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

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),N@?Z^+. The state vector and its derivatives up to the (N-1)th-order derivative can then be stably recovered by successive integration. Moreover, we present our solution method for solving linear-quadratic regulator (LQR) problems as we aim to cover a wider collection of CTOCPs with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature. The proposed numerical scheme fully parameterizes the state and the control variables using Gegenbauer expansion series. For problems with various order time derivatives of the state variables arising in the cost function, dynamical system, or path/terminal constraints, the GTM seeks to fully parameterize the control variables and the highest-order time derivatives of the state variables. The time horizon is mapped onto the closed interval [0,1]. The dynamical system characterized by differential equations is transformed into its integral formulation through direct integration. The resulting problem on the finite interval is then transcribed into a nonlinear programming (NLP) problem through collocation at the Gegenbauer-Gauss (GG) points. The integral operations are approximated by optimal Gegenbauer quadratures in a certain optimality sense. The reduced NLP problem is solved in the Gegenbauer spectral space, and the state and the control variables are approximated on the entire finite horizon. The proposed method achieves discrete solutions exhibiting exponential convergence using relatively small-scale number of collocation points. The advantages of the proposed direct GTM over other traditional discretization methods are shown through four well-studied OC test examples. The present work is a major breakthrough in the area of computational OC theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points, states and controls expansion terms. Moreover, the GTM produces very small-scale NLP problems, which can be solved very quickly using the modern NLP software.