A Universal Birkhoff Theory for Fast Trajectory Optimization

TL;DR

This paper introduces a universal Birkhoff interpolation framework for trajectory optimization that significantly improves computational stability and efficiency over traditional Lagrange-based methods, especially for nonlinear problems.

Contribution

It develops a new Birkhoff interpolation theory that reduces condition number growth and isolates computations to well-conditioned systems, without relying on specific basis functions or grid choices.

Findings

Reduces condition number growth from O(N^2) to O(1).

Isolates primal-dual computations to well-conditioned linear systems.

Theoretically establishes equivalence between Birkhoff methods.

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 highly successful methods, 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 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. In addition, the Birkhoff-specific primal-dual computations are isolated to a well-conditioned linear system even for nonlinear, nonconvex problems. This is part I of a two-part paper. In part I, a new theory is developed on the basis of two hypotheses. Other than these hypotheses, the theoretical development makes no assumptions on the choices of basis functions or the selection of grid points. Several covector mapping theorems are proved to establish the mathematical equivalence between direct and indirect Birkhoff methods. In part II of this paper (with Proulx), it is shown that a select family of Gegenbauer grids satisfy the two hypotheses required for the theory to hold. Numerical examples in part II illustrate the power and utility of the new theory.