Optimal solar sail trajectory approximation with finite Fourier series

The heliocentric transfer of a solar sail-based spacecraft is usually studied from an optimal perspective, by looking for the control law that minimizes the total flight time. The optimal control problem can be solved either with an indirect approach, whose solution is difficult to obtain due to its sensitivity to an initial guess of the costates, or with a direct method, which requires a good estimate of a feasible (guess) trajectory. This work presents a procedure to generate an approximate optimal trajectory through a finite Fourier series. The minimum time problem is solved using a nonlinear programming solver, in which the optimization parameters are the coefficients of the Fourier series and the positions of the spacecraft along the initial and target orbits. Suitable constraints are enforced on the direction and magnitude of the sail propulsive acceleration vector in order to obtain feasible solutions. A comparison with the numerical results from an indirect approach shows that the proposed method provides a good approximation of the optimal trajectory with a small computational effort.