Refinements to the Q-law for low-thrust orbit transfers

Low-thrust orbit transfers around a central body are considered, where specified changes are sought in the orbit elements except true anomaly. The desired changes in the five elements can be arbitrarily large. The Q-law is a Lyapunov feedback control law developed by the author, based on analytic expressions for maximum rates of change of the orbit elements and the desired changes in the elements. Q, the proximity quotient, serves as a candidate Lyapunov function. Three refinements to the Q-law are presented here. First, the concept of relative effectivity, rather than absolute, is introduced for deciding whether to thrust or to coast at any particular point on the transfer. Second, a mechanism is introduced to avoid chatter in the thrust direction when the spacecraft is very near the target orbit and is in an unfavourable location on the osculating orbit. Third, for changing the argument of periapsis, the beneficial effect of out-of-plane thrust, particularly as inclination approaches 0 or 180 degrees, is better utilised. The first two refinements are primarily of use for circle-to-circle transfers, while the latter refinement is of use in orbit transfers involving changes in the argument of periapsis. Two sample orbit transfers demonstrate the utility of the refinements. As before, the Q-law permits a rapid evaluation of the trade-off between propellant mass and flight time, provides reasonable estimates of the flight path and performance of optimal orbit transfers, and also serves as a mechanism for recovering from flight-path disturbances.