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A Reachable Set Analysis Method for Generating Near-Optimal Trajectories of Constrained Multiphase Systems

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Abstract

Few sophisticated problems in the optimal control of a dynamical system can be solved analytically. There are many numerical solution methods, but most, especially those with the most potential accuracy, work iteratively and must be initialized with a guess of the solution. Satisfactory guesses may be very difficult to generate. In this work, a “Reachable Set Analysis” (RSA) method is developed to find near-optimal trajectories for multiphase systems with no a priori knowledge. A multiphase system is a generalization of a dynamical system that includes possible changes on the governing equations throughout the trajectory; the traditional dynamical system where the governing equations do not change is included in the formulation as a special case. The RSA method is based on a combination of metaheuristic algorithms and nonlinear programming. A particularly beneficial aspect of the solution found using RSA is that it satisfies the system governing equations and comes arbitrarily close (to a degree chosen by the planner) to satisfying given terminal conditions. Three qualitatively different multiphase problems, such as a low-thrust transfer from Earth to Mars, a system with chattering arcs in the optimal control, and a motion planning problem with obstacles, are solved using the near-optimal trajectories found by RSA as initial guesses to show the effectiveness of the new method.

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Authors and Affiliations

  1. Department of Aerospace Engineering, University of Illinois, 306 Talbot Laboratory, MC-236, Urbana, IL, 61801, USA

    Christian M. Chilan & Bruce A. Conway

Authors
  1. Christian M. Chilan
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  2. Bruce A. Conway
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Correspondence to Bruce A. Conway.

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Communicated by Kenneth D. Mease.

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Chilan, C.M., Conway, B.A. A Reachable Set Analysis Method for Generating Near-Optimal Trajectories of Constrained Multiphase Systems. J Optim Theory Appl 167, 161–194 (2015). https://doi.org/10.1007/s10957-014-0651-2

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  • DOI: https://doi.org/10.1007/s10957-014-0651-2

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