## 4 Numerical Tests of the Operation of the Algorithm
### Simulation Test 1 (PD Controller)  
Figure 5 presents the results of the simulation showing the waveforms of the implemented angles of rotation of individual mecanum wheels (Figure 5a), implemented angular velocities (Figure 5b), tracking errors of the rotation angles, and angular velocities (Figure 5c,d, respectively). The control signals generated by the PD controller are shown in Figure 5e. The diagram of the desired and implemented motion path is shown in Figure 5f. All of the graphs use the color marking of the mecanum wheel number: red, wheel 1; yellow, wheel 2; blue, wheel 3; and green, wheel 4.  
The values of angular velocity vectors of the wheels (Figure 5b) were the same for the pairs of wheels, 1, 4 and 2, 3, due to the constant angle of the robot's frame orientation. The waveforms of the control signals (Figure 5e) reflected this phenomenon by having the same values for the aforementioned wheels pairs. The influence of the disturbance on the waveform of the control signal occurring at t\({}_{21}=6.5\) [s] and ending at t\({}_{22}=11.2\) [s] was also noticeable (the time of the occurrence of the disturbance in the images is marked with a dashed line).  
The angular velocity error (Figure 5d) was limited, and after stabilization, it tended toward zero. The rotation angle error (Figure 5c) in steady states had a non-zero value. The error value decreased and tended toward zero as a result of the start of the braking process and the end of the motion. The value of the error of mapping the desired motion path (Figure 5f) was small.  
The obtained values of quality indicators were used to compare the quality of the control using a neural control algorithm.  
Figure 4: The desired path of point S of the mobile robot’s frame.  
desired and implemented motion path is shown in Figure 6i. For all the graphs (except for Figure 6h,i), the color markings of the mecanum wheel numbers were used: red, wheel 1; yellow, wheel 2; blue, wheel 3; and green, wheel 4.  
The quality indicators obtained as a result of the simulation are presented in Table 2.  
The implemented motion parameters had similar waveforms to those obtained in Test 1 (Figure 6a,b). The tracking errors were limited and had the greatest values at the beginning and ending phases of motion, when the angular velocity vector of the wheel changed to the opposite value and, to a lesser extent, at the time of the disturbance. The error was minimized to the vicinity of zero for both the angular velocity and the angle of rotation error (Figure 6c,d).  
The control signal PD assumed a non-zero value during the beginning of the acceleration and deceleration phases, during changes in the direction of the rotation of the wheels pairs, and as a result of the disturbance acting on the object (Figure 6e). Neural control (Figure 6f), after partially determining the output weights of the network, intervened as the controller. The values of the NN weights were relatively stable after the initial stabilization (shown as the first wheel in Figure 6h). As found in Test 1, the error of mapping the desired motion path (Figure 6i) was small.  
All quality indicators, with the exception of d\({}_{\rm n}\), had lower values for the control algorithm when the NNs were used, as compared to a PD controller. This difference was apparent, especially for the RMSE of tracking the desired rotation angle \(\varepsilon_{\rm i}\), where the index values decreased by approximately 90% when using intelligent control, as compared to the PD controller itself. Additionally, the results of the neural control were obtained under the most unfavorable conditions, during which the NN output weights had zero initial values. The intelligent algorithm, due to the process of NN weight adaptation, provided more accurate mapping of the motion path, as compared to a classic PD controller.