# Tracking Control of Pneumatic Artificial Muscle-Activated Robot Arm Based on Sliding-Mode Control

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Experimental Setup

#### 2.2. Introduction of a Dynamic Model

#### 2.2.1. Analysis of Motion of Joint Angle and Terminal Point of a Robot Arm System

#### 2.2.2. Dynamic Math and Model of Proportional Flow Control Servo Valve of PAM Cylinder

^{2}) is the moment of inertia of the robot arm; B (kg·mm

^{2}/s) is the system damper; ${\dot{\theta}}_{1}$ is the rotary angular velocity of the robot arm in the theta axis; ${\dot{\theta}}_{1}$ is the rotary angular acceleration of the robot arm in the theta axis; ${F}_{a}$ and ${F}_{b}$ (kg·mm/s

^{2}) are, respectively, the two forces generated when the PAM cylinder is in motion; and r is the rotary radius of the base axis of the robot arm.

#### 2.3. Parameter Identification of the Dynamic Model

#### 2.3.1. Parameter Identification of the Model of the Proportional Flow Control Servo Valve

#### 2.3.2. Using a Genetic Algorithm to Find the Optimum Parameters for a Dynamic Model

#### 2.4. Extended Sliding-Mode Feedback Controller and Parameter Identification

## 3. Results and Discussion

#### 3.1. Experiment and Discussion on the Fixed Moment of Inertia of a Robot Arm

#### 3.1.1. Configuration of Sampling Time

#### 3.1.2. Investigation on the Fixed Moment of Inertia of a Robot Arm Controller

#### 3.2. Circular Trajectory Tracking Based on Robot Arm Control

#### 3.2.1. Outcomes at Zero Load

#### 3.2.2. Outcomes at a Lower Load

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Results of the tracking circular trajectory of a robot arm (the ${\theta}_{2}$ axis uses PID, and the ${\theta}_{1}$ axis uses HOSMC controllers): (

**a**) results of the tracking circular trajectory of a robot arm (the ${\theta}_{2}$ axis uses PID, and the ${\theta}_{1}$ axis uses HOSMC controllers); (

**b**) tracking error of the ${\theta}_{2}$ axis at a lower load; (

**c**) tracking error of the ${\theta}_{1}$ axis at a lower load.

**Figure 6.**Results of the tracking circular trajectory of a robot arm (the ${\theta}_{2}$ axis uses PID, and the ${\theta}_{1}$ axis uses axis uses PID): (

**a**) results of the tracking circular trajectory of a robot arm (the ${\theta}_{2}$ axis uses PID, and the ${\theta}_{1}$ axis uses PID); (

**b**) tracking error of the ${\theta}_{2}$ axis at a lower load; (

**c**) tracking error of the ${\theta}_{1}$ axis at a lower load.

Item | Type | Specifications |
---|---|---|

Proportional flow control servo valve | Developed by Festo (MPYE-5-M5-010-B) | Standard nominal flow rate (L/min): 100 Product weight (g): 290 (not containing connectors) |

PRV | Developed by Festo (VPPM-6L-L-1-G18-0L10H-V1N) | Pressure range: 0 to 10 bar Input voltage range: 0 to 10 V Feedback voltage by pressure range: 0 to 10 V |

PAM cylinder | Developed by Festo (MAS-20-300N-AA-MC-O-ER-BG) | The structure includes a contractile system and a connector of two ends, where the inside of the contractile system is a hose, and the outside is covered by a fabric mesh with high intensity. |

Power sensor | Developed by VPG load cell | A correspondent voltage generated by deformation due to the extension of the load cell can be obtained, where the force measure range is from 0 to 100 kg, and the feedback voltage signal is from 0 to 10 V. |

Pressure sensor | Developed by Festo (SPAB-P10R-G18-NB-K1) | Pressure measure range: 0 to 10 bar Feedback voltage signal: 1 to 5 V |

Optical encoder | Developed by QPhase (TSD-HB-8-1000A-H) | Can be utilized for the analysis of frequency quadruple, where the resolution of one cycle is 8000 Hz. |

Laser rangefinder | Developed by Keyence Type of IL-300 | Measure distance: 300 mm Measure range: 160 to 450 mm Precision of measure repeatability: 30 µm Output voltage range: 0 to 5 V |

