Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function (RBF) interpolation with Galerkin projection to efficiently solve general optimal control problems. The goal is to develop a highly flexible solution to optimal control problems, especially nonsmooth problems involving discontinuities, while accounting for trajectory accuracy and computational efficiency simultaneously. The proposed solution, called the RBF-Galerkin method, offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points. The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker (KKT) conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem, if a set of discrete conditions holds. The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem. In addition, the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

I. Introduction

Direct methods are extensively used for solving optimal control problems, mainly due to their ability to handle path constraints, robustness to initial guess of parameters, and greater radii of convergence compared to indirect methods [1]–[3]. Direct transcription is based on approximating states and/or controls with a specific function with unknown coefficients and discretizing the optimal control problem with a set of proper points (nodes) to transcribe it into a nonlinear programming (NLP) problem. The resulting NLP can then be efficiently solved by NLP solvers available. Many direct methods are collocation-based approaches using either local or global polynomials depending on the type of function used in the approximation. Runge-Kutta methods [4], [5] and Bspline approaches [6], [7] are examples of local collocation methods that leverage low-degree local polynomials for the approximation of states and controls. The main drawback of these methods is their algebraic convergence rate, so their solution is not usually as accurate as the solution of global polynomial methods [3].