Abstract
This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Halme, A., Schönberg, T., and Wang, Y., Motion Control of a Spherical Mobile Robot, in Proc. of the 4th IEEE Internat. Workshop on Advanced Motion Control (Mie, Japan, 1996): Vol. 1, pp. 259–264.
Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972.
Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631–634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398–401.
Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579.
Bizyaev, I.A., Bolsinov, A.V., Borisov, A.V., and Mamaev, I. S., Topology and Bifurcations in Nonholonomic Mechanics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2015, vol. 25, no. 10, 21 pp.
Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.
Borisov, A. V. and Mamaev, I. S., Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.
Borisov, A. V., Fedorov Yu.N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: an Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.
Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics-Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.
Borisov, A. V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., and Sedova J.V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.
Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.
Chase, R. and Pandya, A., A Review of Active Mechanical Driving Principles of Spherical Robots, Robotics, 2012, vol. 1, no. 1, pp. 3–23.
Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh,Pa., 2006. 6 pp.
Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.
Ylikorpi, T. and Suomela, J., Ball-Shaped Robots, in Climbing and Walking Robots: Towards New Applications, H. Zhang (Ed.), Vienna: InTech, 2007, pp. 235–256.
Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.
Svinin, M., Bai, Y., and Yamamoto, M., Dynamic Model and Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, in Proc. of the 2015 IEEE Internat. Conf. on Robotics and Automation (ICRA), pp. 656–661.
Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, arXiv:1511.02655 (2015).
Zhan, Q., Motion Planning of a Spherical Mobile Robot, in Motion and Operation Planning of Robotic Systems, G. Carbone, F. Gomez-Bravo (Eds.), Cham: Springer, 2015, pp. 361–381.
Gajbhiye, S. and Banavar, R. N., Geometric Modeling and Local Controllability of a Spherical Mobile Robot Actuated by an Internal Pendulum, Int. J. Robust Nonlinear Control, 2015.
Borisov, A. V. and Mamaev, I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 191–198.
Morinaga, A., Svinin, M., and Yamamoto, M., A Motion Planning Strategy for a Spherical Rolling Robot Driven by Two Internal Rotors, IEEE Trans. on Robotics, 2014, vol. 30, no. 4, pp. 993–1002.
Fantoni, I. and Lozano, R., Non-Linear Control for Underactuated Mechanical Systems, London: Springer, 2002.
Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1–57.
Borisov, A.V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kilin, A.A., Pivovarova, E.N. & Ivanova, T.B. Spherical robot of combined type: Dynamics and control. Regul. Chaot. Dyn. 20, 716–728 (2015). https://doi.org/10.1134/S1560354715060076
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354715060076