Ethan Foss, Stanford University; Aaron J. Rosengren, University of California, San Diego; Ashley D. Biria, Air Force Research Laboratory
Keywords: Reachability Analysis, Cislunar Astrodynamics, Space Domain Awareness
Abstract:
As presence in cislunar space expands, new methods for characterizing the capabilities of low-thrust spacecraft in the regions beyond geosynchronous orbit (GEO) are becoming more important. Spacecraft reachability analysis, which refers to the determination of the set of states that a spacecraft can travel to or come from using its propulsion system over a given amount of time, is a key aspect to unlocking a better awareness of spacecraft trajectories in this new dynamical topography. Reachability analysis offers multi-faceted utility to the space situational awareness community for tracking, detection, maneuver reconstruction, and evaluation of collision probabilities. However, the computation of reachable sets is a particularly difficult problem in astrodynamics and few methods exist to accurately and rapidly compute reachable sets for low-thrust spacecraft. Furthermore, computing reachable sets in the beyond GEO (xGEO) regime presents unique challenges due to the highly sensitive and chaotic dynamics environment.
Several existing techniques for reachable set computation have been proposed for spacecraft applications. These include, but are not limited to, direct solutions to the Hamilton-Jacobi-Isaacs (HJI) PDE, solving direct minimum-time optimal control problems for a grid of terminal points, and indirect method-based optimization of trajectories. Solutions using the HJI PDE can be prohibitively expensive and difficult to compute, and can only be determined for systems with low dimensionality (5 or less). Likewise, the direct method, minimum-time, optimal-control technique is also computationally expensive. Lastly, the indirect-method technique, which has seen the most success for rapid reachable set computation, still has its limitations. The method does not perform well in sensitive dynamical regimes, such as near multiple gravitational bodies. It also does not provide mathematical guaruntees for computation of non-convex regions of the reachable set and relies on the propagation of a large number of trajectories to accurately outline the contour of the reachable set.
In this work, we propose leveraging state transition tensors and set-based computing to directly propagate the reachable sets of spacecraft in cislunar space. Set-based reachability analysis, which uses set representations like polytopes, zonotopes, Taylor models, and polynomial zonotopes, is a popular method for performing reachability analysis among the cyberphysical systems community. However, such a technique for reachability analysis has not been demonstrated for cislunar space situational awareness. Moreover, typical implementations of set-based reachability computing use inaccurate means of discretization. By leveraging state transition tensors, which have become popular for many astrodynamics applications like uncertainty propagation and optimal control, higher accuracy non-convex approximations of reachable sets can be obtained. We will show how the proposed method performs in comparison to a selection of previously-proposed methods for reachability set computation for families of distant retrograde orbits (DROs) in the circular, restricted, three-body problem (CR3BP). We then show how the method can be easily extended for higher-fidelity reachability set computation using ephemeris models of a cislunar spacecraft in a near-rectilinear halo orbit (NRHO).
Date of Conference: September 17-20, 2024
Best Student Paper Award Winner 2024
Track: Cislunar SDA