Omnidirectional mobile robots have been popularly employed in several application areas. However, the kinematics for these systems have not been clearly identified, specially for the redundantly actuated case which is common in omnidirectional mobile robot such as Nomadic model. For such mobile robot systems, exploitation of redundant actuation as well as singularity analysis has not been extensively addressed. In light of this fact, this paper introduces two different kinematic approaches for omnidirectional mobile robots and examine singularity configurations of such systems. Then, a singular-free load distribution scheme for a redundantly actuated three-wheeled omnidirectional mobile robot is proposed. Through simulation, several advantages of the redundantly actuated mobile robot with respect to singularity avoidance and exploiting several sub-tasks, are presented.