Single-Impulse Reachable Set in Arbitrary Dynamics Using Polynomials

TL;DR

The paper addresses the single-impulse RS problem under arbitrary dynamics, extending RS analysis beyond two-body models. It introduces a framework that combines differential algebra (DA) and automatic domain split (ADS) to produce high-order polynomial maps of impulse parameters, and uses envelope theory to extract RS boundaries. A central contribution is the ensemble of methods (partial map inversion and Newton's iteration) to project RS onto a low-dimensional plane, plus a framework to solve the envelope equation efficiently, including a local polynomial approximation to accelerate boundary prediction. The approach is demonstrated on cislunar CRTBP with NRHO trajectories, achieving RS envelope accuracy better than $0.0658\%$ relative error and reducing envelope-computation time by over 84\% through local polynomial pieces; it also highlights the flexibility to obtain RS boundaries in both state and observation spaces under highly nonlinear dynamics.

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%.

Figures (30)

• Figure 1: An illustration of two computational frameworks Valli2013.
• Figure 2: An illustration of ADS Wittig2015.
• Figure 3: An illustration of the auxiliary plane.
• Figure 4: Framework for solving the polynomial of the envelope equation.
• Figure 5: Analytical pieces for solving the envelope equation.