# A New Adaptive Synergetic Control Design for Single Link Robot Arm Actuated by Pneumatic Muscles

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## Abstract

**:**

## 1. Introduction

- Forming the extended system of differential equations, which reflects different operations such as achieving the set values, coordinating observing, optimization, suppressing the disturbances etc.
- Synthesizing “external” controls which ensure a reduction in the extra degrees of freedom of the extended system with respect to the final manifold. The motion of the representing point is described by the equations of the system’s “internal” dynamics.
- Synthesizing the “internal” controls by means of forming the links between the “internal” coordinates of the system. These links ensure the reaching of the control aim.
- The synergetic controller directs the trajectories of the system to move onto the manifold from any initial points to their corresponding equilibrium points.

- ❑
- To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis.
- ❑
- To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm.
- ❑
- To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis.
- ❑
- To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers’ design parameters towards better performance of controllers.

## 2. The Dynamic Model of the PAM-Actuated Single-Link Robot Arm

## 3. Classical and Adaptive Synergetic Control Design for Single Arm PAM-Actuated Robot

#### 3.1. Synergetic Control Design

**Proof.**

**Theorem**

**1.**

#### 3.2. Design of Adaptive Synergetic Control for Single-Link Robot Arm

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Theorem**

**2.**

## 4. Improvement of Controllers’ Performances Based on PSO Technique

## 5. Computer Simulation

**Scenario I: PAM-actuated Robot Arm based on CSC**

**Scenario II: PAM-actuated Robot Arm Based on ASC**

**Scenario III: Validation of Proposed Controller**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic Diagram of the Proposed Classical and Adaptive Synergetic Controller for Single-link Robot Arm actuated by Pneumatic Artificial Muscles.

**Figure 9.**Behaviors of angular velocities for PAM actuated robot arm system controlled with optimal and non-optimal CSC.

**Figure 11.**Behaviors of linear angular positions for PAM actuated Robot Arm based on optimal and non-optimal ASC.

**Figure 15.**Behaviors of linear angular positions for PAM actuated Robot based on optimal ASC and optimal CSC.

**Figure 16.**Actual and Estimated Viscosity Coefficients for Bicep muscle of PAM actuated Robot based on ASC.

**Figure 17.**Actual and Estimated Spring Coefficient for Bicep muscle of PAM actuated robot based on ASC.

**Figure 18.**The tracking response based on optimal ASC and the compared controller of reference [16].

${Z}_{i}$ | ${f}_{i}$ | ${b}_{i}$ |

${z}_{1}=sin\theta $ | ${f}_{1}=\left(a{F}_{0}+a{F}_{1}{P}_{ob}-MgL\right)/I$ | ${b}_{1}=a{F}_{1}/I$ |

${z}_{2}=sin\theta \left(cos\theta -1\right)$ | ${f}_{2}={a}^{2}\left({K}_{o}+{K}_{1}{P}_{ob}\right)/I$ | ${b}_{2}={a}^{2}{K}_{1}/I$ |

${z}_{3}=si{n}^{2}\theta .\dot{\theta}$ | ${f}_{3}=-{a}^{2}\left({B}_{0b}+{B}_{1b}{P}_{ob}\right)/I$ | ${b}_{3}=-{a}^{2}{B}_{1b}/I$ |

${z}_{4}=1+cos\theta $ | ${f}_{4}=ar\left({K}_{o}+{K}_{1}{P}_{0t}\right)/I$ | ${b}_{4}=-ar{K}_{1}/I$ |

${z}_{5}=sin\theta .\dot{\theta}$ | ${f}_{5}=-ar\left({B}_{0t}+{B}_{1t}{P}_{0t}\right)/I$ | ${b}_{5}=ar{B}_{1t}/I$ |

${z}_{6}=1$ | ${f}_{6}=\left(-r{F}_{0}-r{F}_{1}{P}_{0t}\right)/I$ | ${b}_{6}=r{F}_{1}/I$ |

