3nd Global Trajectory Optimization Competition Workshop Team 9 F. Jiang, Y. Li, K. Zhu, S. Gong, H. Ba oyin, J. Li, etc. School of Aero space Tsinghua University Beijing, China
Feb 01, 2016
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
3 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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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GTOC3 Workshop Torino, Italy, June 27, 2008
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
5 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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
6 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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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GTOC3 Workshop Torino, Italy, June 27, 2008
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
8 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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.
9 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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.
10 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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:
11 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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
12 Team 9
GTOC3 Workshop Torino, Italy, June 27, 2008
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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GTOC3 Workshop Torino, Italy, June 27, 2008
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
Arrival date (MJD): 59371.8310
Departure mass (kg): 1960.6172
Arrival mass (kg): 1807.5461
Position error (km): 909.0563
Velocity error (m/s): 0.1313
Leg 3: From A76 to A49
Departure date (MJD): 59806.8411
Stay-time at A76 (JD): 435.0101
Arrival date (MJD): 60470.0672
Departure mass (kg): 1807.5461
Arrival mass (kg): 1624.7850
Position error (km): 223.0663
Velocity error (m/s): 0.0822
Leg 4: From A49 to the Earth
Departure date (MJD): 61059.06844
Stay-time at A49 (JD): 589.0012
Arrival date (MJD): 61641.9721
Departure mass (kg): 1624.7850
Arrival mass (kg): 1564.6000
Position error (km): 870.5896
Velocity error (m/s): 0.9879
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GTOC3 Workshop Torino, Italy, June 27, 2008
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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GTOC3 Workshop Torino, Italy, June 27, 2008
Solution(3)
The trajectory from asteroid 76 to asteroid 49 The trajectory from asteroid 49 to the Earth
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GTOC3 Workshop Torino, Italy, June 27, 2008
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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GTOC3 Workshop Torino, Italy, June 27, 2008
Thank you for your attention