Path constrained trajectory optimization using sparse sequential quadratic programming

Abstract

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*

Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

“…As a result, we reduced the original problem to a nonlinear programming problem, meaning we obtained a problem of minimization for the scalar function of several (but not tens as in Refs. [24][25][26][27][28][29][30][31][32][33][34][35][36] variables:…”

“…Hargraves and Paris, 24 von Stryk and Bulirsch, 25 Convay and colleagues, 26,27 Betts, 28 Calise and Leung, 29 Hull, 30 and others used the so-called collocation-based or direct transcription methods, which are similar in many respects to Galerkin's procedure. They reduce the initial problem by segmenting the time interval into the 5-20 pieces and representing the solution both for state variables and controls by piecewise polynomials (constants).…”

Aircraft trajectory optimization using nonlinear programming

“…Hargraves et al (1981) present the state trajectories with high-order patched polynomials, whose coefficients are the decision variables. Betts and Huffman (1993) discuss trapetzoidal, HermiteSimpson and Runge-Kutta discretization. In the following we describe two schemes, direct collocation and a recently proposed method based on differential inclusions Seywald (1994).…”

A Hybrid Direct–Indirect Approach for Solving the Singular Optimal Control Problems of Finite and Infinite Order

“…The basic approach to solve the optimal control problem by Euler transcription has been presented in details in (Betts 1994;Betts and Huffman 1993). However, in what follows, we shortly recall Euler method for solving SOCP (1).…”