# Implementation of an Upper-Limb Exoskeleton Robot Driven by Pneumatic Muscle Actuators for Rehabilitation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Design

#### PMA-Actuated Upper-Limb Exoskeleton Robot Building

## 3. Dynamic Equations

#### 3.1. PMA Model

_{0}is the initial radius of PMA. Moreover, the input voltage to the proportional pneumatic pressure regulator and the internal pressure of the PMA is in a linear relation; the relative pressure P can be replaced by the control input voltage for control.

#### 3.2. Dynamics of the Exoskeleton Robot

_{N}Y

_{N}Z

_{N}}, is defined with the origin O being at the intersection point of the three revolute joints of the shoulder. With the reference to the initial configuration, the Z

_{N}, Y

_{N}, and X

_{N}axes are defined along the J

_{1}-, J

_{2}-, and J

_{3}-joint axis, respectively. Two local coordinate systems, l

_{1}-frame {A

_{1}_x

_{1}y

_{1}z

_{1}} and l

_{2}-frame {A

_{2}_x

_{2}y

_{2}z

_{2}} with their corresponding origins A

_{1}and A

_{2}, are attached to the respective upper arm and forearm, in which A

_{1}is coincident with the origin O, and A

_{2}is located at the elbow J

_{4}-joint.

_{i}-frame, and, thus, the kinetic energy T and the potential energy V of the exoskeleton robot including the wearer’s arm are formulated as

_{i}is the mass of the limb i (=1,2),

**L**is the length vector, ${d}_{i}$ is the position of the center of mass of the limb i, ${\mathit{s}}_{\mathbf{i}}$ is the unit vector along the i-th limb axis and is defined as ${\mathit{s}}_{\mathit{i}}={[100]}^{T}$,

_{i}**g**denotes the gravity,

**I**means the mass moment of inertia matrix of the A

_{i}_{i}in the l

_{i}-frame, and the tilde over a vector denotes a skew-symmetric matrix formed from this vector. Besides, ${}_{\mathit{i}}{}^{\mathit{N}}\mathit{R}$ describes the rotation matrix from the body-fixed l

_{i}-frame to the inertial N-frame. Moreover, a spring potential energy is also considered in (3) due to spring with the coefficient k being attached to the joint J

_{3}.

**M**is symmetric and positive definite, $\mathit{H}\dot{\mathit{\theta}}$ is the Coriolis and centrifugal vector,

**G**is the gravity vector, and $\mathit{\tau}={[{\tau}_{1}{\tau}_{2}{\tau}_{3}{\tau}_{4}]}^{T}$ is the joint torques. All the matrices, vectors, and symbol definitions in (4) can be fully referred to in the Appendix A. It is seen that the formulated equations of motion in a matrix–vector form stay compact and closed, making a model-based control design appropriate. Moreover, some useful properties for controller design exist in these matrices; ($\dot{\mathit{M}}-2\mathit{H}$) is a skew-symmetric matrix, and thus for any vector

**x**,

_{i}of the PMA i (i = 1, …4) by the corresponding joint radius r

_{i}. In the contractile force, the relative pressure P

_{i}is linearly proportional to the applied voltage u

_{i}. The contraction length Δl

_{i}of the PMA leads to the joint angle ${\theta}_{i}$, and can be related with Δl

_{i}= r

_{i}${\theta}_{i}$. The joint torque ${\tau}_{i}$ is thus expressed as

_{i}is defined as positive, the applied voltage u

_{i}to a PMA is larger than zero, and the corresponding joint radius r

_{i}is also positive, thus, the input function ${f}_{i}$ is positive.

**B**= diag[f

_{1}… f

_{4}] is a diagonal positive definite matrix. Furthermore, the dynamic equations of the exoskeleton robot are now represented with the input voltages. The controller will be designed for the input voltages.

## 4. Control Design

#### 4.1. Sliding Mode Control

**B**,

**M**,

**H**, and

**G**, and with $\left|{G}_{i}-{\widehat{G}}_{i}\right|\le {\delta}_{{G}_{i}}$, $\left|{M}_{ij}-{\widehat{M}}_{ij}\right|\le {\delta}_{{M}_{ij}}$, and $\left|{H}_{ij}-{\widehat{H}}_{ij}\right|\le {\delta}_{{H}_{ij}}$ bounded. The diagonal control gain components ${b}_{i}$ of

**B**with 0<${b}_{i\left(min\right)}$<b

_{i}<${b}_{i\left(max\right)}$ are unknown but of known bounds. Therefore, the estimated ${\widehat{b}}_{i}$ of gain b

_{i}can be naturally chosen as 0<${\sigma}_{i}^{-1}$<$\text{}{\widehat{b}}_{i}{b}_{i}^{-1}$<${\sigma}_{i}$, which is the geometric mean of the above bounds.

