December  2010, 2(4): 321-342. doi: 10.3934/jgm.2010.2.321

Impulsive control of a symmetric ball rolling without sliding or spinning

1. 

Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET, Argentina

2. 

Laboratorio de Electrónica Industrial, Control e Instrumentación, Facultad de Ingeniería, Universidad Nacional de La Plata and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata., CC 172, 1900 La Plata, Argentina

3. 

Departamento de Mateemática and Instituto de Matemática Bahía Blanca, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET, Argentina

Received  May 2010 Revised  December 2010 Published  January 2011

A ball having two of its three moments of inertia equal and whose center of mass coincides with its geometric center is called a symmetric ball. The free dynamics of a symmetric ball rolling without sliding or spinning on a horizontal plate has been studied in detail in a previous work by two of the authors, where it was shown that the equations of motion are equivalent to an ODE on the 3-manifold $S^2 \times S^1$. In this paper we present an approach to the impulsive control of the position and orientation of the ball and study the speed of convergence of the algorithm. As an example we apply this approach to the solutions of the isoparallel problem.
Citation: Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321
References:
[1]

Andrei A. Agrachev and Yuri L. Sachkov, An intrinsic approach to the control of rolling bodies, In "Proceedings of the 38th Conference on Decision & Control," Phoenix, Arizona USA, December 1999. Google Scholar

[2]

Yasumichi Aiyama and Tamio Arai, A quantitative stability measure for graspless manipulation, In "Proceedings of the 1996 IEEE/RSJ International Conference on Intelligent Robots and Systems 96, IROS 96," volume 2 (1996), 911-916. Google Scholar

[3]

Anthony M. Bloch, P. S. Krishnaprasad, Jerrold E. Marsden and Richard M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[4]

Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461. doi: 10.1007/BF01232676.  Google Scholar

[5]

Hernán Cendra and María Etchechoury, Rolling of a symmetric sphere on a horizontal plane without sliding or spinning, Rep. Math. Phys., 57 (2006), 367-374. doi: 10.1016/S0034-4877(06)80027-3.  Google Scholar

[6]

Hernán Cendra and Sebastián J. Ferraro, A nonholonomic approach to isoparallel problems and some applications, Dyn. Syst., 21 (2006), 409-437. doi: 10.1080/14689360600734112.  Google Scholar

[7]

Hernán Cendra, Ernesto A. Lacomba and Walter Reartes, The Lagrange-d'Alembert-Poincaré equations for the symmetric rolling sphere, In "Proceedings of the Sixth 'Dr. Antonio A. R. Monteiro' Congress of Mathematics (Spanish) (Bahía Blanca, 2001)," pages 19-32. Univ. Nac. Sur Dep. Mat. Inst. Mat., Bahía Blanca, (2001).  Google Scholar

[8]

Hernán Cendra, Jerrold E. Marsden and Tudor S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, In "Mathematics Unlimited--2001 and Beyond," pages 221-273. Springer, Berlin, (2001).  Google Scholar

[9]

Tuhin Das and Ranjan Mukherjee, Exponential stabilization of the rolling sphere, Automatica J. IFAC, 40 (2004), 1877-1889. doi: 10.1016/j.automatica.2004.06.003.  Google Scholar

[10]

Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico," PhD thesis, Universidad Nacional del Sur, 2005. Google Scholar

[11]

Wesley H. Huang, Control strategies for fine positioning via tapping, In "Proceedings of the 5th Symposium on Assembly and Task Planning," Besançon, France, July 2003. doi: 10.1109/ISATP.2003.1217211.  Google Scholar

[12]

Wesley H. Huang, Eric P. Krotkov and Matthew T. Mason, Impulsive manipulation, In "Proceedings of 1995 IEEE International Conference on Robotics and Automation," volume 1 (1995), 120-125. Google Scholar

[13]

Alberto Ibort, Manuel de León, Ernesto A. Lacomba, David Martín de Diego and Paulo Pitanga, Mechanical systems subjected to impulsive constraints, J. Phys. A, 30 (1997), 5835-5854. doi: 10.1088/0305-4470/30/16/024.  Google Scholar

