Trajectory Optimization by Successive Pseudospectral Convexification on Riemannian Manifolds

This paper proposes an intrinsic pseudospectral convexification framework for optimal control problems with manifold constraints. While successive pseudospectral convexification combines spectral collocation with successive convexification, classical pseudospectral methods are not geometry-consistent on manifolds. This is because interpolation and differentiation are performed in Euclidean coordinates. We introduce a geometry-consistent transcription that enables pseudospectral collocation without imposing manifold constraints extrinsically. The resulting method solves nonconvex manifold-constrained problems through a sequence of convex subproblems. A six-degree-of-freedom landing guidance example with unit quaternions and unit thrust-direction vectors demonstrates the practicality of the approach and preserves manifold feasibility to machine precision.