Mapped Chebyshev pseudospectral methods for optimal trajectory planning of differentially flat hypersonic vehicle systems

Abstract

This paper presents a novel trajectory planning algorithm for hypersonic glide vehicles based on differentially flatness theory and mapped Chebyshev pseudospectral method (MCPM). Firstly, the flatness property of a hypersonic glide vehicle dynamics system is extended explicitly by introducing a pseudo control input along the opposite direction of aerodynamics drag. In this way, all the states and controls of the system can be represented by flat outputs and their derivatives, and the trajectory planning problem is formulated into the flat output space with differential constraints entirely eliminated. Secondly, the optimization problem in the flat output space is converted into a nonlinear programming one via the mapped Chebyshev pseudospectral method. The MCPM employs the conformal mapping principle to discretize the flat outputs at the time interval, in terms of which, the closely clustered Chebyshev points near the boundary are stretched and the ill-conditions of the differential matrix is improved. The Barycentric Lagrange interpolation algorithm is then utilized to interpolate the flat output functions, which is seized of better approximation properties than polynomial interpolation. Finally, the transformed nonlinear programming problem are solved by sequence quadratic program technique. The states and controls are then generated by the expressions of the flatness outputs. At the end of this paper, numerical simulations are performed to evaluate the proposed approaches, and comparisons are made with traditional methods. Simulation results demonstrate that the proposed differentially flatness based mapped Chebyshev pseudospectral method possessed better computing efficiency than traditional ones without losing the merits of high precision.