GTOC9: Results and Methods of Team 13 (NPU)

Xun Pan∗, Binfeng Pan, Peiyang Liu, Yangyang Ma, Hui Zeng

Northwestern Polytechnical University, 710072 Xi’an (China)

May 8, 2017

AbstractThis paper presents the methods proposed byteam 13-NPU in the 9th Global Trajectory Op-timization Competition (GTOC9). The de-bris removal sequence in single submissionand the reasonable allocation of all debrisin submissions are designed by branch andbound algorithm and image method. Thena three-impulse transfer method involving adeep space maneuver is used to achieve thetransfer of spacecraft between any two debris.The best solution found has a performance in-dex 878.998 MEUR with 13 submissions.

1 IntroductionIn GTOC 9, the issue of debris pieces removalis investigated [1]. To avoid the threat to orbit-ing satellites, a set of 123 critical debris piecesin Sun-synchronous orbits need to be removedby a series of missions. During the trans-fers, spacecrafts are subject not only the cen-tral force of the Earth’s gravity, but also per-turbed by the main effects of an oblate Earth,

∗Corresponding author. E-mail:xpan2012@gmail.com

i.e. J2 perturbation. The position and ve-locity vectors of each orbiting debris piecesare computed by updating Keplerian elementsusing the mean motion, which [Ω, ω] is per-turbed by J2 perturbation. The only manoeu-ver allowed to control the spacecrafts is im-pulse propulsion to change the velocity vectorinstantaneously. Due to the fuel restrictions, asingle spacecraft cannot clean up all the dis-tributed debris, so multiple missions are de-signed to remove all the debris.

The performance to be minimized is

J =n∑

i=1

Ci =n∑

i=1

[ci + α(m0i −mdry)

2]

(1)

where Ci is the cost charged by the con-tracted launcher supplier for the i-th missionand it is composed of a base cost ci anda term α(m0i −mdry)

2 favouring a lighterspacecraft, m0i and mdry denote the initialspacecraft mass and dry mass of the i-th mis-sion. Each spacecraft initial mass m0i is thesum of dry mass, propellant mass, and theweight of N de-orbit packages: m0i = mdry +mp + Nmde, where mdry = 2000 kg, mp ≤5000 kg, mde = 30 kg. The base cost ci ofeach mission depends on how early the sub-mission submitted to the authorities, and it in-

1

creases linearly during the competition, variesfrom 45 MEUR to 55 MEUR.

The process of launching spacecraft fromthe ground to the first debris of any submissionis not considered, only the transfer trajectoriesfrom one debris to another debris are calcu-lated. After spacecraft arrived target debris,there is a waiting time constraint of 5 days torelease the debris removal device before leav-ing the debris to the next debris. The time be-tween any two successive debris rendezvouswithin the same submission must not exceed30 days, which means the flight time of singletransfer trajectory varies from 0 to 25 days. Itis forbidden to operate different submissionsin parallel, a time of at least 30 days must beaccounted for between any two submissions.To prevent spacecrafts from colliding with theearth, the orbital periapsis rp cannot be smallerthan 6600 km.

The spacecraft is perturbed by J2 perturba-tion during transfers, and the dynamics equa-tion in the geocentric inertial coordinate is de-scribed as

x = −µxr3

[1 +3

2J2

r2eq

r2(1− 5

z2

r2)] (2)

y = −µyr3

[1 +3

2J2

r2eq

r2(1− 5

z2

r2)] (3)

z = −µzr3

[1 +3

2J2

r2eq

r2(3− 5

z2

r2)] (4)

where req denoted the Earth average radius.The position and velocity vectors of debris

are computed by the classical orbital elements[a, e, i,Ω, ω,M ]. The elements osculate dueto the J2 perturbation, and only the long-terminfluence of the ascending node Ω, the argu-ment of perigee ω and the mean anomaly Mare computed as follows

Ω = −3

2J2

r2eq

p2

õ

a3cos i (5)

ω =3

4J2

r2eq

p2

õ

a3(5cos2i− 1) (6)

M =

õ

a3(7)

where p = a(1−e2) denotes the semilatus rec-tum.

The only allowed manoeuver is instanta-neous changes of the spacecraft velocity vec-tor. The fuel consumption of the manoeuver∆V is calculated using Tsiolkovsky equation

mf = mi exp(− ∆V

Ispg0

) (8)

where Isp is the specific impulse, g0 is thestandard acceleration of gravity at sea level,mi is the mass before the manoeuver and mf

is the mass after the manoeuver.

