Spectral Shifted Stabilized Continuation for Indirect Optimal Control

Abstract

Continuation methods provide a solution to the initial guess problem that numerical root solvers and shooting methods often face by iteratively transforming an auxiliary solution into a desired solution through differential corrections. While seemingly straightforward, designing an automated continuation strategy that starts at the auxiliary problem and successfully terminates at the desired problem with minimal user intervention is often challenging especially for optimal trajectory-planning problems with nonlinear motion models subject to state/path and control constraints. One scenario where the auxiliary problem does not connect to the desired problem occurs when the intermediate problems are ill posed (i.e., infeasible). This paper presents a two-layered Jacobian conditioned stabilized continuation algorithm that circumvents these infeasible zones along a user-defined continuation path with little designer intervention. The efficacy of this approach is evaluated in the context of several prototype problems including quadratic root-solving, optimal path planning with a Dubins model, an optimal orbit transfer problem, and an optimal high-speed vehicle trajectory generation problem.