3nd Global Trajectory Optimization
Competition Workshop
Team 9
F. Jiang, Y. Li, K. Zhu, S. Gong, H. Baoyin, J. Li, etc.
School of Aerospace
Tsinghua University
Beijing, China
GTOC3 Workshop
Torino, Italy, June 27, 2008
Outline
Team Composition
Problem Summary
Technical Approach
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Sequence Selection
Global Optimization
Local Optimization
Solution
Conclusions
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Team Composition
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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 11th in GTOC3)
Team 9
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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:
J 
mf
mi
K
m in 
j  1,3
j

 m ax
Where mi and mf are the initial and final mass, respectively; K=0.2;
 max
=10;  j is the stay-time at the j-th asteroid.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Sequence Selection(1)
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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)
 V   V1   V 2
r
 ra 1  
 V1 

 V2 
V i  V f  2V iV f cos i r
2 rp1  2
2
p1
2 rp1  2
2
r
Vi 

2 ra 2  2
Vf 

2 ra 2  1 a 2
p1
r
p1
 ra 2 
 V2
 ra 2 
cos ir  cos i1 cos i 2  sin i1 sin i 2 cos   2   1 
 V1
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Sequence Selection(2)
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Third: Range the potential sequences on the base of orbit phase differences.
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
   i   i  M i    j   j  M
j


Orbit angular velocity difference
 n  n j  ni 

Asteroid i
Initial phase difference, relative to Jan 1, 2016
Synodic time
s a j 
s  k      2k
3

 s ai
3
Asteroid j
Sun
 n , k  0,1, 2,
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.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Sequence Selection(3)
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Synodic times (ST) of potential
sequences
 Expected sequence:
S T E  A 1   0 ,1 0  , S T A 3  E  2 0, S T A 3  E -S T E  A 1  1 0 ;
STA1 A2 -STE  A1 , STA2  A3 -STA1 A2 , ST A3  E -ST A2  A3
are all about 3 years
 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 88-7649.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach(1)
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Astrodynamic model: equinoctial elements
 Accommodate all possible conic orbits except i=180°.
Conversion from classical orbit elements:
p
a 1  e
2

 , f  e cos      , g  e sin     
h  tan  i 2  cos  , k  tan  i 2  sin  , L      
Motion equation:
 p , f , g , h , k , L   function  p , f , g , h , k , L , T , T , T
r
t
n

Though more complicated Cartesian quantities, they are more efficient
in computing.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Global Optimization(2)
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Particle swarm optimization (PSO)
 A population based stochastic optimization technique developed by Dr.
Eberhart and Dr. Kennedy in 1995, inspired by social behavior of bird
flocking or fish schooling
 Formulation
Objective function f  x   f  x1 , x 2 , , x D 
Choose N particles with random initial position xi0 and velocity vi0. The
iteration from the G generation to G+1 generation can be presented as
G 1
vi
G 1
xi
if f
 w  v i  c1  r1   p i  x i
G
G
c
 r2   g  x i
G
2

G 1
 xi  vi
G
 x  <f  p  , p =x
G 1
i
i
i
G 1
i
; if f
 x  <f  g  , g =x
G 1
i
G 1
i
where r1 and r2 are both uniformly distributed random numbers; w, c1 and
c2 should be valued case to case.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Global Optimization(3)
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Differential evolution (DE)
 A population based, stochastic function optimization proposed by Price
and Storn in 1995
 DE/rand/2/exp
Mutation:
G 1
vi
Crossover: u
Selection:
G 1
i, j
G 1
xi
= x r 1  F1   x r 2  x r 3   F2   x r 4  x r 5 
G
G
 v iG, j 1 , for j  n
 G
 x i , j , else
G
D
G
, n 1
D
,
G
, n  L 1
D
1
 u iG  1 , if f  u iG  1   f  x iG 


G
G 1
G
 x i , if f  u i   f  x i 
where F1 and F2 are weighing factors in [0, 1]; the integers rk (k=1,…,5) are
chosen 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];
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Global Optimization(4)
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Hybrid algorithm (PSODE) of PSO and DE
 In every 50 iterations, use PSO in the former 36 iterations, and DE in the
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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, w  0.94 e  N I / 500 ;
Optimize one leg by one leg
 Divide each leg into 10 segments. m f  m f  t i , t f , T1 , T2 ,
obj   m f    h 
T11

 r

6
ch  h, h  
 10 ;  v  ,    c h , c   4
2
 6

1
 Departure time and arrival time are optimized according to synodic time.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Technical Approach: Local Optimization(5)
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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
objective function.
Team 9
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GTOC3 Workshop
Leg 1: From the Earth to A88
Torino, Italy, June 27, 2008
Solution(1)
Leg 2: From A88 to A76
Launch date (MJD):
58090.8510
Departure date (MJD):
58704.1343
Launch velocity (km/s):
[-0.3378, 0.05498, 0.3645]
Stay-time at A88 (JD):
224.9855
Arrival date (MJD):
58479.1488
Arrival date (MJD):
59371.8310
Departure mass (kg):
2000.0000
Departure mass (kg):
1960.6172
Arrival mass (kg):
1960.6172
Arrival mass (kg):
1807.5461
Position error (km):
541.8060
Position error (km):
909.0563
Velocity error (m/s):
0.1578
Velocity error (m/s):
0.1313
Leg 3: From A76 to A49
Leg 4: From A49 to the Earth
Departure date (MJD):
59806.8411
Departure date (MJD):
61059.06844
Stay-time at A76 (JD):
435.0101
Stay-time at A49 (JD):
589.0012
Arrival date (MJD):
60470.0672
Arrival date (MJD):
61641.9721
Departure mass (kg):
1807.5461
Departure mass (kg):
1624.7850
Arrival mass (kg):
1624.7850
Arrival mass (kg):
1564.6000
Position error (km):
223.0663
Position error (km):
870.5896
Velocity error (m/s):
0.0822
Velocity error (m/s):
0.9879
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Solution(2)
J 
mf
mi
K
m in j 1,3 (  j )
 m ax

The trajectory from the Earth to asteroid 88
Team 9
1564.60
2000
 0.2
224.9855
 0.7946
3652.5
The trajectory from asteroid 88 to asteroid 76
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GTOC3 Workshop
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Solution(3)
The trajectory from asteroid 76 to asteroid 49
Team 9
The trajectory from asteroid 49 to the Earth
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GTOC3 Workshop
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Conclusions and Remarks
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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.
Team 9
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GTOC3 Workshop
Torino, Italy, June 27, 2008
Thank you for your attention
Team 9
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