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The minimum-time trajectory planning problem for three-wheeled omnidirectional mobile robots (TOMRs) is solved based on the combined dynamic model of a mobile robot and dc motor actuators, under the constraint of bounded control inputs due to the battery voltage. We constrain that the bounded-curvature path based on a smooth road (which is described as a clothoid) be given for the translational motion of the TOMR and that the reference profile with respect to the path-length parameter be predetermined for the heading motion of the TOMR. The dynamics of the TOMR is transformed into normal and tangent spaces for motion analysis on the bounded-curvature path. We find out the time-optimality condition of the TOMR, which imposes that the input voltage vector of three motors should have at least one extreme component. Based on the optimality condition, we present a systematic way to construct the optimal control input vector. Finally, several examples are analyzed by the use of the proposed method.