Satellite Orbit Transfer optimization by Roberto Bonfiglio

Politecnico di Milano Dipartimento di Scienze e Tecnologie aerospaziali Prova finale: Introduzione all’Analisi di Missioni spaziali A. A. 2015/2016

Docente: Bernelli Franco

Elaborato n. A30 Autori:

Codice Persona

Cognome

Nome

10412356 10456744 10426656

Bonfiglio Bramani Gioia

Roberto Nicolò Andrea

Data di Consegna: Lunedì, 2 Maggio 2016

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

Commenti del docente

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

Contents 1 Introduction

2 Initial orbit characterisation 2.1 Initial orbital parameters determination . . . . . . . . . . . . . . 2.2 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . .

5 5 5

3 Final orbit characterisation 3.1 Final orbital parameters determination . . . . . . . . . . . . . . . 3.2 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . .

7 7 7

4 Transfer trajectory definition and analysis 4.1 The transfer manoeuvres problem . . . . . . 4.2 Strategies discussion . . . . . . . . . . . . . 4.2.1 Strategy n.1 . . . . . . . . . . . . . . 4.2.2 Strategy n.2 . . . . . . . . . . . . . . 4.2.3 Strategy n.3 . . . . . . . . . . . . . . 4.3 Strategy choice . . . . . . . . . . . . . . . . 4.4 Strategies Representation . . . . . . . . . .

. . . . . . .

9 9 10 10 11 12 12 12

5 Conclusions

6 Appendix â&#x20AC;&#x201C; Explanation of StrategyCombinations.m scripts

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Introduction

The assignment’s purpose is the characterization of the initial and the final orbit of a satellite and the detailed analysis of different transfer strategies. First the initial and final orbits and positions have been analysed and characterised with position and velocity vectors, Keplerian parameters (see Figure 1a) and others orbit’s significant characteristics (shown in Figure 1b). In the second part some possible transfer strategies and their manoeuvres have been implemented and analysed, paying particular attention in identifying an optimal strategy that minimises propellant expenditure and reduces transfer time. In conclusion one strategy has been selected and analysed in detail. Satellite ~v

Perigee

~r ~e

~h θ ω

Vernal equinox

Ω

à Plane of re ference

~n i

it Orb

Line of nodes

(a) Draw of the Keplerian elements and the main vectors employed in the discussion.

yPF

~v ~r θ

xPF

C p

(b) Parameters of a generic elliptical orbit.

Figure 1: Tipical orbital mechanics parameters.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

2 2.1

Initial orbit characterisation Initial orbital parameters determination

The initial orbit is characterised by a given point’s position and velocity vectors, expressed in the celestial Cartesian coordinates system (considered inertial in this report):     −9, 05941070  2, 56340000  ~rin = −6, 91334370 · 103 km , ~vin = −3, 18590000 km/s . (1)     −2, 13914020 −4, 05020000 In order to calculate the six Keplerian elements of the initial orbit, a MatR function called RV2ParOrb.m has been developed, that receives as inputs lab the position (relative to the focus of the orbit, taken as the origin of the reference system) and the velocity of the object in space, given in a generic point of the orbit and calculates the output orbit parameters: semi-major axis a, eccentricity e, inclination i, right ascension of ascending node Ω, pericenter’s argument ω, true anomaly θ (see Figures 1a and 1b on the preceding page). The results for the initial orbit are: ain = 1, 11875 · 104 km ein = 0, 11759 iin = 0, 79194 rad = 45, 37492◦ Ωin = 0, 46549 rad = 26, 67080◦

(2)

ωin = 1, 40805 rad = 80, 67539◦ θin = 1, 99575 rad = 114, 34815◦ .

2.2

Discussion of the results

The initial orbit is elliptical, in fact ein < 1 (and 6= 0). This can be seen also on the Figure 2a on the next page, where the orbit has been plotted in its perifocal plane, identified by its normal ~hin and whose plane vectors are ~ein (parallel to xPF axis) and ~nin . It can be classified as a MEO (Medium Earth Orbit, 5000 km < hMEO < 10000 km), in fact, evaluating its pericenter and apocenter distances: rPin = 9872, 02 km (3) rAin = 12503, 06 km and calculating the distances from Earth’s surface: hPin = 3501, 02 km hAin = 6132, 06 km , we obtain: 3600 km < hin < 6200 km . The initial orbit has a period Tin = 3 hours, 16 minutes, 16 seconds .

