# Data-Driven Identification of Crane Dynamics Using Regularized Genetic Programming
## 3 Experiment and Results  
The data used in the identification procedure were obtained from experiments carried out on a laboratory double-girder crane mechanism. The mechanism was driven by two AC gear motors with a gear ratio of 15.5, operating at 1400 rpm and with an output power of 0.18 kW. The AC motors were supplied by two LG iC5 0.4 kW frequency inverters, controlled by a Mitsubishi FX2N series Programmable Logic Controller (PLC). The crane mechanism was equipped with the incremental encoders installed on the wheels to measure the position \(x\) relative to the crane bridge, under the trolley connected to a pair of fork arms within which the hoisting cable passed through to measure the sway angle \(\alpha\) of the cable-suspended payload and on the drum of the lifting mechanism to measure the rope length \(l\). The incremental encoders had a resolution of 400 pulses per rotation (ppr), 2000 ppr,  
Figure 5: NARX-NN architecture considered for comparative study. Figure 6: TSF-ARX architecture considered for comparative study. and 100 ppr to measure the position \(x\), sway, angle \(\alpha\), and rope length \(l\), respectively. The data from the sensors were sampled at 10 Hz using a PC with 16 GB RAM and Quad Core 4 GHz Intel Core i7-6700K CPU running Windows 10 and Matlab R2020. The PLC was used during the experiments to transfer and convert the control and measurement signals exchanged between the PC with IO card and a plant. The control analog voltage signal \(u\) within the range \(\pm\)10 V was sent from the PC with IO card to the PLC, which transmitted this signal to the frequency inverters converting it to the voltage signal 0-10 V and binary signals corresponding to a frequency change within the range 0-50 Hz and the direction of motor rotation, respectively. Figure 7 shows a view of the laboratory overhead crane and the data acquisition equipment used in the identification experiments. The overhead crane dynamics were assumed to be composed of two submodels, comprising the actuated and underactuated parts of the crane system, connected in series as shown in Figure 8. The first model, denoted as the velocity model, presented the relation between control signal \(u\) and crane velocity \(v\), while the second model, denoted as the sway model, presented the relation between crane velocity \(v\) and payload sway \(\alpha\). This assumption was implemented in the identification experiments to derive the G3P-SR models, as well as the NARX-NN and TSF-ARX velocity and sway models used for our comparative analysis. The laboratory overhead crane was not equipped with a sensor to measure the velocity, and, therefore, a constant-velocity Kalman filter was used to estimate the velocity of the trolley. The process noise and measurement noise followed Gaussian distributions \(N\big{(}0,\sigma_{p}^{2}\big{)}\) and \(N\big{(}0,\sigma_{m}^{2}\big{)}\), where \(\sigma_{p}=8\) and \(\sigma_{m}=0.2\). Ten experiments were carried out with a varying payload mass m and rope length l. The experimental data were partitioned into three sets: the training set, which was used to minimize (5), and the MSE on the training set was used as the fitness function in G3P-SR and, therefore, drove the evolutionary process; the validation set, which was used only for model selection, i.e., the model with the lowest MSE on the validation set was stored as the best model; and the testing set, which was used to evaluate the performance of the obtained model. The training and validation sets comprised experiments 1-8, and the data were combined and split randomly with a 70:30 ratio into the training and validation sets, respectively, while  
Figure 7: Experimental setup. experiments 9-10 were used as the testing sets.