# Hierarchical Sliding Mode Control Combined with Nonlinear Disturbance Observer for Wheeled Inverted Pendulum Robot Trajectory Tracking

^{*}

## Abstract

**:**

## Featured Application

**The research presented herein could be used mostly in warehouse logistics transport activities in smart manufacturing.**

## Abstract

## 1. Introduction

- (1)
- A wheeled inverted pendulum robot with a transport platform is envisioned for use in warehouses or other application scenarios to move goods.
- (2)
- The convergence law of hierarchical sliding mode control is improved to mitigate the jitter phenomenon of the sliding mode control system, and an adaptive function is introduced to minimize the system jitter.
- (3)
- By combining a nonlinear disturbance observer and hierarchical sliding mode control to estimate unknown external disturbances as input compensation, the system is made to control more accurately.

## 2. Materials and Methods

#### 2.1. WIPR Model

**Remark**

**1.**

**Assumption**

**1.**

#### 2.2. The Design of the Kinematic Control Law

#### 2.3. The Design of NDO

**Lemma**

**1.**

**Assumption**

**2.**

**Property**

**1.**

**Property**

**2.**

**Theorem**

**1.**

**Proof.**

#### 2.4. The Design of Improved Slide Mode Control

#### 2.5. The Design of the Forward-Rotation Subsystem

#### 2.6. The Design of the Tilt-Angle Subsystem

## 3. Simulation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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

${m}_{w}$ | Mass of each wheel |

$M$ | The total weight of the transport platform plus the pendulum |

${I}_{w}$ | The rotational inertia of each driven wheel |

${I}_{M}$ | The rotational inertia of the transport platform and the pendulum |

$d$ | The distance between the two wheels |

$L$ | The length of the pendulum |

${\tau}_{l}$ | The torque of the left wheel |

${\tau}_{r}$ | The torque of the right wheel |

$v$ | WIPR forward velocity |

$w$ | WIPR rotation velocity |

$\theta $ | WIPR Yaw angle |

$\alpha $ | The tilt angle of the pendulum |

Parameter (Unit) | Value |
---|---|

$M\left(\mathrm{kg}\right)$ | 8 |

$m\left(\mathrm{kg}\right)$ | 0.5 |

${I}_{M}\left(\mathrm{kg}\cdot {\mathrm{m}}^{2}\right)$ | 5 |

${I}_{w}\left(\mathrm{kg}\cdot {\mathrm{m}}^{2}\right)$ | 0.3 |

$d\left(\mathrm{m}\right)$ | 0.5 |

$L\left(\mathrm{m}\right)$ | 0.5 |

$r\left(\mathrm{m}\right)$ | 0.1 |

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**MDPI and ACS Style**

Hou, M.; Zhang, X.; Chen, D.; Xu, Z. Hierarchical Sliding Mode Control Combined with Nonlinear Disturbance Observer for Wheeled Inverted Pendulum Robot Trajectory Tracking. *Appl. Sci.* **2023**, *13*, 4350.
https://doi.org/10.3390/app13074350

**AMA Style**

Hou M, Zhang X, Chen D, Xu Z. Hierarchical Sliding Mode Control Combined with Nonlinear Disturbance Observer for Wheeled Inverted Pendulum Robot Trajectory Tracking. *Applied Sciences*. 2023; 13(7):4350.
https://doi.org/10.3390/app13074350

**Chicago/Turabian Style**

Hou, Ming, Xuedong Zhang, Du Chen, and Zheng Xu. 2023. "Hierarchical Sliding Mode Control Combined with Nonlinear Disturbance Observer for Wheeled Inverted Pendulum Robot Trajectory Tracking" *Applied Sciences* 13, no. 7: 4350.
https://doi.org/10.3390/app13074350