Spacecraft Relative Motion with Respect to a Spinning Chief Body Frame

Relative motion between orbiting spacecraft is commonly modeled in the Hill frame due to its analytical first-order solutions and the intuitive geometry of the resulting trajectories. However, the Hill frame is not ideal for mission scenarios involving constraints defined in the body frame of a spinning chief, such as collision avoidance during close-proximity operations or docking within a keep-in zone. This paper investigates relative motion from the perspective of the rotating body frame of the chief, focusing on three fundamental spin cases. In each case, the body frame is initially aligned with the Hill frame and undergoes constant rotation about one of the Hill frame principal axes: radial, along-track, and cross-track direction. Closed-form solutions to the Clohessy-Wiltshire equations are used to derive analytical expressions for the relative motion in the chief body frame, assuming circular chief orbits and small separation distances. The resulting trajectories are described using geometrically meaningful invariants of motion, providing intuitive insights into the trajectory shapes and locations. These trajectories are characterized as parametric epitrochoid or hypotrochoid curves. The analysis considers both bounded and drifting motion, and includes both resonant cases, where the spin rate equals the orbital rate, and non-resonant spin cases.