We establish a highly feasible algorithm for time-optimal cornering trajectory planning (TP) for car-like mobile robots (CLMRs) based on a dynamic model that contains actuator dynamics. First, we formulate an accurate dynamic model of a robot that contains DC motor actuators; this includes steering braking (caused by the lateral force of the front steering wheel) and two types of friction (viscous and Coulomb) under a nonslip condition. Our TP algorithm can utilize the full power of the DC motor actuators within proper pulse width modulation bounds and generated torque limits. Then, we establish an algorithm for a time-optimal cornering trajectory planning for CLMRs (TOCTP-CLMR). Our algorithm divides the trajectory into five sections comprising three turnings and two translations to minimize the travel distance. Then, we utilize the quickest rotation when turning to construct the time-optimal trajectory that satisfies the bang-bang principle. In addition, simulations are performed to demonstrate the validity of this method. Finally, we conduct open-loop experiments to validate our dynamic model and a trajectory tracking experiment to demonstrate the feasibility of the TOCTP-CLMR trajectory.