# Multi-Gene Genetic Programming-Based Identification of a Dynamic Prediction Model of an Overhead Traveling Crane

^{*}

## Abstract

**:**

## 1. Introduction

- This study uses data from both a simulated model as well as from the experimental stand for obtaining an approximating function to map the input to the output using multi-gene genetic programming. To the best of the authors’ knowledge this is the first study that uses genetic programming for an iterative predictive model of the crane dynamics.
- The Levenberg–Marquardt algorithm is used for minimizing the square errors over the entire prediction horizon using the least squares from the standard MGGP algorithm as an initial point.

## 2. Technical Background

#### 2.1. Multi-Gene Genetic Programming

_{0}is the bias term, θ

_{i}is the i-th gene weight and the vector of outputs is obtained by evaluating the tree-based structure. The least squares method is used to obtain the linear-in-the-parameter gene weights.

#### 2.2. Nonlinear Parameter Estimation

## 3. Crane Dynamic Prediction—A Simulation Study

#### 3.1. Crane Dynamics

^{T}and F, respectively. The equations of motion are given by

#### 3.2. Simulation Results

_{s}= 0.1 s.

^{2}(12).

^{2}of the position and sway prediction for the 10-step ahead and 20-step ahead prediction for the MGGP model and ARX model are given in Table 5. We noticed that the position model obtained by the MGGP has a similar structure to the ARX model with an additional nonlinear term. The MGGP model and ARX model accurately predicted the position of the crane with no significant loss of accuracy as the prediction step increased. However, the ARX model was a poor predictor of the payload sway, and there was a significant loss of accuracy as the prediction step was increased. The mean and standard deviation of the NMSE for the MGGP sway model were 0.0586 and 0.0021 for the 10-step ahead predictor and 0.0518 and 0.0209 for the 20-step ahead predictor, while for the ARX sway model they were 0.2387 and 0.0933 for the 10-step ahead predictor and 0.3714 and 0.1514 for the 20-step ahead predictor. The mean and standard deviation of the NMSE for the MGGP position model were 0.0015 and 2.0616 × 10

^{−4}for the 10-step ahead predictor and 0.0047 and 1.7321 × 10

^{−4}for the 20-step ahead predictor, while for the ARX sway model they were 0.0028 and 5 × 10

^{−5}for the 10-step ahead predictor and 0.0044 and 1.4141 × 10

^{−4}for the 20-step ahead predictor.

## 4. Experimental Results

^{2}of the position and sway prediction for the 10-step ahead and 20-step ahead predictions for the MGGP model and ARX model are given in Table 9. We can observe from the plots that the MGGP sway model is better at predicting the residual oscillations with the transient oscillations being either over or underpredicted. The ARX sway model either over or underpredicts both the transient as well as the residual oscillations. The mean and standard deviation of the NMSE for the MGGP sway model were 0.1471 and 0.0486 for the 10-step ahead predictor and 0.1519 and 0.0428 for the 20-step ahead predictor, while for the ARX sway model they were 0.3277 and 0.0289 for the 10-step ahead predictor and 0.3984 and 0.0812 for the 20-step ahead predictor. The mean and standard deviation of the NMSE for the MGGP position model were 0.0114 and 4.1633 × 10

^{−4}for the 10-step ahead predictor and 0.0167 and 9.6047 × 10

^{−4}for the 20-step ahead predictor, while for the ARX sway model they were 0.0224 and 9.6954 × 10

^{−4}for the 10-step ahead predictor and 0.0339 and 0.0017 for the 20-step ahead predictor, respectively.

## 5. Conclusions

^{2}statistics are also presented. The results show that the model is capable of prediction with varying parameters that may be encountered during everyday operation. We noticed that in both the simulation model and the experimental model the predictor did not take the mass of the payload into account. This can be attributed to the high mechanical impedance of the crane in the experimental stand and M >> m in the simulated model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The 10- and 20-step ahead prediction with the MGGP model of the simulated crane position and payload sway with 0.9 m rope length and 10 kg payload mass.

**Figure 6.**The 10- and 20-step ahead prediction with the MGGP model of the simulated crane position and payload sway with 0.9 m rope length and 40 kg payload mass.

**Figure 7.**The 10- and 20-step ahead prediction with the MGGP model of the simulated crane position and payload sway with 1.3 m rope length and 40 kg payload mass.

**Figure 8.**The 10- and 20-step ahead prediction with the MGGP model of the simulated crane position and payload sway with 1.7 m rope length and 10 kg payload mass.

