Efficiently grasping and releasing objects using robotic grippers is an essential step in robotic assembly. This paper presents a low-cost four-finger adaptive gripper capable of performing stable and reliable grasping operations on irregular-shaped flat objects. Unlike other grasping systems in the fixture-to-fixture robotic assembly, the assembly process using the proposed gripper does not rely on any vision systems or six-axis force/torque sensors. Specifically, assuming that the relative position <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\Lambda }$</tex-math></inline-formula> between the two fixtures is known, such assembly tasks can be accomplished simply by performing a predefined motion of a robot arm only related to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\Lambda }$</tex-math></inline-formula> . This is mainly because the developed gripper not only adapts to the shape and size of the grasped object, but also keeps its position and posture relative to the gripper unchanged throughout the grasping process. A novel underactuated grasping mechanism consisting of an X-shaped differential mechanism and two seesaw ones is proposed to perform adaptive and stable fingertip grasps with more contact points. The kinematics and statics of the gripper-object system are derived for the analysis and design of the gripper. The proposed gripper is fabricated, and a grasping system is built using a commercial robot arm (UR5) to verify its capability for adaptive grasps and assembly of difficult-to-handle flat objects. Experimental results show that it is effective to grasp objects with uncertain shape/size and position, control the grasping force, and accomplish the fixture-to-fixture assembly, etc.