## 1 Introduction  
Balancing mechanisms compensating for a gravity load play an important role in mechanical design in terms of energy efficiency. In an active device, gravity balancing helps to minimize the driving torque, thereby increasing the efficiency of the system. In a passive device, the user can use the equipment without feeling the gravity load. Currently, gravity balancing mechanisms are widely applied in industrial, service, and medical robots, as well as in construction. Research on gravity compensation has been actively carried out, and various applications using gravity compensation have been introduced. Two approaches can be used to design a gravity balancing mechanism, namely a counterweight or a counter spring. Gravity compensation using a counterweight is a method of balancing a mechanism by putting weights on the opposite side of the center of rotation. The gravity compensation method using a spring is a method where the mechanism is balanced using the restoring force of a spring instead of a counterweight. [PERSON] and [PERSON] discussed the advantages and disadvantages of these two methods [1]. Using counterweights to balance a two degrees of freedom (2-DOF) parallel robot for antennas was suggested by [PERSON] and [PERSON] [2]. [PERSON] and [PERSON] [3] proposed a general approach for the equilibration of planar linkages using springs. Gravity balancing of a six degrees of freedom (6-DOF) spatial parallel mechanism by springs was presented by [PERSON] et al. [4]. [PERSON] et al. [5; 6; 7; 8] obtained the gravity balancing conditions of a three degrees of freedom (3-DOF) planar mechanism, 3-DOF spatial parallel mechanism, four degrees of freedom (4-DOF) spatial parallel mechanism, and a 6-DOF parallel mechanism by using both counterweights and springs. Using counterweights and a pantograph, [PERSON] et al. [9] obtained the static balancing conditions of hexapods. [PERSON] and [PERSON] [10] proposed a general method for statically balanced rotational flexure using an idealized hinge and torsion spring. To obtain the balance of a DELTA parallel robot, three zero-free-length elastic systems (springs) for each connecting kinematic chain were proposed by [PERSON] et al. [11]. [PERSON] et al. [12] designed a spatial 6-DOF decoupling parallel mechanism with static balancing using a counterweight and a spring. [PERSON] and [PERSON] [13] proposed a static balancing mechanism for a face robot that uses a gravity compensator (spring). Using the concept of constant potential energy, [PERSON] et al. [14] proposed a statically balanced stackable planar mechanism. Gravity compensation methods are currently applied to various mechanism designs. A gravity-balancing mechanism design with auxiliary parallelograms was investigated by [PERSON] et al. [15]. [PERSON] and [PERSON] [16] also used auxiliary parallelograms to design a reactionless 3-DOF planar parallel mechanism. [PERSON] et al. [17] designed a statically balanced and partial decoupled PAMINSA by adding counterweights at some axes of the pantograph linkages. In the design of a haptic mechanism, [PERSON] et al. [18] designed a five-bar parallel mechanism by using an actuator as a counterweight. In the design of a service robot, [PERSON] and [PERSON] [19] designed a novel multi-DOF gravity balancing robot arm by attaching compression springs at each joint. Employing double parallelograms made it possible to counterbalance a multi-DOF robot arm through the use of springs, and their practical design was also very compact. The need for a gravity balancing mechanisms for medical devices is growing to support the needs of the medical operator, such as feeling less physical fatigue. The medical microscope being used in operation rooms is such an example. [PERSON] et al. [20] designed a medical tool holding arm apparatus that was capable of widely changing the position and angle of a medical tool. [PERSON] et al. [21] also designed a gravity balancing medical stand apparatus with a balancing adjustment in accordance with weight changes on the operating side.