# Data-Driven Identification of Crane Dynamics Using Regularized Genetic Programming
## 3 Experiment and Results
### Sensitivity Analysis  
A sensitivity analysis was used to determine the effect of the rope length and payload mass on the sway model output on for both the NARX-NN with six hidden neurons and the G3P-SR models. The elementary effect (EE) method was applied to perform a global sensitivity analysis. The EE method consists of generating a series of \(R\) trajectories, where each trajectory is composed of \(k+1\) points, where \(k\) is the number of input variables to be varied. The inputs are then perturbed one factor at a time (OAT) by a step \(\Delta\), as shown in (12), and require a total of \(R(k+1)\) model evaluations. \[EE_{i}(x)=\frac{f\Big{(}x_{1}^{j},\dots,x_{i-1}^{j},x_{i}^{j}+\Delta,x_{i+1}^{ j},\dots,x_{k}^{j}\Big{)}-f\Big{(}x_{1}^{j},\dots,x_{i-1}^{j},x_{i}^{j},x_{i+1}^{ j},\dots,x_{k}^{j}\Big{)}}{\Delta} \tag{12}\]  
The sample mean \(\mu_{i}^{*}\) of the absolute value of the elementary effects as well as the standard deviation \(\sigma_{i}\) are computed using (13) and (14), respectively, which gives us a measure of the contribution of each parameter and an estimate of the nonlinear effects of the parameter. \[\mu_{i}^{*}=\frac{1}{R}\sum_{j=1}^{R}\left|EE_{i}\Big{(}x^{j}\Big{)}\right| \tag{13}\]  
\[\sigma_{i}=\sqrt{\sum_{j=1}^{R}\frac{\left(EE_{i}\big{(}x^{j}\big{)}-\mu_{i} \right)^{2}}{R}} \tag{14}\]  
The mean and standard deviation of the elementary effects of the rope length and mass are displayed in Figures 19 and 20, respectively. The mean of the elementary effect of the payload mass was greater in the NARX-NN model compared to the G3P-SR model, indicating that the G3P-SR model was more robust to changes in the payload mass. The mean of the elementary effect of the rope length was slightly larger in the G3P-SR model. In both cases, it was noticeable that the model was less sensitive during the transient state when the trolley was in motion and the residual oscillations were more sensitive to parameter changes. The mean of the elementary effect of both models was greater by about two orders of magnitude than that of the mean elementary effect of the payload mass; thus, we can conclude that both models are robust to changes in the payload mass. The standard deviation of the elementary effect of the rope length indicated there were more nonlinear effects in the G3P-SR model than in the NARX-NN model. In industrial applications of overhead cranes, there is uncertainty involved in the measurement of both rope length and payload mass; therefore, it is necessary to analyze the effect of uncertainty in the operating point on the performance of the identified model, which is achieved by calculating the deviation from the sway trajectory using  
\[S=\sqrt{\frac{\Big{(}\hat{g}_{nominal}-\hat{g}_{perturbed}\Big{)}^{2}}{N}} \tag{15}\]  
Figure 18: Sway model population statistics: (**a**) MSE for training and validation set, (**b**) population mean, and (**c**) population standard deviation. Almost all the models developed in this study exhibited an increase in the deviation from the nominal trajectory when the rope length decreased from 1.7 m to 1.1 m, with the exception of the NARX-NN model with three neurons in the hidden layer, which experienced a decrease in the deviation from the nominal trajectory at 1.7 m compared to 1.1 m. The NARX-NN with six neurons in the hidden layer had a lower deviation from the nominal trajectory at the 1.7 m operating point but a higher deviation from the nominal trajectory at an operating point of 1.1 m compared to the G3P-SR model. The TSF-ARX model experienced the highest standard deviation from the nominal trajectory for both 1.7 m and 1.1 m nominal rope lengths. The deviation from the nominal mass had a significantly smaller effect on the standard deviation from the nominal trajectory for all the models compared with the deviation in the rope length, which was in agreement with the results obtained with the elementary effect method.