Single-Impulse Reachable Set in Arbitrary Dynamics Using Polynomials

TL;DR

This paper introduces a polynomial-based method to efficiently compute the reachable set of spacecraft after a single impulse under arbitrary dynamics, applicable to complex space mission scenarios.

Contribution

It extends reachable set analysis to arbitrary spacecraft dynamics using polynomial envelopes and reduces computational complexity with a local polynomial approximation.

Findings

Successfully applied to near-rectilinear halo orbits in cislunar space

Achieved less than 0.0658% relative error in RS approximation

Reduced envelope equation solving time by over 84%

Abstract

This paper presents a method to determine the reachable set (RS) of spacecraft 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 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 boundary of RS 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 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%.