Two-impulse Rendezvous Planning about Thrusting Spacecraft on $\mathrm{SE}_2(3)$

Abstract

The classical Hill--Clohessy--Wiltshire equations assume an unforced Keplerian reference trajectory, an assumption that is violated by missions requiring continuous thrust. We address this limitation with a relative motion framework on the $\mathrm{SE}_2(3)$ Lie group that encodes position, velocity, and attitude in a unified geometric state. For computational tractability we linearize both the gravity mismatch and the body-frame control mismatch between the two vehicles, deriving tight analytic upper bounds on the neglected higher-order terms in each case. Under circular coasting Keplerian assumptions the framework recovers the Hill--Clohessy--Wiltshire equations exactly, establishing classical rendezvous theory as a special case rather than an independent linearization. For thrusting reference trajectories the state transition matrix acquires off-diagonal attitude--translation coupling blocks absent from classical formulations, and absorbing the control mismatch re-centers the linearization at the mean of the two vehicles' inputs. A two-impulse rendezvous planner derived directly from the state transition matrix accounts for both effects. Numerical simulations confirm recovery of the classical equations to machine precision, demonstrate successful rendezvous about a thrusting reference where classical planners fail, and validate the gravity and control mismatch linearization bounds throughout the transfer.