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, (1999).

[2]

Yasumichi Aiyama and Tamio Arai, A quantitative stability measure for graspless manipulation,, In, 2 (1996), 911.

[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. doi: 10.1007/BF02199365.

[4]

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

[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. doi: 10.1016/S0034-4877(06)80027-3.

[6]

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

[7]

Hernán Cendra, Ernesto A. Lacomba and Walter Reartes, The Lagrange-d'Alembert-Poincaré equations for the symmetric rolling sphere,, In, (2001), 19.

[8]

Hernán Cendra, Jerrold E. Marsden and Tudor S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, In, (2001), 221.

[9]

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

[10]

Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico,", PhD thesis, (2005).

[11]

Wesley H. Huang, Control strategies for fine positioning via tapping,, In, (2003). doi: 10.1109/ISATP.2003.1217211.

[12]

Wesley H. Huang, Eric P. Krotkov and Matthew T. Mason, Impulsive manipulation,, In, 1 (1995), 120.

[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. doi: 10.1088/0305-4470/30/16/024.

[14]

Wang Sang Koon and Jerrold E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101.

[15]

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

[16]

Kevin M. Lynch and Matthew T. Mason, Controllability of pushing,, In, 1 (1995), 112.

[17]

Yuseke Maeda and Tamio Arai, A quantitative stability measure for graspless manipulation,, In, (2002).

[18]

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

[19]

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

[20]

Giuseppe Oriolo, Marilena Vendittelli, Alessia Marigo and Antonio Bicchi, From nominal to robust planning: the plate-ball manipulation system,, In, 3 (2003), 3175.

[21]

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

show all references

References:
[1]

Andrei A. Agrachev and Yuri L. Sachkov, An intrinsic approach to the control of rolling bodies,, In, (1999).

[2]

Yasumichi Aiyama and Tamio Arai, A quantitative stability measure for graspless manipulation,, In, 2 (1996), 911.

[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. doi: 10.1007/BF02199365.

[4]

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

[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. doi: 10.1016/S0034-4877(06)80027-3.

[6]

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

[7]

Hernán Cendra, Ernesto A. Lacomba and Walter Reartes, The Lagrange-d'Alembert-Poincaré equations for the symmetric rolling sphere,, In, (2001), 19.

[8]

Hernán Cendra, Jerrold E. Marsden and Tudor S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, In, (2001), 221.

[9]

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

[10]

Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico,", PhD thesis, (2005).

[11]

Wesley H. Huang, Control strategies for fine positioning via tapping,, In, (2003). doi: 10.1109/ISATP.2003.1217211.

[12]

Wesley H. Huang, Eric P. Krotkov and Matthew T. Mason, Impulsive manipulation,, In, 1 (1995), 120.

[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. doi: 10.1088/0305-4470/30/16/024.

[14]

Wang Sang Koon and Jerrold E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101.

[15]

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

[16]

Kevin M. Lynch and Matthew T. Mason, Controllability of pushing,, In, 1 (1995), 112.

[17]

Yuseke Maeda and Tamio Arai, A quantitative stability measure for graspless manipulation,, In, (2002).

[18]

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

[19]

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

[20]

Giuseppe Oriolo, Marilena Vendittelli, Alessia Marigo and Antonio Bicchi, From nominal to robust planning: the plate-ball manipulation system,, In, 3 (2003), 3175.

[21]

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

[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 - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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 - A, 2017, 37 (8) : 4249-4275. doi: 10.3934/dcds.2017182

[13]

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

[14]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[15]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[16]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69

[17]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389

[18]

Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967

[19]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441

[20]

Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631

2016 Impact Factor: 0.857

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (2)

[Back to Top]