Rapid nonlinear convex guidance via overparameterized monomial coordinates and fundamental solution expansions

This paper introduces a framework by which the nonlinear trajectory optimization problem is posed as a path-planning problem in a space liberated of dynamics. In this space, general state constraints for continuous and impulsive control problems are encoded as linear constraints on the native overparameterized variables. This framework is enabled by nonlinear expansion in the vicinity of a reference in terms of fundamental solutions and a minimal nonlinear basis of mixed monomials in problem initial conditions. The former can be computed using state transition tensors, differential algebra, or analytic approaches, and the latter is computed analytically. Nonlinear guidance schemes are proposed taking advantage of this framework, including a successive convex programming scheme for delta-V minimizing trajectory optimization. This work enables a stable and highly rapid nonlinear guidance implementation without the need for collocation or real-time integration.