# A Study of the Controlled Motion of a Four-wheeled Mecanum Platform

2018, Vol. 14, no. 2, pp.  265-290

The object of the study is the mobile platform of the KUKA youBot robot equipped with four Mecanum wheels. The ideal conditions for the point contact of the wheels and the floor are considered. It is assumed that the rollers of each Mecanum wheel move without slipping and the center of the wheel, the center of the roller axis, and the point of contact of the roller with the floor are located on the same straight line. The dynamics of the system is described using Appel’s equations and taking into account the linear forces of viscous friction in the joints of the bodies. An algorithm for determination of the control forces is designed. Their structure is the same as that of the reactions of ideal constraints determined by the program motion of the point of the platform. The controlled dynamics of the system is studied using uniform circular motion of the platform point as an example: conditions for the existence and stability of steady rotations are found, conditions for the existence of stable-unstable stationary regimes and rotational motions of the platform are obtained. Within the framework of the theory of singular perturbations, an asymptotic analysis of the rotation of the platform is carried out.
Keywords: omniwheel, Mecanum wheel, omnidirectional platform, servo-constraint, youBot, singular perturbation
Citation: Adamov B. I., A Study of the Controlled Motion of a Four-wheeled Mecanum Platform, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp.  265-290
DOI:10.20537/nd180209

## Supplement

This paper proposes a method for implementing the program motion of a point of the platform. The structure of control forces repeats that of reactions of ideal constraints which are given by the program motion. Animations of the motion of the platform along a circle are presented below:

a) The case of stable steady rotation of the platform (the radius of the circle is larger than the critical one, conditions (4.8) are satisfied, see also the results of mathematical modeling in Fig. 4

b) The case of rotational motion (the radius of the circle is smaller than the critical one and conditions (4.8) are violated.

