# Extended State Observer-Based Sliding Mode Control Design of Two-DOF Lower Limb Exoskeleton

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## Abstract

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## 1. Introduction

- (I)
- A linear extended state observer (LESO) is used to estimate the unmeasurable angular velocity of two joints and the model uncertainties in the exoskeleton Lagrangian model, which can avoid the numerical differentiation of the encoder data for the angular velocity estimation. In fact, the LESO is a high-gain observer that guarantees a satisfactory estimation error in the exoskeleton inner-loop by regulating the observer bandwidth.
- (II)
- A sliding mode controller is designed to improve the tracking performance of the passive control mode of human–exoskeleton cooperative motion under model uncertainties and the unknown angular velocity of the exoskeleton. Meanwhile, the sliding mode controller guarantees that the joint tracking error converges to a small-enough zero neighborhood by regulating the control gains, which is easily realized in the experimental bench.

## 2. Exoskeleton Dynamic Model

**Remark**

**1.**

**Assumption**

**1.**

## 3. Linear ESO Design

**Assumption**

**2.**

**Theorem**

**1.**

**Proof.**

## 4. Sliding Mode Control

**Remark**

**2.**

## 5. Simulation

## 6. Experiment Verification

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The hip angle response of simulation (${\theta}_{1d}$: the angle demand, ${\theta}_{1}$: the angle response of exoskeleton, ${\hat{\theta}}_{1}$: the angle estimation of the LESO).

**Figure 4.**The knee angle response of simulation (${\theta}_{2d}$: the angle demand, ${q}_{2}={\theta}_{2}$: the angle response of exoskeleton, ${\hat{q}}_{2}={\hat{\theta}}_{2}$: the angle estimation of the LESO).

**Figure 5.**The lumped disturbance estimation of two joints in simulation (${\Delta}_{i}\{i=1,2\}$: the lumped disturbance of hip and knee, ${\hat{\Delta}}_{i}\{i=1,2\}$: the disturbance estimation of the LESO).

**Figure 6.**The angular velocity estimation of two joints in simulation (${\dot{\theta}}_{i}\{i=1,2\}$: the angular velocity of hip and knee, ${\hat{\dot{\theta}}}_{i}\{i=1,2\}$: the angular velocity estimation of the LESO).

**Figure 7.**The control torque of two joints in simulation (${\tau}_{1}$: the hip control torque, ${\tau}_{2}$: the knee control torque).

**Figure 8.**The 2-DOF lower limb exoskeleton platform, and the color arrows represent the signal transmission between two modules.

**Figure 10.**The hip angle response of experiment (${\theta}_{1d}$: the angle demand, ${\theta}_{1}$: the angle response of exoskeleton, ${\hat{\theta}}_{1}$: the angle estimation of the LESO).

**Figure 11.**The knee angle response of experiment (${\theta}_{2d}$: the angle demand, ${\theta}_{2}$: the angle response of exoskeleton, ${\hat{\theta}}_{1}$: the angle estimation of the LESO).

**Figure 12.**The motor torque of two joints in the experiment (${\tau}_{1}$: the hip motor torque, ${\tau}_{2}$: the knee motor torque).

**Figure 13.**The human–exoskeleton interaction torques of two joints (${\tau}_{1ext}$: the human–exoskeleton interaction torque of hip joint, ${\tau}_{2ext}$: the human–exoskeleton interaction torque of knee joint).

Parameter | Symbol | Parameter | Symbol |
---|---|---|---|

Thigh weight | ${m}_{th}$ | Shank weight | ${m}_{sh}$ |

Thigh length | ${a}_{th}$ | Shank length | ${a}_{sh}$ |

Thigh centroid length | ${l}_{th}$ | Shank centroid length | ${l}_{sh}$ |

Thigh moment of inertia | ${I}_{th}$ | Shank moment of inertia | ${I}_{sh}$ |

Component | Brand | Number |
---|---|---|

Servo motor | GDM1-100N2/120N2 | 2 |

Motor driver | Elmo-G-SOLHOR15/100EE | 2 |

Absolute encoders | INC-4-150/3-125 | 2 |

3-D force sensors | JNSH-2-10kg-BSQ-12 | 4 |

Controller | NI cRIO-9035 | 1 |

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**MDPI and ACS Style**

Zhang, J.; Gao, W.; Guo, Q.
Extended State Observer-Based Sliding Mode Control Design of Two-DOF Lower Limb Exoskeleton. *Actuators* **2023**, *12*, 402.
https://doi.org/10.3390/act12110402

**AMA Style**

Zhang J, Gao W, Guo Q.
Extended State Observer-Based Sliding Mode Control Design of Two-DOF Lower Limb Exoskeleton. *Actuators*. 2023; 12(11):402.
https://doi.org/10.3390/act12110402

**Chicago/Turabian Style**

Zhang, Jiyu, Wei Gao, and Qing Guo.
2023. "Extended State Observer-Based Sliding Mode Control Design of Two-DOF Lower Limb Exoskeleton" *Actuators* 12, no. 11: 402.
https://doi.org/10.3390/act12110402