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This paper presents a motion control method for multi-joint robots, which involves the use of mechanical elastic elements with adjustable stiffness. Elastic elements are installed at each joint of the robots, and the proposed control method adaptively adjusts the stiffness of the elements. This stiffness adaptation minimizes actuator torque, while the control method generates fixed periodic motions. Advantages of the control method are that it works well without requiring the knowledge of the exact values of physical parameters or elaborate numerical calculations. We mathematically show trajectory-tracking performance of the control system, and the effect of torque minimization. Experimental results demonstrate that the proposed control method greatly reduces actuator torque while generating the desired motions.