3nd Global Trajectory Optimization Competition Workshop Team 9

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3nd Global Trajectory Optimization Competition
WorkshopTeam 9

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

School of Aerospace Tsinghua University Beijing,
China
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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
Where mi and mf are the initial and final mass,
respectively K0.2 10 is the
stay-time at the j-th asteroid.
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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. (referenceGTOC2
  Activities and Results of ESA Advanced Concepts
  Team)

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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

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.

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Technical Approach(1)

• Astrodynamic model equinoctial elements
• Accommodate all possible conic orbits except
  i180.

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

Objective function
Choose N particles with random initial position
xi0 and velocity vi0. The iteration from the G
generation to G1 generation can be presented as
where r1 and r2 are both uniformly distributed
random numbers w, c1 and c2 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

Mutation
Crossover
Selection
where F1 and F2 are weighing factors in 0, 1
the integers rk (k1,,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(Lgtm)(CR)m-1, mgt0. CR is the crossover
constant in 0,1
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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 size400, Iteration times1000
• Weighing factors of DE are both 0.8
• Maximum velocity0.5
• Crossover constant0.618
• 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.

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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 objective function.

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Solution(1)
Leg 2 From A88 to A76
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
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 4 From A49 to the Earth
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
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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Solution(2)
The trajectory from the Earth to asteroid 88
The trajectory from asteroid 88 to asteroid 76
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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
  winners 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