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This paper describes the design and implementation of a robot's gripper control system. In order to safely grasp an unknown object using the robot's gripper, the gripper should detect the force of gripping direction and the force of gravity direction, and should perform the force control using the detected forces and the robot's gripper control system. In this paper, the robot's gripper control system is designed and manufactured using DSP(Digital Signal Processor), and the gripper is composed of two 6-axis force/moment sensors which measures the Fx force(force of x-direction), Fy force, Fz force, and the Mx moment(moment of x-direction), My moment, Mz moment at the same time. The response characteristic test of the system is performed to determine the proportional gain Kp and the integral gain Ki of PI controller. As a result, it is shown that the developed robot's gripper control system grasps an unknown object safely.