This paper presents a power-limited low-thrust optimal guidance law for transfer in a central gravitational field. This guidance law is obtained analytically as a particular solution to the case of transfer with free final time. The optimal thrust acceleration is shown to be colinear with the vehicle velocity vector multiplied by a time-varying gain that depends on a constant-guidance parameter. An analysis is performed of the system trajectories as obtained when this guidance law is applied. This guidance law is applied to the case of Earth-to-moon transfer. The initial condition is a low circular Earth orbit. For the final condition two cases are considered: 1) impact on the moon and 2) moon orbit injection. The solution for both cases is obtained by considering the gravity field of one body at a time. The initial Earth-centered trajectory consists of a number of revolutions during which the trajectory spirals away gradually from Earth. In the impact case, by adequately choosing the guidance parameter, it is possible to achieve, at a prescribed range, a moon-relative velocity such that the moon gravitational field will capture the spacecraft. For moon orbit injection, the trajectory is obtained by matching the outward Earth-spiral trajectory with a moon inward spiral. Both spirals are generated by the guidance law with the guidance parameter positive for the Earth outward spiral and negative for the moon inward one.
[1]
M. Kaplan.
Modern Spacecraft Dynamics and Control
,
1976
.
[2]
George Leitmann,et al.
The Calculus of Variations and Optimal Control
,
1982
.
[3]
Jean Pierre Marec,et al.
Optimal Space Trajectories
,
1979
.
[4]
M. Guelman,et al.
Power limited soft landing on an asteroid
,
1994
.
[5]
F. Gobetz.
Errata: "Optimal Variable-Thrust Transfer of a Power-Limited Rocket between Neighboring Circular Orbits"
,
1964
.
[6]
George Leitmann,et al.
Optimization with Vector — Valued Cost
,
1981
.
[7]
W. G. Melbourne,et al.
OPTIMUM INTERPLANETARY RENDEZVOUS WITH POWER-LIMITED VEHICLES
,
1963
.