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How to control Chaplygin’s sphere using rotors. (English) Zbl 1264.37016
The paper studies the controllability of a dynamically non-symmetric balanced sphere moving over a plane. Internal rotors are used as a mechanism for control. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories, explicit tracking algorithms are proposed.

MSC:
37J60 Nonholonomic dynamical systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70E18 Motion of a rigid body in contact with a solid surface
70F25 Nonholonomic systems related to the dynamics of a system of particles
70H45 Constrained dynamics, Dirac’s theory of constraints
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[1] Bobylew, D.K., On Ball with Gyroscope Inside Rolling on Horizontal Plane without Sliding, Mat. sb., 1892, vol. 16, no. 3, pp. 544–581 (Russian).
[2] Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318]. · doi:10.4213/rm9346
[3] Borisov, A.V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2001 (Russian).
[4] Borisov, A. V. and Mamaev, I. S., Rolling of a Heterotgeneous Ball over a Sphere without Sliding and Spinning, Rus. J. Nonlin. Dyn., 2006, vol. 2, no. 4, pp. 445–452 (Russian).
[5] Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Rus. J. Nonlin. Dyn., 2008, vol. 4, no. 3, pp. 223–280 [Regul. Chaotic Dyn., 2008, vol. 13, o. 5, pp. 443–490]. · Zbl 1229.70038
[6] Zhukovsky, N. E., On Gyroscopic Ball of D.K.Bobylev, Collected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 257–289 (Russian).
[7] Markeev, A.P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1986, vol. 21, no. 1, pp. 64–65 [Regul. Chaotic Dyn, 2002, vol. 7, no. 2, pp. 149–151].
[8] Martynenko, Yu.G., Motion Control of Mobile Wheeled Robots, Fundam. Prikl. Mat., 2005, vol. 11, no. 8, pp. 29–80 [J. Math. Sci. (N. Y.), 2007, vol. 147, no. 2, pp. 6569–6606].
[9] Moskvin, A.Yu., Chaplygin’s Ball with a Gyrostat: Singular Solutions, Rus. J. Nonlin. Dyn., 2009, vol. 5, no. 3, pp. 345–356 (Russian).
[10] Rashevsky, P.K., Any Two Points of a Totally Nonholonomic Space May Be Connected by an Admissible Line, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math., 1938, vol. 3, no. 2, pp. 83–94 (Russian).
[11] Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Math. Sb., 1903, vol. 24, no. 1, pp. 139–168 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148].
[12] Agrachev, A.A. and Sachkov, Yu. L., An Intrinsic Approach to the Control of Rolling Bodies, in Proc. of the 38th IEEE Conf. on Decision and Control (Phoenix, AZ, Dec 1999): Vol. 1, pp. 431–435.
[13] Alouges, F., Chitour, Y., and Long, R., A Motion Planning Algorithm for the Rolling-Body Problem, IEEE Trans. on Robotics, 2010, vol. 26, no. 5, pp. 827–836. · doi:10.1109/TRO.2010.2053733
[14] Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, Journal of Systems and Control Engineering, 2003, vol. 217, pp. 457–467.
[15] Armour, R.H. and Vincent, J. F. V., Rolling in Nature and Robotics: A Review, Journal of Bionic Engineering, 2006, vol. 3, no. 4, pp. 195–208. · doi:10.1016/S1672-6529(07)60003-1
[16] Bonnard, B., Contrôlabilité des systèmes nonlinéaires, C. R. Acad. Sci. Paris, Sér. 1, 1981, vol. 292, pp. 535–537. · Zbl 0498.93009
[17] Borisov, A. V. and Mamaev, I. S., The Rolling of Rigid Body on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 177–200. · Zbl 1058.70009 · doi:10.1070/RD2002v007n02ABEH000204
[18] Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Rolling of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–220. · Zbl 1058.70010 · doi:10.1070/RD2002v007n02ABEH000205
[19] Camicia, C., Conticelli, F., and Bicchi, A., Nonholonimic Kinematics and Dynamics of the Sphericle, in Proc. of the 2000 IEEE/RSJ Internat. Conf. on Intelligent Robots and Systems, 2000, pp. 805–810.