θ | d | a | α |
---|---|---|---|

${\theta}_{1}$ | 0 | 0 | 90° |

${\theta}_{2}$ | 0 | ${L}_{2}$ | 0° |

$-2{\theta}_{2}$ | 0 | ${L}_{3}$ | 0° |

Parameter | Units | Value | Range |
---|---|---|---|

${P}_{atm}$ | $\mathrm{kPa}$ | 101.3 | – |

${P}_{s}$ | $\mathrm{kPa}$ | 707 | – |

γ | – | 1.4 | – |

R | $\mathrm{kJ}/\mathrm{kg}\xb7\mathrm{K}$ | 0.287 | – |

T | $\mathrm{K}$ | 293 | – |

${J}_{S}$ | $\mathrm{kg}\xb7{\mathrm{mm}}^{2}$ | – | 0–30,000 |

${J}_{L}$ | $\mathrm{kg}\xb7{\mathrm{mm}}^{2}$ | – | 200,000–400,000 |

B | $\mathrm{kg}\xb7{\mathrm{mm}}^{2}/\mathrm{s}$ | – | 0–10,000 |

Gain | – | – | 0.8–1.2 |

${L}_{0}$ | $\mathrm{mm}$ | – | 250–255 |

D | $\mathrm{mm}$ | – | 20–40 |

b | $\mathrm{mm}$ | – | 250–441.673 |

${C}_{f}$ | – | – | 0–1 |

${C}_{r}$ | – | – | 0.143–1 |

Gene | Range | Optimum Parameter |
---|---|---|

${J}_{S}$ | 0 to 30,000 | 15,960.7 |

${J}_{L}$ | 200,000 to 400,000 | 267,077 |

B | 0 to 10,000 | 7042.55 |

${L}_{0}$ | 250 to 255 | 252.446 |

D | 20 to 40 | 35.406 |

b | 250 to 441.673 | 385.02 |

${C}_{f}$ | 0 to 1 | 0.0889213 |

${C}_{r}$ | 0.14268 to 1 | 0.538054 |

$Gain$ | 0.8 to 1.2 | 0.988137 |

Parameter | Range | Optimum Parameter |
---|---|---|

Λ | 0 to 100 | 17.6953 |

G | 0 to 10,000 | 384.302 |

$\varphi $ | 0 to 10,000 | 776.93 |

Sampling Time | Max Error (Degree) | RMSE (Degree) |
---|---|---|

20 ms | 2.089 | 0.51468 |

10 ms | 0.402 | 0.12895 |

Controller | Controller Parameters |
---|---|

HOSMC | $\lambda :34.1913$, $G:897.268$, $\varphi :500$ |

PID | $kp:200$, $Ti:0.03$, $Td:\mathrm{10,000}$ |

Controller | State | Max Error (Degree) | RMSE (Degree) |
---|---|---|---|

HOSMC | Short | 0.448 | 0.11126 |

Long | 0.381 | 0.090299 | |

PID | Short | 0.523 | 0.25232 |

Long | 0.706 | 0.289 |

**Table 9.**Parameter table of the theta axis controller of a manipulator with fixed rotation inertia control.

Controller | Controller Parameters |
---|---|

HOSMC | $\lambda :34.1913$, $G:897.268$, $\varphi :500$ |

2-SMC | ${C}_{1}:20$, ${C}_{2}:5000$, $G:7000$, $\varphi :420$ |

TSMC | ${\alpha}_{1}:2.02915$, ${\alpha}_{2}:1.00049$, ${\beta}_{1}:4878.4087$, ${\beta}_{2}:64.267$, $\varphi :6.227$, $G:1100.75$ |

PID | ${k}_{p}:200$, $Ti:0.03$, $Td:\mathrm{10,000}$ |

**Table 10.**Comparison table of the tracking results on the circular trajectory at zero load based on theta axis control.

Controller | Part | Max Error | RMSE |
---|---|---|---|

PID/HOSMC | ${\theta}_{1}$ (degree) | 0.524 | 0.14781 |

Contour (mm) | 10.2528 | 3.6808 | |

PID/PID | ${\theta}_{1}$ (degree) | 0.768 | 0.30876 |

Contour (mm) | 10.0909 | 2.9096 |

**Table 11.**Comparison table of the tracking results on the circular trajectory at a lower load of an robot arm.

Controller | Part | Max Error | RMSE |
---|---|---|---|

PID/HOSMC | ${\theta}_{1}$ (degree) | 0.524 | 0.14781 |

Contour (mm) | 10.2528 | 3.6808 | |

PID/PID | ${\theta}_{1}$ (degree) | 0.768 | 0.30876 |

Contour (mm) | 10.0909 | 2.9096 |

**Table 12.**Comparison table of robustness investigation between different robot arm controllers at a lower load.

Controller | Max Error (Degree) | RMSE (Degree) |
---|---|---|

HOSMC | 0.399 | 0.10167 |

PID | 4.086 | 1.6673 |

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**MDPI and ACS Style**

Lin, C.-J.; Sie, T.-Y.; Chu, W.-L.; Yau, H.-T.; Ding, C.-H. Tracking Control of Pneumatic Artificial Muscle-Activated Robot Arm Based on Sliding-Mode Control. *Actuators* **2021**, *10*, 66.
https://doi.org/10.3390/act10030066

**AMA Style**

Lin C-J, Sie T-Y, Chu W-L, Yau H-T, Ding C-H. Tracking Control of Pneumatic Artificial Muscle-Activated Robot Arm Based on Sliding-Mode Control. *Actuators*. 2021; 10(3):66.
https://doi.org/10.3390/act10030066

**Chicago/Turabian Style**

Lin, Chih-Jer, Ting-Yi Sie, Wen-Lin Chu, Her-Terng Yau, and Chih-Hao Ding. 2021. "Tracking Control of Pneumatic Artificial Muscle-Activated Robot Arm Based on Sliding-Mode Control" *Actuators* 10, no. 3: 66.
https://doi.org/10.3390/act10030066