**Figure 9.**The 10- and 20-step ahead prediction with the ARX model of the simulated crane position and payload sway with 0.9 m rope length and 10 kg payload mass.

**Figure 10.**The 10- and 20-step ahead prediction with the ARX model of the simulated crane position and payload sway with 1.3 m rope length and 40 kg payload mass.

**Figure 12.**Excitation input signal (control input), position response and sway response for training data set with rope length of 1.2 m and payload mass of 20 kg.

**Figure 13.**The 10- and 20-step ahead prediction with the MGGP model of the experimental crane position and payload sway with 0.9-meter rope and 10 kg payload mass.

**Figure 14.**The 10- and 20-step ahead prediction with the MGGP model of the experimental crane position and payload sway with 0.9-meter rope and 40 kg payload mass.

**Figure 15.**The 10- and 20-step ahead prediction with the MGGP model of the experimental crane position and payload sway with 1.3-meter rope and 40 kg payload mass.

**Figure 16.**The 10- and 20-step ahead prediction with the MGGP model of the experimental crane position and payload sway with 1.7-meter rope length and 10 kg payload mass.

**Figure 17.**The 10- and 20-step ahead prediction with the ARX model of the experimental crane position and payload sway with 0.9-meter rope and 10 kg payload mass.

**Figure 18.**The 10- and 20-step ahead prediction with the ARX model of the experimental crane position and payload sway with 1.3-meter rope and 40 kg payload mass.

Parameters | Values |
---|---|

M [kg] | 500 |

m [kg] | 10–50 |

l [m] | 0.5–3 |

${f}_{fr0}$ | 4.4 |

$\xi $ | 0.01 |

${k}_{p}$ | 0.05 |

${k}_{r}$ | 0.45 |

Parameters | Simulation |
---|---|

Population size | 200 |

Number of generations | 100 |

Initialization method | PTC2 [31] |

Max tree depth during initialization | 5 |

Max number of genes | 8 |

Terminal set | u(k−1), …, u(k−5), y(k−1), …, y(k−3), m, g, 1/l, t_{s} |

Non-terminal set | +, −, ×, analytic quotient |

High level crossover | 0.2 |

Mutation | 0.14 |

Tournament size | 8 |

Prediction horizon | 20 |

Weights | x(k) | α(k) |
---|---|---|

θ_{1} | −0.9947 | −0.9987 |

θ_{2} | 1.9947 | −0.1042 |

θ_{3} | −8.1689 × 10^{−6} | −1.0723 × 10^{−5} |

θ_{4} | 1.3829 × 10^{−4} | 1.9993 |

θ_{5} | 2.6866 × 10^{−5} | −8.4135 × 10^{−6} |

θ_{6} | −2.9573 × 10^{−8} | −1.5966 × 10^{−4} |

Coefficients | x(k) | α(k) |
---|---|---|

b_{1} | 1.0229 × 10^{−5} | −5.9573 × 10^{−6} |

b_{2} | 9.4147 × 10^{−6} | −6.1440 × 10^{−6} |

a_{1} | 1.9944 | 1.8967 |

a_{2} | −0.9944 | −0.9914 |

Output | Parameters | NMSE | R^{2} | ||||||
---|---|---|---|---|---|---|---|---|---|

10-Step Ahead | 20-Step Ahead | 10-Step Ahead | 20-Step Ahead | ||||||

MGGP | ARX | MGGP | ARX | MGGP | ARX | MGGP | ARX | ||

Position | l = 0.9 m; m = 10 kg | 0.0013 | 0.0028 | 0.0046 | 0.0043 | 0.9999 | 0.9998 | 0.9995 | 0.9996 |

l = 0.9 m; m = 40 kg | 0.0016 | 0.0028 | 0.0047 | 0.0043 | 0.9999 | 0.9998 | 0.9994 | 0.9996 | |

l = 1.3 m; m = 40 kg | 0.0017 | 0.0029 | 0.0050 | 0.0046 | 0.9999 | 0.9998 | 0.9994 | 0.9995 | |

l = 1.7 m; m = 10 kg | 0.0013 | 0.0028 | 0.0047 | 0.0044 | 0.9999 | 0.9998 | 0.9995 | 0.9996 | |

