## 1 Introduction  
Soft actuators, like McKibben pneumatic artificial rubber muscles [1; 2], have a long history of development. They have a high power-to-weight ratio compared to electric motors and are key elements in lightweight robot arms by locating valves and compressors, apart from the link mechanism. Moreover, they are lighter than conventional metal cylinders and are flexible. The applications of these actuators include cooperating robots, for which safety is a critical concern due to their regular contact with human subjects [3; 4], biomimetic robots [5; 6], and even medical robots [7; 8; 9; 10]. 3D printers are currently being used to fabricate soft actuators. This facilitates the manufacturing of actuators with complex mechanisms, which has resulted in the development of a variety of soft robots [11]. The manufacturing process for soft actuators is simple and low cost. This involves the preparation of 3D-printed models that are subsequently filled with liquid silicone rubber. PneuNet [12; 13; 14; 15] is a major invention, which has an array of chambers on a fiber-reinforced base, which bends with the application of compressed air. It is applied to soft grippers [16] and durable multi-leg robots [17]. Linear soft actuators that are simple compared to PneuNets have also been developed. Their simple motion enables the easy design of soft robots since they can be used with conventional rigid-link mechanisms. They also have an advantage compared to metal pneumatic cylinders in that they do not have sliding parts. Working fluid such as air or water is sealed, preventing the leakage of fluid and the generation of dust. Furthermore, they enable high precision positioning [18] because no stick-slip occurs, in combination with an appropriate control system using precise sensors and servo valves. Bellow actuators and reinforced cylindrical actuators are the representative examples of these actuators. Bellows, especially those made of rubber, are used for soft robotics [19]. However, rubber bellows expand radially and bursts when high pressure is applied, and fiber reinforcement is necessary. [PERSON] et al. proposed a simple cylindrical soft actuator [20]. In this case, a fiber is wound around the silicone rubber cylinder. The supply of compressed air into the cylinder causes elongation in the axial direction, while the fiber restricts radial expansion. The cylindrical soft actuator generates a linear motion, but this simple behavior leads to a wide array of potential robotic applications, as an alternative to conventional pneumatic hydraulic cylinders. Moreover, pneumatic actuators can control the output force and estimate the external force using servo valves in conventional rigid link robots. Force control is useful for the manipulation of a variety of materials, especially for food and organs. Pneumatic cylinders are back-drivable and low-friction without reduction gears, and the product of the cross-section of the piston and the air pressure directly derives the external force. In this study, force estimation was implemented in a pneumatically-driven surgical robot [21]. In these highlighted examples, force estimation is realized only for rigid pneumatic robots. In this work, the authors extended the method to a soft robot and developed a system to estimate the grasping force of a surgical robot using a soft actuator [22]. A quasi-static model that describes the force output of a soft actuator is desirable for precise control and external force estimation. Several works have reported on the analytical model of soft actuators, while most soft robots have complex dynamics and rely on numerical analysis. [PERSON] et al. modeled a bending soft actuator reinforced by fiber [23]. [PERSON] et al. analyzed the relationship between the deformation of a cylindrical actuator and the helical angle of the reinforcement fiber [24]. Conventional modeling of cable-driven bending manipulators has also been investigated [25; 26; 27; 28].