# Sparse in the Time Stabilization of a Bicycle Robot Model: Strategies for Event- and Self-Triggered Control Approaches

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## Abstract

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## 1. Introduction

## 2. Mathematical Model of the Considered 2DoF Bicycle Robot Model

## 3. Considered Control Strategies

#### 3.1. Introduction

#### 3.2. Preliminaries—Standard LQR Control

#### 3.3. The Event-Triggered Control Approach

#### 3.4. The Self-Triggered Control Approach

#### 3.5. The Improved Self-Triggered Control Approach

## 4. Simulation Study

- standard LQR control, where the following weighting matrices have been taken: $\mathit{Q}={\mathit{I}}^{4\times 4}$, $R=1$, and the control signal is updated at every step with sampling period ${T}_{S}=0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$,
- event-triggered control, where it is assumed that the control update should be made no less than every 10 sampling periods, which forms an additional triggering condition, ${T}_{S}=0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$, and
- self-triggered control/improved-self triggered control based on prediction from the linearized model, where it is also assumed that control update should be made no less than every 10 sampling periods, ${T}_{S}=0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

## 5. Summary and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic picture of the modeled robot [18].

Symbol | Meaning |
---|---|

$\underline{x}\in {\mathcal{R}}^{n}$ | state vector |

${x}_{1}\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{rad}\right]$ | vertical deflection angle of the robot |

${x}_{2}\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{rad}}{\mathrm{s}}\right]$ | angular velocity of the robot |

${x}_{3}\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{rad}\right]$ | angle of rotation of the reaction wheel |

${x}_{4}\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{rad}}{\mathrm{s}}\right]$ | angular velocity of the reaction wheel |

$u\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{A}\right]\in \mathcal{R}$ | control signal (current of the motor) |

${m}_{r}\left[\mathrm{kg}\right]$ | weight of the robot |

${I}_{I}\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}\right]$ | moment of inertia of the reaction wheel |

${I}_{mr}\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}\right]$ | moment of inertia of the rotor of the motor |

${I}_{rg}\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}\right]$ | moment of inertia of the robot (rel. to the ground) |

${h}_{r}\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{m}\right]$ | distance between the ground and the center of mass of the robot |

$g\phantom{\rule{0.166667em}{0ex}}\left[\frac{\mathrm{m}}{{\mathrm{s}}^{2}}\right]$ | gravity force |

${k}_{m}\phantom{\rule{0.166667em}{0ex}}[-]$ | constant of the motor |

${b}_{r}\phantom{\rule{0.166667em}{0ex}}[-]$ | friction coefficient in rotational movement |

${b}_{I}\phantom{\rule{0.166667em}{0ex}}[-]$ | friction coefficient in the rotation of the reaction wheel |

${P}_{1}$, ${P}_{2}$ | contact points of the wheels with the ground |

${C}_{1}$ | center of the rear wheel |

${C}_{2}$ | center of the front wheel |

LQR | linear-quadratic regulator |

ETC | event-triggered control |

STC | self-triggered control |

ISTC | improved self-triggered control |

Parameter | Value | Description |
---|---|---|

${m}_{r}$ | $3.962\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}$ | weight of the robot |

${I}_{I}$ | $0.0094\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$ | moment of inertia (MOI) of the reaction wheel |

${I}_{mr}$ | $0.001\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$ | MOI of the rotor |

${I}_{rg}$ | $0.0931\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$ | MOI of the robot related to the ground |

${h}_{r}$ | $0.13\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | distance from the ground to the center of mass |

g | $9.8066\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ | gravity constant |

${k}_{m}$ | $0.421$ | motor constant |

${b}_{r}$ | $0.0001$ | friction coefficient in the robot rotation |

${b}_{I}$ | $0.0001$ | friction coefficient of the reaction wheel |

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**MDPI and ACS Style**

Zietkiewicz, J.; Horla, D.; Owczarkowski, A.
Sparse in the Time Stabilization of a Bicycle Robot Model: Strategies for Event- and Self-Triggered Control Approaches. *Robotics* **2018**, *7*, 77.
https://doi.org/10.3390/robotics7040077

**AMA Style**

Zietkiewicz J, Horla D, Owczarkowski A.
Sparse in the Time Stabilization of a Bicycle Robot Model: Strategies for Event- and Self-Triggered Control Approaches. *Robotics*. 2018; 7(4):77.
https://doi.org/10.3390/robotics7040077

**Chicago/Turabian Style**

Zietkiewicz, Joanna, Dariusz Horla, and Adam Owczarkowski.
2018. "Sparse in the Time Stabilization of a Bicycle Robot Model: Strategies for Event- and Self-Triggered Control Approaches" *Robotics* 7, no. 4: 77.
https://doi.org/10.3390/robotics7040077