Two-impulse Rendezvous Planning about Thrusting Spacecraft on SE2(3)

The classical Hill--Clohessy--Wiltshire equations assume an unforced Keplerian reference trajectory, an assumption that is violated by missions requiring continuous thrust. We formulate spacecraft relative motion on the $\mathrm{SE}_2(3)$ Lie group, which encodes position, velocity, and attitude in a single geometric state, linearize the gravity and body-frame control mismatches between the two vehicles, and derive analytic bounds on every neglected term. Under circular coasting Keplerian assumptions the framework recovers the Hill--Clohessy--Wiltshire equations exactly. For thrusting references the state transition matrix acquires thrust-induced attitude--translation coupling blocks that, when the two vehicles share body-frame inputs, are exact in log coordinates and independent of the attitude error, and a mismatch between the vehicles' inputs re-centers the linearization at their mean. A two-impulse rendezvous planner follows directly. A scalar coupling ratio, computable before planning, separates two regimes. Where it exceeds unity, the planner reduces the terminal error of coupling-blind planners by more than an order of magnitude, and somewhat above unity it is also more accurate than a Cartesian planner of the same computational structure; well below unity, simpler planners are adequate or better. Simulations confirm the recovery to machine precision and validate every bound along the transfer.