Simple analytic solution to the optimal reconfigurations in general relative motions

The current paper presents and examines simple analytic solutions to the fuel-optimal reconfiguration problem of multiple satellites governed by various relative equations of motion. The problem is addressed by solving a standard optimal control problem for a linear time-varying system. This paper shows that the optimal thrust vector is directly proportional to the fundamental matrix associated with the given state equations, and other quantities such as the cost function and the state vector during the reconfiguration can be concisely represented as well, if two basic assumptions are met. These two assumptions are very common due to the fact that most relative motion equations are represented in the LVLH frame. The method allows predicting the explicit form of optimal solutions in advance without having to solve the problem and we only need to determine coefficients to satisfy the boundary conditions. A numerical simulator is employed to confirm the brevity and the accuracy of the general analytic solutions developed in the current paper.