# Global Formulation and Motion Planning for a Sphere Rolling on a Smooth Surface

• Mahmut Reyhanoglu
Regular Papers Control Theory and Applications

## Abstract

This paper studies the motion planning problem for a rolling sphere on an arbitrary smooth manifold embedded in ℝ3. The sphere is allowed to roll on the surface without slipping or twisting. A mathematical model describing the kinematics of the sphere is developed in a geometric form so that the model is globally defined without singularities or ambiguities. An algorithm for constructing a path between specified initial and final configurations is presented. The algorithm utilizes the nonholonomic nature of the system. The theoretical results are specialized for two specific surfaces defined in ℝ3, namely a flat surface and the surface of a stationary sphere.

## Keywords

Global formulation kinematics nonholonomic nonlinear

## References

1. [1]
R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.
2. [2]
Z. Li and J. Canny, “Motion of two rigid bodies with rolling constraints,” IEEE Transactions on Automatic Control, vol. 6, no. 1, pp. 62–72, 1990.
3. [3]
C. Liu, J. Gao, and D. Xu, “Lyapunov-based model predictive control for tracking of nonholonomic mobile robots under input constraints,” International Journal of Control, Automation and Systems, vol. 15, no. 5, pp. 2313–2319, 2017.
4. [4]
J. Cheng, B. Wang, Y. Zhang, and Z. Wang, “Backward orientation tracking control of mobile robot with n trailers,” International Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 867–874, 2017.
5. [5]
T. Petrinic, M. Brezak, and I. Petrovic, “Time-optimal velocity planning along predefined path for static formations of mobile robots,” International Journal of Control, Automation and Systems, vol. 15, no. 1, pp. 293–302, 2017.
6. [6]
A. M. Bloch, Nonholonomic Mechanics and Control, Springer, New York, NY, 2003.
7. [7]
A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization of nonholonomic dynamic systems,” IEEE Transactions on Automatic Control, vol. 37, no. 11, pp. 1746–1757, 1992.
8. [8]
A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization of Cˇ aplygin nonholonomic dynamic systems,” Proceedings of IEEE Conference on Decision and Control, pp.1127-1132, 1991.Google Scholar
9. [9]
M. Reyhanoglu, “A general nonholonomic motion planning strategy for Cˇ aplygin systems,” Proceedings of IEEE Conference on Decision and Control, pp.2964-2966, 1994.Google Scholar
10. [10]
R. M. Murray and S. S. Sastry, “Nonholonomic motion planning: steering using sinusoids,” IEEE Transactions on Automatic Control, vol. 38, no. 5, pp. 700–716, 1993.
11. [11]
A. Astolfi, “Discontinuous control of nonholonomic systems,” Systems and Control Letters, vol. 27, pp. 37–45, 1996.
12. [12]
Z. P. Jiang and H. Nijmeijer, “A recursive technique for tracking control of nonholonomic systems in chained form,” IEEE Transactions on Automatic Control, vol. 44, no. 2, pp. 265–279, 1999.
13. [13]
O. J. Sordalen and O. Egeland, “Exponential stabilization of nonholonomic chained systems,” IEEE Transactions on Automatic Control, vol. 40, no. 1, pp. 35–49, 1995.
14. [14]
Y. Zhao, C. Wang, and J. Yu, “Partial-state feedback stabilization for a class of generalized nonholonomic systems with ISS dynamic uncertainties,” International Journal of Control, Automation and Systems, vol. 16, no. 1, pp. 79–86, 2018.
15. [15]
B. D. Johnson, “The nonholonomy of the rolling sphere,” American Mathematical Monthly, vol. 114, no. 6, pp. 500–508, 2007.
16. [16]
V. Jurdjevic, “The geometry of the plate-ball problem,” Archive for Rational Mechanics and Analysis, vol. 124, no. 4, pp. 305–328, 1993.
17. [17]
A. Bicchi, D. Prattichizzo, and S. S. Sastry, “Planning motions of rolling surfaces,” Proceedings of IEEE Conference on Decision and Control, pp. 2812–2817, 1995.Google Scholar
18. [18]
A. Marigo and A. Bicchi, “Rolling bodies with regular surface: controllability theory and applications,” IEEE Transactions on Automatic Control, vol. 45, no. 9, pp. 1586–1599, 2000.
19. [19]
R. Mukherjee, M. Minor, and J. Pukrushpan, “Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-Plate Problem,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 124, no. 4, pp. 502–511, 2002.
20. [20]
M. Svinin and S. Hosoe, “Motion planning algorithms for a rolling sphere with limited contact area,” IEEE Transactions on Robotics, vol. 24, no. 3, pp. 612–625, 2008.
21. [21]
M. Zheng, Q. Zhan, J. Liu, and Y. Cai, “Control of a spherical robot: path following based on nonholonomic kinematics and dynamics,” Chinese Journal of Aeronautics, vol. 24, pp. 337–345, 2011.
22. [22]
C. Camicia, F. Conticelli, and A. Bicchi, “Nonholonomic kinematics and dynamics of the sphericle,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 805–810, 2000.Google Scholar
23. [23]
A. V. Borisov, I. S. Mamaev, and A. A. Kilin, “The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics,” Regular and Chaotic Dynamics, vol. 7, pp. 201–218, 2002.
24. [24]
J. Shen, D. A. Schneider, and A. M. Bloch, “Controllability and motion planning of multibody Chaplygin’s sphere and Chaplygin’s top,” International Journal on Robust and Nonlinear Control, vol. 18, pp. 905–945, 2008.
25. [25]
P. Morin and C. Samson, “Stabilization of trajectories for systems on Lie groups. Application to the rolling sphere,” IFAC Proceedings, pp. 508–513, 2008.Google Scholar
26. [26]
A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “How to control Chaplygin’s sphere using rotors,” Regular and Chaotic Dynamics, vol. 13, pp. 144–158, 2012.
27. [27]
V. Muralidharan and A. Mahindrakar, “Geometric controllability and stabilization of spherical robot dynamics,” IEEE Transactions on Automatic Control, vol. 60, no. 10, pp. 2762–2767, 2015.
28. [28]
M. Kleinsteuber, K. Huper, and F. Silva Leite Complete Controllability of the Rolling n-Sphere-A Constructive Proof, Department of Mathematics, University of Coimbra, Portugal, 2006.Google Scholar
29. [29]
N. H. McClamroch, M. Reyhanoglu, and M. Rehan, “Knife-edge motion on a surface as a nonholonomic control problem,” IEEE Control Systems Letters, vol. 1, no. 1, pp. 26–31, 2017.
30. [30]
T. Lee, M. Leok, and N. H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis, Springer, New York, NY, 2017.
31. [31]
M. Rehan and M. Reyhanoglu, “Control of rolling disk motion on an arbitrary smooth surface,” IEEE Control Systems Letters, vol. 2, no. 3, pp. 357–362, 2018.