To maximize total operating time, an online minimum-energy translational and rotational velocity trajectory planning and control system is presented on a straight-line path for three-wheeled omnidirectional mobile robots (TOMRs). We suggest an efficient online trajectory planning algorithm, which minimizes a practical cost function as the energy that is drawn from batteries to motors, which is based on an accurate TOMR dynamic model that includes both the Coriolis force and the actuator motor dynamics. Using Pontryagin's minimum principle, we find the minimum-energy rotational velocity trajectory in analytic form. Then, the minimum-energy translational velocity trajectory is found using a novel algorithm that has a time complexity of O(n), which is based on a linearity condition on the state transition of TOMRs. Moreover, a trajectory control system is implemented using the resolved-acceleration control to validate the actual performance. Simulation and experiment results show that these minimum-energy trajectories can save energy up to 4.76% compared with the energy-efficient trapezoidal velocity profile and up to 5.29% compared with the loss-minimization control.