# Performance Analysis of a Robust Controller with Neural Network Algorithm for Compliance Tendon–Sheath Actuation Lower Limb Exoskeleton

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description and Modeling of CLLE

#### 2.1. Dynamic Model and Friction Model

#### 2.2. Model of CLLE

## 3. Controller Design

#### 3.1. SMC Design

#### 3.2. Neural Network Framework

#### 3.3. Control Law

#### 3.4. Proof of Lyapunov Stability

## 4. Model and Controller Identification

#### 4.1. Model Parameters Identification

#### 4.2. Design of SMPIC and RASMC

## 5. Parameters and Numerical Simulations

## 6. Results and Discussions

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The simplified mechanic structure schematic of CLLE consists of two active DOFs and one passive DOF in each leg. The hip joint drives the thigh, the knee joint drives the shank, and the ankle joint is passively driven by the foot for rehabilitation training. Where ${l}_{1}$ and ${l}_{c1}$ are the length of thigh and the distance of centroid position from knee joint, ${l}_{2}$ and ${l}_{c2}$ are the length of shank and the distance of centroid position from knee joint. ${m}_{1}$ and ${m}_{2}$ are the mass of thigh linkage and shank linkage, ${\theta}_{1}$ and ${\theta}_{2}$ are the initial angle of joints.

**Figure 2.**The schematic diagram of a double tendon–sheath. In the figure, tendons a and b are attached to the input side pulley and output side pulley with a pull-pull configuration ${\tau}_{in}$ and ${\tau}_{out}$ are the input and output torque, ${\theta}_{in}$ and ${\theta}_{out}$ are the input and output angles, ${\omega}_{in}$ and ${\omega}_{out}$ are the angular velocities of input and output, and ${r}_{1}$ and ${r}_{2}$ are the radius of the pulleys. The tendon transmits the force and torque by mechanical displacement in the inner tendon.

