## 4 Numerical Tests of the Operation of the Algorithm  
The results of the simulation of WMR motion controlled by the proposed algorithm are presented below. The use of a robot dynamics model enabled approximate mapping of the response of the dynamic system to a set excitation, which would verify the accuracy of the control system. In addition, numerical tests analyzed the quality of the dynamic object motion with the use of various control methods, using the adopted quality indicators. The simulation tests were conducted in the MATLAB/Simulink package. The Runge-Kutta integration with a constant time discretization step equal to \(\mathrm{t}_{\mathrm{d}}=0.01\) [\(\mathrm{s}\)] was used. The disturbances introduced in the simulation were implemented by changes in the value of the rolling friction coefficient from the initial value \(\mathrm{f}_{\mathrm{n}}=0.046\) [\(\mathrm{m}\)] to the value \(\mathrm{f}_{\mathrm{d}}=0.055\) [\(\mathrm{m}\)] during \(6.5\leq\mathrm{t}\leq 11.2\) [\(\mathrm{s}\)]. The coefficient was changed for all of the mecanum wheels. In the simulation, the controlled object parameters were used to replicate the real parameters of [PERSON]'s Panther mobile robot. The following geometric parameters of the robot were adopted: \(\mathrm{R}=0.085\) [\(\mathrm{m}\)], \(\mathrm{r}=0.0165\) [\(\mathrm{m}\)], \(\mathrm{s}_{\mathrm{x}}=0.22\) [\(\mathrm{m}\)], and \(\mathrm{s}_{\mathrm{y}}=0.351\) [\(\mathrm{m}\)]. Dynamic parameters of the robot were the following: \(\mathrm{A}=0.01404\) [\(\mathrm{kg}\cdot\mathrm{m}^{2}\)], \(\mathrm{B}=0.03767\) [\(\mathrm{kg}\cdot\mathrm{m}^{2}\)], and \(\mathrm{C}=0.01193\) [\(\mathrm{kg}\cdot\mathrm{m}^{2}\)]. Numerical tests of the WMR motion were conducted using a PD controller (Test 1) with the gain of the proportional term \(\mathrm{K}_{\mathrm{P}\mathrm{i}\mathrm{j}}=2\) and the derivative term \(\mathrm{K}_{\mathrm{D}\mathrm{i}\mathrm{j}}=1\). In the intelligent control algorithm test (Test 2), the same PD controller gain values were used as in Test 1 while in the nonlinearity compensation term of the controlled object, four NNs were used, each of which consisted of \(\mathrm{h}=4\) inputs (including the threshold value), \(\mathrm{m}=10\) neurons in the hidden layer, and one output. The choice of \(\mathrm{m}=10\) neurons in the hidden layer resulted from numerous simulation studies for a different number of neurons, in which a slight increase in the quality of tracking was observed with more than 10 neurons while the NN complexity increased proportionally to the number of neurons. A coefficient value of \(\gamma=4\) and matrix of learning gain coefficients \(\mathrm{F}=\mathrm{diag}(9)\) were adopted. The parameters of the PD and NN controllers were selected by trial and error so as to obtain the best quality of motion execution. The waveform of the velocity vector of the characteristic point S of the robot was adopted in accordance with the following relationship:  
\[\mathrm{v_{S}}=\mathrm{v_{max}}\bigg{(}\frac{1}{1+\mathrm{e^{-c(t-t_{e})}}}-\frac {1}{1+\mathrm{e^{-c(t-t_{h})}}}\bigg{)}\big{[}\mathrm{m/s}\big{]} \tag{30}\]  
where \(\mathrm{v_{max}}=0.5\) [m/s] is the maximum velocity in steady state, \(\mathrm{c}=9\) [1/s] is the rate of velocity change in the acceleration and deceleration phases, \(\mathrm{t_{r}}=2\) [s] is the mean acceleration time, and \(\mathrm{t_{h}}=13\) [s] is the mean braking time. To assess the quality of motion implementation with the use of a given control algorithm, the following quality indicators were adopted:  
1. The maximum value of the rotation angle error of the mecanum wheel: \[\mathrm{e_{imax}}=\mathrm{max}|\mathrm{e_{ik}}|[\mathrm{rad}]\] (31) where i = 1,2,3,4 is the mecanum wheel number and k is the number of consecutive discrete measurements. 2. The maximum value of the angular velocity error of the mecanum wheel: \[\mathrm{\dot{e}_{imax}}=\mathrm{max}|\mathrm{\dot{e}_{ik}}|[\mathrm{rad/s}]\] (32)
3.