OPTIMAL CONTROL OF A CIRCULAR SPACECRAFT FORMATION SUBJECT TO GRAVITATIONAL PERTURBATION

摘要

In recent years, spacecraft formations have been studied extensively as platforms for distributed sensor systems. The linearized equations of motion allow a natural circular formation which would appear particularly well-suited for Earth-sensing applications. Unfortunately, this formation is not stable when gravitational perturbations are considered. Although it may be possible to design workable sensor systems that work with the natural motion, a better understanding of the optimal fuel cost of maintaining the circular formation will enable more informed decisions in making the system trade-offs between orbit control requirements and sensor system requirements. This paper develops the optimal open-loop control for a deputy satellite to maintain a circular formation about a chief satellite. The equations of motion for the two spacecraft formation include the nonlinear two-body and J_2 gravitational terms. The chief satellite is assumed to be in a circular, inclined orbit. The desired relative motion of the deputy satellite is a circular path around the chief, with an angular rate equal to the mean motion of the primary orbit. The optimal trajectory is found using a traditional approach based on the calculus of variations for a continuous nonlinear system. This approach requires the numerical solution of the two-point boundary value problem, in which convergence to a solution is notoriously dependent upon the choice of boundary conditions for the states and costates. It is shown that the using initial states for a circular formation as derived analytically from linearized equations provides reasonable results. For an 800-km orbit, the delta-v required to maintain the circular formation is dependent upon exact initial conditions and error tolerances, but are approximately 40 m/s/year.