Particle Swarm Optimization of Multiple-Burn Rendezvous Trajectories

The particle swarm algorithm is a population-based heuristic method successfully applied in several fields of research, only recently to aerospace trajectories. It represents a very intuitivemethodology for optimization, inspired by the behavior of bird flocks while searching for food. In this work, the method is applied to (impulsive and finitethrust) multiple-burn rendezvous trajectories. First, the technique is employed to determine the globally optimal four-impulse rendezvous trajectories for two challenging test cases. Second, the same problems are solved under the assumption of using finite thrust. In this context, the control function is assumed to be a linear combination of B-splines. Themethod at hand is relatively straightforward to implement anddoes not require an initial guess, unlike gradient-based solvers. Despite its simplicity and intuitiveness, the particle swarm methodology proves to be quite effective in finding the optimal solution to orbital rendezvous optimization problems with considerable numerical accuracy.

[1]  Massimiliano Vasile,et al.  On testing global optimization algorithms for space trajectory design , 2008 .

[2]  Hans Seywald,et al.  Trajectory optimization based on differential inclusion , 1994 .

[3]  Bruce A. Conway,et al.  Optimal Low-Thrust Trajectories to the Interior Earth-Moon Lagrange Point , 2010 .

[4]  O. Weck,et al.  A COMPARISON OF PARTICLE SWARM OPTIMIZATION AND THE GENETIC ALGORITHM , 2005 .

[5]  John E. Prussing,et al.  Optimal two- and three-impulse fixed-time rendezvous in the vicinityof a circular orbit , 1970 .

[6]  Zbigniew Michalewicz,et al.  Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization , 1999, Evolutionary Computation.

[7]  Bruce A. Conway,et al.  Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming , 1991 .

[8]  Anil V. Rao,et al.  Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method , 2010, TOMS.

[9]  Bruce A. Conway,et al.  Method and solution for the 2009 Global Trajectory Optimization Contest , 2010 .

[10]  B. Prasad,et al.  A class of generalized variable penalty methods for nonlinear programming , 1981 .

[11]  Andries Petrus Engelbrecht,et al.  A Cooperative approach to particle swarm optimization , 2004, IEEE Transactions on Evolutionary Computation.

[12]  P. Eberhard,et al.  Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering , 2006 .

[13]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[14]  P. Fourie,et al.  The particle swarm optimization algorithm in size and shape optimization , 2002 .

[15]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[16]  R. Eberhart,et al.  Comparing inertia weights and constriction factors in particle swarm optimization , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[17]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[18]  Bruce A. Conway,et al.  A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems , 2011, Journal of Optimization Theory and Applications.

[19]  R. Bhattacharya,et al.  Nonlinear Trajectory Generation Using Global Local Approximations , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[20]  Xiaohui Hu,et al.  Engineering optimization with particle swarm , 2003, Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706).

[21]  Jaroslaw Sobieszczanski-Sobieski,et al.  Particle swarm optimization , 2002 .

[22]  J. O. Ramsay,et al.  Functional Data Analysis (Springer Series in Statistics) , 1997 .

[23]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[24]  Peter J. Angeline,et al.  Evolutionary Optimization Versus Particle Swarm Optimization: Philosophy and Performance Differences , 1998, Evolutionary Programming.

[25]  I. M. Levitt.  Advances in astronautical sciences, vol.: edited by Horace Jacobs. 460 pages, illustrations, 612 × 934 in. New York, Plenum Press, Inc., 1959. Price, $8.00 , 1960 .

[26]  Anil V. Rao,et al.  Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method , 2006 .

[27]  B. Conway,et al.  Numerical Trajectory Optimization with Swarm Intelligence and Dynamic Assignment of Solution Structure , 2012 .

[28]  Zbigniew Michalewicz,et al.  Evolutionary Algorithms for Constrained Parameter Optimization Problems , 1996, Evolutionary Computation.

[29]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[30]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[31]  M. Clerc,et al.  The swarm and the queen: towards a deterministic and adaptive particle swarm optimization , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[32]  Kaijian Zhu,et al.  Satellite scheduling considering maximum observation coverage time and minimum orbital transfer fuel cost , 2010 .

[33]  Masao Arakawa,et al.  Adaptive range particle swarm optimization , 2006 .

[34]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[35]  Lorenzo Casalino,et al.  Cooperative evolutionary algorithm for space trajectory optimization , 2009 .

[36]  Russell C. Eberhart,et al.  A new optimizer using particle swarm theory , 1995, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

[37]  Dario Izzo,et al.  Spacecraft Trajectory Optimization: Global Optimization and Space Pruning for Spacecraft Trajectory Design , 2010 .

