Hybrid jacobian computation for fast optimal trajectories generation

Nowadays the new, increased capabilities of CPUs have constantly encouraged researchers and engineers towards the investigation of numerical optimization as an analysis and synthesis tool in order to generate optimal trajectories and the controls to track them. In particular, one of the most promising techniques is represented by direct collocation methods. Among these, Pseudospectral Methods are gaining popularity for their straightforward implementation and some useful properties, like the possibility to remove the Runge phenomenon present in traditional interpolation techniques and the "spectral" convergence observable in the case of smooth problems. Experience shows that the quality of the results and the computation time are strongly affected by the jacobian matrix describing the transcription of the optimal control problem as an NLP. In this paper, the structure of the Jacobian matrix is analyzed, taking advantage of the sparse nature of such matrices. Additionally, its systematic "hybridization" will be discussed and implemented in order to speed up the simulations. Two different problems will be then described and solved with this approach and the results will be shown. Finally, a quantitative analysis of the performances deriving from the use of the hybrid jacobian compared to a traditional numerical technique will be shown as well.