An analytic solution of full-sky spherical geometry for satellite relative motions

Herein, an exact and efficient analytic solution for an unperturbed satellite relative motion was developed using a direct geometrical approach. The derivation of the relative motion geometrically interpreted the projected Keplerian orbits of the satellites on a sphere (Earth and celestial spheres) using the solutions of full-sky spherical triangles. The results were basic and computationally faster than the vector and plane geometry solutions owing to the advantages of the full-sky spherical geometry. Accordingly, the validity of the proposed solution was evaluated by comparing it with other analytic relative motion theories in terms of modeling accuracy and efficiency. The modeling accuracy showed an equivalent performance with Vadali’s nonlinear unit sphere approach, which is essentially equal to the Yan–Alfriend nonlinear theory. Moreover, the efficiency was demonstrated by the lowest computational cost compared with other models. In conclusion, the proposed modeling approach illustrates a compact and efficient closed-form solution for satellite relative motion dynamics.

Reachable domain for satellite relative motion along elliptic orbit with uncertainty and process noise

For satellite close-range relative motion under uncertainty and process noise, the relative reachable domain is the confidence region, upon providing a user-defined probability level, of the occurrence of a flying trajectory. The relative reachable domain is a risk-assessing tool used to prevent inadvertent collisions. This study conducted a follow-up relative reachable domain investigation with two improvements: (1) the reference orbit of the close-range relative motion is generalized to an arbitrary elliptic orbit and (2) the process noise during the relative motion is incorporated. The equation of the relative reachable domain envelope, the boundary of the relative reachable domain, is derived. An algorithm is proposed to solve for the relative reachable domain envelope. By comparing the solved relative reachable domain envelopes with the results of a direct Monte Carlo simulation, the accuracy of the proposed algorithm is verified. Moreover, a relative reachable domain-based collision detection method is presented to screen out encounters that have a low chance of collision. The proposed collision detection method exhibits efficiency and reliability in deciding whether a passive flyby mission is safe or not.