Development of Multiphase Radau Pseudospectral Method for Optimal Control Problems

Abstract

In this work, the flipped Radau pseudospectral method is employed in order to solve multiphase optimal con-trol problems. The application of interest is the generation of multi-stage rocket trajectories, both in ascent toorbit and descent to vertical landing. The method is implemented in the MATLAB tool SPARTAN, developed atDLR, with the use of the NLP solver IPOPT. Two relevant numerical examples are implemented for validationof the algorithm. The examples are appropriate because they contain either mutations in the equations of mo-tion from one phase to the next or jump discontinuities in the states (or both). The first problem is composedof three distinct phases and is concerned with the recovery of the main booster of the orbital launcher Falcon9 via a boost-back manoeuvre and a vertical landing near the launch site. The second problem is composedof four phases and is concerned with the launch, ascent and orbit insertion of the multiple-staged solid fuelorbital rocket Delta III. The algorithm is validated both, by the comparison of the results to the correspondingreference solutions, and by the analysis of the dual variables and of the Hamiltonian associated with the tra-jectories. The results show that the method is able to generate optimal trajectories with accuracy comparableto state of the art solvers.