# American Institute of Mathematical Sciences

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.

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##### 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.
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