In this work, we present a method involving the fast grasping of an unknown object by a mobile robot with a parallel-jaw gripper. A 2D range sensor is installed on the robot at an inclined angle to acquire partial shape information regarding the unknown objects. The candidate grasping points for an unknown object can then be determined by directly extracting features from this partial shape information. After object grasping and lifting, whether the object can be lifted is judged. If the robot fails to lift the object, then detects other candidate grasping points and performs grasping trials until the object is lifted. The path for the robot to find the grasping points is also designed. The grasping time can be decreased without the need for acquiring and processing all of the object information. Experiments are conducted, and the results illustrate the validity of the proposed algorithm.