Continuous-Thrust Reachable Set for Linear Relative Motion Near Elliptical Orbits

Continuous-thrust relative reachable set is presented for the spacecraft linear relative motion near elliptical orbits. Two cases with different constraints are considered: the energy-constrained case and the fuel-constrained case. Based on the distance-fields-over-grids method, the envelopes of the relative reachable sets for both cases are approximated by a number of grid points on it, and these grid points are obtained from a series of optimization problems. Based on the continuous-thrust analytical state propagation, the original optimization problems for both cases are converted into the nonlinear programming problems. Furthermore, for the energy-constrained case, the analytical solution is obtained by using the Lagrange multiplier method; whereas for the fuel-constrained case, the problem is converted into a mixed-integer linear programming problem to rapidly obtain solutions. Several numerical examples are provided to verify that the developed methods for solving the spacecraft relative reachable sets are feasible and computationally efficient.