In this paper, we propose a computationally efficient online local motion planning algorithm for mobile robots in unknown cluttered dynamic environments. The algorithm plans a trajectory incrementally up to the finite horizon in state-time space. Incremental planning method is capable of fast computation but has poor obstacle avoidance performance. To compensate for the drawbacks of incremental planning, a partial trajectory modification scheme is used that sets an interim goal and then plans a trajectory to pass through the interim goal. The interim goal is a temporary desired goal to prevent the robot from falling into the inevitable collision state. By using incremental planning and partial trajectory modification, it is possible to plan collision-free trajectory with small computation even in cluttered dynamic environment. To generate smooth trajectories around given waypoints and goal, we systematize bidirectional trajectory planning with three kinds of trajectories: 1) a forward trajectory from the current robot state; 2) a backward trajectory from the state of current target waypoint; 3) and a connecting trajectory between the forward and backward trajectories. A smooth trajectory is generated around the way point and goal by setting the state of the current target waypoint (or goal), while taking into account the positional relationship among the planned forward trajectory, the target waypoint and the next waypoint. Performances of the proposed algorithm are validated through extensive simulations and experiment with two types of mobile robots: 1) a holonomic mobile robot and 2) a differential drive mobile robot.