Virtual Attractive-Repulsive Potentials Control Theory: A Review and an Extension to Riemannian Manifolds

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds.