# Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator

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## Abstract

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## 1. Introduction

- Six controller structures are suggested by combining the proportional, integral, and derivate operations and neural networks.
- Suggest a new objective function to make the tuning process produces a controller that has a minimum chattering in the control signal.
- Applying a strong competition between the proposed controllers, especially for robustness, among the proposed controllers that integrate the specifications of the PID controller and neural networks.

## 2. Dynamic Model of 2-LRRM

## 3. Artificial Gorilla Troops Optimizer (GTO)

#### 3.1. Exploration Phase

_{r}(t) parameter is used for anyone in the troop nominees taken randomly from them and GX

_{r}(t) will be the position of that random one. UL and LL are the upper and lower variables levels, ${\mathrm{r}}_{1}$, ${\mathrm{r}}_{2}$ and ${\mathrm{r}}_{3}$ are assumed to be random items [0] to [1]. The variables a, L and H is calculated below:

#### 3.2. Exploitation Phase

#### 3.2.1. Following the Adult Silverback Leader

#### 3.2.2. Competition for Adult Females

_{5}is a value taken randomly between [0, 1] as struggle happens, Equation (26) was adopted to find the violence value during struggles, and A is the vector used for utilizing to solve the violence level by using Equation (24). Several parameters will be preset previously, like a, β, and E, which are regarded as an effect of violence on choice levels. The fitness function of all GX (t) is found, as the fitness function of GX (t) < X (t), the GX (t) will use the X (t) as an optimal solution, the best decision taken from the troop is taken as a new silverback leader. Figure 2 below explains all algorithm steps.

## 4. The Structures of the Proposed Controllers

#### 4.1. Conventional PID Controller (Con-PID)

#### 4.2. Conventional Fractional Order PID Controller (Con-FOPID)

#### 4.3. Self-Tuning Neural Network PID Controller (STNN-PID)

#### 4.4. Self-Tuning Neural Network FOPID Controller (STNN-FOPID)

#### 4.5. Neural Network PID Controller (NN-PID)

#### 4.6. Neural Network FOPID controller (NN-FOPID)

## 5. Simulation Results

#### 5.1. Robustness Tests

#### 5.1.1. Change Initial Position

#### 5.1.2. Disturbance Addition

#### 5.1.3. Parameters Variations

#### 5.1.4. All Previous Tests Together

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**GTO algorithm steps [21].

**Figure 4.**The block diagram of the control system with Conventional. PID or FOPID controller structures.

**Figure 11.**Block diagram of the feedback control system with neural network PID or FOPID controller.

**Figure 12.**(

**a**) Desired and actual ${\theta}_{1}$, (

**b**) Desired and actual ${\theta}_{2}$, (

**c**) Torque ${T}_{1}$, (

**d**) Torque ${T}_{2}$, (

**e**) Desired and actual paths of end-effector.

**Figure 13.**Desired and actual trajectories (

**a**) for ${\mathsf{\theta}}_{1}$, (

**b**) for ${\mathsf{\theta}}_{2}$, (

**c**) Desired and actual paths of end-effector based on initial positions (0.15, 0.15).

**Figure 14.**Desired and actual trajectories (

**a**) for ${\mathsf{\theta}}_{1}$, (

**b**) for ${\mathsf{\theta}}_{2}$, (

**c**) Desired and actual paths of end-effector with disturbance term [sin (50t), sin (50t)] and the initial position (0, 0).

**Figure 15.**Desired and actual trajectories (

**a**) for ${\mathsf{\theta}}_{1}$, (

**b**) for ${\mathsf{\theta}}_{2}$, and (

**c**) Desired and actual paths of end-effector for 5% increasing in both masses and initial position (0, 0).

**Figure 16.**Desired and actual (

**a**) for ${\mathsf{\theta}}_{1}$, (

**b**) for ${\mathsf{\theta}}_{2}$, (

**c**) performance of all controllers for Torque ${T}_{1}$ (

**d**) performance of all controllers for Torque ${T}_{2}$, and (

**e**) Desired and actual paths of end-effector when using initial position (0.15, 0.15), adding disturbance sin (50t) for both links torques and increasing 5% in masses of both links.

Parameters | Nominal Value |
---|---|

m_{1} | 0.1 kg |

m_{2} | 0.1 kg |

l_{1} | 0.8 m |

l_{2} | 0.4 m |

g | 9.81 m/s^{2} |

Controller | Total Number of Controller Parameters | Range of PID Gains K _{p}, K_{i}, K_{d} | Corner Frequency of Derivative Filter N | Range of Fractional Parameters | All Other Parameters Range |
---|---|---|---|---|---|

Con-PID | 8 | −150 to 150 | 10 to 100 | μ ≡ 1 λ ≡ 1 | -------- |

Con-FOPID | 12 | −150 to 150 | 10 to 100 | μ ≡ 0 to 2 λ ≡ 0 to 2 | -------- |

STNN-PID | 122 | −150 to 150 | 10 to 100 | μ ≡ 1 λ ≡ 1 | V ≡ −5 to 5 W≡ −1 to 1 |

