Reachable Set Approximation in Cislunar Space with Pseudospectral Method and Homotopy

As humanity seeks to expand its presence in outer space, the area beyond geosynchronous orbit to approximately sixty-five thousand kilometers beyond the moon's orbit, known as cislunar space, is the focus of significant effort. A key consideration for upcoming missions in this space is determining, given a current position, velocity, and thrust capability, what vehicle states, i.e. position and velocity, can be reached in some amount of time. This research investigates using a pseudospectral method for approximating reachable sets in cislunar space, specifically in the Planar Circular Restricted Three-Body Problem (CRTBP), for a spacecraft using low-thrust Solar Electric Propulsion (SEP). Reachable sets are approximated by repeatedly solving a minimum-time optimal control problem (OCP) from an arbitrary starting state to a large number of terminal positions using continuous thrust. From this data, contours are generated that approximate the reachable sets after specified increments of time. The OCP is solved using an hp-adaptive Legendre-Gauss-Radau orthogonal collocation method, where
$h$
denotes segment widths and
$p$
denotes polynomial degree, which are established simultaneously in the method. To enable the solution process, a homotopy (also known as continuation) is employed. Results for a common cislunar initial location is presented: the L1 Lagrange point. The reachable sets are visualized as time contours of the terminal positions.