On the normal force and static friction acting on a rolling ball actuated by internal point masses. (English) Zbl 1433.37064
Summary: The goal of this paper is to investigate the normal and tangential forces acting at the point of contact between a horizontal surface and a rolling ball actuated by internal point masses moving in the ball’s frame of reference. The normal force and static friction are derived from the equations of motion for a rolling ball actuated by internal point masses that move inside the ball’s frame of reference, and, as a special case, a rolling disk actuated by internal point masses. The masses may move along one-dimensional trajectories fixed in the ball’s and disk’s frame. The dynamics of a ball and disk actuated by masses moving along one-dimensional trajectories are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed.
MSC:
| 37J60 | Nonholonomic dynamical systems |
| 70E18 | Motion of a rigid body in contact with a solid surface |
| 70E60 | Robot dynamics and control of rigid bodies |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
Software:
RODASReferences:
| [1] | 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. · Zbl 1264.37016 |
| [2] | Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1-2, pp. 144-158. · Zbl 1303.37021 |
| [3] | Burkhardt, M. R. and Burdick, J.W., Reduced Dynamical Equations for Barycentric Spherical Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (Stockholm, Sweden, 2016), pp. 2725-2732. |
| [4] | Kilin, A.A., Pivovarova, E.N., and Ivanova, T.B., Spherical Robot of Combined Type: Dynamics and Control, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 716-728. · Zbl 1339.93081 |
| [5] | Gajbhiye, S. and Banavar, R. N., Geometric Modeling and Local Controllability of a Spherical Mobile Robot Actuated by an Internal Pendulum, Internat. J. Robust Nonlinear Control, 2016, vol. 26, no. 11, pp. 2436-2454. · Zbl 1346.93271 |
| [6] | Das, T.; Mukherjee, R.; Yuksel, H., Design Considerations in the Development of a Spherical Mobile Robot (2001) |
| [7] | Javadi, A. H.A.; Mojabi, P., Introducing August: A Novel Strategy for an Omnidirectional Spherical Rolling Robot, 3527-3533 (2002) |
| [8] | 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 |
| [9] | Ilin, K. I., Moffatt, H.K., and Vladimirov, V.A., Dynamics of a Rolling Robot, Proc. Natl. Acad. Sci. USA, 2017, vol. 114, no. 49, pp. 12858-12863. · Zbl 1404.70023 |
| [10] | Putkaradze, V. and Rogers, S.M., On the Dynamics of a Rolling Ball Actuated by Internal Point Masses, Meccanica, 2018, vol. 53, no. 15, pp. 3839-3868. |
| [11] | Editorial Discussion on Some Papers by G.M.Rosenblat, Nelin. Dinam., 2009, vol. 5, no. 4, pp. 621-624 (Russian). |
| [12] | Ivanova, T. B. and Pivovarova, E. N., Comments on the Paper by A.V.Borisov, A.A.Kilin, I. S.Mamaev “How To Control the Chaplygin Ball Using Rotors: 2”, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 140-143. · Zbl 1353.70059 |
| [13] | Vas’kin, V. V. and Naimushina, O. S., A study of the Motion of Axisymmetric Sphere with a Shifted Center of mass on a Rough Plane, Vestn. Udmurtsk. Univ. Fiz. Khim., 2012, vol. 2, pp. 10-17 (Russian). |
| [14] | Wagner, A., Heffel, E., Arrieta, A. F., Spelsberg-Korspeter, G., and Hagedorn, P., Analysis of an Oscillatory Painlevé-Klein Apparatus with a Nonholonomic Constraint, Differ. Equ. Dyn. Syst., 2013, vol. 21, nos. 1-2, pp. 149-157. · Zbl 1333.70013 |
| [15] | Ivanova, T. B. and Mamaev, I. S., Dynamics of a Painlevé-Appel System, J. Appl. Math. Mech., 2016, vol. 80, no. 1, pp. 7-15; see also: Prikl. Mat. Mekh., 2016, vol. 80, no. 1, pp. 11-23. · Zbl 1434.70039 |
