSST '97), 1997 4 Haykin, S., Adaptive Filter Theory, 3rd Edition, Prentice Hall, New Jersey, USA, 1996. 5 Sidi, M. J., Spacecraft Dynamics and Control, Cambridge University Press, Cambridge, 1997.

11 American Institute of Aeronautics and Astronautics AIAA-2009-2068