Single-Impulse Reachable Set in Arbitrary Dynamics Using Polynomials

This paper presents a method to accurately determine the reachable set (RS) of spacecraft with deterministic state after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current single-impulse RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variable parameterized polynomials in the differential algebra scheme. Then, using the envelope theory, the RS boundary is identified by solving the envelope equation. A framework is proposed to reduce the complexity of solving the envelope equation by converting it to the problem of searching for the root of a one-variable polynomial. Moreover, a high-order local polynomial approximation for the RS envelope is derived to improve computational efficiency. The method successfully determines the RSs of two near-rectilinear halo orbits in the cislunar space. Simulation results show that the RSs in both state and observation spaces can be accurately approximated under the three-body dynamics, with relative errors of less than 0.0658%. In addition, using the local polynomial approximation, the computational time for solving the envelope equation is reduced by more than 84%.