Approximate initial Lagrange costates for continuous-thrust spacecraft

A method is presented to obtain approximate initial costate values and flight time for the optimal control of a continuous-thrust spacecraft on a coplanar, circle-to-circle transfer. The approximate initial costates are then used as starting values for the associated boundary-value problem to match the desired final states. The exact, nonlinear differential equations are integrated to solve the boundary-value problem with a shooting method. The approximate expressions for the initial costates and flight time are useful when the thrust acceleration is greater than or equal to the change in orbital radius, in canonical units. Numerical examples are provided for a geocentric and an Earth-Mars orbital transfer. EW propulsion technologies have raised interest in the space community for continuous-thrust orbital missions. In military applications, this could mean more responsive deployment of space assets and longer on-orbit lifetimes. Tactical repositioning of a satel- lite using chemical propulsion can consume large amounts of the available fuel mass per maneuver. This is certainly unacceptable for many reasons including lifetime, reliability, and cost. A space- craft propelled by high-efficiency thrusters could accomplish many more maneuvers for the same amount of fuel mass as a chemical propulsion system. Another interesting application of continuous-th rust propulsion is interplanetary space travel. A permanently orbiting space station could serve as a launch point for solar system exploration. Maintain- ing cryogenic fuels for this purpose, however, would be technically difficult and extremely costly. These problems would be less signif- icant if the orbiting space station were used as an assembly point for an exploration vehicle propelled by high-efficien cy continuous thrusters. Such a vehicle could be made reusable much more easily in terms of reliability and cost than a chemically propelled space- craft. Also, a vehicle using continuous thrust with existing technol- ogy could shorten travel times compared to fuel-optimal impulsive maneuvers, then could return to Earth orbit for reconditioning. Although the optimization of impulsive transfers has a direct solution,1 none has been found for the continuous-thrust case. This problem may be solved numerically, and many examples of this are to be found in the literature.2 Optimization of a continuous- thrust trajectory involves the simultaneous solution of an optimal control problem and a boundary-value problem. The initial and final states are normally known, but there is usually no infor- mation available for the initial values of the Lagrange costates. This presents quite a problem, since the optimal control law is often a function of the Lagrange costates that must be initialized for numerical integration. The usual approach is to make an edu- cated guess for the initial values, then update them by solving the boundary-value problem. Trussing,3 Broucke,4 and others2 have re- cast the boundary-value problem in terms of other variables, but the initial values of these must be guessed and refined as well. Prussing3 incorporated the second derivative of the primer vec- tor into a fourth-order dynamics equation, thus eliminating the control variables. Once this is accomplished, four constants of