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Dynamics of a rolling robot

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JournalProceedings of the National Academy of Sciences of the United States of America
DateAccepted/In press - 7 Nov 2017
DateE-pub ahead of print (current) - 20 Nov 2017
Number of pages6
Early online date20/11/17
Original languageEnglish

Abstract

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modelled as a point mass mounted inside a spherical shell, and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, non-holonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency towards chaotic behaviour
as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realised
while contact between the shell and the plane is maintained. The predicted behaviour has been observed in our experiments.

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