Hermite-Legendre-Gauss-Lobatto direct transcription methods in trajectory optimization

Among the popular methods for direct trajectory optimization are the Hermite-Simpson and Legendre pseudospectral methods. In this paper, a framework is developed for implementing arbitrary higher order direct methods that belong in the class of Hermite-Simpson methods. This class has been termed Hermite-Legendre-Gauss-Lobatto. In these methods, a Hermite interpolating polynomial is used to construct approximations to the state trajectories using a set of Legendre-Gauss-Lobatto (LGL) points. The state equations are enforced at an additional set of collocation points, belonging to a different set of LGL points. It is shown that the Lagrange multipliers obtained from the NLP do not suffer from defects at the boundaries, as in the standard Legendre pseudospectral method, and hence the multipliers provide excellent costate information.