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

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.