STNN-FOPID | 162 | −150 to 150 | 10 to 100 | μ ≡ 0 to 2 λ ≡ 0 to 2 | V ≡ −5 to 5 W≡ −1 to 1 |

NNPID | 66 | −150 to 150 | ------ | μ ≡ 1 λ ≡ 1 | −1 to 1 |

NNFOPID | 70 | −150 to 150 | ------ | μ ≡ 0 to 2 λ ≡ 0 to 2 | −1 to 1 |

Controller | ITSE | No. of Slop Sign Change in All Control Signals |
---|---|---|

Con-PID | 3.729543 × 10^{−4} | 93 |

Con-FOPID | 2.227023 × 10^{−4} | 47 |

STNN-PID | 3.075515 × 10^{−4} | 91 |

STNN-FOPID | 3.883774 × 10^{−4} | 45 |

NNPID | 0.954084 × 10^{−4} | 85 |

NNFOPID | 0.748071 × 10^{−4} | 94 |

**Table 4.**The performance of the proposed controllers when the initial position (0.1745, 0.1745) is used.

Controller Type | Link No. | Rise Time | Over Shoot % | Settling Time | ITSE ×10 ^{−4} |
---|---|---|---|---|---|

Con-PID | L1 | 0.070 | 6.6 | 0.684 | 1.47752 |

L2 | 0.012 | 5.95 | 0.188 | 0.64646 | |

Con-FOPID | L1 | 0.074 | 3.27 | 0.584 | 1.30261 |

L2 | 0.054 | 1.40 | 0.131 | 0.10037 | |

STNN-PID | L1 | 0.069 | 6.14 | 0.594 | 1.12337 |

L2 | 0.012 | 7.41 | 0.394 | 0.83790 | |

STNN-FOPID | L1 | 0.081 | 4.40 | 6.430 | 1.25247 |

L2 | 0.026 | 1.05 | 0.166 | 1.00217 | |

NN-PID | L1 | 0.081 | 1.60 | 0.134 | 0.34509 |

L2 | 0.042 | 2.84 | 0.103 | 0.07345 | |

NN-FOPID | L1 | 0.076 | 1.80 | 0.123 | 0.31060 |

L2 | 0.043 | 0.47 | 0.043 | 0.03249 |

Controller | ITSE |
---|---|

Con-PID | 1.82669 × 10^{−4} |

Con-FOPID | 1.17614 × 10^{−4} |

STNN-PID | 1.61995 × 10^{−4} |

STNN-FOPID | 27.3328 × 10^{−4} |

NN-PID | 1.05251 × 10^{−4} |

NN-FOPID | 0.24644 × 10^{−4} |

**Table 6.**The ITSE of the proposed controllers based on disturbances sin (50t) for each link and initial position (0.0, 0.0).

Controller | ITSE |
---|---|

Con-PID | 5.54533 × 10^{−4} |

Con-FOPID | 1.43023 × 10^{−4} |

STNN-PID | 191.4245 × 10^{−4} |

STNN-FOPID | Unstable |

NN-PID | 2.1375 × 10^{−4} |

NN-FOPID | 0.092827 × 10^{−4} |

**Table 7.**The ITSE of the proposed controllers when increasing both masses of two links by 5% & Initial position (0.0, 0.0).

Controller | ITSE |
---|---|

Con-PID | 1.183509 × 10^{−4} |

Con-FOPID | 0.691371 × 10^{−4} |

STNN-PID | 0.743505 × 10^{−4} |

STNN-FOPID | Unstable |

NN-PID | 0.196180 × 10^{−4} |

NN-FOPID | 0.005068 × 10^{−4} |

**Table 8.**The ITSE of the proposed controllers based on initial position (0.15, 0.15), adding disturbances sin (50t) for both links torques and increasing 5% in masses of both links.

Controller | ITSE |
---|---|

Con-PID | 6.54278 × 10^{−4} |

Con-FOPID | 2.09812 × 10^{−4} |

STNN-PID | Unstable |

STNN-FOPID | Unstable |

NN-PID | 4.26462 × 10^{−4} |

NN-FOPID | 0.447529 × 10^{−4} |

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## Share and Cite

**MDPI and ACS Style**

Mohamed, M.J.; Oleiwi, B.K.; Abood, L.H.; Azar, A.T.; Hameed, I.A.
Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator. *Fractal Fract.* **2023**, *7*, 693.
https://doi.org/10.3390/fractalfract7090693

**AMA Style**

Mohamed MJ, Oleiwi BK, Abood LH, Azar AT, Hameed IA.
Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator. *Fractal and Fractional*. 2023; 7(9):693.
https://doi.org/10.3390/fractalfract7090693

**Chicago/Turabian Style**

Mohamed, Mohamed Jasim, Bashra Kadhim Oleiwi, Layla H. Abood, Ahmad Taher Azar, and Ibrahim A. Hameed.
2023. "Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator" *Fractal and Fractional* 7, no. 9: 693.
https://doi.org/10.3390/fractalfract7090693