(4)

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

(a) Plot of the initial orbit in its perifocal reference system (the plot has been done by the function DrawOrbitIn_CE_PF.m).

(b) Plot of the initial orbit in celestial reference system (the plot has been done by the function DrawOrbitIn_CE_PF.m).

Figure 2: Plots of the initial orbit. 6

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Final orbit characterisation

3.1

Final orbital parameters determination

The final orbit is characterised by Keplerian elements: afin = 3, 51511484 · 104 km efin = 0, 119700000 ifin = 2, 97160000 rad = 170, 26014◦ Ωfin = 0, 729700000 rad = 26, 67080◦

(5)

ωfin = 2, 81670000 rad = 161, 38502◦ θfin = 1, 77900000 rad = 101, 92919◦ . In order to calculate the position and velocity’s vectors of the final orbit, a R function called ParOrb2RV.m has been developed. The results are: Matlab     −26266, 47420  1, 90213  23164, 02786 2, 73213 ~rfin = · 103 km , ~vfin = km/s . (6)     −5969, 32878 −0, 13190

3.2

Discussion of the results

The final orbit is also elliptical, in fact efin < 1 (and 6= 0). This can be seen also on the Figure 3a on the following page, where the orbit has been plotted in its perifocal plane, identified by its normal ~hfin and whose plane vectors are ~efin (parallel to xPF axis) and ~nfin . Evaluating its pericenter and apocenter distances: rPfin = 30943, 55594 km rAfin = 39358, 74086 km

(7)

and calculating the distances from Earth’s surface: hPfin = 24572, 55594 km hAfin = 32987, 74086 km , we obtain: 25000 km < hfin < 33000 km . The initial orbit has a period Tfin = 18 hours, 13 minutes, 7 seconds .

(8)

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

(a) Plot of the final orbit in its perifocal reference system (the plot has been done by the function DrawOrbitFin_CE_PF.m).

(b) Plot of the final orbit in celestial reference system (the plot has been done by the function DrawOrbitFin_CE_PF.m).

Figure 3: Plots of the final orbit.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

a e i Ω ω θ T

(km) (-) (◦ ) (◦ ) (◦ ) (◦ ) (hours)

Initial orbit

Final orbit

11188 0.1176 45.38 26.67 80.68 114.35 3.27

35151 0.1197 170.26 41.81 161.39 101.93 18.22

Table 1: Keplerian parameters of the initial and final orbits.

4 4.1

Transfer trajectory definition and analysis The transfer manoeuvres problem

The final position and velocity can be achieved with some different possible strategies. The initial and final orbits have no common points so there must be certainly at least one transfer orbit that has a point in common with both the initial and the final orbit. This transfer strategy with only one transfer orbit requires only two impulsive manoeuvres but it is of hardly (or even impossible) identification. A typical strategy in astrodynamics, taken in consideration in this essay, consists in selecting a series of manoeuvres each of which changes only a few orbital parameters.

Figure 4: Plot of the two orbits and their main characterizing vectors.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Strategy n.1

Strategy n.2

Strategy n.3

Plane change Pericenter’s anomaly change Shape change

Shape change Plane change Pericenter’s anomaly change

First shape change Plane change Pericenter’s anomaly change Second shape change

Table 2: Sequences for a preliminary analysis. In particular three main kinds of manoeuvres can be identified: • plane change manoeuvre: changes the plane (represented by i and Ω) where the orbit lies. It has been implemented in PlaneChange.m. • Pericenter’s anomaly change manoeuvre: changes the pericenter’s orientation (identified by ω). It has been implemented in PerAnomalyChange.m. • Shape change manoeuvre: changes orbit’s shape and size (identified by a and e). It has been implemented in OrbitShapeChange.m.