#### 4.2. Fuzzy Sliding Mode Control Design

**s**and the associated discontinuous reaching control always generate chattering phenomena. To eliminate the jitter problem, a fuzzy sliding mode controller (FSMC) [38] is used instead of the SMC, and thus, the reaching control with a fuzzy type is represented as

#### 4.3. Stability Analysis

**u**(t) =${\mathit{u}}_{\mathit{e}\mathit{q}}\left(t\right)$+$\text{}{\mathit{u}}_{\mathit{r}}\left(t\right)$ for the rehabilitation, and control is thus implementable.

**u**for the PMAs are synthesized based on the aforementioned FSMC design, and, thus, the PMAs produce the corresponding actuating forces to drive the exoskeleton robot to track the rehabilitation trajectory.

## 5. Realization and Discussion

#### 5.1. Rehabilitation Trajectory Planning

#### 5.2. Realization on Shoulder Joints J1, J2 and Elbow Joint J4 Rehabilitation

#### 5.3. Rehabilitation for Shoulder Internal/External Rotation

#### 5.4. Rehabilitation while Suffering from Instant Spasm and Tremor

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{1}-frame{A

_{1}_x

_{1}y

_{1}z

_{1}}, being defined as

_{2}-frame {A

_{2}_x

_{2}y

_{2}z

_{2}}.

**S**(i = 1,2) is the unit vectors, respectively, along the upper arm axis (i = 1) and the forearm axis (i = 2); ${\mathit{z}}_{\mathbf{1}}={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{T}$ defines the unit vector along the z-axis of the l

_{i}_{1}-frame.

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**Figure 6.**Membership function of the assignment of the fuzzy set for (

**a**) input variables $\left({s}_{i},{\dot{s}}_{i}\right)$; (

**b**) output variable $FSM{C}_{i}$.

DOF | Man [18] | Proposed |
---|---|---|

Elbow flexion/extension | 135°/0° | 120°/0° |

Shoulder internal/external rotation | 180°/60° | 150°/45° |

Shoulder flexion/extension | 135°/45° | 110°/30° |

Shoulder abduction/adduction | 180°/45° | 150°/30° |

FSMC | s | ||||||
---|---|---|---|---|---|---|---|

$\dot{\mathit{s}}$ | NB | NM | NS | ZO | PS | PM | PB |

NB | PB | PB | PB | PB | PM | PS | ZO |

PM | PB | PB | PB | PM | PS | ZO | NS |

NS | PB | PB | PM | ZO | ZO | NS | NM |

ZO | PB | PM | PS | ZO | NS | NM | NB |

PS | PM | PS | ZO | ZO | NM | NB | NB |

PM | PS | ZO | NS | NM | NB | NB | NB |

PB | ZO | NS | NM | NB | NB | NB | NB |

RMSE | Exercise 1 | Exercise 2 | ||
---|---|---|---|---|

$\mathbf{Shoulder}\text{}\mathbf{Joint}\text{}{\mathit{J}}_{1}$$({\mathit{\theta}}_{1})$ | $\mathbf{Shoulder}\text{}\mathbf{Joint}\text{}{\mathit{J}}_{2}$$({\mathit{\theta}}_{2})$ | $\mathbf{Elbow}\text{}\mathbf{Joint}\text{}{\mathit{J}}_{4}$$({\mathit{\theta}}_{4})$ | $\mathbf{Shoulder}\text{}\mathbf{Joint}\text{}{\mathit{J}}_{3}$$({\mathit{\theta}}_{3})$ | |

PID | 17.099 | 10.7635 | 53.3137 | 18.4418 |

SMC | 3.4161 | 5.2894 | 9.4021 | 17.5915 |

FSMC | 2.5087 | 3.5783 | 8.2311 | 12.3227 |

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**MDPI and ACS Style**

Chen, C.-T.; Lien, W.-Y.; Chen, C.-T.; Wu, Y.-C. Implementation of an Upper-Limb Exoskeleton Robot Driven by Pneumatic Muscle Actuators for Rehabilitation. *Actuators* **2020**, *9*, 106.
https://doi.org/10.3390/act9040106

**AMA Style**

Chen C-T, Lien W-Y, Chen C-T, Wu Y-C. Implementation of an Upper-Limb Exoskeleton Robot Driven by Pneumatic Muscle Actuators for Rehabilitation. *Actuators*. 2020; 9(4):106.
https://doi.org/10.3390/act9040106

**Chicago/Turabian Style**

Chen, Chun-Ta, Wei-Yuan Lien, Chun-Ting Chen, and Yu-Cheng Wu. 2020. "Implementation of an Upper-Limb Exoskeleton Robot Driven by Pneumatic Muscle Actuators for Rehabilitation" *Actuators* 9, no. 4: 106.
https://doi.org/10.3390/act9040106