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Journal of Vibration Testing and System Dynamics 1(4) (2017) 319--331 | DOI:10.5890/JVTSD.2017.12.003

Soufiane Haddout$^{1}$, Zhiyi Chen$^{2}$, Mohamed Ait Guennoun$^{1}$

$^{1}$ Department of Physics, Faculty of Science, Ibn Tofail University, B.P 242, 14000 Kenitra, Morocco

$^{2}$ University of British Columbia, 1935 Lower Mall, Vancouver, BC V6T 1X1 Canada

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Abstract

Mechanical systems have traditionally provided a fertile area of study for researchers interested in nonlinear control, due to the inherent nonlinearities and the Lagrangian structure of these systems. Recently, a great deal of emphasis has been placed on studying systems with nonholonomic constraints, including mobile wheeled robots and multiple-trailer vehicles, where the wheels provide a no-slip velocity constraint. In this paper, a new methods in non-holonomic mechanics are applied to a problem of an ackerman steered car motion for the first time. This method of the geometrical theory of general nonholonomic constrained systems on fibered manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupkov´a in 1990’s. The relevance of this theory for general types of nonholonomic constraints, not only linear or affine ones, was then verified on appropriate models. Frequently considered constraints on real physical systems are based on rolling without sliding, i.e. they are holonomic, or semi-holonomic, i.e. integrable. Moreover, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. On the other hand, the equations of motion of an ackerman steered car are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupkov´a approach. The results of numerical solutions of constrained equations of motion, derived within the theory, are presented and thus they open the possibility of direct application of the theory to practical situations in engineers.

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