[14]

Wang Sang Koon and Jerrold E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics (Calgary, AB, 1997).  Google Scholar

[15]

Zexiang Li and John Canny, Motion of two rigid bodies with rolling constraint, IEEE Transactions on Robotics and Automation, 6 (1990), 62-72. doi: 10.1109/70.88118.  Google Scholar

[16]

Kevin M. Lynch and Matthew T. Mason, Controllability of pushing, In "Proceedings of the 1995 IEEE International Conference on Robotics and Automation," volume 1 (1995), 112-119. Google Scholar

[17]

Yuseke Maeda and Tamio Arai, A quantitative stability measure for graspless manipulation, In "Proceedings of the 2002 IEEE International Conference on Robotics and Automation," Washington DC, USA, May 2002. Google Scholar

[18]

Jerrold E. Marsden and Tudor S. Ratiu, "Introduction to Mechanics and Symmetry," volume 17 of "Texts in Applied Mathematics," Springer-Verlag, New York, 1994.  Google Scholar

[19]

Richard Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys., 128 (1990), 565-592. doi: 10.1007/BF02096874.  Google Scholar

[20]

Giuseppe Oriolo, Marilena Vendittelli, Alessia Marigo and Antonio Bicchi, From nominal to robust planning: the plate-ball manipulation system, In "ICRA'03, IEEE International Conference on Robotics and Automation," volume 3 (2003), 3175-3180. Google Scholar

[21]

Alexander P. Veselov and Lidia V. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 604-619. doi: 10.1007/BF01158420.  Google Scholar

show all references

References:
[1]

Andrei A. Agrachev and Yuri L. Sachkov, An intrinsic approach to the control of rolling bodies, In "Proceedings of the 38th Conference on Decision & Control," Phoenix, Arizona USA, December 1999. Google Scholar

[2]

Yasumichi Aiyama and Tamio Arai, A quantitative stability measure for graspless manipulation, In "Proceedings of the 1996 IEEE/RSJ International Conference on Intelligent Robots and Systems 96, IROS 96," volume 2 (1996), 911-916. Google Scholar

[3]

Anthony M. Bloch, P. S. Krishnaprasad, Jerrold E. Marsden and Richard M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[4]

Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461. doi: 10.1007/BF01232676.  Google Scholar

[5]

Hernán Cendra and María Etchechoury, Rolling of a symmetric sphere on a horizontal plane without sliding or spinning, Rep. Math. Phys., 57 (2006), 367-374. doi: 10.1016/S0034-4877(06)80027-3.  Google Scholar

[6]

Hernán Cendra and Sebastián J. Ferraro, A nonholonomic approach to isoparallel problems and some applications, Dyn. Syst., 21 (2006), 409-437. doi: 10.1080/14689360600734112.  Google Scholar

[7]

Hernán Cendra, Ernesto A. Lacomba and Walter Reartes, The Lagrange-d'Alembert-Poincaré equations for the symmetric rolling sphere, In "Proceedings of the Sixth 'Dr. Antonio A. R. Monteiro' Congress of Mathematics (Spanish) (Bahía Blanca, 2001)," pages 19-32. Univ. Nac. Sur Dep. Mat. Inst. Mat., Bahía Blanca, (2001).  Google Scholar

[8]

Hernán Cendra, Jerrold E. Marsden and Tudor S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, In "Mathematics Unlimited--2001 and Beyond," pages 221-273. Springer, Berlin, (2001).  Google Scholar

[9]

Tuhin Das and Ranjan Mukherjee, Exponential stabilization of the rolling sphere, Automatica J. IFAC, 40 (2004), 1877-1889. doi: 10.1016/j.automatica.2004.06.003.  Google Scholar

[10]

Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico," PhD thesis, Universidad Nacional del Sur, 2005. Google Scholar

[11]

Wesley H. Huang, Control strategies for fine positioning via tapping, In "Proceedings of the 5th Symposium on Assembly and Task Planning," Besançon, France, July 2003. doi: 10.1109/ISATP.2003.1217211.  Google Scholar

