# Disturbance-Estimated Adaptive Backstepping Sliding Mode Control of a Pneumatic Muscles-Driven Ankle Rehabilitation Robot

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## Abstract

**:**

## 1. Introduction

## 2. The Ankle Rehabilitation Robot

## 3. Control Strategy

#### 3.1. Backstepping Sliding Mode Control

**Step 1:**Define the tracking error ${e}_{1}={q}_{1}-{q}_{d}$, then ${\dot{e}}_{1}={\dot{q}}_{1}-{\dot{q}}_{d}={q}_{2}-{\dot{q}}_{d}$, and define the Lyapunov function as

**Step 2:**Define the switch function as

#### 3.2. Adaptive Backstepping Sliding Mode Control

#### 3.3. Stability Analysis

**Remark**

**1.**

**Proof.**

**Remark**

**2.**

**Proof.**

## 4. Experimental and Results Discussion

#### 4.1. Step Response

#### 4.2. Sine Trajectory Tracking Experiment (without Subject)

#### 4.3. Robustness Test with Human Subjects

#### 4.4. Sudden External Disturbance

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Kinematics of the designed 2-DOF ankle rehabilitation robot: (

**a**) structure model, (

**b**) geometrical diagram.

**Figure 5.**Actuator position tracking results and errors in step response experiment with robot controlled by BS-SMC and ABS-SMC respectively.

**Figure 8.**Actuator position tracking results with subject 1: (

**a**) actuator position tracking results; (

**b**) the actuator tracking errors; (

**c**) the estimated external torque (using ABS-SMC) and (

**d**) the control output tuning processing via ABS-SMC disturbance estimation.

**Table 1.**Statistical analysis of actuator position tracking errors under different control methods (without subject).

Methods | Maximum Error (mm) | Average Error (mm) | |||||
---|---|---|---|---|---|---|---|

A1 | A2 | A3 | A1 | A2 | A3 | ||

Position tracking results | ABS-SMC | 0.84 | 1.05 | 0.93 | 0.39 | 0.47 | 0.46 |

BS-SMC | 1.48 | 1.64 | 1.55 | 0.64 | 0.72 | 0.75 |

**Table 2.**Statistical analysis of end-effector angle tracking errors under different control methods (without subject).

Methods | Maximum Error (°) | Average Error (°) | |||
---|---|---|---|---|---|

θ | φ | θ | φ | ||

Angle tracking results | ABS-SMC | 0.69 | 0.68 | 0.19 | 0.20 |

BS-SMC | 1.48 | 1.41 | 0.44 | 0.44 |

Participants | Gender | Age | Height (cm) | Weight (kg) |
---|---|---|---|---|

Subject 1 | male | 23 | 175 | 65 |

Subject 2 | male | 22 | 178 | 64 |

Subject 3 | female | 23 | 160 | 49 |

Subject 4 | female | 24 | 165 | 50 |

Subject 5 | male | 25 | 180 | 70 |

**Table 4.**Statistical analysis of actuator position tracking errors under different control methods (with five subjects).

Participants | Methods | Maximum Error (mm) | Average Error (mm) | |||||
---|---|---|---|---|---|---|---|---|

A1 | A2 | A3 | A1 | A2 | A3 | |||

Position tracking results | Subject 1 | ABS-SMC | 1.10 | 1.13 | 1.33 | 0.43 | 0.47 | 0.49 |

BS-SMC | 2.71 | 3.60 | 3.24 | 1.30 | 1.48 | 1.56 | ||

Subject 2 | ABS-SMC | 1.52 | 2.07 | 1.76 | 0.39 | 0.47 | 0.37 | |

BS-SMC | 3.71 | 4.67 | 4.20 | 1.19 | 1.43 | 1.07 | ||

Subject 3 | ABS-SMC | 1.53 | 2.02 | 1.81 | 0.40 | 0.47 | 0.37 | |

BS-SMC | 3.90 | 5.01 | 4.19 | 1.17 | 1.46 | 1.10 | ||

Subject 4 | ABS-SMC | 1.77 | 2.07 | 1.88 | 0.39 | 0.48 | 0.38 | |

BS-SMC | 3.86 | 5.22 | 5.30 | 1.22 | 1.27 | 1.29 | ||

Subject 5 | ABS-SMC | 1.39 | 1.97 | 1.66 | 0.39 | 0.47 | 0.37 | |

BS-SMC | 3.74 | 4.96 | 4.63 | 1.14 | 1.34 | 1.09 |

Participants | Methods | Maximum Error (°) | Average Error (°) | |||
---|---|---|---|---|---|---|

θ | φ | θ | φ | |||

Angle tracking results | Subject 1 | ABS-SMC | 0.90 | 0.99 | 0.20 | 0.39 |

BS-SMC | 2.04 | 2.50 | 0.54 | 0.75 | ||

Subject 2 | ABS-SMC | 1.12 | 0.99 | 0.29 | 0.28 | |

BS-SMC | 2.25 | 2.18 | 0.50 | 0.78 | ||

Subject 3 | ABS-SMC | 1.21 | 1.18 | 0.29 | 0.34 | |

BS-SMC | 2.91 | 2.36 | 0.67 | 0.78 | ||

Subject 4 | ABS-SMC | 1.41 | 1.13 | 0.43 | 0.34 | |

BS-SMC | 3.32 | 2.75 | 0.63 | 0.66 | ||

Subject 5 | ABS-SMC | 1.14 | 0.89 | 0.27 | 0.28 | |

BS-SMC | 2.97 | 2.17 | 0.92 | 0.94 |

Man-Made Resistance | Size (N) | Duration (s) | |
---|---|---|---|

Phase i (P i) | None | 0 | 0 |

Phase ii (P ii) | Applied | 10 | 2 |

Phase iii (P iii) | Applied | 30 | 2 |

Phase iv (P iv) | Applied | 30 | 3 |

**Table 7.**Comparison of existing control methods and the proposed method for PMs-driven parallel rehabilitation robot. (*, unknown).

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## Share and Cite

**MDPI and ACS Style**

Ai, Q.; Zhu, C.; Zuo, J.; Meng, W.; Liu, Q.; Xie, S.Q.; Yang, M.
Disturbance-Estimated Adaptive Backstepping Sliding Mode Control of a Pneumatic Muscles-Driven Ankle Rehabilitation Robot. *Sensors* **2018**, *18*, 66.
https://doi.org/10.3390/s18010066

**AMA Style**

Ai Q, Zhu C, Zuo J, Meng W, Liu Q, Xie SQ, Yang M.
Disturbance-Estimated Adaptive Backstepping Sliding Mode Control of a Pneumatic Muscles-Driven Ankle Rehabilitation Robot. *Sensors*. 2018; 18(1):66.
https://doi.org/10.3390/s18010066

**Chicago/Turabian Style**

Ai, Qingsong, Chengxiang Zhu, Jie Zuo, Wei Meng, Quan Liu, Sheng Q. Xie, and Ming Yang.
2018. "Disturbance-Estimated Adaptive Backstepping Sliding Mode Control of a Pneumatic Muscles-Driven Ankle Rehabilitation Robot" *Sensors* 18, no. 1: 66.
https://doi.org/10.3390/s18010066