Rapid Nonlinear Convex Guidance Using a Monomial Method

This paper addresses the challenge of accommodating nonlinear dynamics and constraints in rapid trajectory optimization with impulsive maneuvers, envisioned for use in the context of onboard guidance. We present a novel framework that uniquely employs overparameterized monomial coordinates and precomputed fundamental solution expansions to facilitate rapid optimization while minimizing real-time computational requirements. The fundamental solution expansions are precomputed about an uncontrolled reference trajectory using differential algebra. Unlike traditional approaches that repeatedly evaluate the nonlinear dynamics and constraints as part of complex shooting or collocation-based schemes, this method replaces the nonlinearity inherent to dynamics and constraint functions entirely with a computationally simpler manifold constraint. With this approach, trajectory optimization is posed efficiently as a path-planning problem on the manifold. This problem is entirely convex except for the manifold constraint, readily lending itself to solution via sequential convex programming. We demonstrate the effectiveness of our approach in computing fast and accurate Delta V optimal solutions for long-range spacecraft rendezvous, including problems with nonlinear state constraints.