| [16] | Ivanov, A.P., On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 355-368. · Zbl 1229.70015 |
| [17] | Ivanov, A.P., Geometric Representation of Detachment Conditions in Systems with Unilateral Constraints, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 435-442. · Zbl 1229.70016 |
| [18] | Blau, P. J., Friction Science and Technology: From Concepts to Applications, 2nd ed., Boca Raton, Fla.: CRC, 2008. |
| [19] | Kozlov, V.V., On the Dry-Friction Mechanism, Dokl. Phys., 2011, vol. 56, no. 4, pp. 256-257; see also: Dokl. Ross. Akad. Nauk, 2011, vol. 437, no. 6, pp. 766-767. |
| [20] | Kozlov, V.V., Friction by Painlevé and Lagrangian Mechanics, Dokl. Phys., 2011, vol. 56, no. 6, pp. 355-358; see also: Dokl. Ross. Akad. Nauk, 2011, vol. 438, no. 6, pp. 758-761. |
| [21] | Balandin, D. V., Komarov, M.A., and Osipov, G. V., A Motion Control for a Spherical Robot with Pendulum Drive, J. Comput. Sys. Sc. Int., 2013, vol. 52, no. 4, pp. 650-663; see also: Izv. Ross. Akad. Nauk. Teor. Sist. Upr., 2013, no. 4, pp. 150-163. · Zbl 1308.93147 |
| [22] | Holm, D.D., Geometric Mechanics: P.2. Rotating, Translating and Rolling, 2nd ed., London: Imperial College Press, 2011. · Zbl 1381.70001 |
| [23] | Putkaradze, V. and Rogers, S.M., On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses, arXiv:1708.03829v5 (2018). |
| [24] | Bai, Y., Svinin, M., and Yamamoto, M., Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, Regul. Chaotic Dyn., 2018, vol. 23, no. 4, pp. 372-388. · Zbl 1411.70021 |
| [25] | Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119-130. · Zbl 1058.70002 |
| [26] | Rozenblat, G.M., On the Separation-Free Motions of a Rigid Body on a Plane, Dokl. Phys., 2007, vol. 52, no. 8, pp. 447-449; see also: Dokl. Ross. Akad. Nauk, 2007, vol. 415, no. 5, pp. 622-624. · Zbl 1379.70030 |
| [27] | Ascher, U.M., Mattheij, R. M. M., and Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Philadelphia,Pa.: SIAM, 1995. · Zbl 0843.65054 |
| [28] | Hairer, E. and Wanner, G., Solving Ordinary Differential Equations:2. Stiff and Differential-Algebraic Problems, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Berlin: Springer, 1996. · Zbl 0859.65067 |
| [29] | Squire, W. and Trapp, G., Using Complex Variables to Estimate Derivatives of Real Functions, SIAM Rev., 1998, vol. 40, no. 1, pp. 110-112. · Zbl 0913.65014 |
| [30] | Martins, J. R.R. A.; Sturdza, P.; Alonso, J. J., The Connection between the Complex-Step Derivative Approximation and Algorithmic Differentiation (2001) |
| [31] | Martins, J. R. R. A., Sturdza, P., and Alonso, J. J., The Complex-Step Derivative Approximation, ACM Trans. Math. Software, 2003, vol. 29, no. 3, pp. 245-262. · Zbl 1072.65027 |
| [32] | ASM Handbook: Vol.18. Friction, Lubrication, and Wear Technology, G. E.Totten (Ed.), ASM, 2017. |
| [33] | Schröder, D., Transferring the Bearing Using a Strapdown Inertial Measurement Unit, in Applications of Geodesy to Engineering, K. Linkwitz, V.Eisele, H. J.Mönicke (Eds.), Berlin: Springer, 1993, pp. 25-38. |
| [34] | Stuelpnagel, J., On the Parametrization of the Three-Dimensional Rotation Group, SIAM Rev., 1964, vol. 6, no. 4, pp. 422-430. · Zbl 0126.27203 |
| [35] | Frisvad, J.R., Building an Orthonormal Basis from a 3D Unit Vector without Normalization, J. Graph. Tools, 2012, vol. 16, no. 3, pp. 151-159. |
| [36] | Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Nonlinear Dynamics & Mobile Robotics, 2013, vol. 1, no. 1, pp. 71-85. |
| [37] | Ivanova, T. B., Kilin, A.A., and Pivovarova, E. N., Controlled Motion of a Spherical Robot with Feedback: 2, J. Dyn. Control Syst., 2019, vol. 25, no. 1, pp. 1-16. · Zbl 1441.70007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
