Optimal power-limited rendezvous for linearized equations of motion

The form of solution of the optimal power-limited rendezvous problem for linear equations of motion is made to conform to previous developments by the author. The solution is then applied to the problem of rendezvous of a spacecraft with an object in the vicinity of a nominal Keplerian orbit. For the case in which the nominal orbit is circular, the controllability matrix can be inverted symbolically, resulting in a closed-form solution for the thrusting function. This result can be expressed as a closed-loop feedback law. For noncircular nominal orbits, a feedback law can be approximated by repeated numerical matrix inversions and reinitialization of the problem. PTIMAL power-limited terminal rendezvous of a spacecraft based on linearized equations of motion offers the mathematical advantage of closed-form or nearly closed-form solutions of the linearized trajectory optimization problem. In many cases this leads to thrust control in the form of optimal feedback laws, or at least, control functions that can be easily reinitialized on a regular basis. This gain in adaptivity may, to some extent, compensate for the loss in accuracy due to the linearization. There are many potential applications of a linearized analysis of optimal power-limited rendezvous. These typically involve the maneuvering of a low-thrust spacecraft near a real or fictitious object in a nominal orbit. Examples include maneuvers in the vicinity of artificial satellites, space stations, small comets, or asteroids. Examples where the object is fictitious include low-thrust satellite station keeping or interplanetar y missions in which the usual high-thrust midcourse maneuver is replaced by a continuous long-term lowthrust correction toward the nominal orbital position and velocity. Early work on the computation of power-limited , variable exhaust-velocity spacecraft trajectories was performed by Irving,1 Ross and Leitman,2 Edelbaum,3 Saltzer and Fetheroff,4 Melbourne,5 and Melbourne and Sauer.6'7 Immediately after these initial studies, investigators began to look at various linear models.8"12 Billik8 and Gobetz9 used linearization about nominal circular orbits, Edelbaum10 used linearization with respect to the orbital parameters of an ellipse, Gobetz11 linearized about an ellipse of low eccentricity, and Euler12 used the Tschauner-Hempel13'14 linearization for general elliptical orbits. More general discussions that include linear and nonlinear models for both power-limited and constant-exhaust-velocity problems can be found in the works of Edelbaum15 and Marec.16 Recently, optimal power-limited low-thrust trajectories to return to an original station in orbit after an impulsive maneuver away from this orbit were examined by Lembeck and Trussing17 using equations linearized about a circular orbit. The same linearized equations were used by Coverstone-Carroll and Trussing18 to investigate the cooperative power-limited rendezvous of two spacecraft near a nominal circular orbit. Power-limited trajectories in an inverse-square gravity field have also been considered by Prussing.19 In the present paper, we consider the fixed-time linear, powerlimited rendezvous problem. We utilize a somewhat more general cost function than usual, an integral of the product of a weighting