Theoretical and Computational Advances in Birkhoff Pseudospectral Transcription for Aerospace Optimal Control
This paper develops theoretical and numerical extensions of Birkhoff pseudospectral transcription for aerospace optimal control problems. Building on the recently proposed body of knowledge, a Birkhoff integration operator is derived for flipped Legendre Gauss Radau (fLGR) grids, drawing connections to regular LGR grids and to the favorable conditioning properties that make Birkhoff interpolants particularly appealing. We also propose the enforcement of differential constraints via pseudospectral operators, and the primal and dual equivalence of this approach to state augmentation is established for control-rate limits. The numerical developments include the formulation and solution of heterogeneous multiphase optimal control problems, parameterized non-affine time mappings, composition of affine scaling and equilibration methods, and a hybrid derivative strategy to combine linear pseudospectral operators with numerical derivatives for nonlinear functions. These methods are implemented in the RAPTOR (Rapid Algorithm for Pseudospectral Trajectory Optimization and Robustification) transcription framework and assessed through feature-oriented benchmarks: a one-dimensional landing problem, a planar orbit-raising scenario, an atmospheric entry problem, and a multiphase reusable launch vehicle problem with simultaneous booster-recovery and upper-stage-insertion trajectories.