Optimal Low-Thrust Rendezvous Control

A study is made of the coplanar rendezvous maneuver between a thrusting vehicle and a passive vehicle in an elliptic orbit. In the terminal phase of rendezvous, where the equations of relative motion can be linearized, an analytic solution for the fuel optimal thrust program is found for power-limited vehicles when the time to maneuver is fixed. The calculations are carried out in polar coordinates where the solution is found to be closer to the exact nonlinear motion than the solution in Cartesian coordinates used in previous analyses. The optimal power-limited thrust program for a practical example is compared to the exact numerical solution obtained using a Newton-Rhapson algorithm. I. Introduction T HE problem of rendezvous between a maneuvering space vehicle and a nonmaneuvering target has received considerable attention in recent years. The first investigations were of the free motion using linearized equations1'2 and perturbation theory to take into account nonlinear effects.3"5 Fuel optimal maneuvers have been studied to find both the structure of these maneuvers and the absolute bound on fuel consumption for comparisons with suboptimal schemes used in practice because of severe mission restrictions. As in general orbital transfer theory, the optimal control problem has been studied for both low-thrust (power-limited) and high-thrust, constant-exhaust velocity propulsion systems. Of particular importance has been the terminal rendezvous maneuver when the vehicles are relatively close together. Tschauner and Hempel6 and Tschauner7 have presented some analytical results for high-thrust rendezvous for both circular and elliptic target orbits and Paiwonsky and Woodrow8 and Goldstein, Greene, and Johnson9 have presented numerical results for the time optimal problem. For the low-thrust case, Edelbaum10 has found solutions for small changes in all orbital parameters when the thrust is of the same magnitude as the small orbital changes. Billik11 has derived the terminal low-thrust optimal control for circular target orbits and Gobetz12 has done the same for elliptic target orbits using linearized equations in Lagrange's planetary variables. Numerical studies of optimal low-thrust planetary transfer trajectories have been carrried out by Melbourne and Sauer.13 In this study, we shall extend the work of Billik to elliptic target orbits and complement the work of Gobetz, where the linearizing assumptions are different. Also, another new set of coordinates is used such that the solution to the linearized equations better approximates the nonlinear motion. The solution to the homogeneous linear equations of motion2 for elliptic target orbits permits the reduction of the low-thrust rendezvous optimal control problem to quadrature. When the eccentricity is small, a power series expansion can be made in this parameter to evaluate the resulting definite integrals.