# Admittance-Controlled Teleoperation of a Pneumatic Actuator: Implementation and Stability Analysis

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## Abstract

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## 1. Introduction

## 2. Experimental Setup

## 3. Control Architecture

#### 3.1. Admittance Control

#### 3.2. Position Controller

## 4. Simulation Studies

## 5. Stability Analysis

#### 5.1. Calculation of Lyapunov Exponents

#### 5.2. Stability Analysis of the Proposed Control System

#### 5.3. Parametric Stability Analysis

## 6. Experimental Results

**Experiment 1:**In this experiment, the operator moves the master and, at the same time, a human subject located at the slave side applies a force to the pneumatic actuator. The external force passes through the admittance model (10), having parameters set as $M=10\text{}\mathrm{Kg}$, $B=50\text{}\mathrm{Ns}/\mathrm{m}$ and $K=250\text{}\mathrm{N}/\mathrm{m}$. The desired position trajectory originating from the master, ${x}_{m}$, is shown in Figure 6a. Figure 6b shows the external force. The modified desired trajectory is shown in Figure 6c. It is evident that the admittance control module effectively adjusts the primary desired trajectory according to the imposed external force. This figure also compares the modified desired trajectory with the actual position of the actuator and illustrates their reasonable agreement. Figure 6d shows the control signal, which is also reasonable and unsaturated. This experiment shows successful application of admittance control with soft stiffness.

**Experiment 2:**In this experiment, the parameters of the admittance model are set to $M=10\text{}\mathrm{Kg}$, $B=50\text{}\mathrm{Ns}/\mathrm{m}$ and $K={10}^{4}\text{}\mathrm{N}/\mathrm{m}$. The goal is to study the behaviour of the system when the stiffness of admittance model is set high. The actuator is subject to an external force with the magnitude of 100 N applied by a human subject. As Figure 7a shows, the primary desired position is fixed during the experiment. The external force is shown in Figure 7b. The admittance model in (10) converts the external force to small displacement as shown in Figure 7c. For external force with 100 N magnitude, the change in primary desired trajectory is about 0.01 m. By comparing Figure 7c and Figure 6c, the effect of changes in the parameters of the admittance control module can be studied. Figure 7c also shows the actual position of slave actuator, ${\mathit{x}}_{\mathit{s}}$. Figure 7d shows the control signal corresponding to the position tracking shown in Figure 7c.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Test setup of unilateral admittance-controlled of the pneumatic actuator; (

**b**) schematic diagram of pneumatic system [21].

**Figure 3.**Admittance control variables pertaining to step tracking: (

**a**) primary desired trajectory provided by the master manipulator, ${x}_{m}$; (

**b**) external force, ${F}_{ext}$; (

**c**) displacement corresponding to the external force, ${x}_{ext}$; (

**d**) desired trajectory achieved from admittance model, ${x}_{d}$.

**Figure 4.**Step tracking with SMC: (

**a**) piston position; (

**b**) chamber pressures; (

**c**) control signal; (

**d**) position error.

**Figure 6.**Experimental study of low Stiffness admittance model while tracking a human-guided trajectory: (

**a**) primary desired position by master manipulator, ${x}_{m}$; (

**b**) external force, ${F}_{ext}$; (

**c**) modified desired trajectory, ${x}_{d}$, versus position of actuator, ${x}_{s}$; (

**d**) control signal, $u$.

**Figure 7.**Experimental study of high stiffness admittance model while tracking a human-guided trajectory: (

**a**) primary desired position by the master manipulator, ${x}_{m}$; (

**b**) external force, ${F}_{ext}$; (

**c**) modified desired trajectory, ${x}_{d}$, versus position of actuator, ${x}_{s}$; (

**d**) control signal, $u$.

**Table 1.**Parameters of Pneumatic Actuator [26].

Parameter | Symbol (Unit) | Value |
---|---|---|

Actuator stroke | $L\text{}\left(\mathrm{m}\right)$ | $0.5$ |

Piston annulus area | $A$ (${\mathrm{cm}}^{2}$) | $10.6$ |

Valve time constant | $\tau \text{}\left(\mathrm{ms}\right)$ | $4.2$ |

Valve orifice area gradient | $w\text{}\left({\mathrm{mm}}^{2}/\mathrm{mm}\right)$ | $22.6$ |

Supply pressure | ${P}_{s}$ ($\mathrm{bar}$) | 5 |

Valve coefficient of discharge | ${C}_{d}$ | $0.7$ |

Static friction | ${F}_{s}\text{}\left(\mathrm{N}\right)$ | 38.5 |

Coulomb friction | ${F}_{c}\text{}\left(\mathrm{N}\right)$ | 32.9 |

Ratio of specific heats | $\gamma $ | $1.4$ |

Ideal gas constant | $R\text{}\left(\mathrm{J}/\mathrm{kgK}\right)$ | $287$ |

Valve critical pressure ratio | ${P}_{cr}$ | $0.2$ |

Valve spool gain | ${K}_{v}\text{}\left(\mathrm{mm}/\mathrm{V}\right)$ | $0.25$ |

