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.