4.2

Strategies discussion

In the transfer strategy analysis there are two main degrees of freedom: • manoeuvres’ order, that is the sequence chosen to do the different manoeuvres, that can largely affect transfer’s efficiency in terms of time and propellant cost; • manoeuvres’ collocation, in fact most of the manoeuvres (e.g. plane change, pericenter’s anomaly change, shape change) can be done in different positions, and an optimal sequence of manoeuvre’s points can largely improve the transfer strategy. In the choice of manoeuvres’ order there is a bond about the order in which plane change and pericenter’s anomaly change are performed: the former must precede the latter, because the plane change manoeuvre modifies not only i and Ω, but also ω. Therefore in a preliminary analysis two simple manoeuvres’ sequences (shown in Table 2) has been considered1 . Then a bi-elliptic approach for the plane change has been analysed, in order to minimise ∆v (proportional to propellant cost) not increasing overly the duration of the transfer operation. 4.2.1

Strategy n.1

8 possible transfer strategies have been analysed for the first set of manoeuvres and the results obtained are shown in Table 3a on the following page. This number of different possibilities depends on where manoeuvres are done. 1 The sequence: plane change-shape change-pericenter’s anomaly change is not taken into account, since the small ∆v saving for the single manoeuvre (the pericenter’s anomaly’s change requires a minimum propellant cost, at least one order of magnitude lower than the total cost of the transfer, as shown in Table 5 on page 15) due to the major distance from the focus of the orbit can be associated with an high ∆t cost, making the whole strategy less efficient than strategy n.1.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Table 3: Strategies ∆v and ∆t costs. (a) Results of the strategy n.1.

Strategy’s Name

N11

N12

N13

N14

N15

N16

N17

N18

∆v (km/s) ∆t (hours)

14.269 13.671

14.047 25.329

14.269 13.387

14.047 21.173

13.729 16.658

13.507 25.045

13.729 13.671

13.507 25.329

(b) Results of strategy n.2.

Strategy’s Name

N21

N22

N23

N24

N25

N26

N27

N28

∆v (km/s) ∆t (hours)

9.156 50.484

9.628 31.554

8.935 43.923

9.248 40.557

9.157 30.653

9.268 15.603

8.935 24.092

9.248 24.606

N29

N30

N31

N32

N33

N34

N35

N36

8.846 30.653

8.958 33.821

8.625 42.311

8.938 24.606

8.846 32.266

8.958 31.554

8.625 43.923

8.938 22.339

∆v (km/s) ∆t (hours)

a1BE

2 · afin

4 · afin

5 · afin

2 · afin

4 · afin

5 · afin

Strategy’s Name

N53-2

N53-4

N53-5

N71-2

N71-4

N71-5

∆v (km/s) ∆t (hours) ∆t (days)

8.4827 78.321 3.2634

7.856 177.08 7.3781

7.6282 237.1 9.879

8.2063 101.54 4.2308

7.5375 261.49 10.896

7.2946 359.19 14.966

The ∆v required for the different strategies are almost the same, but there is an important difference in the time required. The most interesting manoeuvres are: N13 which is the faster one, N16 which requires the minimum ∆v and N17 which seems to be a good compromise between the previous ones. 4.2.2

Strategy n.2

16 possible transfer strategies have been analysed for these set of manoeuvres and the results obtained are shown in Table 3b. This high number of manoeuvres is due to the possibility to do manoeuvres in different points and in different ways. In particular with this different manoeuvres order is possible to achieve a reduced ∆v required, thanks to the initial resize of orbit’s dimensions. All these new transfer trajectories have a reduced propulsion cost, due to the fact that the plane change manoeuvre is done after the shape and size change, so further from orbit’s focus (in the examined case: Earth) where manoeuvre’s cost is lower. From the Table 3b seems particular interesting the N26, N31 and N36 strategies: N26 is the fastest one, N31 requires the minimum ∆v and N36 is a good compromise between the two. However N26 seems to be the most attracting strategy due to its low ∆v cost

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

and relatively short transfer time. 4.2.3

Strategy n.3 – A bi-elliptic approach to plane change

The main goal of the bi-elliptic approach to plane change is to minimise the ∆v cost and to find a low-thrust orbit transfer strategy. The initial orbit is enlarged 2 in order to perform the plane change manoeuvre at a greater distance from Earth. The plane change manoeuvre is the most expensive one since there is an important ∆i ≈ 120◦ . The pericenter’s anomaly change is done after plane change and then final shape change is done in order to reach the final orbit. This strategy requires 4 manoeuvres with a total of six ∆v. From Table 3c on the preceding page it can be seen that there is a little propellant saving and an important increase in transfer time required. This strategy has so achieved the goal to reduce ∆v, but it has a relevant time transfer increase which makes it seem not so attractive.

