# Soft-Computing-Based Estimation of a Static Load for an Overhead Crane

^{*}

## Abstract

**:**

## 1. Introduction

_{0}penalty is added for model selection, which results in reduced complexity and decreases the probability of overfitting. The subsequent sparse regression problem is solved by using the monotonically accelerated proximal gradient descent algorithm [40]. To the best of the authors’ knowledge, the only other evolutionary sparse regression algorithm is based on an elastic net regularization [41]. The contribution of this paper can be summarized as follows:

- Develop a novel genetic programming variant called G3PSR that can be used for symbolic regression problems that can be expressed as a linear in the parameters model.
- Apply genetic programming variants, namely G3PSR and MGGP, to identify a mathematical relationship between the payload mass and the trolley position and girder strain.
- Compare the genetic programming models for mass estimation with a method proposed in the literature [39].

## 2. Methodology

#### 2.1. Multi-Gene Genetic Programming

#### 2.2. Grammar-Guided Genetic Programming with Sparse Regression

Algorithm 1: mAPG. |

Input: $\varphi ,y,\lambda $Initialize: $\rho <1,\delta ,{\mathit{z}}_{1}={\mathit{\theta}}_{1}={\mathit{\theta}}_{0},{t}_{1}=1,{t}_{0}=0,k=1$ while not converged do$k\leftarrow k+1$ ${w}_{k}={\theta}_{k}+\frac{{t}_{k-1}}{{t}_{k}}\left({z}_{k}-{\theta}_{k}\right)+\frac{{t}_{k-1}-1}{{t}_{k}}\left({\theta}_{k}-{\theta}_{k-1}\right)$ Initialize step size ${\eta}_{w}$ and ${\eta}_{\theta}$ using Barzilai-Borwein method while $F\left({z}_{k+1}\right)\ge F\left({w}_{k}\right)-\delta {\parallel {z}_{k+1}-{w}_{k}\parallel}_{2}^{2}$ do${z}_{k+1}={\mathrm{prox}}_{{\eta}_{w}\lambda}\left({z}_{k}-{\eta}_{w}{\varphi}^{T}\left(\varphi {z}_{k}-y\right)\right)$ ${\eta}_{w}=\rho {\eta}_{w}$ end while while $F\left({v}_{k+1}\right)\ge F\left({\theta}_{k}\right)-\delta {\parallel {v}_{k+1}-{\theta}_{k}\parallel}_{2}^{2}$ do${v}_{k+1}={\mathrm{prox}}_{{\eta}_{\theta}\lambda}\left({v}_{k}-{\eta}_{\theta}{\varphi}^{T}\left(\varphi {v}_{k}-y\right)\right)$ ${\eta}_{\theta}=\rho {\eta}_{\theta}$ end while ${t}_{k+1}=\frac{\sqrt{4{t}_{k}^{2}+1}+1}{2}$ ${\theta}_{k+1}=\{\begin{array}{c}{z}_{k+1}\\ {v}_{k+1}\end{array}\begin{array}{l}\mathrm{if}F\left({z}_{k+1}\right)\le F\left({v}_{k+1}\right)\hfill \\ \mathrm{otherwise}\hfill \end{array}$ end whileOutput: $\theta $ |

#### 2.3. TS Fuzzy Model

_{1}is the potential of the data point chosen as the first cluster center. Thus, the cluster centroid is either accepted if ${P}_{i}{\xi}_{1}{P}_{1}$, or rejected if ${P}_{i}<{\xi}_{2}{P}_{1}$, and the algorithm is terminated. If condition (11) is satisfied, the shortest distance ${d}_{min}$ between ${d}_{k}$ and all previously found centroids is verified using condition (12). The data point is accepted as the centroid if condition (12) is satisfied, otherwise, the data point with the next highest potential is tested, and the algorithm is terminated if all data points violate condition (12).

