We propose an efficient time-optimal trajectory planning algorithm for differential-driven wheeled mobile robots with bounded motor control inputs in the environment with cranked road where two corners with arbitrary angles exist. Based on dynamics for differential-driven wheeled mobile robots including actuating motors as well as on the bang-bang principle for time-optimality, we plan the time-optimal free path trajectory which unifies path planning and trajectory generation. Thus it improves total time significantly while handling obstacle avoidance. We divide our trajectory into three sections and then divide each section into an appropriate number of subsections to make five subsections in total. We introduce the concept of transition angle to solve the problem with formulating search loops efficiently. Numerical results are provided to validate the effectiveness of the proposed algorithm.