Optimal splicing of multi-segment analytical trajectories for electric sails

Abstract

In this paper, the analytical approximation, and the corresponding modified solution of the two-dimensional propelled trajectory of the electric sail with fixed pitch angle are spliced for the optimal design of the transfer orbit. Based on the linear perturbation theory, the analytical approximate propelled trajectory of the electric sail launched from a general heliocentric elliptical orbit can be obtained. Interestingly, this semi-analytic solution is no longer limited to the circular orbit assumption. Therefore, the optimal transfer trajectories between different heliocentric orbits can be obtained by splicing the analytical propelled trajectories with different pitch angles. Significantly, for such a problem that requires a large amount of iterative computation, compared to numerical integration, the analytical solution can greatly reduce the calculation time. In addition, the method of using pitch angles as the main optimization variables can greatly improve optimization efficiency. Moreover, based on the high accuracy of the modified analytical approximation, the optimal trajectory obtained by the proposed method is extremely close to the optimal real solution obtained by the shooting method and numerical integration. For the electric sail with a characteristic acceleration of 0.03 mm/s2, the approximation error of the optimal trajectories in the multi-target mission obtained by the proposed method and the integral shooting method is as low as 10−6∼10−4. For multiple variable targets, the difference of indicators is approximately 10−5∼10−4. Furthermore, compared with numerical integration, the proposed analytical method can reduce the calculation time of multi-target optimization problems by at least 80 %. Notably, with extremely low accuracy loss and calculation time, the proposed method is applicable to the rapid design of optimal transfer trajectory of electric sail-based spacecraft in multi-target mission scenarios.