###### Abstract  
Flexible surgical robots are emerging as advanced tools for minimally invasive surgeries, offering greater versatility compared to traditional rigid robots. Unlike commercial endoscopes, the overtube should remain fixed to maintain stability, ensure a clear field of view, and allow surgical tools to perform tasks efficiently. While constant curvature bending of the overtube is sufficient for some lesions, certain lesions require the overtube to bend into specific shapes to achieve appropriate positioning. Various methods for creating different bending shapes have been proposed in previous research, typically involving connecting multiple segments. However, this approach complicates control and reduces both space and cost efficiency. This study proposed a conceptual method for adding a shaping tendon to control the bending shape and mathematically analyzed the effect of this shaping tendon, inserted along an arbitrary path in addition to the main driving tendons for constant curvature bending, on the bending shape of the hyper-redundant manipulator with rolling joints. The overall system was modeled and analyzed from an energy perspective, and the validity of the proposed mathematical modeling was verified through comparison with results obtained from physical experiments. In addition, it was identified that the design parameter determining the tendon path is a significant element in defining the bending shape of the overtube. t +
Footnote †: journal: Applied sciences  
1
Footnote †: journal: Applied sciences  
1
Footnote 1: email: [EMAIL_ADDRESS]. To ensure flexibility and clinical safety, most flexible surgical robots are controlled using a tendon-driven mechanism. This mechanism operates the robot by using multiple tendons (or cables) that run along the length of the robot body. By pulling or releasing these tendons, the robot can bend in different directions. Considering the complexity of analyzing the dynamics and geometry of hyper-redundant manipulators with numerous joints, tendons are typically inserted parallel within the robot. For practical design, ease of control, and analysis, the main motion of the robot is commonly determined by main driving tendons inserted parallel within the robot. Control is then achieved by deriving kinematic equations based on constant curvature bending assumption [9; 10; 11; 12; 13; 14; 15]. Flexible surgical robots can be generally divided into an overtube, which determines the approximate position within the task space, and surgical tools, which perform specific tasks such as injection, marking, cutting, and dissection. Unlike commercial endoscopes, the overtube is fixed when performing tasks to provide a stable environment and maintain an appropriate position, ensuring a clear field of view and allowing surgical tools to perform tasks efficiently [6; 7; 8; 9; 10; 16; 17; 18; 19]. However, since tasks are generally performed in constrained and confined spaces within the human body, relying solely on the constant curvature bending of the overtube to achieve an appropriate field of view is limited. Generally, endoscopic submucosal dissection (ESD) is used as a standard treatment for early-stage gastrointestinal cancer [20; 21; 22]. As shown in Figure 1, the constant curvature bending of the overtube is sufficient to achieve an appropriate pose for performing tasks on some lesions in the large intestine. However, for certain lesions located in the direction of the gravity of the robot, it is necessary to bend the overtube in a specific shape, such as a cobra-like shape. To achieve the ideal pose of the overtube end tip for effective task performance within the human body, non-constant curvature bending may be required in certain situations (Figure 1). The systems proposed in previous studies have slightly different joint designs (rolling joints, pin joints), but they all operate using a tendon-driven mechanism [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. The design methods of these systems have focused solely on miniaturizing the robot.