Navigation Solution to Maneuver a Spacecraft Relative to a Sphere Centered on a Cooperative Satellite

• Matrix algorithm to solve for constrained maneuver relative to another satellite.
• Presented results with a phasing maneuver tailored by the user׳s application.
• Determined empirical solution for optimum ΔVΔV based on semi-major axis ratio.
• Demonstrated use of navigation algorithm for relative spacecraft motion.

A differential correction algorithm is presented to deliver an impulsive maneuver to a satellite to place it within a sphere, with a user defined radius, centered around a non-maneuvering satellite within a constrained time. The differential correction algorithm develops and utilizes the State Transition Matrix along with the Equations of Motion and multiple satellite׳s state information to determine the optimum trajectory to achieve the desired results. The results from the differential correction algorithm are very accurate for prograde orbits, as presented. The results allow for orbit design trade-offs, including satellites׳ initial inclinations, semi-major axes, as well as the ballistic coefficients. The results also provide an empirical method to determine the optimum ΔVΔV solution for the provided problem. Understanding that the minimum fuel solution lies with a semi-major axis ratio of 1, a very accurate empirical approximation is presented for semi-major axis ratio values less than and greater than 1. This work ultimately provides the generalized framework for applying the algorithm to a unique user defined maneuvering spacecraft scenario.