[20] Campion, G. and Chung, W., Wheeled Robots, in Handbook of Robotics, B. Siciliano and O. Khatib (Eds.), Berlin: Springer, 2008, pp. 391–410.
[21] Chow, W. L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 1939, vol. 117, pp. 98–105. · JFM 65.0398.01
[22] Chung, W., Nonholonomic Manipulators, Springer Tracts in Advanced Robotics, vol. 13, Berlin: Springer, 2004. · Zbl 1074.93001
[23] Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh,PA, 2006.
[24] Crouch, P.E., Spacecraft Attitude Control and Stabilization: Applications of Geometric Control Theory to Rigid Body Models, IEEE Trans. Autom. Contr., 1984, vol. AC-29, no. 4, pp. 321–331. · Zbl 0536.93029 · doi:10.1109/TAC.1984.1103519
[25] Duistermaat, J. J., Chaplygin’s sphere, arXiv:math/0409019v1 [math.DS] 1 Sep 2004.
[26] Goncharenko, I., Svinin, M., and Hosoe, S., Dynamic Model, Haptic Solution, and Human-inspired Motion Planning for Rolling-based Manipulation, Journal of Computing and Information Science in Engineering, 2009, vol. 9, no. 1, 011004, 10 pp.
[27] Johnson, B.D., The Nonholonomy of the Rolling Sphere, Amer. Math. Monthly, 2007, vol. 114, no. 6, pp. 500–508. · Zbl 1189.70036 · doi:10.1080/00029890.2007.11920439
[28] Joshi, V. A. and Banavar, R.N., Motion Analysis of a Spherical Mobile Robot, Robotica, 2009, vol. 27, no. 3, pp. 343–353. · doi:10.1017/S0263574708004748
[29] Joshi, V.A., Banavar, R.N., and Hippalgaonkar, R., Design and Analysis of a Spherical Mobile Robot, Mech. Mach. Theory, 2010, vol. 45, pp. 130–136. · Zbl 1379.70015 · doi:10.1016/j.mechmachtheory.2009.04.003
[30] Karimpour, H., Keshmiri, M., and Mahzoon, M., Stabilization of an Autonomous Rolling Sphere Navigating in a Labyrinth Arena: A Geometric Mechanics Perspective, Systems Control Lett., 2012, vol. 61, pp. 495–505. · Zbl 1250.93087 · doi:10.1016/j.sysconle.2012.01.014
[31] Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306. · Zbl 1074.70513 · doi:10.1070/RD2001v006n03ABEH000178
[32] Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152. · Zbl 1229.37089 · doi:10.1134/S1560354707020025
[33] Koshiyama, A. and Yamafuji, K., Design and Control of an All-direction Steering Type Mobile Robot, Int. J. Robot. Res., 1993, vol. 12, no. 5, pp. 411–419. · doi:10.1177/027836499301200502
[34] Levi, M., Geometric Phases in the Motion of Rigid Bodies, Arch. Ration. Mech. Anal., 1993, vol. 122, pp. 213–229. · Zbl 0782.70005 · doi:10.1007/BF00380255
[35] Lewis, A. D., Ostrowski, J.P., Burdickz, J.W., and Murray, R. M., Nonholonomic Mechanics and Locomotion: The Snakeboard Example, in Proc. of the 1994 IEEE Internat. Conf. on Robotics and Automation, 1994, 16 pp.
