A quadratic homotopy method for fuel-optimal low-thrust trajectory design

Solving fuel-optimal low-thrust trajectory problems is a long-standing challenging topic, mainly due to the existence of discontinuous bang–bang controls and small convergence domain. Homotopy methods, the principle of which is to embed a given problem into a family of problems parameterized by a homotopic parameter, have been widely applied to address this difficulty. Linear homotopy methods, the homotopy functions of which are linear functions of the homotopic parameter, serve as useful tools to provide continuous optimal controls during the homotopic procedure with an energy-optimal low-thrust trajectory optimization problem as the starting point. However, solving energy-optimal problem is still not an easy task, particularly for the low-thrust orbital transfers with many revolutions or asteroids flyby, which is typically solved by other advanced numerical optimization algorithms or other homotopy methods. In this paper, a novel quadratic homotopy method, the homotopy function of which is a quadratic function of the homotopic parameter, is presented to circumvent this possible difficulty of solving the initial problem in the existing linear homotopy methods. A fixed-time full-thrust problem is constructed as the starting point of this proposed quadratic homotopy, the analytical solution of which can be easily obtained under a modified linear gravity approximation formulation. The criterion of energy-optimal problem is still involved in the homotopic procedure to provide continuous optimal controls until the original fuel-optimal problem is solved. Numerical demonstrations in an Earth to Venus rendezvous problem, a geostationary transfer orbit (GTO) to geosynchronous orbit (GEO) orbital transfer problem with many revolutions, and an Earth to Mars rendezvous problem with an asteroid flyby are presented to illustrate the applications of this proposed homotopy method.