[12]

Wesley H. Huang, Eric P. Krotkov and Matthew T. Mason, Impulsive manipulation, In "Proceedings of 1995 IEEE International Conference on Robotics and Automation," volume 1 (1995), 120-125. Google Scholar

[13]

Alberto Ibort, Manuel de León, Ernesto A. Lacomba, David Martín de Diego and Paulo Pitanga, Mechanical systems subjected to impulsive constraints, J. Phys. A, 30 (1997), 5835-5854. doi: 10.1088/0305-4470/30/16/024.  Google Scholar

[14]

Wang Sang Koon and Jerrold E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics (Calgary, AB, 1997).  Google Scholar

[15]

Zexiang Li and John Canny, Motion of two rigid bodies with rolling constraint, IEEE Transactions on Robotics and Automation, 6 (1990), 62-72. doi: 10.1109/70.88118.  Google Scholar

[16]

Kevin M. Lynch and Matthew T. Mason, Controllability of pushing, In "Proceedings of the 1995 IEEE International Conference on Robotics and Automation," volume 1 (1995), 112-119. Google Scholar

[17]

Yuseke Maeda and Tamio Arai, A quantitative stability measure for graspless manipulation, In "Proceedings of the 2002 IEEE International Conference on Robotics and Automation," Washington DC, USA, May 2002. Google Scholar

[18]

Jerrold E. Marsden and Tudor S. Ratiu, "Introduction to Mechanics and Symmetry," volume 17 of "Texts in Applied Mathematics," Springer-Verlag, New York, 1994.  Google Scholar

[19]

Richard Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys., 128 (1990), 565-592. doi: 10.1007/BF02096874.  Google Scholar

[20]

Giuseppe Oriolo, Marilena Vendittelli, Alessia Marigo and Antonio Bicchi, From nominal to robust planning: the plate-ball manipulation system, In "ICRA'03, IEEE International Conference on Robotics and Automation," volume 3 (2003), 3175-3180. Google Scholar

[21]

Alexander P. Veselov and Lidia V. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 604-619. doi: 10.1007/BF01158420.  Google Scholar

[1]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61

[2]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595

[3]

Andrew D. Lewis. Nonholonomic and constrained variational mechanics. Journal of Geometric Mechanics, 2020, 12 (2) : 165-308. doi: 10.3934/jgm.2020013

[4]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527

[5]

María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1

[6]

Thomas Hagen, Andreas Johann, Hans-Peter Kruse, Florian Rupp, Sebastian Walcher. Dynamical systems and geometric mechanics: A special issue in Honor of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : i-iii. doi: 10.3934/dcdss.20204i

[7]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[8]

Luis C. García-Naranjo, Mats Vermeeren. Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics. Journal of Computational Dynamics, 2021, 8 (3) : 241-271. doi: 10.3934/jcd.2021011

[9]

Gianne Derks. Book review: Geometric mechanics. Journal of Geometric Mechanics, 2009, 1 (2) : 267-270. doi: 10.3934/jgm.2009.1.267

[10]

Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019

[11]

François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1229-1242. doi: 10.3934/dcdss.2020071

[12]

Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220

[13]

Kurt Ehlers. Geometric equivalence on nonholonomic three-manifolds. Conference Publications, 2003, 2003 (Special) : 246-255. doi: 10.3934/proc.2003.2003.246

[14]

Alexis Arnaudon, So Takao. Networks of coadjoint orbits: From geometric to statistical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 447-485. doi: 10.3934/jgm.2019023

[15]

Waldyr M. Oliva, Gláucio Terra. Improving E. Cartan considerations on the invariance of nonholonomic mechanics. Journal of Geometric Mechanics, 2019, 11 (3) : 439-446. doi: 10.3934/jgm.2019022

[16]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137

[17]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[18]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204

[19]

Luis C. garcía-Naranjo, Fernando Jiménez. The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4249-4275. doi: 10.3934/dcds.2017182

[20]

François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (3)

[Back to Top]