The paper deals with the dynamics of a mobile robot with four Mecanum wheels. For such a system the kinematical rolling conditions lead to non‐holonomic constraints. From the framework of non‐holonomic mechanics Chaplygin's equation is used to obtain the exact equation of motion for the robot. Solving the constraint equations for a part of generalized velocities by using a pseudoinverse matrix the mechanical system is transformed to another system that is not equivalent to the original system. Limiting the consideration to certain special types of motions, e.g., translational motion of the robot or its rotation relative to the center of mass, and impose appropriate constraints on the torques applied to the wheels, the solution obtained by means of the pseudoinverse matrix will coincide with the exact solution. In these cases, the constraints imposed on the system become holonomic constraints, which justifies using Lagrange's equations of the second kind. Holonomic character of the constraints is a sufficient condition for applicability of Lagrange's equations of the second kind but it is not a necessary condition. Using the methods of non‐holonomic mechanics a greather class of trajectories can be achieved.