2 Methods

There are two main difficulties in solving thecompetition: how to allocate the debris to dif-ferent submissions rationally to minimize thenumber of submissions, and for each submis-sion, how to optimize the debris removal se-quence to minimize the total manoeuver mag-nitude, so as to reduce the initial spacecraftmass.

Before the optimization of debris removalsequence, it is necessary to select a perfor-mance index about manoeuver of transfers be-tween any two debris which can be calculatedquickly. The transfer of the spacecraft be-tween any two debris is considered as a lam-bert problem, and the double-pulse lamberttransfer is the most common method. Afterthe initial position and final position fixed, andthe transfer time determined, the velocity in-crement required for the transfer can be calcu-

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lated by lambert method. However, the trans-fer time is free before the debris removal se-quence determined, and it takes times to opti-mize the transfer time to get a suitable transfertrajectory, so the lambert method is not suit-able to measure the velocity increment whendetermining the debris removal sequence.

According to the knowledge of orbital me-chanics, it takes much larger ∆V to change thespacecraft orbital plane than change the shapein the plane, so it is reasonable to make the de-bris with close orbital plane in the same sub-mission. In the six classical orbital elements,the semi-major axis a and the eccentricity edescribe the shape in orbital plane, the inclina-tion i and the ascending node Ω describe theorbital plane, the argument of perigee ω andthe mean anomaly M describe the spacecraftposition in the orbit. All of the given 123 de-bris are located in Sun-synchronous with theconstant inclination i ∈ [1.6976, 1.7640] radand the osculating ascending node Ω ∈ [0, 2π].Due to the J2 perturbation, the average os-cillation period of Ω is 364.7326 days, andthe Ω changes about 8 cycles in 8 years mis-sion time, so different rendezvous time havegreat influence on the angle between two de-bris orbital planes. According to the sphericaltrigonometry, the angle θ between any two or-bital plane can be calculated as

cos θ = cos i1 cos i2+sin i1 sin i2 cos(Ω2−Ω1)(9)

But in the competition, we use a simpler for-mula as

∆α= |i2 − i1|+ |Ω2 − Ω1| (10)

The ∆V required to accomplish the transferbetween the two debris is assumed to be pos-itively correlated to ∆α. The performancewhen determine the debris removal sequence

is to minimizem∑j=1

∆αj .

First, the debris removal sequence of eachsubmission is designed by branch and boundalgorithm. In order to minimize the numberof submissions, a single submission should re-move debris as many as possible. The trans-fer time between two debris varies 0∼25 days,and the ∆Ω also varies with time. Thereforeboth the order of debris and the time of de-bris rendezvous should be optimized in the se-quence. The minimum ∆Ω within allowabletransfer time range between any two debris iscomputed by the different rate Ω, then the ren-dezvous time achieved. Two alternative debrisare select as the next target debris, and up to213 sequence are selected in single mission,i.e. in the branch and bound algorithm, eachbranch has two branches, and the number ofreserved sequence is up to 213, then the restbranches are deleted. It should be noted thatin all submissions, only a few of the submis-sions are able to clean up more than 13 debris,and the debris numbers of others submissionsare less than 13, then the sequence numberswill be reduced. The branch and bound algo-rithm is shown in Fig. 1, where each node inthe sequence represents a alternate debris andthe rendezvous time.

The branch and bound algorithm can onlyoptimize the debris sequence in single submis-sion, but cannot be used to achieve the solutionof minimum number of submissions. For theglobal optimization problem, the approach weadopted is modifying the starting and endingtime of each submissions and the associateddebris to obtain the solution with minimumsubmission number. Since the ∆Ω betweendebris varies greatly over time, the intersec-tions of the element Ω of any two debris arefound and plotted as Fig. 2, where the x-axis is

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Figure 1: The branch and bound algorithm

the time from the 23467 MJD2000, the y-axisis the value of Ω, every red circle represents amoment that two debris have the same Ω. Asshowed in Fig. 2, the distribution of the inter-section of Ω is asymmetrical, the time intervalwhen many debris planes close to each othershould be selected to be submission time, i.e.the section where the densest area of intersec-tions should be selected. In Fig. 2, the inter-sections near the blue line is intensive, a se-quence of 20 debris is selected along the blueline, and every blue circle represents a debrisin the sequence and the rendezvous time. Af-ter several submissions determined, the inter-sections of the remaining debris are scatteredand sparse, and the figure of the intersectionbecomes useless to select the sequences. Thenthe curve of remaining debris Ω over time isplotted to design the remaining submissions,as showed in Fig. 3.

