# American Institute of Mathematical Sciences

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. [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. [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. [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. [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. [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. [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). [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). [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. [10] Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico," PhD thesis, Universidad Nacional del Sur, 2005. [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. [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. [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. [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). [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. [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. [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. [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. [19] Richard Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys., 128 (1990), 565-592. 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 "ICRA'03, IEEE International Conference on Robotics and Automation," volume 3 (2003), 3175-3180. [21] Alexander P. Veselov and Lidia V. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 604-619. 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 "Proceedings of the 38th Conference on Decision & Control," Phoenix, Arizona USA, December 1999. [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. [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. [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. [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. [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. [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). [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). [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. [10] Sebastián José Ferraro, "Reducción de Sistemas Lagrangianos Dependientes de un Parámetro y el Problema Isoholonómico," PhD thesis, Universidad Nacional del Sur, 2005. [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. [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. [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. [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). [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. [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. [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. [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. [19] Richard Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys., 128 (1990), 565-592. 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 "ICRA'03, IEEE International Conference on Robotics and Automation," volume 3 (2003), 3175-3180. [21] Alexander P. Veselov and Lidia V. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 604-619. doi: 10.1007/BF01158420.
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