Power-Law Truncation Correction for the Relative Orbital Element State Transition Matrix in Active Debris Removal

In active debris removal missions in low Earth orbit, the semi-major axis difference between a service platform and its target can be large. Analytical relative dynamics models used in formation-flying applications typically retain only the first-order expansion in the orbital element differences; at large separations, the discarded higher-order terms—particularly the power-law dependence on the semi-major axis—introduce systematic along-track drift that degrades the propagation accuracy. This paper derives the power-law truncation correction, a closed-form additive vector that exactly compensates the truncated semi-major-axis power-law remainder, together with a consistent Jacobian correction for the extended Kalman filter covariance prediction. The state dimension and state transition matrix structure remain unchanged. Propagation tests over semi-major axis differences of 36–146 km yield ten-revolution terminal position errors of 0.008–0.065 km for the corrected models, compared with tens to hundreds of kilometers for the uncorrected first-order models and 0.1–8 km for the second-order state transition tensor. In 500-run Monte Carlo angles-only filtering experiments, the corrected filter reduces the median terminal position error by one to nearly three orders of magnitude relative to the uncorrected model. A process noise sensitivity study confirms robustness to calibration uncertainty across two orders of magnitude at a lower computational cost and with simpler implementation than higher-order tensor methods.