## References

[1] Adamov, B. I. and Kobrin, A. I., “Parametric Identification of the Mathematical Model of the Omnidirectional Mobile Robot KUKA youBot”, Mekhatronika, Avtomatizatsiya, Upravlenie, 19:4 (2018), 251–258 (Russian)
[2] Adamov, B. I. and Orlov, I. V., “Control of a Mobile Manipulator in Cylindrical Coordinate System”, Vestn. Mosk. Energ. Inst., 2012, no. 1, 28–35 (Russian)
[3] Beghin H., Étude théorique des compas gyrostatiques Anschütz et Sperri, Thése, Faculté des sciences de Paris, Paris, 1922, 132 pp.
[4] Borisov, A. V., Kilin, A. A., and Mamaev, I. S., “An Omni-Wheel Vehicle on a Plane and a Sphere”, Nelin. Dinam., 7:4 (2011), 785–801 (Russian)
[5] Gantmacher, F. R., The Theory of Matrices, v. 1, 2, Chelsea, New York, 1959
[6] Golubev, Yu. F., Appel’s Function in the Dynamics of Rigid Body Systems, Preprint № 49, KIAM, Moscow, 2014 (Russian)
[7] Gradshtein, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th ed., Acad. Press, Amsterdam, 2007, 1200 pp.
[8] Zobova, A. A., “Application of Laconic Forms of the Equations of Motion in the Dynamics of Nonholonomic Mobile Robots”, Nelin. Dinam., 7:4 (2011), 771–783 (Russian)
[9] Kilin, A. A. and Bobykin, A. D., “Control of a Vehicle with Omniwheels on a Plane”, Nelin. Dinam., 10:4 (2014), 473–481 (Russian)
[10] Kilin A. A., Karavaev Yu. L., Klekovkin A. V., “Kinematic Control of a High Manoeuvrable Mobile Spherical Robot with Internal Omni-Wheeled Platform”, Nelin. Dinam., 10:1 (2014), 113–126 (Russian)
[11] Kozlov, V. V., “Principles of Dynamics and Servo-Constraints”, Nelin. Dinam., 11:1 (2015), 169–178 (Russian)
[12] Kozlov, V. V., “The Dynamics of Systems with Servoconstraints: 1”, Regul. Chaotic Dyn., 20:3 (2015), 205–224
[13] Kozlov, V. V., “The Dynamics of Systems with Servoconstraints: 2”, Regul. Chaotic Dyn., 20:4 (2015), 401–427
[14] Markeev, A. P., Theoretical Mechanics, R&C Dynamics, Institute of Computer Science, Izhevsk, 2007, 592 pp. (Russian)
[15] Markeev, A. P., “On the Gauss Principle”, Collection of Articles on Theoretical Mechanics, v. 23, ed. A. N. Kolesnikov, Mosk. Gos. Univ., Moscow, 2000, 29–44 (Russian)
[16] Martynenko, Yu. G., “Motion Control of Mobile Wheeled Robots”, J. Math. Sci. (N. Y.), 147:2 (2007), 6569–6606      ; Fundam. Prikl. Mat., 11:8 (2005), 29–80 (Russian)
[17] Martynenko, Yu. G., Orlov, I. V., “Motion Control Software for Telescopic Manipulator on the Mobile Platform”, Vestn. Mosk. Energ. Inst., 2003, no. 5, 60–70 (Russian)
[18] Martynenko, Yu. G. and Formal’skii, A. M., “On the Motion of a Mobile Robot with Roller-Carrying Wheels”, J. Comput. Syst. Sci. Int., 46:6 (2007), 976–983        ; Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2007, no. 6, 142–149 (Russian)
[19] Martynenko, Yu. G., “Stability of Steady Motions of a Mobile Robot with Roller-Carrying Wheels and a Displaced Centre of Mass”, J. Appl. Math. Mech., 74:4 (2010), 436–442        ; Prikl. Mat. Mekh., 74:4 (2010), 610–619 (Russian)
[20] Mukhametzyanov, I. A., “The Principle of Feedback on the Quasi-Accelerations for Unstressed Stabilization in Finite Time of Given Manifolds of Mechanical and Generalized Systems”, Vestn. RUDN. Mat. Inform. Fiz., 2014, no. 3, 107–114 (Russian)
[21] Mukharlyamov, R. G., “Simulation of Control Processes, Stability and Stabilization of Systems with Program Constraints”, J. Comput. Syst. Sci. Int., 54:1 (2015), 13–26        ; Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2015, no. 1, 15–28 (Russia)
[22] Mukharlyamov, R. G. and Abramov, N. V., “Dynamic Control of Manipulator with Program Constraints”, Problems of Mechanics and Control: Nonlinear Dynamical Systems, v. 43, Perm State Univ., Perm, 2011, 90–102 (Russian)
[23] Neimark, Ju. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., 33, AMS, Providence, R.I., 1972, 518 pp.
[24] Novozhilov, I. V., Fractional Analysis. Methods of Motion Decomposition, Birkhäuser, Basel, 1997, X+232 pp.
[25] Tikhonov, A. N., “Systems of Differential Equations Containing Small Parameters in the Derivatives”, Mat. Sb. (N. S.), 31(73):3 (1952), 575–586 (Russian)
[26] Adascalitei, F. and Doroftei, I., “Practical Applications for Mobile Robots Based on Mecanum Wheels: A Systematic Survey”, The Romanian Review Precision Mechanics, Optics & Mechatronics, 2011, no. 40, 21–29
[27] Gfrerrer, A., “Geometry and Kinematics of the Mecanum Wheel”, Comput. Aided Geom. Des., 25:9 (2008), 784–791
[28] Ilon, B. E., Wheels for a Course Stable Selfpropelling Vehicle Movable in Any Desired Direction on the Ground or Some Other Base: Patent Sweden, B60B 19/12 (20060101); B60b 019/00, REF/3876255, November 13, 1972
[29] Indivery, G., Paulus, J., and Ploge, P.-G., “Motion Control of Swedish Wheeled Mobile Robot in the Presence of Actuator Saturation”, RoboCup 2006: Robot Soccer World Cup X, Lecture Notes in Comput. Sci., 4434, eds. G. Lakemeyer, E. Sklar, D. G. Sorrenti, T. Takahashi, Springer, New York, 2007, 29–44
[30] Ivanov, A. P., “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448
[31] Li, X. and Zell, A., “Motion Control of an Omnidirectional Mobile Robot”, Informatics in Control, Automation and Robotics: Selected Papers from the International Conference on Informatics in Control, Automation and Robotics (2007): Part 2, Lecture Notes in Electrical Engineering, 24, eds. J. Filipe, J. A. Cetto, J. L. Ferrier, Springer, Berlin, 2009, 181–193
[32] Lin, L.-C. and Shih, H.-Y., “Modeling and Adaptive Control of an Omni-Mecanum-Wheeled Robot”, Intelligent Control and Automation, 4:2 (2013), 166–179
[33] Planitz, M., “Inconsistent Systems of Linear Equations”, Math. Gazette, 63:425 (1979), 181–185
[34] Plumpton, J. J., Hayes, M. J. D., Langlois, R. G., and Burlton, B. V., “Atlas Motion Platform Mecanum Wheel Jacobian in the Velocity and Static Force Domains”, CCToMM Mechanisms, Machines, and Mechatronics Symposium, 2013, 10
[35] Purwin, O. and D'Andrea, R., “Trajectory Generation and Control for Four Wheeled Omnidirectional Vehicles”, Robot. Auton. Syst., 54:1 (2006), 13–22
[36] Zimmermann, K., Zeidis, I., and Behn, C., Mechanics of Terrestrial Locomotion: With a Focus on Non-pedal Motion Systems, Springer, London, 2010, 292 pp.
[37] http://www.youbot-store.com