Solution for the Third Global Trajectory Optimization Competition

1
Solution for the Third Global Trajectory
Optimization Competition
Workshop 3 GTOC 27 June 2008 TORINO

• Team Politecnico di Milano - M.Massari, R.
  Armellin, G. Bellei, P. Di Lizia, M. Lavagna, G.
  Mingotti, S. Tonetti, F. Topputo
• Department of Aerospace Engineering
• Politecnico di Milano

2
Outline

• Optimization strategy
• Problem Analysis
• Global Optimization
• Solution Refinement
• Results

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Two Phase Approach

• The proposed problem is not a typical global
  trajectory optimization, it is a sequence of
• a combinatorial problem on the asteroids sequence
  
• a continuous problem on the trajectory between
  asteroids
• A two phase approach has been followed
• The Asteroid Selection and preliminary trajectory
  definition
• An asteroids pruning
• A Stochastic algorithm has been used
• The Trajectory Refinement
• Parametric Optimization Problem
• Multiple Shooting and Collocation
• A first look to the objective function allows to
  identify the conditions that would characterize
  the global optimum
• the stay time needs to be maximized
• the thrust time needs to be reduced.

4
Problem AnalysisAsteroid selection

• The list of asteroids has been pruned by
  analyzing
• Semi-major axis AU
• Eccentricity
• Inclination deg.
• The constraints on the orbital parameters can
  guide to a smart selection of the asteroids to
  visit.
• After the pruning process only six asteroids
  remain 49, 61, 76, 85, 88, 96.

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Problem ModelizationPreliminary Trajectory
Definition

• The Problem of Global Trajectory Optimization has
  been modeled using for each transfer
• Multiple Revolution Lamberts problem solution
• Lamberts problem for Exponential Sinusoids
  solution (for multi-revolution transfer)
• Solution of an optimal control problem by means
  of an indirect method formulation.
• No gravity assists of the Earth have been
  considered in the preliminary phase.
• The Problem is completely identified considering
• selection of the asteroids IDs
• determination of the departure epoch, the four
  transfer times, and the three stay times
• choice of the number of revolutions for the
  Lambert and the exponential sinusoid first guess
• the exponential sinusoid characterization
  parameter k2.

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Global Optimization Particle Swarm Optimization

• The Problem has been solved using a stochastic
  search method.
• The Particle Swarm Optimization
• Is based on the idea of swarms
• Initially the particles are randomly initialized
• Particles move in the search space on the basis
  of a velocity which is influenced by
• Inertia
• Personal Best Solution
• Global Best Solution

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Solution Refinement

• The solution refinement is necessary as the
  solution found by the global optimizer does not
  satisfy the problem constraints
• The trajectory refinement can be formulated as
• the solution of an optimal control problem in
  which the objective function must be maximized,
• subject to
• the differential constraints given by the
  dynamics,
• The boundary constraints deriving from rendezvous
  conditions,
• the path constraints deriving from the threshold
  on the available thrust.
• The solution found with the global optimization
  can be used as initial guess in the local
  optimization process

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Solution RefinementOptimal Control Problem

• The Optimal control problem has been solved
• Transcribing the continuous variables in
  parametric variables
• Expressing the differential constraints as
  algebraic constraints on the parametric variables
• Solving the resulting NLP problem with a
  Sequencial Quadratic Programming Solver
• Two different transcription techniques have been
  applied
• Multiple Shooting method
• Collocation method
• Earths flyby have been included based on
  energetic considerations.

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Results

• Solution sequence
• Phase 1 Earth Asteroid 2001 GP2 (GTOC3 N. 96)
• Phase 2 Asteroid 2001 GP2 Earth
• Phase 3 Earth Asteroid 1991 VG (GTOC3 N. 88)
• Phase 4 Asteroid 1991 VG Asteroid 2000 SG344
  (GTOC3 N. 49)
• Phase 5 Asteroid 2000 SG344 Earth
• Departure epoch 58169 MJD
• Arrival epoch 61693.21 MJD
• Total time of flight 9.6487 years
• Minimum stay time 100.0 days
• Initial s/c mass 2000 kg
• Final s/c mass 1663.1148536687001 kg
• Propellant mass used 336.89 kg
• Objective function value 0.83758069856604

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Results
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Results

• Phase 1
• Hyperbolic excess velocity 0.4999 km/s
• Departure epoch 58169.0 MJD
• Time of flight 569.65 days
• Arrival epoch 58738.65 MJD
• Initial s/c mass 2000 kg
• Final s/c mass 1907.70 kg

Phase 3 Fly-By radius 6878.63 km Departure
epoch 59142.33 MJD Time of flight 531.67
days Arrival epoch 59674 MJD Initial s/c mass
1884.87 kg Final s/c mass 1776.05 kg
Phase 2 Stay time at Asteroid 2001 GP2 110.34
days Departure epoch 58849 MJD Time of flight
293.33 days Arrival epoch 59142.33 MJD Initial
s/c mass 1907.70 kg Final s/c mass 1884.87 kg
Phase 4 Stay time at Asteroid 1991 VG 110
days Departure epoch 59784 MJD Time of flight
390 days Arrival epoch 60174 MJD Initial s/c
mass 1776.05 kg Final s/c mass 1709.37 kg
Phase 5 Stay time at Asteroid 2000 SG344 110
days Departure epoch 60284.00 MJD Time of
flight 1409.21 days Arrival epoch 61693.21
MJD Initial s/c mass 1709.37 kg Final s/c mass
1663.148536687001 kg
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Results
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Team 21 - Politecnico di Milano

• Mauro Massari
• Roberto Armellin
• Gabriele Bellei
• Pierluigi Di Lizia
• Michéle Lavagna
• Giorgio Mingotti
• Stefania Tonetti
• Francesco Topputo
• Aerospace Engineering Department
• Politecnico di Milano
• Via La Masa, 34
• 20156 Milano, Italy
• Ph. 39 02 2399 8308
• Fax [phone]
• Contact Person
• Mauro Massari