We perform an in-depth analysis on the energy efficiency of a mobile robot employing a spherical type continuously variable transmission (S-CVT). The S-CVT permits the motor to operate in its most power-efficient regime of high-speed and low-torque, and allows for smooth operation of the mobile robot in all phases of forward, neutral, and reverse motion without the use of any brakes or clutches. Changes in steering direction are achieved by a novel pivoting device that enables the robot to rotate about its center. The current study takes into account features of the DC motors, the S-CVT, and the mobile robot dynamics to perform a comprehensive study on the overall system performance. We develop optimal control laws for both the reduction gear unit and S-CVT equipped mobile robots, and compare their energy efficiency through numerical studies. We then present numerical results that demonstrate the power savings possible front the use of the S-CVT mechanism over standard reduction gear units.