## 3. Results of Identification Experiments

**,**while the G3PSR model is slightly less affected by strain signal perturbation $\epsilon -\sigma $, since the RMSE is 2.0835, 2.2169, and 2.2169 for G3PSR, MGGP, and TSF models, respectively. Taking into account the nominal and deviated strain signal, the uncertainty of the models’ output (estimated payload weight) is expressed by standard error calculated according to (14), and the results are presented in Table 9 and Figure 14 (where the dotted line is a linear interpolation between testing points). The general tendency observed in Figure 14 and Table 9 is that the standard error decreases with the increase in the payload weight from 30 to 90 kg, from 2.2297 to 1.7460 for the G3PSR model, and from 2.0670 to 1.7786 and from 2.2066 to 1.7740 for MGGP and TSF models, with the exception of 50 kg.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Symbols and Definitions

Variable | Definition |
---|---|

$b,c,d$ | Squash factor, cluster center, distance between cluster centers |

$m$ | Suspended payload mass |

${p}_{i1},{p}_{i2},{p}_{i3}$ | Rule consequent parameters |

${r}_{a}$ | Cluster radius |

${w}_{i}$ | Weights of i-th rule |

$x$ | Trolley position |

$N$ | Set of all nonterminal symbols |

$P$ | Set of production rules (also potential of chosen datapoint as cluster center and probability of selection) |

$S$ | Start symbol |

$\epsilon $ | Strain |

${\zeta}_{1},{\zeta}_{2}$ | Accept ratio, reject ratio |

$\eta $ | Step size |

$\theta $ | Model term coefficients |

$\overline{\theta}$ | Normalized coefficients |

$\lambda $ | Sparsification parameter |

$\varphi $ | Regressor matrix |

$\mathsf{\Sigma}$ | Set of all terminal symbols |

Abbreviation | Definition |
---|---|

GP | Genetic programming |

G3PSR | Grammar guided genetic programming with sparse regression |

mAPG | Monotone accelerated proximal gradient descent |

MGGP | Multi-gene genetic programming |

PTC2 | Probability tree creation 2 |

TSF | Takagi–Sugeno fuzzy |

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**Figure 6.**Experimental data partition into training and validation data (solid line), and testing data (dashed line).

Parameters | Settings |
---|---|

Function set | ×, √, inv |

Terminal set | x, ε |

Population size | 100 |

Number of generations | 500 |

Initialization | Ramped Half-and-Half |

Maximum number of genes | 25 |

Maximum tree depth | 5 |

Tournament size | 2 |

Crossover probability | 0.84 |

Mutation probability | 0.14 |

Direct reproduction | 0.02 |

Parameters | Settings |
---|---|

Set of nonterminal symbols N | ×, √, inv |

Set of terminal symbols Σ | x, ε |

Population size | 100 |

Number of generations | 500 |

Initialization | Probability tree creation 2 (PTC2) |

Number of candidate model terms | 25 |

Maximum tree depth during initialization | 8 |

Tournament size | 2 |

Subtree crossover probability | 0.75 |

High-level crossover probability | 0.15 |

Mutation probability | 0.1 |

Sparsification parameter λ | 0.001 |

〈S〉 | ::= | 〈exp〉 |

〈exp〉 | ::= | 〈op_{b}〉 〈exp〉 〈exp〉 | 〈op_{u}〉 〈exp〉 | 〈T〉 |

〈op_{b}〉 | ::= | × |

〈op_{u}〉 | ::= | √ | inv |

〈T〉 | ::= | x | ε |

G3PSR | MGGP | ||
---|---|---|---|

Model Coefficients | Model Terms | Model Coefficients | Model Terms |

−0.0058 | $\frac{1}{\epsilon}$ | −5.2140 × 10^{5} | 1 |

4.5510 × 10^{22} | $\frac{{x}^{5}{\epsilon}^{6}}{\sqrt{\epsilon}}$ | 879.6109 | $x{\epsilon}^{2}$ |

