Universal Birkhoff Theory for Fast Trajectory Optimization

Abstract

Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these successes, a new starting point for trajectory optimization is proposed. This starting point is based on the recently developed concept of universal Birkhoff interpolants. The new first-principles’ approach offers a substantial computational upgrade to the Lagrange theory in completely flattening the rapid growth of the condition numbers from O(N2) to O(1), where N is the number of grid points. Furthermore, a Birkhoff discretization offers the theoretical possibility of an infinite-order rate of convergence. In addition, the Birkhoff-specific primal-dual computations are isolated to a well-conditioned linear/convex system even for nonlinear, nonconvex problems. This is Part I of a two-part paper. In Part I, the new theory is developed on the basis of two hypotheses. Other than these hypotheses, the theory makes no assumptions on the choices of basis functions, be they polynomials or otherwise, or even the distribution of grid points. Several theorems are proved to establish the mathematical equivalence between direct and indirect Birkhoff methods. In Part II of this paper, it is shown that a select family of Gegenbauer grids satisfy the two hypotheses required for the theory to hold. The theory developed in Part I is used in Part II to produce the fast computational implementation.