### 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 modeled 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, nonholonomic, 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 toward chaotic behavior 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 realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

Original language | English |
---|---|

Pages (from-to) | 12858-12863 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 114 |

Issue number | 49 |

DOIs | |

Publication status | Published - Dec 5 2017 |

### Fingerprint

### Keywords

- Chaotic rolling
- Internal rotor
- Nonholonomic system
- Rolling robot

### ASJC Scopus subject areas

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*114*(49), 12858-12863. https://doi.org/10.1073/pnas.1713685114

**Dynamics of a rolling robot.** / Ilin, K. I.; Moffatt, H. K.; Vladimirov, V. A.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 114, no. 49, pp. 12858-12863. https://doi.org/10.1073/pnas.1713685114

}

TY - JOUR

T1 - Dynamics of a rolling robot

AU - Ilin, K. I.

AU - Moffatt, H. K.

AU - Vladimirov, V. A.

PY - 2017/12/5

Y1 - 2017/12/5

N2 - 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 modeled 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, nonholonomic, 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 toward chaotic behavior 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 realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

AB - 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 modeled 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, nonholonomic, 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 toward chaotic behavior 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 realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

KW - Chaotic rolling

KW - Internal rotor

KW - Nonholonomic system

KW - Rolling robot

UR - http://www.scopus.com/inward/record.url?scp=85037034298&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037034298&partnerID=8YFLogxK

U2 - 10.1073/pnas.1713685114

DO - 10.1073/pnas.1713685114

M3 - Article

VL - 114

SP - 12858

EP - 12863

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 49

ER -