**Figure 3.**The block diagram of CLLE. (

**a**) DC motor and encoder of the electrical part; (

**b**) the transmission system of the CTSA; (

**c**) the block diagram of mechanical part.

**Figure 6.**The tracking response of the joints. (

**a**) the response of ${\theta}_{1}$ under case 0 condition; (

**b**) the response of ${\theta}_{2}$ under case 0 condition.

**Figure 7.**The tracking error of the joints. (

**a**) the tracking error of ${\theta}_{1}$ under case 0 condition; (

**b**) the tracking error of ${\theta}_{2}$ under case 0 condition.

**Figure 8.**The time history of the joints. (

**a**) The time history of hip torque under case 0 condition and (

**b**) The time history of knee torque under case 0 condition.

**Figure 9.**The tracking response of the joints. (

**a**) the response of ${\theta}_{1}$ under case 1 condition; (

**b**) the response of ${\theta}_{2}$ under case 1 condition.

**Figure 10.**The tracking error of the joints. (

**a**) the tracking error of ${\theta}_{1}$ under case 1 condition; (

**b**) the tracking error of ${\theta}_{2}$ under case 1 condition.

**Figure 11.**The time history of the joints. (

**a**) The time history of hip torque under case 1 condition and (

**b**) The time history of knee torque under case 1 condition.

**Figure 12.**The tracking response of the joints. (

**a**) the response of ${\theta}_{1}$ under case 2 condition; (

**b**) the response of ${\theta}_{2}$ under case 2 condition.

**Figure 13.**The tracking error of the joints. (

**a**) the tracking error of ${\theta}_{1}$ under case 2 condition; (

**b**) the tracking error of ${\theta}_{2}$ under case 2 condition.

**Figure 14.**The time history of the joints. (

**a**) The time history of hip torque under case 2 condition and (

**b**) The time history of knee torque in case 2 condition.

**Figure 15.**2-D trajectory tracking of the hip and knee joints. (

**a**) 2-D trajectory tracking of the hip and knee under case 0; (

**b**) 2-D trajectory tracking under case 1; (

**c**) 2-D trajectory tracking under case 2.

**Figure 16.**The tracking response of SMPIC of the joints under Case Ⅲ. (

**a**) the response of ${\theta}_{1}$ and (

**b**) the response of ${\theta}_{2}$ under the disturbances with different magnitudes.

**Figure 17.**The tracking response of RASMC of the joints under Case Ⅲ. (

**a**) the response of ${\theta}_{1}$ and (

**b**) the response of ${\theta}_{2}$ under the disturbances with different magnitudes.

**Figure 18.**The tracking response of RBFNN-SMC of the joints under Case Ⅲ. (

**a**) the response of ${\theta}_{1}$ and (

**b**) the response of ${\theta}_{2}$ under the disturbances with different magnitudes.

**Figure 19.**Comparison of controller response speed under step signals (

**a**) the response speed of ${\theta}_{1}$ and (

**b**) the response speed of ${\theta}_{2}$.

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

${m}_{1}$ | 8 (kg) | ${l}_{\mathrm{c}1}$ | 0.25 (m) |

${m}_{2}$ | 4 (kg) | ${l}_{\mathrm{c}2}$ | 0.2 (m) |

${l}_{1}$ | 0.5 (m) | $g$ | $9.8(\mathrm{kg}\xb7\mathrm{m}/{\mathrm{s}}^{2}$) |

${l}_{2}$ | 0.4 (m) |

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

${T}_{0}$ | 80 (N-m) | $\mu $ | 0.1333 |

$\mathsf{\Theta}$ (initial) | Pi/2 (rad) | ${r}_{1}$ | 65 (mm) |

${r}_{2}$ | 65 (mm) |

Condition Case 0 | MAPE | MSE | MAE | RMSE | ISE |
---|---|---|---|---|---|

proposed method | (0.0087, 0.0074) | (0.0001, 0.0001) | (0.0025, 0.0012) | (0.0106, 0.0097) | (0.0112, 0.0094) |

SMPIC | (0.0479, 0.0804) | (0.0003, 0.0003) | (0.0051, 0.0052) | (0.0189, 0.0198) | (0.0359, 0.0356) |

RASMC | (0.0908, 0.1106) | (0.0004, 0.0005) | (0.0115, 0.0114) | (0.0208, 0.0226) | (0.0433, 0.0511) |

Condition Case 1 | MAPE | MSE | MAE | RMSE | ISE |
---|---|---|---|---|---|

proposed method | (0.0230, 0.0277) | (0.0001, 0.0020) | (0.0035, 0.0039) | (0.0100, 0.0457) | (0.0101, 0.2091) |

SMPIC | (0.0504, 0.1575) | (0.0004, 0.0105) | (0.0049, 0.0201) | (0.0210, 0.1127) | (0.0443, 1.0549) |

RASMC | (0.1074, 0.1107) | (0.0003, 0.0081) | (0.0072, 0.0184) | (0.0178, 0.0902) | (0.0316, 0.8191) |

Condition Case 2 | MAPE | MSE | MAE | RMSE | ISE |
---|---|---|---|---|---|

proposed method | (0.0291, 0.0255) | (0.0001, 0.0020) | (0.0040, 0.0039) | (0.0121, 0.0561) | (0.0146, 0.3147) |

SMPIC | (0.0700, 0.2569) | (0.0004, 0.0105) | (0.0061, 0.0201) | (0.0321, 0.1772) | (0.0486, 1.4189) |

RASMC | (0.1160, 0.1600) | (0.0004, 0.0081) | (0.0094, 0.0184) | (0.0216, 0.1103) | (0.0467, 1.2163) |

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## Share and Cite

**MDPI and ACS Style**

He, H.; Xi, R.; Gong, Y.
Performance Analysis of a Robust Controller with Neural Network Algorithm for Compliance Tendon–Sheath Actuation Lower Limb Exoskeleton. *Machines* **2022**, *10*, 1064.
https://doi.org/10.3390/machines10111064

**AMA Style**

He H, Xi R, Gong Y.
Performance Analysis of a Robust Controller with Neural Network Algorithm for Compliance Tendon–Sheath Actuation Lower Limb Exoskeleton. *Machines*. 2022; 10(11):1064.
https://doi.org/10.3390/machines10111064

**Chicago/Turabian Style**

He, Haimin, Ruru Xi, and Youping Gong.
2022. "Performance Analysis of a Robust Controller with Neural Network Algorithm for Compliance Tendon–Sheath Actuation Lower Limb Exoskeleton" *Machines* 10, no. 11: 1064.
https://doi.org/10.3390/machines10111064