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## Archive of Issues

Russia Yekaterinburg
Year
2019
Volume
29
Issue
3
Pages
408-421
 Section Mechanics Title Quaternion model of programmed control over motion of a Chaplygin ball Author(-s) Mityushov E.A.a, Misyura N.E.a, Berestova S.A.a Affiliations Ural Federal Universitya Abstract This paper deals with the problem of program control of the motion of a dynamically asymmetric balanced ball on the plane using three flywheel motors, provided that the ball rolls without slipping. The center of mass of the mechanical system coincides with the geometric center of the ball. Control laws are found to ensure the motion of the ball along the basic trajectories (line and circle), as well as along an arbitrarily given piecewise smooth trajectory on the plane. In this paper, we propose a quaternion model of ball motion. The model does not require using the traditional trigonometric functions. Kinematic equations are written in the form of linear differential equations eliminating the disadvantages associated with the use of Euler angles. The solution of the problem is carried out using the quaternion function of time, which is determined by the type of trajectory and the law of motion of the point of contact of the ball with the plane. An example of ball motion control is given and a visualization of the ball-flywheel system motion in a computer algebra package is presented. Keywords quaternions, control, nonholonomic connection, geometric dynamics, smooth movement, spherical robot UDC 531.133.1, 531.36, 514.758.3 MSC 70Q05, 34H05, 93C15 DOI 10.20537/vm190310 Received 27 June 2019 Language Russian Citation Mityushov E.A., Misyura N.E., Berestova S.A. Quaternion model of programmed control over motion of a Chaplygin ball, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 408-421. References Bolotin S.V. The problem of optimal control of a Chaplygin ball by internal rotors, Nelin. 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Movement of the ball with three internal flywheel engines in a straight line, Mathematical Modeling in UrFU. https://youtu.be/kUta7xlhB-o Movement of the ball with three internal engines around the circle, Mathematical Modeling in UrFU. https://youtu.be/I8BvP84_YcI Movement of the ball with three internal engines on the trajectory of “slalom”, Mathematical Modeling in UrFU. https://youtu.be/aHbboXhw1ZY Movement of the ball with three internal engines along the clothoid, Mathematical Modeling in UrFU. https://youtu.be/-r566MX3keg Movement of the ball with three internal engines along the astroid, Mathematical Modeling in UrFU. https://youtu.be/7mo-XR0QebI Animation of a ball moving on a plane without slipping, Mathematical Modeling in UrFU. https://youtu.be/9PVqnH1Ap9A Full text