Parameters of PSO Technique | Value |
---|---|

The inertia coefficient $w$ | $1.4$ |

The personal acceleration coefficient ${C}_{1}$ | $2$ |

The social acceleration coefficient ${C}_{2}$ | $2$ |

The swarm size (population size) | 30 |

The number of iteration | 300 |

Coefficient Description | Value |
---|---|

Nominal force exerted by PAM ${F}_{0}$ | $0.986\times {10}^{2}\mathrm{N}$ |

Variation in force exerted by PAM ${F}_{1}$ | $0.803\mathrm{N}$ |

Bicep/nominal viscosity coefficient ${B}_{0b}$ | $1.35\left(\mathrm{N}.\mathrm{s}/\mathrm{m}\right)$ |

Bicep/variation in viscosity coefficient ${B}_{1b}$ | $4.66\times {10}^{-3}\left(\mathrm{N}.\mathrm{s}/\mathrm{m}\right)$ |

Tricep/nominal viscosity coefficient ${B}_{0t}$ | $4.03\times {10}^{-1}\left(\mathrm{N}.\mathrm{s}/\mathrm{m}\right)$ |

Tricep/variation in viscosity coefficient ${B}_{1t}$ | $12.0\times {10}^{-4}\left(\mathrm{N}.\mathrm{s}/\mathrm{m}\right)$ |

Nominal spring coefficient ${k}_{0}$ | $6.51\left(\mathrm{N}/\mathrm{m}\right)$ |

Variation in spring coefficient ${k}_{1}$ | $2.12\times {10}^{-2}\left(\mathrm{N}/\mathrm{m}\right)$ |

Nominal bicep pressure ${P}_{ob}$ | $510.4\mathrm{kPa}$ |

Nominal tricep pressure ${P}_{ot}$ | $400\mathrm{kPa}$ |

Mass M | $20\mathrm{kg}$ |

The distance from mass center to the joint L | $0.46\mathrm{m}$ |

The distance from PAM attached point to the joint axis a | $0.0762\mathrm{m}$ |

Pulley radius r | $0.0508\mathrm{m}$ |

Gravity Acceleration $g$ | $9.8\mathrm{m}/{\mathrm{s}}^{2}$ |

Controller | Optimal Values | Trial and Error Values | ||
---|---|---|---|---|

Coefficient | Value | Coefficient | Value | |

$\mathrm{CSC}$ | ${c}_{c}$ | $0.15$ | ${c}_{c}$ | $1$ |

$\mathrm{ASC}$ | ${c}_{a}$ | $3.6533\times {10}^{-7}$ | ${c}_{a}$ | $1\times {10}^{-5}$ |

Controller | PSO | Trial and Error Procedure |
---|---|---|

$\mathrm{CSC}$ | 0.1686 | $0.2563$ |

$\mathrm{ASC}$ | 0.0483 | $0.0530$ |

Controller | PSO | Trial and Error Procedure |
---|---|---|

$\mathrm{CSC}$ | $1.1367\times {10}^{5}$ | $1.1095\times {10}^{5}$ |

$\mathrm{ASC}$ | $7.5494\times {10}^{6}$ | $1.2366\times {10}^{6}$ |

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**MDPI and ACS Style**

Humaidi, A.J.; Ibraheem, I.K.; Azar, A.T.; Sadiq, M.E. A New Adaptive Synergetic Control Design for Single Link Robot Arm Actuated by Pneumatic Muscles. *Entropy* **2020**, *22*, 723.
https://doi.org/10.3390/e22070723

**AMA Style**

Humaidi AJ, Ibraheem IK, Azar AT, Sadiq ME. A New Adaptive Synergetic Control Design for Single Link Robot Arm Actuated by Pneumatic Muscles. *Entropy*. 2020; 22(7):723.
https://doi.org/10.3390/e22070723

**Chicago/Turabian Style**

Humaidi, Amjad J., Ibraheem Kasim Ibraheem, Ahmad Taher Azar, and Musaab E. Sadiq. 2020. "A New Adaptive Synergetic Control Design for Single Link Robot Arm Actuated by Pneumatic Muscles" *Entropy* 22, no. 7: 723.
https://doi.org/10.3390/e22070723