Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

TL;DR

This paper demonstrates that using Legendre and Chebyshev grid points in Birkhoff-based trajectory optimization enables fast, accurate solutions with built-in optimality testing, handling discontinuities and bang-bang controls effectively.

Contribution

It extends the universal Birkhoff theory to practical grid points, enabling efficient, reliable trajectory optimization with new optimality verification methods.

Findings

Grid points from Legendre and Chebyshev families satisfy the theory's hypotheses.

Optimality can be tested without solving boundary value problems.

Birkhoff methods handle discontinuities and bang-bang controls without Gibbs phenomenon.

Abstract

This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner.