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A class of methods for the numerical solution of optimal control problems is analyzed and applied to the optimization of finite-thrust spacecraft trajectories. These methods use discrete approximations to the state and control histories, and a discretization of the equations of motion to derive a mathematical programming problem which approximates the optimal control problem, and which is solved numerically. This conversion is referred to as "transcription." Transcription methods were developed in the sixties and seventies; however, for nonlinear problems, they were used in conjunction with a "state-elimination" procedure which propagated the integration forward over the trajectory, and resulted in a relatively unconstrained programming problem which was easier to handle. Recent advances in nonlinear programming, however, have made it feasible to solve the original heavily-constrained nonlinear programming problem, which is referred to as the "direct transcription" of the optimal control problem. This method is referred to as "direct transcription and nonlinear programming."
A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables and enforces the equations of motion via a collocation procedure, resulting in a nonlinear programming problem, which is solved numerically. This method is identified as being of the general class of direct transcription methods described above. Also, a new direct transcription method which discretizes the equations of motion using a parallel-shooting approach is developed. For both methods, the relationship between the original optimal control problem and the approximating nonlinear programming problem is investigated by comparing the optimal control necessary conditions with the optimality conditions for the discretized problem.
Both methods are applied to thrust-limited spacecraft trajectory problems, including finite-thrust transfer, rendezvous, and orbit insertion, a low-thrust escape, and a low-thrust Earth-moon transfer. The basic methods have been modified to accurately model discontinuities in the optimal control, to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., coast arcs of the two-body problem, and to allow the simultaneous use of several coordinate systems.