Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

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 families of node points, be it Lobatto, Radau, or Gauss, then the resulting collection of trajectory optimization methods satisfies the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an [Formula: see text] 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 totality of necessary conditions that result from an application of Pontryagin’s principle. Numerical problems are solved to illustrate all these ideas in addition to demonstrating near-exponential rates of convergence. 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.