Workshop 3° GTOC
27 June 2008
TORINO
Solution for the Third Global Trajectory Optimization Competition
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
Outline
 Optimization strategy
 Problem Analysis
 Global Optimization
 Solution Refinement
 Results
Team Politecnico di Milano – M. Massari
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.
Team Politecnico di Milano – M. Massari
Problem Analysis
Asteroid selection
The list of asteroids has been pruned by analyzing:
Semi-major axis a [0.95,1.05] AU;
 Eccentricity e [0, 0.12] ;
 Inclination i [0, 4] 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.
Team Politecnico di Milano – M. Massari
Problem Modelization
Preliminary Trajectory Definition
The Problem of Global Trajectory Optimization has been modeled using for each
transfer:
 Multiple Revolution Lambert’s problem solution
 Lambert’s 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.
Team Politecnico di Milano – M. Massari
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
Team Politecnico di Milano – M. Massari
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
Team Politecnico di Milano – M. Massari
Solution Refinement
Optimal 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
Earth’s flyby have been included based on energetic considerations.
Team Politecnico di Milano – M. Massari
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:
Arrival epoch:
Total time of flight:
Minimum stay time:
Initial s/c mass:
Final s/c mass:
Propellant mass used:
58169 MJD
61693.21 MJD
9.6487 years
100.0 days
2000 kg
1663.1148536687001 kg
336.89 kg
Objective function value:
0.83758069856604
Team Politecnico di Milano – M. Massari
Results
Team Politecnico di Milano – M. Massari
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
Team Politecnico di Milano – M. Massari
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
Results
Team Politecnico di Milano – M. Massari
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: +39 02 2399 8028
Contact Person:
Mauro Massari
e-mail: [email protected]
Team Politecnico di Milano – M. Massari