## 3 Results
The control input \(u\), determined by the NMPC-GP controller, as well as the crane velocity, position, and payload sway angle are depicted in Figures 6 and 7a. The crane position rise time is within 5.4-5.8 s, while the settling time range is within 6.4-7.2 s; after this time, the residual oscillation remains within the expected range (the linear payload deviation from equilibrium point is less than 0.02 m). The maximum value of the payload sway angle varies with rope length, while the influence of the payload mass is minor within the range tested during the experiments. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Rope Length**} & **Payload** & \multirow{2}{*}{**Rise Time**} & **Settling** & **Maximum** & **Maximum** & **Residual** & **Residual** \\  & **Mass** & & **Time** & **Sway Angle** & **Deviation** & **Sway** & **Deviation** \\  & \(m\)[kg] & & \(t_{s}\)[s] & \(\alpha_{max}\)[rad] & \(d_{max}\)[m] & \(\alpha_{r}\)[rad] & \(d_{r}\)[m] \\ \hline \multirow{4}{*}{0.8} & 10 & 5.7 & 7.2 & 0.080 & 0.064 & 0.004 & 0.0032 \\  & 30 & 5.5 & 6.5 & 0.079 & 0.063 & 0.012 & 0.0096 \\  & 50 & 5.4 & 6.4 & 0.076 & 0.061 & 0.006 & 0.0048 \\  & 10 & 5.5 & 6.5 & 0.052 & 0.073 & 0.001 & 0.0014 \\
1.4 & 30 & 5.7 & 7.1 & 0.058 & 0.081 & 0.001 & 0.0014 \\  & 50 & 5.8 & 7.0 & 0.055 & 0.077 & 0.001 & 0.0014 \\  & 10 & 5.7 & 6.6 & 0.044 & 0.088 & 0.003 & 0.006 \\
2.0 & 30 & 5.7 & 6.5 & 0.044 & 0.088 & 0.008 & 0.016 \\  & 50 & 5.8 & 6.6 & 0.046 & 0.092 & 0.008 & 0.016 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of NMPC-GP in terms of rope length and payload mass variation. Figure 6: NMPC-GP performance for \(m=\{10,30,50\}\) kg and (a) \(l=0.8\) m and (b) \(l=1.4\) m.  
In Case 2, the performance of the NMPC-GP was verified in the presence of an external disturbance affecting the pendulum. It is presented in Figure 7b. The experiment was carried out for a rope length \(l\) = 2.0 m and a payload mass \(m\) = 10 kg. In the time interval between 1.8 and 2.8 s, the pendulum is displaced from \(-\)0.04 rad to 0.15 rad, the control signal varies between \(-\)4.8 V and 8 V (corresponding to a velocity change between \(-\)0.09 m/s and 0.24 m/s), and the crane and payload positions are settled at the desired value with a settling time of 9 s with the suppression of the suspended payload oscillation within the allowable tolerance range of below 0.01 rad. In Case 3, the proposed NMPC-GP is compared with the DFC developed using the pole placement method in Section 2.3. The comparison was carried out for two operating points not included in the training of the GP-NARX model: rope length \(l\) = 1.1 m and \(l\) = 1.7 m, and payload mass \(m\) = 10 kg. Using input-output data obtained from the experiment carried out with a rope length \(l\) = 1.1 m, a discrete-time output-error model identification was performed to parametrize models (13) and (14), as follows:  
\[v(k)=\frac{0.0031z^{-1}}{1-0.8942z^{-1}}u(k),\alpha(k)=\frac{-0.1409z^{-1}+0.14 23z^{-2}}{1-1.9302z^{-1}+0.9999z^{-2}}u(k) \tag{26}\]  
The parameters of control law (15) were determined using the pole placement method as \(k_{1}\) = 162.471, \(k_{2}\) = \(-\)141.906, \(k_{3}\) = 394.426, \(k_{4}\) = \(-\)364.362, and \(k_{5}\) = 0.1078, by choosing the desired closed-loop poles as \(\lambda_{i}\) = 0.7865 (\(i\) = 1, 2, \(\ldots\), 5) to obtain a similar settling time for the crane position as in the case of the NMPC-GP. The control performances are compared in Table 5 and Figure 8. The DFC performs similarly to the NMPC-GP in the experiment with \(l\) = 1.1 m in terms of settling time of crane position and payload residual oscillation. However, the NMPC-GP outperforms the DFC in the reduction in payload oscillations in the transient state. The first two amplitudes of oscillation are reduced by the  
Figure 7: NMPC-GP performance for (**a**) \(m\) = {10, 30, 50} kg and \(l\) = 2.0 m, and (**b**) in the presence of a disturbance (between 1.8 and 2.8 s, the pendulum is displaced from \(-\)0.04 rad to 0.15 rad).