Journal of Vibration Testing and System Dynamics
The Ackerman Steered Car NonHolonomic Lagrangian Mechanics System: Mathematics Problem Treatment of the Geometrical Theory
Journal of Vibration Testing and System Dynamics 1(4) (2017) 319331  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
Download Full Text PDF
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 multipletrailer vehicles, where the wheels provide a noslip velocity constraint. In this paper, a new methods in nonholonomic 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 socalled Chetaevtype 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 semiholonomic, i.e. integrable. Moreover, there exist some practical examples of systems subjected to true (nonintegrable) 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.
References

[1]  Swaczyna, M. (2011), Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19(1), 2756. 

[2]  Bullo, F. and Lewis, A.D. (2004), Geometric Control of Mechanical Systems, Springer Verlag, New York, Heidelberg, Berlin. 

[3]  Cardin, F. and Favreti, M. (1996), On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18, 295325. 

[4]  Carinena, J.F. and Raada, M.F. (1993), Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26, 13351351. 

[5]  Cortes, J., de Leon, M., Marrero, J.C., and Martinez, E. (2009), Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst. A, 24, 213271. 

[6]  de Leon, M., Marrero, J.C., and de Diego, D.M. (1997), Nonholonomic Lagrangian systems in jet manifolds, J. Phys. A: Math. Gen., 30, 11671190. 

[7]  de Leon, M., Marrero, J.C., and de Diego, D.M. (1997), Mechanical systems with nonlinear constraints, Int. Journ. Theor. Phys., 36(4), 979995. 

[8]  Giachetta, G. (1992), Jet methods in nonholonomic mechanics, J. Math. Phys., 33, 16521655. 

[9]  Janová, J. and Musilová, J. (2009), Nonholonomic mechanics mechanics: A geometrical treatment of general coupled rolling motion, Int. J. NonLinear Mechanics, 44, 98105. 

[10]  Czudková, L. and Musilová, J. (2013), Nonholonomic mechanics. A practical application of the geometrical theory on fibred manifolds to a planimeter motion, International Journal of NonLinear Mechanics, 50, 1924. 

[11]  Krupková, O. (1997a), Mechanical systems with nonholonomic constraints, J. Math. Phys., 38, 5098. 

[12]  Krupková, O. (2000), Higher order mechanical systems with constraints, J. Math. Phys., 41, 5304. 

[13]  Krupková, O. and Musilová, J. (2001), The relativistic particle as a mechanical system with nonlinear constraints, J. Phys. A: Math. Gen., 34, 3859. 

[14]  Chetaev, N.G. (19321933), On the Gauss principle, Izv. Kazan. Fiz.Mat. Obsc., 6, 323 (in Russian). 

[15]  Krupková, O. and Musilová, J. (2005), Nonholonomic variational systems, Rep. Math. Phys., 55, 211. 

[16]  Swaczyna, M. (2005), Variational aspects of nonholonomic mechanical systems. Ph.D. Thesis, Palack′y University, Olomouc. 

[17]  Brdicka, M. and Hladík, J. (1987), Theoretical Mechanics (Teoretická mechanika, in Czech), Academia, Praha. 

[18]  Neimark, J.I. and Fufaev, N.A. (1972), Dynamics of nonholonomic systems, 33, of Translations of Mathematical Monographs. Providence, Rhode Island: AMS. 

[19]  Koon, W.S. and Marsden, J.E. (1997), The Hamiltonian and Lagrangian approach to the dynamics of nonholonomic systems, Reports in Mathematical Physics, 40, 2162. 

[20]  Krupková, O. (1998), On the geometry of nonholonomic mechanical systems, in: Proceedings of the Conference on Differential Geometry and its Applications, 533. 

[21]  Krupková, O. (1997b), The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics, Springer, Berlin. 

[22]  Conner, D. (2007), Integrating Planning and Control for Constrained Dynamical Systems[thesis], Robotic Institute, Carnegie Mellon University, Pittsburgh, Pennsylvania, December. 

[23]  Lipták, T., Kelemen, M., Gmiterko, A., Virgala, I., Miková, L’, and Hroncová, D. (2016), Comparison of Different Approaches of Mathematical Modelling of Ackerman Steered Carlike System, Journal of Automation and Control, 4(2), 1521. 