Bang-bang optimal control for differentially flat systems using mapped pseudospectral method and analytic homotopic approach

Abstract

Summary The bang‐bang type optimal control problems arising from time‐optimal or fuel‐optimal trajectory planning in aerospace engineering are computationally intractable. This paper suggests a hybrid computational framework that utilizes differential flatness and mapped Chebyshev pseudospectral method to generate a related but smooth trajectory, from which the original non‐smooth solutions are achieved continuously by the analytic homotopic algorithm. The flatness allows for transcribing the original problem into an integration‐free flat outputs optimization problem with reduced number of decision variables. Chebyshev pseudospectral method is applied to parameterizing the flat outputs, and the numerical accuracy for the derivatives of flat outputs at collocation nodes, which are readily computed using differentiation matrices, is greatly enhanced by conformal map and barycentric rational interpolation techniques. Based on the obtained smooth trajectory, the analytic homotopic approach constructs an auxiliary optimal control problem whose costates are simply zero, avoiding the estimation of initial costates. The hybrid framework successfully addresses the difficulties of pseudospectral method and homotopic approach when they are applied separately. Numerical simulations of time‐optimal trajectory planning for spacecraft relative motion and attitude maneuver are presented, validating the performance of the hybrid computational framework. Copyright © 2016 John Wiley & Sons, Ltd.

A mesh refinement method for low‐thrust orbit transfer problems

“…The Pontryagin's principle‐based indirect method solves low‐thrust transfer problems 12,13 well, but is usually very sensitivity to the initial guess of optimization parameters and costate variables. The sensitivity can be tackled by various methods, such as multi‐shooting methods, 14 adjoint control transformation methods, 15 and homotopic methods 16,17 . In order to tackle the low‐thrust orbit transfer problem, Shen 18 addressed the boundary value problem by proposing a novel shooting procedure based on Newton's method.…”

“…The sensitivity can be tackled by various methods, such as multi-shooting methods, 14 adjoint control transformation methods, 15 and homotopic methods. 16,17 In order to tackle the low-thrust orbit transfer problem, Shen 18 addressed the boundary value problem by proposing a novel shooting procedure based on Newton's method. Zhu et al 19 developed a transition strategy by gradually decreasing the magnitude of thrust and generated the low-thrust transfer scheme from bi-impulse and finite thrust solutions.…”

Semi-analytical minimum time solutions with velocity constraints for trajectory following of vehicles

“…minimum time in our case). It is known from [13] that bang-bang problems are not particularly suitable for this method because of the intrinsic discontinuous nature of the solution, unless continuity is enforced. However, the main drawback of those solutions is that it is not possible to avoid purely numeric computations to deal with constraints, that are computationally slower than analytic or semi-analytic solutions.…”