Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method

Abstract

This paper reports a novel direct Gegenbauer (ultraspherical) transcription method (GTM) for solving continuous-time optimal control (OC) problems (CTOCPs) with linear/nonlinear dynamics and path constraints. In (Elgindy et al. 2012) [1], we presented a GTM for solving nonlinear CTOCPs directly for the state and the control variables, and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment. This article extends the GTM to deal further with problems including higher-order time derivatives of the states by solving the CTOCP directly for the control u(t) and the highest-order time derivative x(N)(t),NZ+. The state vector and i..