# Dynamics Modeling of Industrial Robotic Manipulators: A Machine Learning Approach Based on Synthetic Data

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dynamics Modeling

#### 2.1.1. Kinematic Model

#### 2.1.2. Lagrange–Euler

#### 2.1.3. Newton–Euler

- ${T}_{0}^{0}=I$,
- ${f}^{n+1}=-{f}^{tool}$,
- ${n}^{n+1}=-{n}^{tool}$,
- ${v}^{0}=0$,
- $\frac{d{v}^{0}}{dt}=-g$,
- ${\omega}^{0}=0$,
- $\frac{d{\omega}^{0}}{dt}=-0$, and
- $i=0$.

#### 2.2. Dataset Generation

#### 2.3. Machine Learning Approach

- Hidden layer size—the number of neurons and layers, given as a tuple in which each value presents a number of neurons in a given layer;
- Activation function—activation function to be used within all of the model’s neurons;
- Initial learning rate—the learning rate of the model;
- Learning rate type—the manner in which the learning rate is adjusted through the training process, inversely to the elapsed training iterations, kept constant, or adapted to the model error;
- Solver—the algorithm used for weight adjustment during training. The possible solvers are Adam, Stochastic Gradient Descent (SGD), and Limited-Memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS);
- L2 regularization parameter—the value that controls the influence of the individual inputs, preventing a single input from having too much influence on the output, to provide models that have better generalization.

#### Model Evaluation

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AI | Artificial Intelligence |

ANN | Artificial Neural Network |

DH | Dennavit–Hartenberg |

LBFGS | Limited-Memory Broyden–Fletcher–Goldfarb–Shanno |

LE | Lagrange–Euler |

LSTM | Long short-term memory |

MAPE | Mean Absolute Percentage Error |

ML | Machine Learning |

LSTM | Long short-term memory |

MAPE | Mean Absolute Percentage Error |

ML | Machine Learning |

MLP | Multilayer Perceptron |

NE | Newton–Euler |

RS | Random Search |

${R}^{2}$ | Coefficient of determination |

SGD | Stochastic Gradient Descent |

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**Figure 1.**The modeled robotic manipulator ABB IRB 120 in isometric view [16].

**Figure 2.**Distributions of the synthetically generated outputs: (

**a**–

**f**) the distributions of the generated values for individual joints; (

**g**) the total torque of the robotic manipulator.

**Table 1.**The upper and lower bounds for all the randomly determined values during the process of dataset generation.

Value | Symbol | Lower Boundary | Upper Boundary | Unit |
---|---|---|---|---|

Angle of joint 1 | ${q}_{1}$ | −2.88 | 2.88 | rad |

Angle of joint 2 | ${q}_{2}$ | −1.92 | 1.92 | rad |

Angle of joint 3 | ${q}_{3}$ | −1.22 | 1.92 | rad |

Angle of joint 4 | ${q}_{4}$ | −2.79 | 2.79 | rad |

Angle of joint 5 | ${q}_{5}$ | −2.09 | 2.09 | rad |

Angle of joint 6 | ${q}_{6}$ | 0 | 6.28 | rad |

Speeds * | ${\omega}_{i},\dot{{q}_{i}}$ | −1.00 | 1.00 | rad/s |

Accelerations * | $\dot{{\omega}_{i}},\ddot{{q}_{i}}$ | −1.00 | 1.00 | rad/s${}^{2}$ |

**Table 2.**Possible hyperparameter values. In cases where the hyperparameter is selected from the list (Hidden Layer Sizes, Activation Function, Learning Rate Type, and Solver), the possible values are given, while in the cases where the values are selected randomly from a range (Initial Learning Rate, L2 Regularization), the lower and upper bound are given.

Hyperparameter | Values | |
---|---|---|

Hidden Layer Sizes | (288), (288, 288), (288, 288, 288), | |

(288, 288, 288, 288), (288, 288, 288, 288, 288), | ||

(144), (144, 144), (144, 144, 144), | ||

(144, 144, 144, 144), (144, 144, 144, 144, 144), | ||

(72), (72, 72), (72, 72, 72), | ||

(72, 72, 72, 72), (72, 72, 72, 72, 72), | ||

(36), (36, 36), (36, 36, 36), | ||

(36, 36, 36, 36), (36, 36, 36, 36, 36), | ||

(18), (18, 18), (18, 18, 18), | ||

(18, 18, 18, 18), (18, 18, 18, 18, 18) | ||

Activation Function | ReLU, Logistic, Identity | |

Initial Learning Rate | 0.0001 | 0.5 |

Learning Rate Type | Constant, Adaptive, Inverse Scaling | |

Solver | Adam, LBFGS, SGD | |

L2 Regularization Parameter | 0.0001 | 0.5 |

**Table 3.**The best results achieved for all the torque targets, with the model hyperparameters used in the best-performing models.

