Exact hybrid jacobian computation for optimal trajectory generation via dual number theory

In this paper the effects of the use of the dual-based hybrid Jacobian computation in combination with the Pseudospectral Methods are thoroughly inspected. The dual-step dif- ferentiation method is implemented in SPARTAN (SHEFEX-3 Pseudospectral Algorithm for Re-entry Trajectory ANalysis), a tool based on the use of the global Flipped Radau Pseudospectral method for the transcription of optimal control problems. The dual number theory is exploited to provide an exact computation of the Jacobian matrix associated with the NonLinear Programming (NLP) problem to be solved. The dual-step differentiation method is compared to standard differentiation schemes (the central difference and the complex-step approximations) and applied in the solution of two examples of opti- mal control problem using two different off-the-shelf NLP solvers (SNOPT and IPOPT). Differentiation based on dual number theory is proved to be a valid alternative to the tradi- tional, well-known, differentiation schemes as its use improves, for the problems analysed, the accuracy of the results, especially in combination with SNOPT.