This paper presents a method for optimally actuating a novel omnidirectional wheeled mobile platform with three driven ball wheels. This platform, which will be integrated with a manipulator, is designed for use in unstructured and congested environments such as those of highway maintenance and construction work sites. The designed ball wheel mobile platform can move in all directions on the plane, instantaneously and isotropically. For accurate motion control with parametric uncertainty in the dynamic model, an adaptive controller is applied for trajectory tracking control of the three-ball wheel platform. Each ball wheel has two active drive mechanisms which makes the platform redundantly actuated for in-plane motion. As such, a pseudo-inverse method is used for redundancy resolution and optimal torque distribution. Simulation results demonstrate the effectiveness of the approach.