15.6505 | $\frac{1}{\sqrt[4]{\epsilon}}$ | −1.2297 × 10^{4} | $x\sqrt[4]{\epsilon}$ |

1.1976 × 10^{5} | ${x}^{3}\epsilon $ | −5.0552 × 10^{5} | $\sqrt{x}$ |

1.6163 × 10^{−7} | $\frac{\sqrt[4]{\epsilon}}{{\epsilon}^{2}x\sqrt{x}}$ | 8.3531 × 10^{4} | $x$ |

−161.6078 | $x$ | 8.2332 × 10^{5} | $\frac{\epsilon}{x}$ |

4.4107 × 10^{−4} | $\frac{x\epsilon}{\sqrt{x}}$ | 2.0059 × 10^{4} | $\frac{\sqrt{\epsilon}}{x}$ |

18.2220 | ${x}^{2}$ | 1.5669 × 10^{4} | $\sqrt{x}\sqrt[4]{\epsilon}$ |

−3.2326 × 10^{13} | ${\epsilon}^{3}x$ | −1.3150 × 10^{5} | $x\sqrt{\epsilon}$ |

2.3790 × 10^{−8} | $\frac{x}{{\epsilon}^{2}}$ | 3.3940 × 10^{5} | $\sqrt{x}\sqrt{\epsilon}$ |

14.1349 | $\frac{{x}^{4}}{\sqrt[8]{\epsilon x}}$ | −3.0590 × 10^{5} | $\sqrt{\epsilon}$ |

−3.2074 | ${x}^{8}\sqrt{x}$ | 1.2318 × 10^{5} | ${x}^{2}\sqrt{\epsilon}\sqrt[4]{\epsilon}$ |

9.7761 × 10^{7} | $\sqrt{{\epsilon}^{3}}$ | 9.8117 × 10^{3} | $\frac{\sqrt[4]{\epsilon}}{\sqrt{x}}$ |

−5.7385 × 10^{−9} | $\frac{1}{{\epsilon}^{2}x\sqrt[8]{{x}^{5}}}$ | 1.1880 × 10^{5} | $\frac{1}{\sqrt[4]{x}}$ |

8.2782 × 10^{5} | $\sqrt[4]{x}$ | ||

−3.7918 × 10^{3} | ${x}^{2}$ | ||

−982.1465 | $\frac{\sqrt[4]{\epsilon}}{{x}^{2}}$ |

Rule Number | Antecedent (Gaussian) Parameters | Consequent (Linear Function) Parameters | |
---|---|---|---|

i | $\left[{\sigma}_{xi},{x}_{i}\right]$ | $\left[{\sigma}_{\epsilon i},{\epsilon}_{i}\right]\times {10}^{-5}$ | $\left[{p}_{1i},{p}_{2i},{p}_{3i}\right]$ |

1 | [0.250, 0.8652] | [2.447, 5.8133] | $\left[-60.65,3.97\times {10}^{3},1.288\times {10}^{2}\right]$ |

2 | [0.250, 0.8523] | [2.447, 9.4161] | $\left[-286.01,5.605\times {10}^{5},286.19\right]$ |

3 | [0.250, 1.2646] | [2.447, 5.2689] | $\left[32.11,-1.4716\times {10}^{6},228.17\right]$ |

4 | [0.250, 0.4850] | [2.447, 5.0614] | $\left[5.81\times {10}^{3},-2.467\times {10}^{8},1.824\times {10}^{4}\right]$ |

5 | [0.250, 1.2048] | [2.447, 3.3429] | $\left[119.45,-1.365\times {10}^{5},-270.93\right]$ |

6 | [0.250, 1.2379] | [2.447, 9.1502] | $\left[2.16,1.211\times {10}^{6},34.67\right]$ |

7 | [0.250, 0.5311] | [2.447, 3.4812] | $\left[3.99\times {10}^{3},-1.309\times {10}^{8},-8.82\times {10}^{3}\right]$ |

