Abstract
In the last decades, heuristic techniques have become established as suitable approaches for solving optimal control problems. Unlike deterministic methods, they do not suffer from locality of the results and do not require any starting guess to yield an optimal solution. The main disadvantages of heuristic algorithms are the lack of any convergence proof and the capability of yielding only a near optimal solution, if a particular representation for control variables is adopted. This paper describes the indirect swarming method, based on the joint use of the analytical necessary conditions for optimality, together with a simple heuristic technique, namely the particle swarm algorithm. This methodology circumvents the previously mentioned disadvantages of using heuristic approaches, while retaining their advantageous feature of not requiring any starting guess to generate an optimal solution. The particle swarm algorithm is chosen among the different available heuristic techniques, due to its apparent simplicity and the recent promising results reported in the scientific literature. Two different orbital maneuvering problems are considered and solved with great numerical accuracy, and this testifies to the effectiveness of the indirect swarming algorithm in solving low-thrust trajectory optimization problems.
Similar content being viewed by others
References
Miele, A., Mancuso, S.: Optimal trajectories for Earth-Moon-Earth flight. Acta Astron. 49(2), 59–71 (2001)
Miele, A., Wang, T.: Optimal trajectories for Earth-to-Mars flight. J. Optim. Theory Appl. 95(3), 467–499 (1997)
Miele, A., Wang, T.: Multiple-subarc gradient-restoration algorithm, Part 1: Algorithm structure. J. Optim. Theory Appl. 116(1), 1–17 (2003)
Miele, A., Wang, T.: Multiple-subarc gradient-restoration algorithm, Part 2: Application to a multistage launch vehicle design. J. Optim. Theory Appl. 116(1), 19–39 (2003)
Brown, K.R., Harrold, E.F., Johnson, G.W.: Rapid Optimization of Multiple-Burn Rocket Flights. NASA CR 1430 (1969)
Brusch, R.G., Vincent, T.L.: Numerical implementation of a second-order variational endpoint condition. AIAA J. 8, 2230–2235 (1970)
Kluever, C.A., Pierson, B.L.: Optimal low-thrust Earth-Moon three-dimensional trajectories. J. Guid. Control Dyn. 18, 830–837 (1995)
Hull, D.: Initial Lagrange multipliers for the shooting method. J. Guid. Control Dyn. 31(5), 1490–1492 (2008)
McAdoo, S., Jezewski, D.J., Dawkins, G.S.: Development of a Method for Optimal Maneuver Analysis of Complex Space Missions. NASA TN D-7882 (1975)
Redding, D.C.: Optimal Low-Thrust Transfers to Geosynchronous Orbits. Stanford University Guidance and Control Lab, report SUDAAR 539, Stanford (1983)
Betts, J.D.: Optimal interplanetary orbit transfers by direct transcription. J. Astronaut. Sci. 42, 247–326 (1994)
Enright, P.J., Conway, B.A.: Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. J. Guid. Control Dyn. 15, 994–1002 (1992)
Enright, P.J., Conway, B.A.: Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming. J. Guid. Control Dyn. 14, 981–985 (1991)
Seywald, H.: Trajectory optimization based on differential inclusion. J. Guid. Control Dyn. 17, 480–487 (1994)
Coverstone-Carroll Williams, S.N.: Optimal low thrust trajectories using differential inclusion concepts. J. Astronaut. Sci. 42, 379–393 (1994)
Zondervan, K.P., Wood, L.J., Caughey, T.K.: Optimal low-thrust, three-burn orbit transfers with large plane changes. J. Astronaut. Sci. 32, 407–427 (1984)
Gao, Y., Kluever, C.: Low-thrust interplanetary orbit transfer using hybrid trajectory optimization method with multiple shooting. Paper AIAA 2004-5088 (2004)
Conway, B.A.: A survey of methods available for the numerical optimization of continuous dynamic systems. J. Optim. Theory Appl. 152, 271–306 (2012)
Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21, 193–207 (1998)
Goldberg, D.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, New York (1989)
Pontani, M., Conway, B.A.: Particle swarm optimization applied to space trajectories. J. Guid. Control Dyn. 33(5), 1429–1441 (2010)
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942–1948 (1995)
Venter, G., Sobieszczanski-Sobieski, J.: Particle swarm optimization. AIAA J. 41(8), 1583–1589 (2003)
Eberhart, R.C., Shi, Y.: Comparison Between Genetic Algorithms and Particle Swarm Optimization. Evolutionary Programming VII. Lecture Notes in Computer Science, vol. 1447, pp. 611–616. Springer, Berlin (1998)
Parsopoulos, K.E., Vrahatis, N.M.: On the computation of all global minimizers through particle swarm optimization. IEEE Trans. Evol. Comput. 8(3), 211–224 (2004)
Pontani, M., Conway, B.A.: Swarming Theory Applied to Space Trajectory Optimization. Spacecraft Trajectory Optimization, pp. 263–293. Cambridge University Press, New York (2010)
Pontani, M., Conway, B.A.: Particle swarm optimization applied to impulsive orbital transfers. Acta Astron. 74, 141–155 (2012)
Pontani, M., Ghosh, P., Conway, B.A.: Particle swarm optimization of multiple-burn rendezvous trajectories. J. Guid. Control Dyn. 35(4), 1192–1207 (2012)
Bryson, A.E., Ho, Y.C.: Applied Optimal Control, pp. 71–89. Hemisphere, New York (1975)
Hull, D.: Optimal Control Theory for Applications, pp. 247–257. Springer, New York (2003)
Hu, X., Shi, Y., Eberhart, R.: Recent advances in particle swarm. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2004), Portland, OR, pp. 90–97 (2004)
Michalewicz, Z., Schoenauer, M.: Evolutionary algorithms for constrained parameter optimization problems. Evol. Comput. 4(1), 1–32 (1996)
Lawden, D.F.: Optimal Trajectories for Space Navigation. Butterworths, London (1963)
Hull, D.G.: Optimal guidance for quasi-planar lunar ascent. J. Optim. Theory Appl. 151(2), 353–372 (2011)
Hull, D.G., Harris, M.W.: Optimal solutions for quasi-planar ascent over a spherical moon. J. Guid. Control Dyn. 35(4), 1018–1023 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pontani, M., Conway, B. Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method. J Optim Theory Appl 162, 272–292 (2014). https://doi.org/10.1007/s10957-013-0471-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0471-9