# Nonlinear Dynamics and Control of a Cube Robot

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Cube Robot Prototype

#### 2.1. System Dynamics

^{2}is the gravitational acceleration. The other parameters can be obtained from the well-known Newton’s law of motion and the friction coefficient experiment.

#### 2.2. The Braking System

#### 2.3. Signal Processing Units

## 3. Estimation and Control

#### 3.1. Attitude and Heading Reference System

#### 3.2. Balancing Control

#### 3.3. System Controllability

#### 3.4. System Controller

#### 3.5. Bouncing Control

## 4. Realization of the System

#### 4.1. Bouncing Procedure

#### 4.2. Bouncing Up and Balancing Procedure

## 5. Experimental Results

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 16.**Time response of the tilt angle angular velocity ${\dot{\theta}}_{P}$ of the cube robot balancing on its edge.

**Figure 17.**Time response of the motor speed ${\dot{\theta}}_{w}$ of the cube robot balancing on its edge.

**Figure 18.**Time response of the tilt angle ${\theta}_{P}$ of the cube robot bouncing and balancing on its edge.

**Figure 19.**Time response of the motor speed ${\dot{\theta}}_{w}$ of the cube robot bouncing and balancing on its edge.

**Figure 20.**Time response of the tilt angle angular velocity ${\dot{\theta}}_{P}$ of the cube robot bouncing and balancing on its edge.

Coefficient | Value |
---|---|

$g$ (m/s^{2}) | 9.81 |

${m}_{P}$ (kg) | 0.723 |

${m}_{w}$ (kg) | 0.162 |

${I}_{P}^{o}$ (kg $\xb7{\mathrm{m}}^{2}$) | $1.37\times {10}^{-2}$ |

${I}_{w}$ (kg $\xb7{\mathrm{m}}^{2}$) | $0.3267\times {10}^{-3}$ |

${l}_{w}(\mathrm{m})$ | 0.11 |

${l}_{p}(\mathrm{m})$ | 0.095 |

${C}_{P}$ (kg $\xb7{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{2}$) | $1.02\times {10}^{-3}$ |

${C}_{w}$ (kg $\xb7{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{2}$) | 0.6 $\times {10}^{-3}$ |

${K}_{t}$ (N $\xb7\mathrm{m}\xb7{\mathrm{A}}^{-1}$) | $38.6\times {10}^{-3}$ |

${R}_{m}(\mathsf{\Omega})$ | 0.8158 |

${L}_{a}$ (H) | 3.6 × ${10}^{-3}$ |

${K}_{e}$ (v $\xb7{\mathrm{web}}^{-1}\xb7\mathrm{s}$) | $1.78\times {10}^{-2}$ |

$n$ | 30 |

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**MDPI and ACS Style**

Liao, T.-L.; Chen, S.-J.; Chiu, C.-C.; Yan, J.-J. Nonlinear Dynamics and Control of a Cube Robot. *Mathematics* **2020**, *8*, 1840.
https://doi.org/10.3390/math8101840

**AMA Style**

Liao T-L, Chen S-J, Chiu C-C, Yan J-J. Nonlinear Dynamics and Control of a Cube Robot. *Mathematics*. 2020; 8(10):1840.
https://doi.org/10.3390/math8101840

**Chicago/Turabian Style**

Liao, Teh-Lu, Sian-Jhe Chen, Cheng-Chang Chiu, and Jun-Juh Yan. 2020. "Nonlinear Dynamics and Control of a Cube Robot" *Mathematics* 8, no. 10: 1840.
https://doi.org/10.3390/math8101840