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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 differentiation 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 optimal 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 traditional, well-known, differentiation schemes as its use improves, for the problems analysed, the accuracy of the results, especially in combination with SNOPT.

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Soft landing is one of the most critical phases for space missions which require landing a spacecraft on the surface of a body like asteroids or planets, as well as concepts like reusable launch vehicles. In order to ensure safety and success for the terminal landing phase, in addition to hazard maps obtained by on-board sensors, there is also a need for a map which characterizes the attainable landing area that the lander can achieve by obeying constraints within the presence of uncertainties. This paper proposes a method to obtain the attainable landing area of a lunar lander with uncertainties by reachability analysis. The method obtains the set of achievable states for a dynamical system starting from an initial condition with given admissible control inputs of the system. Nonconvex reachable sets (RS) are computed using optimal control. The candidate landing area on the Moon surface is represented by equidistant grid points and for each point an optimal control problem (OCP) is defined. The corresponding OCP is transcribed into a finite dimensional Nonlinear Programming Problem (NLP) by using Pseudospectral Methods (PSM). The solution of the NLP leads to the RS of the dynamical system. A Riccati equation-based controller is designed to track the reference trajectories. Monte Carlo simulations are carried out to obtain the safely attainable landing area of the lunar lander as probability maps.