[38]  Bruce A. Conway,et al.  Optimal space trajectories via particle swarm technique , 2010 .

[39]  Slawomir J. Nasuto,et al.  Search space pruning and global optimisation of multiple gravity assist spacecraft trajectories , 2007, J. Glob. Optim..

[40]  Haiyan Lu,et al.  Self-adaptive velocity particle swarm optimization for solving constrained optimization problems , 2008, J. Glob. Optim..

[41]  Michael N. Vrahatis,et al.  On the computation of all global minimizers through particle swarm optimization , 2004, IEEE Transactions on Evolutionary Computation.

[42]  W. Williamson.  Use of Polynomial Approximations to Calculate Suboptimal Controls , 1971 .

[43]  M. Gerdts.  Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems , 2003 .

[44]  C. Hargraves,et al.  DIRECT TRAJECTORY OPTIMIZATION USING NONLINEAR PROGRAMMING AND COLLOCATION , 1987 .

[45]  David B. Spencer,et al.  Identifying Optimal Interplanetary Trajectories through a Genetic Approach , 2006 .

[46]  Luciano Blasi,et al.  Particle Swarm Approach in Finding Optimum Aircraft Configuration , 2007 .

[47]  Russell C. Eberhart,et al.  chapter seven – The Particle Swarm , 2001 .

[48]  Bruce A. Conway,et al.  Particle Swarm Optimization Applied to Space Trajectories , 2010 .

[49]  Manas Khurana,et al.  Application of swarm approach and aritficial neural networks for airfoil shape optimization , 2008 .

[50]  Russell C. Eberhart,et al.  Recent advances in particle swarm , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[51]  José Neves,et al.  The fully informed particle swarm: simpler, maybe better , 2004, IEEE Transactions on Evolutionary Computation.

[52]  John E. Prussing,et al.  Optimal Two- and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit , 2003 .

[53]  Yuhui Shi,et al.  Particle swarm optimization: developments, applications and resources , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[54]  R. Storn,et al.  Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces , 2004 .

[55]  Guido Colasurdo,et al.  Indirect Optimization Method for Impulsive Transfers , 1994 .

[56]  E. Mooij,et al.  Performance Evaluation of Global Trajectory Optimization Methods for a Solar Polar Sail Mission , 2009 .

[57]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[58]  Pradipto Ghosh,et al.  A direct method for trajectory optimization using the particle swarm approach , 2011 .

[59]  Bradley J. Wall,et al.  Shape-Based Approach to Low-Thrust Rendezvous Trajectory Design , 2009 .

[60]  Hitoshi Iba,et al.  Particle swarm optimization with Gaussian mutation , 2003, Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706).

[61]  R. Bhattacharya,et al.  OPTRAGEN: A MATLAB Toolbox for Optimal Trajectory Generation , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[62]  John E. Prussing,et al.  Optimal four-impulse fixed-time rendezvous in the vicinity of a circular orbit. , 1969 .

[63]  Q. Henry Wu,et al.  A bacterial swarming algorithm for global optimization , 2007, 2007 IEEE Congress on Evolutionary Computation.

[64]  I. Michael Ross,et al.  Direct Trajectory Optimization by a Chebyshev Pseudospectral Method ; Journal of Guidance, Control, and Dynamics, v. 25, 2002 ; pp. 160-166 , 2002 .

[65]  Bruce A. Conway,et al.  Genetic Algorithm Preprocessing for Numerical Solution of Differential Games Problems , 2004 .

[66]  Bruce A. Conway,et al.  New numerical methods for determining periodic orbits in the circular restricted three-body problem , 2010 .

[67]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[68]  Riccardo Poli,et al.  Particle swarm optimization , 1995, Swarm Intelligence.

[69]  Kaijian Zhu,et al.  Trajectory Optimization of Multi-Asteroids Exploration with Low Thrust , 2009 .

[70]  Zbigniew Michalewicz,et al.  Evolutionary algorithms , 1997, Emerging Evolutionary Algorithms for Antennas and Wireless Communications.

[71]  Goldberg,et al.  Genetic algorithms , 1993, Robust Control Systems with Genetic Algorithms.

[72]  Eliot Winer,et al.  Implementation of Digital Pheromones in Particle Swarm Optimization for Constrained Optimization Problems , 2008 .

[73]  John E. Prussing,et al.  Optimal multiple-impulse time-fixed rendezvous between circular orbits , 1984 .