Mobile robots with independently steerable wheels possess many high maneuverability features of omnidirectional robots while benefiting from better performance and capability of moving on rough terrains. However, motion control of such robots is a challenging task due to presence of singular configurations and unboundedly large steering velocities in the neighborhood of those singularities. Many proposed approaches rely on numerical solutions that keep the robot out of bulky regions around the singular points and hence lose some of the robot maneuverability. Based on a class of traditional path followers we design a new globally stable path following controller that exploits the high maneuverability of the platform. This design allows us to derive a set of closed-form analytical functions that describe the robot base velocity as a function of the wheels driving and steering velocities while abide to the robot non-holonomic constraints. Those functions are then utilized to find the maximum instantaneous velocity of the body that keeps the wheels velocities under the pre-specified bounds no matter how much the robot gets close or far from its singular configurations. The control algorithms developed in this paper have been evaluated on iMoro, a four wheel independently steered mobile manipulator designed and developed at IHA/TUT. Experimental data is also shown that show efficacy of the method.