Scenario-Dependent Closed-Loop Validity of the Clohessy-Wiltshire Model for Proportional-Derivative-Controlled Low Earth Orbit Rendezvous
Advik Sushanth
25/09/2026
Space rendezvous and docking are critical for orbital debris removal, deep-space exploration, and in-orbit refueling. The Clohessy-Wiltshire (CW) model is widely employed for Low Earth Orbit (LEO) rendezvous due to its simplicity in estimating the relative motion of a chaser spacecraft. However, most existing studies assess the accuracy of the CW model in open-loop motion, leaving its performance under continuous feedback largely unexplored. While model discrepancies have the potential to accumulate during open-loop propagation, it remains unclear if these simplified models remain reliable during actively controlled maneuvers. To address this gap, this study propagates CW and nonlinear trajectories independently using an identical Proportional-Derivative (PD) controller and gain set. The analysis was conducted across various initial separations and offset directions, quantifying the maximum trajectory difference over a single orbit. By utilizing a two-body nonlinear reference, the linearization error was isolated, while the effects of J2 perturbations and atmospheric drag were investigated separately. Validation via NASA’s General Mission Analysis Tool (GMAT) demonstrated a discrepancy of less than 1 μm in two-body relative motion. Results indicate that radial trajectories remained within 1 m for initial separations up to 5.5 km under feedback, compared to 95 m in free drift. The corresponding closed-loop thresholds were approximately 7.7 km along-track and 10 km cross-track, as opposed to the 600 m and 850 m in free drift. These thresholds are scenario-specific and shift with the selected error criterion, available thrust, controller bandwidth and radial energy matching, and must not be interpreted as universal CW validity bounds. The findings reveal that in the tested radial cases, available thrust became insufficient to counteract the induced along-track drift. Under the tested conditions, continuous feedback significantly extended the agreement between the linear and nonlinear models relative to free drift.