Robots used in hazardous environments need a high degree of mobility with good precision and robustness to actuator failures. In this paper, a novel gear train is proposed to satisfy the requirement for such a mechanism. The gear train is developed based on the following principles: (i) the mechanism is omnidirectional for a high degree of mobility, (ii) the mechanism uses only conventional tirewheels for high precision, and (iii) the mechanism uses only three motors for no actuation redundancy; at the same time, it has robustness to actuator failure so that when any one of the motors is not functioning properly, regardless of which one is not and regardless of the configuration at the moment of failure, the posture is controllable with the other two working actuators. By virtue of the proposed gear train, in an actuator failure, the entire structure becomes similar to a differentially driven two-wheeled mobile robot subject to nonholonomic constraints. The nonholonomic constraints, inherent in the mechanism and stemming from an actuator failure, are crucial to maintain the controllability of the robot posture when omnidirectional mobility is lost due to actuator failure. Controllability is proved and control laws are presented for both the omnidirectional mode, when all three motors are functioning, and the non-omnidirectional mode, when one of the motors is locked due to a failure. The omnidirectional mobility and robustness to an actuator failure is verified by experimental results.