Closed form parametric solutions of nonlinear Abel-type and Riccati-type spacecraft relative motion

In comparison to the conventional approach of using Cartesian coordinates to describe spacecraft relative motion, the relative orbit description using Keplerian orbital elements provides a better visualization of the relative motion due to the benefit of having only one term (anomaly) that changes with time out of the six orbital elements leading to the reduction of the number of terms to be tracked from six, as in the case of Hill coordinates, to one. In this paper, under certain assumptions and transformations, the spacecraft relative equations of motion, in terms of orbital element differences, is approximated into the nonlinear first kind Abel-type and Riccati-type differential equations. Furthermore, we present methodologies for the formulation of the close form analytical solutions of the approximated equations. As shown by the numerical simulations, the closed form solutions and the nonlinear equations are in conformity with Riccati-type equations having higher errors than the Abel-type equations. This shows that the Abel-type equation, a third order polynomial, approximated the relative motion better than the Riccati-type equation, a second order polynomial. The resulting new analytical solutions gave better insight into the relative motion dynamics and can be used for the analysis of spacecraft formation flying, proximity and rendezvous operations.