[36] Li, Z. and Canny, J., Motion of Two Rigid Bodies with Rolling Constraint, Robotics and Automation, 1990, vol. 6, no. 1, pp. 62–72. · doi:10.1109/70.88118
[37] Michaud, F. and Caron, S., Roball, the Rolling Robot, Autonomous Robots, 2002, vol. 12, pp. 211–222. · Zbl 1057.68695 · doi:10.1023/A:1014005728519
[38] Marigo, A. and Bicchi, A., Rolling Bodies with Regular Surface: Controllability Theory and Applications, IEEE Trans. on Automatic Control, 2000, vol. 45, no. 9, pp. 1586–1599. · Zbl 0986.70002 · doi:10.1109/9.880610
[39] Mukherjee, R., Minor, M.A., and Pukrushpan, J. T., Simple Motion Planning Strategies for Spherobot: A Spherical Mobile Robot, in Proc. of the 38th IEEE Conf. on Decision and Control (Phoenix, AZ, Dec 1999): Vol. 3, pp. 2132–2137.
[40] Mukherjee, R., Minor, M.A., and Pukrushpan, J. T., Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-plate Problem, J. Dyn. Systems, Measurement, and Control, 2002, vol. 124, pp. 502–511. · doi:10.1115/1.1513177
[41] Nakashima, A., Nagase, K., and Hayakawa, Y., Control of a Sphere Rolling on a Plane with Constrained Rolling Motion, in Proc. of the 44th IEEE Conf. on Decision and Control, and the European Control Conference (2005), pp. 1445–1452.
[42] Ostrowski, J.P., Desai, J.P., and Kumar, V., Optimal Gait Selection for Nonholonomic Locomotion Systems, Int. J. Robot. Res., 2000, vol. 19, no. 3, pp. 225–237. · Zbl 05422431 · doi:10.1177/02783640022066833
[43] Reza Moghadasi, S., Rolling of a Body on a Plane or a Sphere: A Geometric Point of View, Bull. Austral. Math. Soc., 2004, vol. 70, pp. 245–256. · Zbl 1061.93027 · doi:10.1017/S0004972700034468
[44] Sang, S., Zhao, J., Wu, H., Chen, S., and An, Q., Modeling and Simulation of a Spherical Mobile Robot, Computer Science and Information Systems, 2010, vol. 7, no. 1, pp. 51–62. · doi:10.2298/CSIS1001051S
[45] Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of Multibody Systems with Nonholonomic Constraints, in Proc. of the 42nd IEEE Conf. on Decision and Control (Maui, Hawaii USA, Dec 2003): Vol. 53, pp. 4369–4374.
[46] Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of a Multibody Chaplygin’s Sphere and Chaplygin’s Top, Internat. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905–945. · Zbl 1284.93044 · doi:10.1002/rnc.1259
[47] Sugiyama, Y. and Hirai, S., Crawling and Jumping by a Deformable Robot, Int. J. Robot. Res., 2006, vol. 25, pp. 603–620. · Zbl 05422569 · doi:10.1177/0278364906065386
[48] Suomela, J. and Ylikorpi, T., Ball-shaped Robots: An Historical Overview and Recent Developments at TKK, Field and Service Robotics, 2006, vol. 25, pp. 343–354. · doi:10.1007/978-3-540-33453-8_29
[49] Tomik, F., Nudehi, S., Flynn, L. L., and Mukherjee, R., Design, Fabrication and Control of Spherobot: A Spherical Mobile Robot, J. Intell. Robot. Syst., 2012 (DOI: 10.1007/s10846-012-9652-2).
[50] Wilson, J. L., Mazzoleni, A.P., DeJarnette, F.R., Antol, J., Hajos, G.A., and Strickland, C. V., Design, Analysis, and Testing of Mars Tumbleweed Rover Concepts, Journal of Spacecraft and Rockets, 2008, vol. 45, no. 2, pp. 370–382. · doi:10.2514/1.31288
[51] Yoon, J.-C., Ahn, S.-S., and Lee, Y.-J., Spherical Robot with New Type of Two-Pendulum Driving Mechanism, in Proc. of the 15th IEEE Internat. Conf. on Intelligent Engineering Systems (2011), pp. 275–279.
[52] Zhan, Q., Cai, Y., Yan, C., Design, Analysis and Experiments of an Omni-directional Spherical Robot, IEEE Internat. Conf. on Robotics and Automation (Shanghai, China, May 2011), pp. 4921–4926.
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