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

An alternative framework for arbitrary higher-order methods named Hermite-Legendre-Gauss-Lobatto (HLGL) direct transcription in trajectory optimization, suitable for implementation on digital computers and in a reusable form, is proposed. The method uses Hermite interpolating polynomials to approximate the state equations, which employ the Legendre-Gauss-Lobatto points as interpolation and collection points. The discrete approximation to the costates for the HLGL discretization is obtained at the collocation points from the corresponding Lagrange multipliers. This approach is a direct extension to higher orders of the Hermite-Simpson and a fifth-order method proposed earlier. The approach is found to offer possibilities for improving mesh refinement procedures by choosing different polynomial orders within particular regions, in combination with increasing the number of intervals.