Optimization-Based Reachability Analysis for Landing Scenarios

Kurzfassung

In this work, optimization-based algorithms are presented to approximate reachability sets. A reachability set comprises all dynamic system states that admissible controls can reach for a given initial state. Approaches based on the theory of optimization and optimal control have the advantage here that further constraints may be included in addition to the dynamics and the boundary conditions. However, solving nonlinear optimization tasks is considered costly in terms of time, so this should be done in a directed manner that yields as much insight as possible. For this reason, the presented algorithms embed the necessary optimization runs in a geometric framework. In particular, convex sets and properties of polytopes are discussed in this work, which allows a structural notion of geometric objects in higher dimensions. The optimization theory is fundamental to this work in treating nonlinear and convex programs. In the dynamic context, optimal control processes are introduced, minimizing a cost functional subject to a first-order ordinary differential equation and other constraints. The parametric sensitivity analysis is an essential part of this work. It is a post-optimal tool to acquire additional knowledge from a solved optimization without significant effort. Three algorithms for set approximation are presented. They are based on grids and polytopes. This work uncovers the parallelization potential and mathematical properties of the outcome of the algorithms. A boundary reconstruction of smooth convex sets through a second-order interpolation is possible with these properties. Furthermore, they allow different initial guess strategies to aim for fast convergence of trajectory recalculations. The methods are not merely theoretical but can be applied to real-world scenarios. The mathematical model describing the behavior of a spacecraft during landing is formulated to prove this. For this scenario, the reachability set is calculated with the methods of this work and visualized. In the process, convexification is performed. The computation times and advantages of the presented algorithms are elaborated to consider the application of the results of this work in future missions.