## 3 Synthesis of the Neural Control Algorithm  
NNs are used in neural control algorithms and act as universal approximators of functions. They enable the approximation of the nonlinearity of the control object and its compensation in the control signal. The output from the network is determined in real time, which enables NN weight adaptation in response to changes in the parameters of the robot model. An additional advantage of this method is that it does not require detailed knowledge of the parameters of the mathematical model of the controlled object. The disadvantage of using NNs in the control algorithm is the large number of calculations that must be performed by the microprocessor robot control system in a given iteration step of the algorithm, which limits their application in real-time control algorithms. This problem has been especially visible in extensive network structures, such as those with a large number of neurons and many input signals. Adaptive intelligent control was based on the model of the control system resulting from the combination of nonlinearity compensating control implemented with the use of NN and the PD controller. The scheme of the algorithm is shown in Figure 3.  
The WMR nonlinearity compensation algorithm consisted of four RVFL NNs, one for each of the mecanum wheels. A bipolar sigmoidal neuron activation function was assumed in the form of the following equation:  
\[\Phi(x)=S\Big{(}V^{T}x\Big{)}=\frac{2}{1+e^{-\gamma V^{T}x}}-1 \tag{11}\]  
where \(\gamma\) is the inclination factor of the activation function, \(V\) is the constant weight matrix of the input layer selected randomly in the network initialization process, \(x=[1,\dot{v},\dot{q}_{d}]\) is the vector of network input values, \(\dot{v}=\ddot{q}_{d}-\lambda\dot{e}\) is the auxiliary variable simplifying the notation, and a value of \(1\) in vector \(x\) represents the constant input signal used to determine the threshold value. The output from the NN was described by the relationship  
\[y(x)=W^{T}\Phi(x) \tag{12}\]  
where \(W\) is the matrix of weights of the output layer with zero initial values. The control signal of the PD controller had the following form:  
\[u_{PD}=K_{D}s=K_{D}\big{(}\dot{e}+\Lambda e\big{)} \tag{13}\]  
where \(K_{D}\) is a diagonal matrix of the gain coefficients of the derivative term, \(K_{P}=K_{D}\Lambda\) is a diagonal matrix of the gain coefficients of the proportional term, and \(\Lambda\) is a diagonal matrix with positive elements \(\lambda_{i,i}\). The tracking error \(e\) and the generalized error \(s\) were defined as  
\[e=q_{d}-q \tag{14}\]  
Figure 3: Diagram of the intelligent control algorithm. \[s=\dot{e}+\Lambda e \tag{15}\]  
where \(q_{d}\) is the vector of desired rotation angles of mecanum wheels, and \(q\) is the vector of implemented rotation angles. Based on the above model, the law of NN weight adaptation was determined so that the analyzed system was stable. The WMR dynamics Equation (4) was adopted in the following form:  
\[M\dot{q}+N(\dot{q})+\eta_{d}(t)=u \tag{16}\]  
where \(n_{d}(t)\) is a vector of bounded, time-varying disturbance signals, and \(u\) is a vector of control signals. By differentiating Equation (15) with respect to time and considering the dynamics of Equation (16), a description of the closed control system as a function of the generalized error was obtained as the following:  
\[M\dot{s}=-u+f(x)+\eta_{d}(t) \tag{17}\]  
where \(f(x)=M\dot{v}+N(\dot{q})\) is the nonlinear function approximated by the NN. The control law was adopted as the following form:  
\[u=\hat{f}(x)+K_{D}s \tag{18}\]  
where \(\hat{f}(x)\) is an estimate of the function \(f(x)\). Assuming an approximation of the nonlinearity of the object dynamics using NN, the following was obtained:  
\[\Phi(x)=S\Big{(}V^{T}x\Big{)}=\frac{2}{1+e^{-\gamma V^{T}x}}-1 \tag{11}\]0  
where \(W\) is the constant matrix of ideal NN output weights and \(\varepsilon\) is the approximation error.