4.3

Strategy choice

The most attracting transfer orbits analysed seem to be N17, N26, N36 (see Figure 5a on the next page and the summarizing Table 4). Strategy n.3 has not been considered for its high transfer time and its lightly reduced ∆v. The first one (N17) is particularly fast since final position and velocity are achieved in 13 hours, but it requires a total manoeuvres’ ∆v of 13, 7 km/s. The second (N26) has a 32, 5% reduced ∆v required (respect to N17), saving about 4, 46 km/s with 1 hour and 56 minutes more in time transfer. The third (N36) has a 35% reduced ∆v required (respect to N17) saving about 4, 79 km/s with 8 hours and 40 minutes more in time transfer. In this assignment we have considered N26 as a good compromise between ∆v and transfer time. N17 can be implemented if there is a strict requirement about time transfer, vice versa N36 is interesting if propellant saving is very important for the mission. Strategy’s Name

N17

N26

N36

∆v (km/s) ∆t (hours)

13.729 13.671

9.268 15.603

8.938 22.339

Table 4: Final considerations of the analysis.

4.4

Strategies Representation

The considered strategies are shown in Figure 6 on page 14 and all the costs details are printout in Table 5 on page 15. The most optimised strategy appears to be the N26, in fact it requires 15 hours and 36 minutes and ∆v ≈ 9, 3 km/s. Its most expensive manoeuvre is the plane change which costs about 66% of total ∆v required. 2 The resizing of the initial orbit is parametrised by a , increasing a1 , the semi-major fin BE axis of the first ellipse of the bi-elliptic manoeuvre.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Figure 5: Costs optimisation graphs. (a) Strategy n.1 and n.2.

Strategy n.1 Strategy n.2

N13 N11

N14

N17

N15

N12 N16 N18

∆v (km/s)

13 12 11 10 N26

N28 N36N27 N32

N22 N25 N34 N30 N29N33

N24 N23 N31N35

N21

8 10

30 ∆t (hours)

(b) All the strategies considered.

15 Strategy n.1 Strategy n.2 Strategy n.3

∆v (km/s)

13 12 11 10 9 8 7 0

15 ∆t (days)

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

The particular transfer orbits sequence permits to reduce time spent along orbits. The longest â&#x2C6;&#x2020;t are the shape change manoeuvre and the awaiting for the plane change manoeuvre which require about 76% of the total transfer time.

(a) Plot of the strategy N17.

(b) Plot of the strategy N26.

Figure 6: Plots of the considered strategies.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

N17

∆t (hours)

IN → TPC TPC TPC → TPAC TPAC TPAC → T1OSC T1OSC 1 TOSC → T2OSC T2OSC 2 TOSC → FIN

3.157

Total

∆v (km/s) 10.402

1.224 0.818 0.397 1.435 4.426 1.075 4.468 13.671

13.729

(a) N17 details.

∆t (hours)

N26 T1OSC

IN → T1OSC 1 TOSC → T2OSC T2OSC 2 TOSC → TPC TPC TPC → TPAC TPAC TPAC → FIN

0.712

Total

15.603

∆v (km/s) 1.652

5.772 0.776 5.847 6.177 1.033 0.662 2.239 9.268

(b) N26 details.

∆t (hours)

N36 T1OSC

IN → T1OSC 1 TOSC → T2OSC T2OSC 2 TOSC → TPC TPC TPC → TPAC TPAC TPAC → FIN

10.215

Total

22.339

∆v (km/s)

2.348 1.107 4.029 1.302 4.494 5.867 1.253 0.662 8.938

Table 5: Details of the considered strategies’ costs.

