Optimal two-impulse rendezvous using constrained multiple-revolution Lambert solutions

A solution to the fixed-time minimum-fuel two-impulse rendezvous problem for the general non-coplanar elliptical orbits is provided. The optimal transfer orbit is obtained using the constrained multiple-revolution Lambert solution. Constraints consist of lower bound for perigee altitude and upper bound for apogee altitude. The optimal time-free two-impulse transfer problem between two fixed endpoints implies finding the roots of an eighth order polynomial, which is done using a numerical iterative technique. The set of feasible solutions is determined by using the constraints conditions to solve for the short-path and long-path orbits semimajor axis ranges. Then, by comparing the optimal time-free solution with the feasible solutions, the optimal semimajor axis for the two fixed-endpoints transfer is identified. Based on the proposed solution procedure for the optimal two fixed-endpoints transfer, a contour of the minimum cost for different initial and final coasting parameters is obtained. Finally, a numerical optimization algorithm (e.g., evolutionary algorithm) can be used to solve this global minimization problem. A numerical example is provided to show how to apply the proposed technique.