Riemannian optimization for spacecraft trajectory design

Many of the equations of motion appearing in the aerospace field are nonlinear, and the problem of input optimization under this equation of motion is important. There are two methods for solving the nonlinear optimal control problem: direct method and indirect method. In the direct method, the equation of motion is discretized, and the problem is solved as a nonlinear programming problem with motion equations as constraints. In the direct method, it is possible to solve the problem by adding various constraints, but the solution becomes complicated and it is difficult to guarantee the convergence to the optimal solution. In this study, we consider a set of unknowns that satisfy constraints as Riemannian manifolds, and treat the problem as an unconstrained optimization problem on Riemannian manifolds. A simplified rocket trajectory optimization problem illustrates the proposed method.