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One of the most effective numerical techniques for the solution of trajectory optimization and optimal control problems is the direct transcription method. This approach combines a nonlinear programming algorithm with a discretization of the trajectory dynamics. The resulting mathematical programming problem is characterized by matrices which are large and sparse. Constraints on the path of the trajectory are then treated as algebraic inequalities to be satisfied by the nonlinear program. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by the transcription formulation. Numerical experience is reported for trajectories with both state and control variable equality and inequality path constraints. nt roduct ion nary differential equations -the first set defined by the vehicle dynamics and the second set (of adjoint differential equations) defined by the optimality conditions. Boundary conditionz are imposed from the problem requirements as well as the optimality criteria. By discretizing the dynamic variables, this boundary value problem can be reduced to the solution of a set of nonlinear algebraic equations. This approach has been successfully utilized (cf. [I], [SI, [12], [13]: [24],) for applications without path constraints. Since the approach requires the adjoint equations it is subject to a number of difficulties. First, the adjoint equations are often very nonlinear and cumbersome to obtain for complex vehicle dynamics, especially when thrust and aerodynamic forces are given by tabular data. Second, the iterative procedure requires an initial guess for the adjoint variables, and this can be quite difficult because they lack a physical interpretation. Third, convergence of the iterations is often quite sensitive t o the accuracy of the adIt is well known that the solution of an op-joint guess. Finally, the adjoint variables may timal control or trajectory optimization probl em can be posed as ,.he solution of a two-be discontinuous when the solution enters or lem requires solving a set of nonlinear ordiDifficulties associated with adjoint equapoint boundary value problem. This p r o b leaves an inequality Path constraint*

One of the most effective numerical techniques for the solution of trajectory optimization and optimal control problems is the direct transcription method. This approach combines a nonlinear programming algorithm with discretization of the trajectory dynamics. The resulting mathematical programming problem is characterized by matrices that are large and sparse. Constraints on the path of the trajectory are then treated as algebraic inequalities to be satisfied by the nonlinear program. This paper describes a nonlinear programming algorithm that exploits the matrix sparsity produced by the transcription formulation. Numerical experience is reported for trajectories with both state and control variable equality and inequality path constraints.