Application of a Finite-Difference Newton-Raphson Algorithm to Problems of Low-Thrust Trajectory Optimization

Abstract

A new numerical approach to the solution of nonlinear two-point boundary value problems is demonstrated with two applications to optimum low-thrust space trajectories. This approach combines the generalized Newton-Raphson iteration with an implicit finite-difference method for solving the associated linear two-point boundary value problems. Thus full advantage of the quadratic convergence of Newton's method is realized since each iteration involves a problem that is both stable and easily solved. With the use of this method, optimum Earth-Mars round-trip trajectories for the 1980 opposition with variable-thrust, power-limited propulsion have been computed. The existence of multiple stationary solutions and the effect of various initial approximations are demonstrated. Also, this method has been applied to low-thrust trajectories that have known hyperbolic excess velocity magnitudes at launch and arrival. The characteristics of such mixed systems are shown by comparison to both an all low-thrust trajectory and a Hohmann transfer. The computing time per trajectory typically ranges from 2 to 5 sec. on an IBM 7094, apparently offering significant savings over previous methods.