Then the transfer trajectory is designed forspacecraft to leave last debris and rendezvousnext debris. At the rendezvous moment, thephase angle of spacecraft should be the sameas the target debris. Only the i and Ω con-straints about orbital plane are accounted, andthe ω and M constraints about phase angle inthe orbit are neglected. If two-impulse transfer

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

∆MJD

Ω/r

ad

Figure 2: The intersections of the element Ωof any two debris

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

∆MJD

Ω/r

ad

Figure 3: The cure of remaining debris Ω

is used directly to accomplish the rendezvousat the rendezvous time in the sequence, a re-markable ∆V is needed to eliminate the ef-fect of phase difference between spacecraftand target debris. The interval of two ren-dezvous time in sequence varies 5∼30 days.Once de-orbit package delivered, a small im-pulse is used to leave the debris and changethe semi-major axis , thus the rate of the M inEq.(1.5) is also changed. After many revolu-tions, spacecraft reaches the next debris at therendezvous moment, then two-impulse lam-bert is used to accomplish the rendezvous.

The two-impulse lambert transfer is com-

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puted without J2 perturbation, and the correc-tion is required to take J2 perturbation into ac-count. Besides, the first impulse applied to ad-just the phase angle may not take spacecraftreach the optimal position at the rendezvoustime, so the first impulse is also modified. Inthe case of initial states and final states fixed,there are 7 optimization variables, includingthe first impulse [∆vx1,∆vy1,∆vz1], the sec-ond impulse [∆vx2,∆vy2,∆vz2] and the timefor second impulse t2, the terminal constraintsare the same position of spacecraft with nextdebris, and the minimum performance index is

the sum of three velocity increments:3∑

i=1∆Vi.

The function fmincon in MATLAB is used foroptimization. Take the transfer from debris 48to debris 7 as example, three impulses are re-quired: the first impulse to leave debris 48 andadjust the phase angle, the second and thirdimpulse to accomplish the rendezvous. At therendezvous time, the difference of phase an-gle between debris 48 and debris 7 is 1.1519rad. By a small impulse of 6.3420 m/s leavingdebris 48 after de-orbit package delivered, thespacecraft arrives at a location close to debris7with similar phase angle, then the second andthird impulse are used to accomplish the ren-dezvous. Figure 4. shows the transfer trajec-tory, the transfer time is 3.5067 days, and thetotal ∆V is 110.3 m/s.

3 Results

By the methods introduced above, we got afeasible solution with 13 submissions. Theperformance index of the final result is J =878.998 MEUR. Table 1 shows the de-tailed information of the result, including the

−8000 −6000 −4000 −2000 0 2000 4000 6000 8000−5000

0

5000−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

x/kmy/km

z/km final position

position of DSM

initial position

Figure 4: The transfer trajectory from debris48 to debris 7

removal debris number, ∆V , and the initialspacecraft mass of each submission. Figure5 shows the Ω curves of spacecraft in the mis-sion, which contains 13 curves, and each circlerepresents a debris removed in the moment.

Table 1: The final result with 13 submissionsSubmissions debris number ∆V /(m/s) m0i/kg

1 20 2738 54682 4 2710 47083 19 3218 62544 14 3067 57185 14 2620 50096 7 2405 44567 9 3002 53968 3 733 25919 8 3156 562710 8 2300 433911 2 447 235312 9 3214 570113 6 2023 3911

4 Conclusions

In this paper, the methods that Team 13-NPUused for GTOC9 are introduced. The differ-ence between any two debris orbital planesis used to approximate the impulsive veloc-ity change, then the debris removal sequenceof single submission is selected by branch and

5

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

∆MJD

Ω/r

ad

Figure 5: The Ω curve of spacecraft in the mis-sion

bound algorithm, and the figures of intersec-tions of Ω and the Ω curve over time areused to modify the starting and ending time ofeach submissions to make minimum numberof submissions. A transfer method with 3 im-pulsive manoeuver is adopted to make space-craft accomplish transfer between any two de-bris in the sequence: the first impulse to mod-ify spacecraft phase angle, the second andthird are lambert transfer. Then the 3-impulsetransfers under J2 perturbation are modifiedby the fmincon function in MATLAB. We ulti-mately got a solution with 13 submissions andthe performance is 878.998MEUR.

References[1] Dario Izzo. Problem description for the

9th Global Trajectory Optimisation Com-petition, May 2017.

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