Prashant R. Patel, Institute for Defense Analyses; Daniel J. Scheeres, University of Colorado Boulder
Keywords: reachable set, electric propulsion, maneuver
Abstract:
Estimating the reachability of electric propulsion spacecraft is a difficult problem. The increased use of electric propulsion means that orbits can continuously vary and are defined by a large number of controls. Identifying the reachable set requires solving a two-point boundary value problem or large dimensional optimal control problem for every point. This is difficult due to the need for the initial guess to be in the region of convergence. Guessing Lagrange multipliers is particularly challenging because they are chaotic thus small initial differences can lead to large errors at the terminal state. Furthermore, commercial, civil, and militaries are looking to expand beyond earth and into cis-lunar and beyond. This increases the complexity as it introduces the need to handle multi-body dynamics. Optimizing with transient multi-body dynamics remains a difficult problem with only a few programs able to deal with them reliably. Finally, there are a large number of satellites in orbit, and even more are planned.
Together these drive the need automatic, fast, and can scalable algorithms that can estimate the reachable set of spacecrafts. Scale is needed to ensure the algorithms can assess many satellites. Speed is needed to make sure the algorithms can support near-real time planning and the eventual goal or on-orbit maneuver. Autonomous algorithms are needed to ensure that the solutions can operate reliably without human intuition.
If we can rapidly compute the reachable set of electric propulsion spacecraft then it will enable better space domain awareness (SDA), rapid constellation management and reconfiguration, supports pursuit-evasion games, designing of robust trajectories, context-based orders, and others. For space domain awareness this would enable optimized search volume-time based tasking and better sensor location investments.
Prior work in this area highlighted how reachable sets can be estimated using forward shooting techniques, which require the user to provide an initial guess of the co-states. Our approach eliminates the need for a user supplied initial guess and eliminates the need for solving a two-point boundary value problem. This allows our approach to estimate the reachable set quickly and automatically.
We present a novel method for rapidly estimating the reachable set of electric propulsion spacecraft. The algorithm works by preforming a single integration along the reference trajectory and computes the state transition matrix or state transition tensors. We then employ a novel optimization algorithm that generates a linear and second order control update that also satisfies constraints to the second order. By selecting the reference trajectory to be non-thrusting and carefully crafting the cost function we avoid the need for a user supplied thrust or co-state initial guess. The algorithm only requires is an estimate of the initial states (e.g., mean estimate or error ellipsoid of state vector), initial mass, max thrust of thruster, and specific impulse of engine.
We demonstrate that the algorithm works remarkably well at estimating the entire reachable set despite only requiring a single integration to estimate all the potential control laws. We show that the estimates are accurate even when there are multi-body effects (i.e., planetary fly-bys). In addition, we present stressing cases where the resultant reachable set is contorted and includes multi-body effects. The algorithm performs well in these cases and can estimate the shape of the reachable set. We check our results by solving for the exact reachable set and show that it matches closely our algorithms estimated results.
Date of Conference: September 27-20, 2022
Track: Astrodynamics