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

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