## 1 Introduction
The conducted research showed that the proposed adaptive-sliding algorithm provided the best control quality among the tested algorithms (e.g., PID controller and sliding control with linearization). The studies in [56; 57; 58] presented methods of motion control of a holonomic WMR with the use of a fuzzy control algorithm. In [56], the orientation angle of a robot (i.e., yaw heading) was determined using an inertial measurement unit (IMU LSM303 DLH). Then, a fuzzy WMR motion control algorithm was applied. The fuzzy controller (FLC) contained a base of 25 rules and had two inputs for the state vector error and the error between current and the previous iteration of the state vector; a Sugeno model with a Mamdani implication was used. The output of the FLC control algorithm determined the work cycle of the PWM signal that controlled the angular velocity of the mecanum wheel motor. A simulation of controlling the robot's motion along a path in the shape of a circle, an ellipse, and a spiral was conducted. The results were presented in the form of a graph of the motion path implemented by the WMR, using only the solution of the inverse kinematics problem, and for a waveform using the IMU system and fuzzy control algorithm. Unfortunately, it was difficult to assess the quality of control due to the lack of state vector error graphs and quality indicators. Additionally, the presented results did not provide a comparison between the fuzzy control algorithm and another algorithm. In [57], a control algorithm with a fuzzy PID controller was used. The authors focused a significant portion of the research on the description of the hardware structure of the test platform used and the communication procedures between the components of the robot. The rules for the behavior of the FLC, in the absence of measurements of the angular motion parameters of a given mecanum wheel, were considered. There was no description of the quality of the implementation of the desired motion path. Only the conceptual implementation of the obstacle avoidance behavior by the WMR and the possible translation and rotation of the robot platform in the case of controlling the rotation of two mecanum wheels were presented. In [58], as opposed to the traditional method of a mathematical description of a robot model by solving the kinematics and dynamics problems, simulation modelling was applied. A physical model, built in Solidworks, was transferred to the Adams environment, where the robot dynamics model was determined by defining constraints, actuators, and sensors. Then, the simulation was performed using both the Adams program and the MATLAB/Simulink development environment. A fuzzy control algorithm was used that consisted of three FLCs (one controller each for two translations and one rotation), each using a base of 25 rules with two inputs for error and error difference from the current and previous iteration, one output, and five membership functions for each variable. To implement the FLC, the MATLAB fuzzy logic toolbox with the Mamdani implication was used. The results of the simulation tests were presented; however, no quality indicators were used to assess the quality of motion implementation by the WMR control algorithm. To verify the correctness of the operation of the proposed method of intelligent control for a mobile robot with mecanum wheels and to assess the quality of the motion implementation, a synthesis of a neural algorithm was conducted. An RVFL (random vector functional link) NN was used [59; 60; 61; 62]; the law of weight adaptation was determined based on the mathematical model of a robot using the Lyapunov stability theory. Next, numerical simulations and verification tests were conducted, the results of which enabled the evaluation of the control quality of the tested method in comparison to a PD controller. Our study evaluated a neural tracking control algorithm of a WMR with the use of the RVFL NNs, demonstrating the accuracy of the motion via numerical tests and studies on a real object, particularly when disturbed.