Sway | l = 0.9 m; m = 10 kg | 0.0578 | 0.2840 | 0.0776 | 0.3823 | 0.9868 | 0.6988 | 0.9762 | 0.4702 |

l = 0.9 m; m = 40 kg | 0.0577 | 0.3226 | 0.0320 | 0.4466 | 0.9866 | 0.6136 | 0.9958 | 0.2662 | |

l = 1.3 m; m = 40 kg | 0.0572 | 0.1082 | 0.0380 | 0.1562 | 0.9868 | 0.9568 | 0.9941 | 0.9126 | |

l = 1.7 m; m = 10 kg | 0.0618 | 0.2402 | 0.0595 | 0.5006 | 0.9850 | 0.7876 | 0.9862 | 0.0787 |

Parameters | Simulation |
---|---|

Population size | 200 |

Number of generations | 100 |

Initialization method | PTC2 [31] |

Max tree depth during initialization | 5 |

Max number of genes | 8 |

Terminal set | u(k−5), …, u(k−9), y(k−1), …, y(k−3), m, g, 1/l, t_{s} |

Non-terminal set | +, −, ×, analytic quotient |

High level crossover | 0.2 |

Mutation | 0.14 |

Tournament size | 8 |

Prediction horizon | 20 |

Weights | x(k) | α(k) |
---|---|---|

θ_{1} | 0.1025 | −0.0753 |

θ_{2} | −0.1025 | 1.4887 |

θ_{3} | −1.2238 × 10^{−4} | −0.5138 |

θ_{4} | −0.0062 | 9.0125 × 10^{−4} |

θ_{5} | 1.6469 | 1.0233 × 10^{−5} |

θ_{6} | - | −0.0010 |

Coefficients | x(k) | α(k) |
---|---|---|

b_{1} | 0.0031 | −2.1083 × 10^{−4} |

b_{2} | 0.0423 | 8.2345 × 10^{−5} |

a_{1} | 1.8249 | 1.8980 |

a_{2} | −0.8249 | −0.9626 |

Output | Parameters | NMSE | R^{2} | ||||||
---|---|---|---|---|---|---|---|---|---|

10-Step Ahead | 20-Step Ahead | 10-Step Ahead | 20-Step Ahead | ||||||

MGGP | ARX | MGGP | ARX | MGGP | ARX | MGGP | ARX | ||

Position | l = 0.9 m; m = 10 kg | 0.0109 | 0.0214 | 0.0169 | 0.0320 | 0.9964 | 0.9876 | 0.9891 | 0.9662 |

l = 0.9 m; m = 40 kg | 0.0113 | 0.0229 | 0.0169 | 0.0353 | 0.9962 | 0.9861 | 0.9895 | 0.9600 | |

l = 1.3 m; m = 40 kg | 0.0115 | 0.0218 | 0.0154 | 0.0330 | 0.9959 | 0.9869 | 0.9907 | 0.9634 | |

l = 1.7 m; m = 10 kg | 0.0119 | 0.0235 | 0.0177 | 0.0353 | 0.9959 | 0.9857 | 0.9888 | 0.9615 | |

Sway | l = 0.9 m; m = 10 kg | 0.0799 | 0.2851 | 0.0893 | 0.3898 | 0.9753 | 0.7384 | 0.9640 | 0.5694 |

l = 0.9 m; m = 40 kg | 0.1450 | 0.3355 | 0.1599 | 0.4245 | 0.9185 | 0.6375 | 0.8771 | 0.4721 | |

l = 1.3 m; m = 40 kg | 0.1722 | 0.3416 | 0.1770 | 0.2925 | 0.8726 | 0.7235 | 0.7911 | 0.6647 | |

l = 1.7 m; m = 10 kg | 0.1911 | 0.3487 | 0.1815 | 0.4868 | 0.8404 | 0.6222 | 0.7561 | 0.3685 |

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## Share and Cite

**MDPI and ACS Style**

Kusznir, T.; Smoczek, J.
Multi-Gene Genetic Programming-Based Identification of a Dynamic Prediction Model of an Overhead Traveling Crane. *Sensors* **2022**, *22*, 339.
https://doi.org/10.3390/s22010339

**AMA Style**

Kusznir T, Smoczek J.
Multi-Gene Genetic Programming-Based Identification of a Dynamic Prediction Model of an Overhead Traveling Crane. *Sensors*. 2022; 22(1):339.
https://doi.org/10.3390/s22010339

**Chicago/Turabian Style**

Kusznir, Tom, and Jaroslaw Smoczek.
2022. "Multi-Gene Genetic Programming-Based Identification of a Dynamic Prediction Model of an Overhead Traveling Crane" *Sensors* 22, no. 1: 339.
https://doi.org/10.3390/s22010339