## 3 Inverse Kinematic Model from Joint Space to Wire-Length Space  
The inverse kinematics of a wire-driven robotic arm represent the physical relationship between the task space and the wire-length space. Inverse kinematics are divided into two steps: one from the task space to the joint space and one from the joint space to the wire-length space. First, the inverse kinematic model from the joint space to the wire-length space is derived from a geometric approach. The conventional curvature model starts with an assumption that the arc length at the center is always conserved. However, in the joint structure, the virtual arc length at the center varies with the bending angle. Figure 7 shows a comparison between the curvature model and the joint model. The wire length at the center is \(l_{c}\), and the outer and inner wire lengths of the bending joint are \(l_{r}\) and \(l_{p}\), respectively. The distance between the joint axis and the wire is \(r\), and the length of all wires is the same as the linkage height \(2h\) before the joint bends. Each wire length is calculated based on the conventional curvature model shown in Equation (7). The wire length during pulling and that during releasing are symmetric. In this study, \(h\) is 1 mm, and \(r\) is 2.45 mm. In the joint structure, the wire length model has to be modified, as shown in Equation (8). The wire lengths during pulling and releasing are not symmetric with respect to the initial wire length in this case. If the wires are driven in accordance with the curvature model, the wires are pulled less and released more compared to the modified model, as shown in Figure 8. Then, slack occurs in the driving wires, and the joint cannot bend to the desired angle or secure stiffness. The modified model with asymmetric lengths for the joint structure guarantees more accurate motion without slack. Figure 6: Comparison of forward kinematics between curvature model and joint model: (**a**) curvature model; (**b**) joint model; (**c**) position difference; (**d**) orientation difference. Figure 7: Diagram of wire length in a single joint: (**a**) conventional curvature model; (**b**) modified model with joint structure. The wire length at the center becomes shorter according to the bending angle. In the general case, wires to control the jaws of the end-effector are located at the center of the robotic arm. The jaws of the end-effector are used to not fully close or open in an arm configuration with a large bending angle. This model shows that the driving wire length at the center to actuate the end-effector needs compensation for the bending angle of the joints:  
\[ \begin{array}{l}l_{r}=2 lt+r\theta\\ l_{p}=2 lt-r\theta\\ l_{c}=2 lt\end{array} , \tag{7}\]  
\[ \begin{array}{l}l_{r}=\left(\frac{h}{\tan(\theta/2)}+r\right)\theta\\ l_{p}=\left(\frac{h}{\tan(\theta/2)}-r\right)\theta\\ l_{c}=\frac{h\theta}{\tan(\theta/2)}\end{array} . \tag{8}\]  
Another advantage of this model is that the compensation length for the coupling problem is solved at the same time. In the stacked joint structure, the wires affect all joints that they pass through. When the wire length to actuate a joint is calculated, the bending angles of all joints that the wire has previously passed through have to be considered. Since this wire length model is a function of only the joint angles, all wire lengths are naturally decided according to the joint configuration. Figure 9 shows views of the distal section from the example in Figure 5. In this case, the arrangement of joints is A-B (proximal)/C-D-D-C (distal). The distal joints are assembled pairs. The two pairs of blue boxes indicate joints, and the red dashed-dotted line across the two blue boxes represents the joint axis. Wires 1, 3, 5, and 7 are connected to proximal joints. Wires 2, 4, 6, and 8 are connected to distal joints through the proximal joints.