Target | ${\mathit{R}}^{2}$ | ${\mathit{\sigma}}_{{\mathit{R}}^{2}}$ | MAPE | ${\mathit{\sigma}}_{\mathit{MAPE}}$ | Hyperparameters | |
---|---|---|---|---|---|---|

${\tau}_{1}$ | 0.95774 | 0.01285 | 1.17815 | 0.03527 | Hidden Layer Sizes | 288, 288, 288 |

Activation | Logistic | |||||

Initial Learning Rate | Adaptive | |||||

Learning Rate Type | 0.00923 | |||||

Solver | Adam | |||||

Regularization | 0.12142 | |||||

${\tau}_{2}$ | 0.98306 | 0.04280 | 1.15615 | 0.02649 | Hidden Layer Sizes | 144, 144, 144 |

Activation | Logistic | |||||

Initial Learning Rate | 0.01656 | |||||

Learning Rate Type | Adaptive | |||||

Solver | Adam | |||||

Regularization | 0.01189 | |||||

${\tau}_{3}$ | 0.95162 | 0.03831 | 1.59342 | 0.03402 | Hidden Layer Sizes | 288, 288, 288, 288 |

Activation | ReLU | |||||

Initial Learning Rate | 0.01432 | |||||

Learning Rate Type | Constant | |||||

Solver | Adam | |||||

Regularization | 0.09456 | |||||

${\tau}_{4}$ | 0.96318 | 0.03493 | 1.80749 | 0.07908 | Hidden Layer Sizes | 288, 288, 288 |

Activation | Logistic | |||||

Initial Learning Rate | 0.00997 | |||||

Learning Rate Type | Inverse Scaling | |||||

Solver | Adam | |||||

Regularization | 0.010375 | |||||

${\tau}_{5}$ | 0.91787 | 0.04833 | 1.90698 | 0.01564 | Hidden Layer Sizes | 144, 144, 144, 144 |

Activation | Tanh | |||||

Initial Learning Rate | 0.04375 | |||||

Learning Rate Type | Constant | |||||

Solver | LBFGS | |||||

Regularization | 0.00184 | |||||

${\tau}_{6}$ | 0.92712 | 0.02718 | 1.93007 | 0.02965 | Hidden Layer Sizes | 144, 144 |

Activation | Tanh | |||||

Initial Learning Rate | 0.01992 | |||||

Learning Rate Type | Adaptive | |||||

Solver | LBFGS | |||||

Regularization | 0.12729 | |||||

${\tau}_{all}$ | 0.89479 | 0.03945 | 2.04094 | 0.02421 | Hidden Layer Sizes | 288, 288, 288 |

Activation | Logistic | |||||

Initial Learning Rate | 0.00951 | |||||

Learning Rate Type | Inverse Scaling | |||||

Solver | Adam | |||||

Regularization | 0.10276 |

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**MDPI and ACS Style**

Baressi Šegota, S.; Anđelić, N.; Šercer, M.; Meštrić, H. Dynamics Modeling of Industrial Robotic Manipulators: A Machine Learning Approach Based on Synthetic Data. *Mathematics* **2022**, *10*, 1174.
https://doi.org/10.3390/math10071174

**AMA Style**

Baressi Šegota S, Anđelić N, Šercer M, Meštrić H. Dynamics Modeling of Industrial Robotic Manipulators: A Machine Learning Approach Based on Synthetic Data. *Mathematics*. 2022; 10(7):1174.
https://doi.org/10.3390/math10071174

**Chicago/Turabian Style**

Baressi Šegota, Sandi, Nikola Anđelić, Mario Šercer, and Hrvoje Meštrić. 2022. "Dynamics Modeling of Industrial Robotic Manipulators: A Machine Learning Approach Based on Synthetic Data" *Mathematics* 10, no. 7: 1174.
https://doi.org/10.3390/math10071174