Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization

Abstract

D IRECT methods have been widely applied for solving trajectory optimization problems [1–5]. Among the popular methods are the Hermite–Simpson method [6] and the Legendre pseudospectral (PS)method [7,8]. There has been some considerable interest in developing theory related to the Legendre PS method due to its high accuracy, although the Hermite–Simpson method continues to be applied to large-scale practical problems [9]. The main characteristic of the Hermite–Simpson method is the combination of reasonable accuracy with a highly sparse constraint Jacobian and Hessian matrix [10]. The PS method offers impressive accuracy (spectral accuracy) for smooth problems, but the constraint Jacobians are much denser than other methods. The sparsity of the constraint Jacobians can be increased in the PS method by using knots [11]. In addition to the Hermite–Simpson method, additional highorder methods have been proposed by Herman and Conway [2]. These methods are attractive from the point of view of accuracy, but in the framework proposed by Herman and Conway, they require detailed derivation when extended to arbitrary higher orders. For instance, in [2], the form of the constraints was derived via the symbolic manipulation software MAPLE. The purpose of this Note is to provide an alternative framework for arbitrary higher-order methods suitable for implementation on digital computers and in a reusable form. The optimal control problem is approximated by a discrete nonlinear programming problem (NLP) by expanding the state trajectories using local Hermite interpolating polynomials. For high accuracy, the collocation points are selected from the family of Gauss–Lobatto points. This also allows the integral performance index to be approximated viaGauss– Lobatto quadrature rules. For optimal control problems of the Bolza form, the natural choice of quadrature points are the Legendre– Gauss–Lobatto (LGL) points, because they are derived on the basis of a unity weight function, giving the highest accuracy for polynomial integrands. The generalization of the approach is referred to as the Hermite–Legendre–Gauss–Lobatto (HLGL) approach throughout the remainder of this Note.