## 1 Introduction
The robot arm has a tiny structure with a diameter of less than 6 mm. The robot arm operates inside the body, and it is hard to attach encoders or sensors to the arm to measure joint angles or trace the end-effector's position and orientation. This type of robot is usually controlled by open-loop control based on a mathematical model, unlike general industrial robot arms. The wire driving length has to be calculated from the robot model and a path generation algorithm according to an arbitrary input from the master device in real-time without any feedback from the joint configuration. Therefore, a comparison study about modeling approaches for tendon-actuated continuum robots was carried out [10]. Most snake-like robot arms have been analyzed with a constant-curvature model for convenience [11]. Recently, a variable-curvature model was derived to improve the accuracy of the curvature model [12]. The curvature model works well for a continuum body such as the slit structure, but it does not fit well with a joint structure. The driving wire length is calculated with modeling error, and wire slack behavior may occur between the joints, where the joints are not able to bend to the desired angle or maintain stiffness. In this paper, a more accurate kinematic model was derived to fit an articulated robot arm with a joint structure, and effective path generation algorithms are proposed to solve the inverse kinematics. One challenging problem is how to solve inverse kinematics from the task space to the wire-length space and generate an actuation path in real-time. A kinematic model of a wire-driven robot arm usually consists of three spaces called the task space, joint space, and wire-length space, as shown in Figure 3. The task space includes the position and orientation of the end-effector, which are given from the master device with the surgeon in real-time. The joint space deals with the joint configuration, which includes the joint angles, translation, and rotation of the robot arm. The wire-length space includes the wire-driving length from the motors. The general approach with the curvature model is to solve the inverse kinematics from the task space to the wire-length space at once. When the coupling problem is solved in multiple segments, however, this approach is restricted by geometrical assumptions [13; 14]. In addition, an accurate kinematic model requires more complicated calculations. The numerical method is an effective way to solve inverse kinematics due to the complexity of the forward kinematic model with a joint structure. The sequential quadratic programming (SQP) method is one of the most successful general methods to solve the nonlinearly constrained minimization problem [15]. A combination of SQP and the iterative Newton-Raphson algorithm was used to solve the inverse kinematics of a humanoid robot with redundancy [16]. Recently, an open-source library (TRAC-IK) was developed and distributed for solving inverse kinematics with SQP [17]. As an alternative to the optimization path generator, a path generator was designed using a PID feedback control algorithm. In this method, solving for inverse kinematics is considered to be a feedback control problem, and the analytical solution is derived by inverting the differential kinematics. A short calculation time can be guaranteed because there is no iteration. The stability of the closed-loop inverse kinematics algorithm for redundant robots  
Figure 3: Relation between three spaces of kinematics of wire-driven robotic arm. was verified in the discrete time domain [18]. The closed-loop inverse kinematics algorithm showed advantages in singular value filtering and task priority strategy in comparative experiments of a whole-arm manipulator (WAM) robot [19]. In this study, the solution for inverse kinematics was 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, an accurate forward kinematic model from the joint space to the task space was derived while considering the joint structure.