Practical Techniques for Low-Thrust Trajectory Optimization with Homotopic Approach

This paper concerns the application of the homotopic approach, which solves the fuel-optimal problem of low-thrust trajectory by starting from the related and easier energy-optimal problem. To this end, some effective techniques are presented to reduce the computational time and increase the probability of finding the globally optimal solution. First, the optimal control problem is made homogeneous to the Lagrange multipliers by multiplying the performance index by a positive unknown factor. Hence, normalization is applicable to restrict the unknown multipliers on a unit hypersphere. Second, the switching function's first- and second-order derivatives with respect to time are derived to detect switching. The switching detection is embedded in the fourth-order Runge-Kutta algorithm with fixed step size to ensure. integration accuracy for bang-hang control. Third, combined with the techniques of normalization and switching detection, the particle swarm optimization with well-chosen parameters considerably increases the probability of finding the approximate initial values of the globally optimal solution. Moreover, intermediate gravity assist, which brings complex inner constraints, is considered. To determine the approximate gravity assist date, analytical formulas are presented to evaluate the minimal maneuver impulse based on the results of Lambert problems. The first-order necessary conditions for gravity assist constraints are derived analytically. The optimal solution can be rapidly obtained by applying the techniques presented to solve the shooting function. The unknowns are far less than with direct methods, and the computational effort is also far lower. Two examples of fuel-optimal rendezvous problems from the Earth directly to Venus and from the Earth to Jupiter via Mars gravity assist are given to substantiate the perfect efficiency of these techniques.