The homotopy method has long served as a useful tool in solving optimal control problems, particularly highly nonlinear and sensitive ones for which good initial guesses are difficult to obtain, such as some of the well-known problems in aerospace trajectory optimization. However, the traditional homotopy method often fails midway: a fact that occasional practitioners are not aware of, and a topic which is rarely investigated in aerospace engineering. This paper first reviews the main reasons why traditional homotopy fails. A new double-homotopy method is developed to address the common failures of the traditional homotopy method. In this approach, the traditional homotopy is employed until it encounters a difficulty and stops moving forward. Another homotopy originally designed for finding multiple roots of nonlinear equations takes over at this point, and it finds a different solution to allow the traditional homotopy to continue on. This process is repeated whenever necessary. The proposed method overcomes some of the frequent difficulties of the traditional homotopy method. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional low-thrust orbital transfer problem are presented to illustrate the applications of the method.

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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.

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