3nd Global Trajectory Optimization Competition Workshop Team 9

3nd Global Trajectory Optimization Competition Workshop

Team 9F. Jiang, Y. Li, K. Zhu, S. Gong, H. Baoyin, J. Li, etc.

School of Aerospace Tsinghua University

Beijing, China

2 Team 9

GTOC3 Workshop Torino, Italy, June 27, 2008

Outline

Team Composition Problem Summary Technical Approach

Sequence Selection Global Optimization Local Optimization

Solution Conclusions

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Team Composition

The Team: Comes from the Institute of Dynamics and Control, School of Aerospace, Tsinghua University, China.

Members: One professor, one associate professor, three Ph.D. Candidates, and some Master Candidates

Main Competence Areas: Liquid sloshing in spacecraft container, deep space exploration, spacecraft formation flying

A team not professional in optimization, though have participated to all three GTOCs. (11-th in GTOC1, 10-th in GTOC2, and 11-th in GTOC3)

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Problem Summary

Maximum excess velocity 0.5 km/s

Year of launch 2016-2025

Minimum stay time 60 d

Maximum flight time 10 y

Initial mass 2000 kg

Specific impulse 3000 s

Maximum thrust 0.15 N

Position and velocity constraints 1000 km, 1 m/s

Objective function:

1,3

max

min jf j

i

mJ K

m

Where mi and mf are the initial and final mass, respectively; K=0.2; =10; is the stay-time at the j-th asteroid.max

j

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Technical Approach: Sequence Selection(1) First: Prune these asteroids (about 2/3) with relatively

large orbit inclination or eccentricity in advance. Second: Range the potential sequences on the base of

orbit energy differences. (reference:GTOC2 Activities and Results of ESA Advanced Concepts Team)

1 2

1 1 1 1 1 1 2

2 22

2 1 2

2 2

1 2 1 2 2 1

2 2 2 2

2 cos

2 2

2 1

cos cos cos sin sin cos

p p a p p a

i f i f r

i a p a

f a

r

V V V

V r r r r r r

V V V VV i

V r r r

V r a

i i i i i

1V

2V

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Technical Approach: Sequence Selection(2) Third: Range the potential sequences on the base of orbit phase differences.

Initial phase difference, relative to Jan 1, 2016

Orbit angular velocity difference

Synodic time

i i i j j jM M

3 3j i s j s in n n a a

2 , 0,1,2,s k k n k

Sun

Asteroid i

Asteroid j

Asteroid i moves faster than asteroid j by (i, j) degrees per year, while its initial phase lags that of asteroid j by (j, i) degrees.

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Technical Approach: Sequence Selection(3)

Synodic times (ST) of potential sequences Expected sequence:

Actual sequence:

By computing the synodic times of potential sequences, no one satisfies absolutely.

We select some sequences with a little inconsistent synodic times, such as 88-76-49.

A3 E A3E E 1A A1 E, ST 2ST 0 0, ST -ST,1 ;0 10

A1 A2 E A1 A2 A3 A1 A2 A3 E A2 A3ST -ST , ST -ST , ST -ST

are all about 3 years

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Technical Approach(1) Astrodynamic model: equinoctial elements

Accommodate all possible conic orbits except i=180°.

, , , , , function , , , , , , , ,r t np f g h k L p f g h k L T T T

21 , cos , sin

tan 2 cos , tan 2 sin ,

p a e f e g e

h i k i L

Conversion from classical orbit elements:

Motion equation:

Though more complicated Cartesian quantities, they are more efficient in computing.

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Technical Approach: Global Optimization(2)

Particle swarm optimization (PSO) A population based stochastic optimization technique developed by Dr. E

berhart and Dr. Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling

FormulationObjective function 1 2, , , Df f x x x x

11 1 2 2

1 1

1 1 1 1if < , = ; if < , =

G G G Gi i i i i

G G Gi i i

G G G Gi i i i i i

w c r c r

f f f f

v v p x g x

x x v

x p p x x g g x

Choose N particles with random initial position xi0 and velocity vi

0. Theiteration from the G generation to G+1 generation can be presented as

where r1 and r2 are both uniformly distributed random numbers; w, c1 andc2 should be valued case to case.

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Technical Approach: Global Optimization(3) Differential evolution (DE)

A population based, stochastic function optimization proposed by Price and Storn in 1995

DE/rand/2/exp

11 1 2 3 2 4 5

1,1

,,

1 1

1

1

, for , 1 , , 1 1

,else

, if

, if

G G G G G Gi r r r r r

Gi jG D D D

i j Gi j

G G Gi i iG

i G G Gi i i

F F

j n n n L

f f

f f

v = x x x x x

vu

x

u u xx

x u x

where F1 and F2 are weighing factors in [0, 1]; the integers rk (k=1,…,5) arechosen randomly in [1, N] and should be different from i; Index n is a randomly chosen integer in [1,D]; Integer L is drawn from [1,D] with the

probability Pr(L>=m)=(CR)m-1, m>0. CR is the crossover constant in [0,1];

Mutation:

Crossover:

Selection:

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Technical Approach: Global Optimization(4) Hybrid algorithm (PSODE) of PSO and DE

In every 50 iterations, use PSO in the former 36 iterations, and DE in the latter 14 iterations.

Population size:400, Iteration times:1000; Weighing factors of DE are both 0.8; Maximum velocity:0.5; Crossover constant:0.618; c1 and c2 of PSO are both 0.5, ;

Optimize one leg by one leg Divide each leg into 10 segments.

Departure time and arrival time are optimized according to synodic time.

1 2 11, , , ,f f i fm m t t T T T

/ 5000.94 NIw e

61obj , 10 ; , , 4

2 6fm c c c rh h h h v h

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Technical Approach: Local Optimization(5)

The toolbox of Matlab: Pattern search Search around the solution obtained by global optimization to satisfy

the constraints on position and velocity. Increase the weight of constraints on position and velocity in objecti

ve function.

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Solution(1)Leg 1: From the Earth to A88

Launch date (MJD): 58090.8510

Launch velocity (km/s): [-0.3378, 0.05498, 0.3645]

Arrival date (MJD): 58479.1488

Departure mass (kg): 2000.0000

Arrival mass (kg): 1960.6172

Position error (km): 541.8060

Velocity error (m/s): 0.1578

Leg 2: From A88 to A76

Departure date (MJD): 58704.1343

Stay-time at A88 (JD): 224.9855






Leg 3: From A76 to A49








Leg 4: From A49 to the Earth








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Solution(2)

The trajectory from the Earth to asteroid 88 The trajectory from asteroid 88 to asteroid 76

1,3

max

min ( ) 1564.60 224.98550.2 0.7946

2000 3652.5f j j

i

mJ K

m

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Solution(3)

The trajectory from asteroid 76 to asteroid 49 The trajectory from asteroid 49 to the Earth

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Conclusions and Remarks

Sequence selection based on orbit energy difference and phase difference is available.

The hybrid algorithm of particle swarm optimization and differential evolution seems feasible.

We obtained only one full solution. It is too few, and lacks of comparison. The result of the winner’s sequence 49-37-85 without using gravity assist is worthy to study.

Our team should make great efforts to catch up with top-ranking teams. Up to now, to learn is more than to compete for us. We are trying to develop professional software by FORTRAN, and to be familiar with gravity assist. Wish to do better in the future.

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Thank you for your attention

3nd Global Trajectory Optimization Competition Workshop Team 9

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