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

On the control of non holonomic systems by active constraints

Alberto Bressan 1, , Ke Han 2, and  Franco Rampazzo 3,

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.
Keywords: Impulsive control, non-holonomic system, active constraints..
Mathematics Subject Classification: Primary: 93C15, 37J60; Secondary: 70F25, 34A3.
Citation: Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & 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.  Google Scholar

[2]

A. M. Bloch, "Nonholonomic Mechanics and Control," Springer Verlag, 2003. doi: 10.1007/b97376.  Google Scholar

[3]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333.  Google Scholar

[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.  Google Scholar

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matem. Italiana, 2-B (1988), 641-656.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, 2004.  Google Scholar

[13]

F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds, Dynam. Cont. Discr. Imp. Systems, 4 (1998), 1-21.  Google Scholar

[14]

E. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Ann. Soc. Polonaise Math., 6 (1927), 1-7. Google Scholar

[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.  Google Scholar

[16]

M. Levi, Geometry of vibrational stabilization and some applications, Int. J. Bifurc. Chaos, 15 (2005), 2747-2756. doi: 10.1142/S0218127405013745.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Nash, The imbedding problem for Riemannian manifolds, Annals of Math., 63 (1956), 20-63.  Google Scholar

[22]

H. Nijmejer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, Differential Geometry and Control, Proc. Sympos. Pure Math., (AMS, Providence) (1999), 279-296.  Google Scholar

[26]

B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132.  Google Scholar

[27]

B. L. Reinhart, "Differential Geometry of Foliations. The Fundamental Integrability Problem," Springer-Verlag, Berlin, 1983.  Google Scholar

[28]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6 (1978), 17-41.  Google Scholar

show all references

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.  Google Scholar

[2]

A. M. Bloch, "Nonholonomic Mechanics and Control," Springer Verlag, 2003. doi: 10.1007/b97376.  Google Scholar

[3]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333.  Google Scholar

[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.  Google Scholar

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matem. Italiana, 2-B (1988), 641-656.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, 2004.  Google Scholar

[13]

F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds, Dynam. Cont. Discr. Imp. Systems, 4 (1998), 1-21.  Google Scholar

[14]

E. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Ann. Soc. Polonaise Math., 6 (1927), 1-7. Google Scholar

[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.  Google Scholar

[16]

M. Levi, Geometry of vibrational stabilization and some applications, Int. J. Bifurc. Chaos, 15 (2005), 2747-2756. doi: 10.1142/S0218127405013745.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Nash, The imbedding problem for Riemannian manifolds, Annals of Math., 63 (1956), 20-63.  Google Scholar

[22]

H. Nijmejer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, Differential Geometry and Control, Proc. Sympos. Pure Math., (AMS, Providence) (1999), 279-296.  Google Scholar

[26]

B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132.  Google Scholar

[27]

B. L. Reinhart, "Differential Geometry of Foliations. The Fundamental Integrability Problem," Springer-Verlag, Berlin, 1983.  Google Scholar

[28]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6 (1978), 17-41.  Google Scholar

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