American Institute of Mathematical Sciences

August  2013, 33(8): 3329-3353. doi: 10.3934/dcds.2013.33.3329

On the control of non holonomic systems by active constraints

 1 Department of Mathematics, Penn State University, University Park, Pa.16802 2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States 3 Dipartimento di Matematica Pura ed Applicata, Università di Padova, Padova 35141, Italy

Received  April 2012 Revised  August 2012 Published  January 2013

The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop a general framework to analyze these problems, deriving the equations of motion and studying the continuity properties of the "control-to-trajectory" maps. Various geometric characterizations are provided, in order that the equations be affine w.r.t. the time derivative of the control. In this case the system is fit for jumps, and the evolution is well defined also in connection with discontinuous control functions. The classical Roller Racer provides an example where the non-affine dependence of the equations on the derivative of the control is due only to the non-holonomic constraint. This is a case where the presence of quadratic terms in the equations can be used for controllability purposes.
Citation: Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
References:
 [1] J. Baillieul, The geometry of controlled mechanical systems, in "Mathematical Control Theory" (Eds. J. Baillieul and J. C. Willems), Springer Verlag, New York, (1998), 322-354. [2] A. M. Bloch, "Nonholonomic Mechanics and Control," Springer Verlag, 2003. doi: 10.1007/b97376. [3] A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333. [4] A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1. [5] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matem. Italiana, 2-B (1988), 641-656. [6] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory & Appl., 71 (1991), 67-84. doi: 10.1007/BF00940040. [7] A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangian mechanics, SIAM J. Control, 31 (1993), 1205-1220. doi: 10.1137/0331057. [8] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Rational Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [9] Aldo Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 19 (1989), 195–-246. (1991). [10] Aldo Bressan, On some control problems concerning the ski or swing, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 1 (1991), 147-196. [11] Aldo Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 9 (1993), 5-30. [12] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, 2004. [13] F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds, Dynam. Cont. Discr. Imp. Systems, 4 (1998), 1-21. [14] E. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Ann. Soc. Polonaise Math., 6 (1927), 1-7. [15] P. S. Krishnaprasad and D. P. Tsakiris, Oscillations, SE(2)-snakes and motion control: A study of the roller racer, Dynamical Systems, 16 (2001), 347-397. doi: 10.1080/14689360110090424. [16] M. Levi, Geometry of vibrational stabilization and some applications, Int. J. Bifurc. Chaos, 15 (2005), 2747-2756. doi: 10.1142/S0218127405013745. [17] M. Levi and Q. Ren, Geodesics on vibrating surfaces and curvature of the normal family, Nonlinearity, 18 (2005), 2737-2743. doi: 10.1088/0951-7715/18/6/017. [18] W. S. Liu and H. J. Sussmann, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, in "Proc. 30-th IEEE Conference on Decision and Control" IEEE Publications, New York, (1991), 437-442. [19] C. Marle, Géométrie des systèmes mécaniques à liaisons actives, in "Symplectic Geometry and Mathematical Physics" (1991), 260-287, (Eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman), Birkhäuser, Boston. [20] B. M. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estim. Control, 4 (1994), 385-388. [21] J. Nash, The imbedding problem for Riemannian manifolds, Annals of Math., 63 (1956), 20-63. [22] H. Nijmejer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer Verlag, New York, 1990. [23] F. Rampazzo, On Lagrangian systems with some coordinates as controls, Atti Accad. Naz. Lincei, Classe di Scienze Mat. Fis. Nat. Serie 8, 82 (1988), 685-695. [24] F. Rampazzo, On the Riemannian structure of a Lagrangean system and the problem of adding time-dependent coordinates as controls, European J. Mechanics A/Solids, 10 (1991), 405-431. [25] F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, Differential Geometry and Control, Proc. Sympos. Pure Math., (AMS, Providence) (1999), 279-296. [26] B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132. [27] B. L. Reinhart, "Differential Geometry of Foliations. The Fundamental Integrability Problem," Springer-Verlag, Berlin, 1983. [28] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6 (1978), 17-41.

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References:
 [1] J. Baillieul, The geometry of controlled mechanical systems, in "Mathematical Control Theory" (Eds. J. Baillieul and J. C. Willems), Springer Verlag, New York, (1998), 322-354. [2] A. M. Bloch, "Nonholonomic Mechanics and Control," Springer Verlag, 2003. doi: 10.1007/b97376. [3] A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333. [4] A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1. [5] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matem. Italiana, 2-B (1988), 641-656. [6] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory & Appl., 71 (1991), 67-84. doi: 10.1007/BF00940040. [7] A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangian mechanics, SIAM J. Control, 31 (1993), 1205-1220. doi: 10.1137/0331057. [8] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Rational Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [9] Aldo Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 19 (1989), 195–-246. (1991). [10] Aldo Bressan, On some control problems concerning the ski or swing, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 1 (1991), 147-196. [11] Aldo Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 9 (1993), 5-30. [12] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, 2004. [13] F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds, Dynam. Cont. Discr. Imp. Systems, 4 (1998), 1-21. [14] E. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Ann. Soc. Polonaise Math., 6 (1927), 1-7. [15] P. S. Krishnaprasad and D. P. Tsakiris, Oscillations, SE(2)-snakes and motion control: A study of the roller racer, Dynamical Systems, 16 (2001), 347-397. doi: 10.1080/14689360110090424. [16] M. Levi, Geometry of vibrational stabilization and some applications, Int. J. Bifurc. Chaos, 15 (2005), 2747-2756. doi: 10.1142/S0218127405013745. [17] M. Levi and Q. Ren, Geodesics on vibrating surfaces and curvature of the normal family, Nonlinearity, 18 (2005), 2737-2743. doi: 10.1088/0951-7715/18/6/017. [18] W. S. Liu and H. J. Sussmann, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, in "Proc. 30-th IEEE Conference on Decision and Control" IEEE Publications, New York, (1991), 437-442. [19] C. Marle, Géométrie des systèmes mécaniques à liaisons actives, in "Symplectic Geometry and Mathematical Physics" (1991), 260-287, (Eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman), Birkhäuser, Boston. [20] B. M. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estim. Control, 4 (1994), 385-388. [21] J. Nash, The imbedding problem for Riemannian manifolds, Annals of Math., 63 (1956), 20-63. [22] H. Nijmejer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer Verlag, New York, 1990. [23] F. Rampazzo, On Lagrangian systems with some coordinates as controls, Atti Accad. Naz. Lincei, Classe di Scienze Mat. Fis. Nat. Serie 8, 82 (1988), 685-695. [24] F. Rampazzo, On the Riemannian structure of a Lagrangean system and the problem of adding time-dependent coordinates as controls, European J. Mechanics A/Solids, 10 (1991), 405-431. [25] F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, Differential Geometry and Control, Proc. Sympos. Pure Math., (AMS, Providence) (1999), 279-296. [26] B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132. [27] B. L. Reinhart, "Differential Geometry of Foliations. The Fundamental Integrability Problem," Springer-Verlag, Berlin, 1983. [28] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6 (1978), 17-41.
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