8 | [0.250, 0.4473] | [2.447, 7.4674] | $\left[-230.99,3.9165\times {10}^{7},5.935\times {10}^{3}\right]$ |

G3PSR | MGGP | TSF | |
---|---|---|---|

RMSE | 1.7813 | 1.8069 | 1.8875 |

MRE | 0.0285 | 0.0283 | 0.0294 |

No. of parameters | 14 | 17 | 56 |

Mean execution time (ms) ± standard deviation | $5.2\times {10}^{-3}$ $\pm 5.8\times {10}^{-7}$ | $80.4\times {10}^{-3}\pm 5.5\times {10}^{-3}$ | $367.5\times {10}^{-3}\pm 7.6\times {10}^{-3}$ |

G3PSR | MGGP | TSF | ||||
---|---|---|---|---|---|---|

Payload Mass (kg) | MRE | max RE | MRE | max RE | MRE | max RE |

30 | 0.0502 | 0.1523 | 0.0449 | 0.1570 | 0.0499 | 0.1108 |

50 | 0.0302 | 0.0883 | 0.0320 | 0.0862 | 0.0344 | 0.1033 |

70 | 0.0200 | 0.0585 | 0.0219 | 0.0606 | 0.0207 | 0.0802 |

90 | 0.0148 | 0.0395 | 0.0156 | 0.0388 | 0.0142 | 0.0402 |

G3PSR | MGGP | TSF | |||||||
---|---|---|---|---|---|---|---|---|---|

ε | ε + σ | ε − σ | ε | ε + σ | ε − σ | ε | ε + σ | ε − σ | |

RMSE | 1.7813 | 2.2115 | 2.0835 | 1.8069 | 2.2104 | 2.2169 | 1.8875 | 2.2889 | 2.2169 |

Payload mass (kg) | MRE | ||||||||

$\epsilon $ | $\epsilon +\sigma $ | $\epsilon -\sigma $ | $\epsilon $ | $\epsilon +\sigma $ | $\epsilon -\sigma $ | $\epsilon $ | $\epsilon +\sigma $ | $\epsilon -\sigma $ | |

30 | 0.0502 | 0.0793 | 0.0504 | 0.0449 | 0.0748 | 0.0469 | 0.0499 | 0.0771 | 0.0532 |

50 | 0.0302 | 0.0255 | 0.0487 | 0.0320 | 0.0246 | 0.0518 | 0.0344 | 0.0348 | 0.0474 |

70 | 0.0200 | 0.0296 | 0.0155 | 0.0219 | 0.0318 | 0.0172 | 0.0207 | 0.260 | 0.0196 |

90 | 0.0148 | 0.0161 | 0.0169 | 0.0156 | 0.0172 | 0.0166 | 0.0142 | 0.0177 | 0.0153 |

G3PSR | MGGP | TSF | |
---|---|---|---|

Payload Mass (kg) | Standard Error (kg) | ||

30 | 2.2297 | 2.0670 | 2.2066 |

50 | 2.1200 | 2.1813 | 2.4352 |

70 | 1.9005 | 2.0492 | 2.0698 |

90 | 1.7460 | 1.7786 | 1.7740 |

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## Share and Cite

**MDPI and ACS Style**

Kusznir, T.; Smoczek, J.
Soft-Computing-Based Estimation of a Static Load for an Overhead Crane. *Sensors* **2023**, *23*, 5842.
https://doi.org/10.3390/s23135842

**AMA Style**

Kusznir T, Smoczek J.
Soft-Computing-Based Estimation of a Static Load for an Overhead Crane. *Sensors*. 2023; 23(13):5842.
https://doi.org/10.3390/s23135842

**Chicago/Turabian Style**

Kusznir, Tom, and Jaroslaw Smoczek.
2023. "Soft-Computing-Based Estimation of a Static Load for an Overhead Crane" *Sensors* 23, no. 13: 5842.
https://doi.org/10.3390/s23135842