Path-constrained trajectory optimization using sparse sequential quadratic programming

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. T is well known that the solution of an optimal control or trajectory optimization problem can be posed as the solution of a two-point boundary value problem. This problem requires solving a set of nonlinear ordinary differential equations; the first set defined by the vehicle dynamics and the second set (of adjoint differential equations) by the optimality conditions. Boundary conditions 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 utilized1'5 for applications without path constraints. Since the approach requires 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 to the accuracy of the adjoint guess. Finally, the adjoint variables may be discontinuous when the solution enters or leaves an inequality path constraint. Difficulties associated with adjoint equations are avoided by the direct transcription or collocation methods.6'10 In this approach, the dynamic equations are discretized, and the optimal control problem is transformed into a nonlinear program, which can be solved directly. The nonlinear programming problem is large and sparse and a method for solving it is presented in Ref. 7. This paper extends the method of Ref. 7 to efficiently handle inequality constraints and presents a nonlinear programming algorithm designed to exploit the properties of the problem that results from direct transcription of the trajectory optimization application.