Damping coefficient of bristle | ${\sigma}_{1}\left(\mathrm{N}/\mathrm{m}/\mathrm{s}\right)$ | 93.13 |

Spring constant of bristle | ${\sigma}_{0}\text{}\left(\mathrm{N}/\mathrm{m}\right)$ | 4500 |

Mass of moving parts | $m$ ($\mathrm{kg}$) | $1.91$ |

Compressibility correction factor | $\alpha $ | $1.2$ |

Temperature of air | $T\text{}\left(\mathrm{K}\right)$ | $300$ |

Atmospheric pressure | ${P}_{a}\text{}\left(\mathrm{bar}\right)$ | 1 |

Viscous damping coefficient | b (Ns/m) | 70 |

Stribeck velocity | ${v}_{sv}\text{}\left(\mathrm{m}/\mathrm{s}\right)$ | 0.02 |

Cylinder inactive volume | ${V}_{0}\text{}\left({\mathrm{m}}^{3}\right)$ | $1.64\times {10}^{-4}$ |

${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ |
---|---|---|---|---|

0.0 | 0.0 | −0.1 | −17.8 | −18.0 |

${\mathit{K}}_{\mathit{e}\mathit{x}\mathit{t}}\left(\mathbf{N}/\mathbf{m}\right)$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ | ${\mathit{\lambda}}_{6}$ | ${\mathit{\lambda}}_{7}$ |
---|---|---|---|---|---|---|---|

10 | 0.0 | 0.0 | −0.1 | −17.7 | −18.0 | −75.7 | −229.3 |

50 | 0.0 | 0.0 | −0.1 | −17.8 | −18.0 | −75.7 | −229.3 |

100 | 0.0 | 0.0 | −0.1 | −17.8 | −18.0 | −75.5 | −229.4 |

150 | 0.0 | 0.0 | −0.1 | −17.8 | −18.1 | −75.5 | −229.5 |

200 | 0.0 | 0.0 | −0.1 | −17.9 | −18.1 | −75.4 | −229.6 |

300 | 0.0 | 0.0 | −0.1 | −17.9 | −18.1 | −75.3 | −229.7 |

600 | 0.0 | 0.0 | −0.1 | −18.1 | −18.6 | −74.8 | −230.1 |

$\mathit{\delta}\text{}(\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{s})$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ | ${\mathit{\lambda}}_{6}$ | ${\mathit{\lambda}}_{7}$ |
---|---|---|---|---|---|---|---|

20 | 0.0 | 0.0 | −0.1 | −19.8 | −64.6 | −64.7 | −191.4 |

30 | 0.0 | 0.0 | −0.1 | −26.0 | −56.1 | −56.1 | −202.3 |

40 | 0.0 | 0.0 | −0.1 | −34.3 | −47.8 | −47.8 | −210.7 |

60 | 0.0 | 0.0 | −0.1 | −31.7 | −31.9 | −55.0 | −222.2 |

80 | 0.0 | 0.0 | −0.1 | −17.8 | −18.1 | −75.5 | −229.4 |

100 | 0.0 | 0.0 | −0.4 | −13.2 | −14.5 | −83.2 | −230.0 |

120 | 0.0 | 0.0 | −0.7 | −7.5 | −34.4 | −70.9 | −228.0 |

140 | 56.6 | - | - | - | - | - | - |

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**MDPI and ACS Style**

Garmsiri, N.; Sun, Y.; Sekhavat, P.; Yang, C.X.; Sepehri, N. Admittance-Controlled Teleoperation of a Pneumatic Actuator: Implementation and Stability Analysis. *Actuators* **2020**, *9*, 103.
https://doi.org/10.3390/act9040103

**AMA Style**

Garmsiri N, Sun Y, Sekhavat P, Yang CX, Sepehri N. Admittance-Controlled Teleoperation of a Pneumatic Actuator: Implementation and Stability Analysis. *Actuators*. 2020; 9(4):103.
https://doi.org/10.3390/act9040103

**Chicago/Turabian Style**

Garmsiri, Naghmeh, Yuming Sun, Pooya Sekhavat, Cai Xia Yang, and Nariman Sepehri. 2020. "Admittance-Controlled Teleoperation of a Pneumatic Actuator: Implementation and Stability Analysis" *Actuators* 9, no. 4: 103.
https://doi.org/10.3390/act9040103