Far-Field Minimum-Fuel Spacecraft Rendezvous using Koopman Operator and $\ell_2/\ell_1$ Optimization

Abstract: We propose a method to compute approximate solutions to the minimum-fuel far-field rendezvous problem for thrust-vectoring spacecraft. It is well-known that the use of linearized spacecraft rendezvous equations may not give sufficiently accurate results for far-field rendezvous. In particular, as the distance between the active and the target spacecraft becomes significantly greater than the distance between the target spacecraft and the center of gravity of the planet, the accuracy of linearization-based control design approaches may decline substantially. In this paper, we use a nonlinear state space model which corresponds to more accurate description of dynamics than linearized models but at the same time poses the known challenges of nonlinear control design. To overcome these challenges, we utilize a Koopman operator based approach with which the nonlinear spacecraft rendezvous dynamics is lifted into a higher dimensional space over which the nonlinear dynamics can be approximated by a linear system which is more suitable for control design purposes than the original nonlinear model. An Iteratively Recursive Least Squares (IRLS) algorithm from compressive sensing is then used to solve the minimum fuel control problem based on the lifted linear system. Numerical simulations are performed to show the efficacy of the proposed Koopman operator based approach.