Attainable Landing Area Computation of a Lunar Lander with Uncertainty by Reachability Analysis

Abstract

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.

Exact Hybrid Jacobian Computation for Optimal Trajectories via Dual Number Theory

“…In this paper we focus on SPARTAN, an optimal-control package developed by the German Aerospace Center, which has already been used in literature [7][8][9][10][11][12][13] . SPARTAN uses the flipped version of the RPM: the global Flipped Radau Pseudospectral Method (FRPM), based on the flipped distribution of points w.r.t.…”

“…where (10) and t j are the roots of linear combinations of Legendre Polynomials P n (t) and P n−1 (t). Formal definitions of the Legendre Polynomials can be found in 15 , while an example of Legendre Polynomials of order 0 − 5 is plotted in Fig.…”