# Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators

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## Abstract

**:**

## 1. Introduction

- manipulation robotic systems;
- mobile robots;
- information and control robotic systems.

## 2. Related Work

## 3. Methodology

#### 3.1. Dataset Preparation

#### 3.2. ML-Based Architecture

#### 3.2.1. Radial Basis Function Neural Network (RBFNN)

#### 3.2.2. Cooperative Search Optimisation Algorithm (CSOA)

**a.**- Team Building:

**b.**- Team Communication Phase:

**c.**- Reflective Learning Phase:

Algorithm 1: Pseudo-code for CSOA algorithm. |

Data: Random data in search spaceResult: Output the best final solutionInitialise objective function and random population on search space; Evaluate the fitness of all particles and create I and M vectors; whiletermination criteria not metdoend |

## 4. Results

#### 4.1. CSOA-RBFNN Model

#### 4.1.1. X-axis Prediction Results

#### 4.1.2. Y-axis Prediction Results

#### 4.1.3. Z-axis Prediction Results

#### 4.2. Comparative Study

## 5. Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Types of joints: (

**a**) articulated manipulator, (

**b**) spherical manipulator, (

**c**) SCARA manipulator, (

**d**) cylindrical manipulator.

**Figure 2.**Frame transformation: frame A is translated along A

_{P}and rotated along one of the axes to produce frame B.

**Figure 4.**Kinematic model of the 3-DOF manipulator designed in ROS-RVIZ. JA represents the the joint angles while RA represents the arm length.

**Figure 6.**(

**a**) Sketch map of an enterprise hierarchy. (

**b**) Working mechanism of CSOA to converge towards global optimum solution.

**Figure 8.**RBFNN model for training and testing. Joint angle JA is used as the input and the Cartesian coordinates of the end-effector is the output.

**Figure 9.**Comparison of actual x-axis position and predicted position from the meta-heuristic RBFNN models.

**Figure 11.**Comparison of actual y-axis position and predicted position from the meta-heuristic RBFNN models.

**Figure 13.**Comparison of actual z-axis position and predicted position from the meta-heuristic RBFNN models.

JA-1 | JA-2 | JA-3 | x-axis | y-axis | z-axis |
---|---|---|---|---|---|

1.675 | 0.558 | 0 | 0.989 | −1.4 | 1.844 |

6.2831 | 6.143 | 4.468 | −0.978 | −0.5 | 4.049 |

1.851 | 0.558 | 0.698 | 0.74 | −0.893 | 1.375 |

1.675 | 5.724 | 0.139 | 1.176 | −1.505 | 3.628 |

1.256 | 5.585 | 5.724 | 0.607 | −0.91 | 4.63 |

1.117 | 6.143 | 0.698 | 1.161 | −1.528 | 3.077 |

Function Name | Mathematical Form |
---|---|

Thin-plate spline | $G\left(x\right)={(x-\mu )}^{2}log(x-\mu )$ |

Multi-quadratic | $G\left(x\right)=\sqrt{{(x-\mu )}^{2}+{\sigma}^{2}}$ |

Inverse multi-quadratic | $G\left(x\right)=\frac{1}{\sqrt{{(x-\mu )}^{2}+{\sigma}^{2}}}$ |

Gaussian | $G\left(x\right)=exp{}^{(-\frac{{(x-\mu )}^{2}}{{\sigma}^{2}})}$ |

Technique | Parameters | Value |
---|---|---|

CSOA | $\alpha $ | 0.1 |

$\beta $ | 0.15 | |

PSO | C1 | 1.5 |

C2 | 1.5 | |

GWO | a | 2.0 |

Technique | Optimal Spread Value | Best Cost | Runtime (s) |
---|---|---|---|

CSOA-RBFNN | 2.200 | 0.0048 | 12.54 |

GWO-RBFNN | 2.700 | 0.0412 | 13.19 |

PSO-RBFNN | 2.900 | 0.2230 | 13.43 |

Technique | Mean RE | NMSE | MAE |
---|---|---|---|

CSOA-RBFNN | 0.0098 | 0.0051 | 0.0334 |

GWO-RBFNN | 0.3015 | 0.0486 | 0.0571 |

PSO-RBFNN | 0.856 | 0.1421 | 0.0591 |

Technique | Optimal Spread Value | Best Cost | Runtime (s) |
---|---|---|---|

CSOA-RBFNN | 1.900 | 0.0083 | 16.05 |

GWO-RBFNN | 2.100 | 0.0225 | 17.23 |

PSO-RBFNN | 2.500 | 0.0861 | 16.78 |

Technique | Mean RE | NMSE | MAE |
---|---|---|---|

CSOA-RBFNN | 0.0355 | 0.0081 | 0.0064 |

GWO-RBFNN | 0.0860 | 0.0141 | 0.0072 |

PSO-RBFNN | 0.3616 | 0.0782 | 0.0331 |

Technique | Optimal Spread Value | Best Cost | Runtime (s) |
---|---|---|---|

CSOA-RBFNN | 1.400 | 0.0321 | 11.54 |

GWO-RBFNN | 2.100 | 0.1426 | 13.22 |

PSO-RBFNN | 7.200 | 0.4307 | 12.43 |

Technique | Mean RE | NMSE | MAE |
---|---|---|---|

CSOA-RBFNN | 0.0121 | 0.0329 | 0.0468 |

GWO-RBFNN | 0.0287 | 0.1426 | 0.0861 |

PSO-RBFNN | 0.0671 | 0.7281 | 0.1122 |

Axis | Technique | Mean RE | NMSE | MAE |
---|---|---|---|---|

x-axis | RBFNN-CSOA | 0.0355 | 0.0081 | 0.0064 |

ANN | 0.103 | 0.092 | 0.044 | |

SVR | 0.068 | 0.015 | 0.0093 | |

y-axis | RBFNN-CSOA | 0.0098 | 0.0051 | 0.0334 |

ANN | 0.031 | 0.104 | 0.024 | |

SVR | 0.0120 | 0.095 | 0.089 | |

z-axis | RBFNN-CSOA | 0.0355 | 0.0081 | 0.0064 |

ANN | 0.095 | 0.0605 | 0.0112 | |

SVR | 0.076 | 0.0299 | 0.0092 |

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**MDPI and ACS Style**

Moosavi, S.K.R.; Zafar, M.H.; Sanfilippo, F. Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators. *Robotics* **2022**, *11*, 43.
https://doi.org/10.3390/robotics11020043

**AMA Style**

Moosavi SKR, Zafar MH, Sanfilippo F. Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators. *Robotics*. 2022; 11(2):43.
https://doi.org/10.3390/robotics11020043

**Chicago/Turabian Style**

Moosavi, Syed Kumayl Raza, Muhammad Hamza Zafar, and Filippo Sanfilippo. 2022. "Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators" *Robotics* 11, no. 2: 43.
https://doi.org/10.3390/robotics11020043