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

Conclusions

Three different possible strategies have been analysed with a particular attention to their transfer time and ∆v. In brief: • strategy n.1 can ensure a little transfer time (less than a day) with relevant ∆v required. • strategy n.2 can ensure a little or medium transfer time (1/2 days) with medium ∆v required. • strategy n.3 leads to a long transfer time (many days) with little/medium ∆v required. The third strategy (with a bi-elliptic approach for the plane change) seemed not attractive for its long transfer time and not so much reduced ∆v. The most attractive transfers come from strategies 1 and 2. Strategy N26 (from the n.2 set) has been chosen since it seems a good compromise between ∆t and ∆v costs. Others interesting strategies are N17 and N36, the former for its low transfer time, the latter for its ∆v. In general there is not a best transfer strategy in absolute, but only an optimal one in relation with mission’s requirements and characteristics.

Appendix – Explanation of StrategyCombinations.m scripts

Different scripts and functions have been developed in order to have a broad view on the different possible strategies, based on the manoeuvres sequences shown in Table 2 on page 10. They have been implemented with a general approach, in order to be able to test them with different sets of project parameters, simply updating the AA2_GivenOrbitsData.m set of variables and running only one script to perform the whole analysis and plotting routine: AA0_GeneralScript.m. In this Appendix, the principal scripts employed in the strategies examine will be rapidly explained, in order to clarify the operation of the pivot R functions of this paper: StrategyCombinations.m. Matlab StrategyCombinations.m scripts refers to three functions that implement the three basic orbital mechanics manoeuvres: PlaneChange.m This function computes all the parameters useful to describe a plane change manoeuvre, starting from an initial orbit’s Keplerian parameters and the desired ∆i and ∆Ω. The function in absolutely general: it can work with all the ∆i and ∆Ω configurations and –the most particular and characterising thing– it can be switched from a non-optimised mode to an optimising propellant cost mode, computing the ∆v cost of the two possible solutions (shown in Figure 7a on page 18) and choosing the best one (if the optimisation is enabled).

Autori: Bonfiglio Roberto, Bramani Nicolò, Gioia Andrea (Gruppo n.30)

PerAnomalyChange.m This function performs the pericenter’s anomaly change manoeuvre, starting from an initial orbit’s Keplerian parameters and the desired ∆ω. As the PlaneChange.m function, it is present an optimising mode, that in this case (when enabled) permits the function to choose the manoeuvre’s true anomaly in order to minimise the ∆t cost (computing it with TimeCalc.m function). Propellant saving cannot be performed in this manoeuvre, since the ∆v required is the same for both the positions (see Figure 7b on the next page) where the manoeuvre is defined. OrbitShapeChange.m This function implements a bi-tangent generalised Hohmann’s manoeuvre, taking as inputs the Keplerian parameters of an initial orbit. The transfer towards next orbit can be performed in four different ways, shown in Figure 7c on the following page. It is important to notice that the Apocenter-Apocenter and the Pericenter-Pericenter manoeuvres are binding if the previous pericenter’s anomaly change has taken to an orbit that is out of phase of π with respect to the final orbit and instead are not recommended (otherwise a new pericenter’s anomaly change will be required) if the starting orbit has already been aligned to the final orbit. This is not still valid for strategy n.2, where the shape change takes place before the pericenter’s anomaly change and there is no such a constrain about the manoeuvre layout. Also this function is provided with an optimisation mode, that selects automatically the less propellant costing positions to perform the impulses required. The operation of these functions is very simple: they are compound by a series of nested for cycles, that switch from non-optimised mode, to optimised mode each of the three manoeuvres functions previously described, organising the computed results in a set of strategy’s matrices, in order to export the data and use them for plots and further analysis.

Autori: Bonfiglio Roberto, Bramani NicolĂ˛, Gioia Andrea (Gruppo n.30)

(a) Two transfer possibilities for plane change.

(b) Two transfer possibilities for pericenterâ&#x20AC;&#x2122;s anomaly change.

Figure 7: Plots of the different manoeuvres positions available: (a) is taken from strategy n.1, (b) and (c) from strategy n.2. 18