## 2 Forward Kinematic Model from Joint Space to Task Space  
The forward kinematic model was derived using a classical approach with rigid body transformation. Moving frames were attached to each joint, and the relative position and orientation between frames can be described with a homogeneous transformation matrix. The arm has multiple one-DoF revolute joints that are stacked. The axes of the stacked joints are arranged orthogonally to increase the DoFs and secure the wire path [20]. The forward kinematic model can be obtained from the sequential product of the transformation matrices. Figure 4 shows a diagram of a single-joint section. The relationship between the \(i-1\)th frame and the \(i\)th frame is described by Equation (1) for frames from 1 to 3, where \(\phi\) is the assigned angle of the joint axis, whose parameter is initially determined; \(\theta\) is the bending angle of the joint, which changes with the joint configuration; and \(h\) is the height of the joint axis from the base plane. To produce  
the transformation matrix of the next joint independently, the assigned angle \(\phi\) has to return to its initial value:  
\[\mathbf{T}^{i-1,j}(\phi,\theta)=\mathbf{T}^{i-1,1}(\phi)\mathbf{T}^ {1,2}(\theta)\mathbf{T}^{2,3,i}(-\phi) \tag{1}\] \[= \left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\phi&-\sin\phi&0\\ 0&\sin\phi&\cos\phi&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{cccc}\cos\theta&0&\sin\theta&h\\ 0&1&0&0\\ -\sin\theta&0&\cos\theta&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&0&0&h\\ 0&\cos(-\phi)&-\sin(-\phi)&0\\ 0&\sin(-\phi)&\cos(-\phi)&0\\ 0&0&1\end{array} \right]\] \[= \left[ \begin{array}{cccc}\cos\theta&-\sin\phi\sin\theta&\cos \phi\sin\theta&h(\cos\theta+1)\\ \sin\phi\sin\theta&\cos^{2}\phi+\cos\theta\sin^{2}\phi&-\frac{\sin 2\phi(\cos\theta-1)} {2}&h\sin\phi\sin\theta\\ -\cos\phi\sin\theta&-\frac{\sin 2\phi(\cos\theta-1)}{2}&\cos^{2}\phi\cos \theta-\cos^{2}\phi+1&-h\cos\phi\sin\theta\\ 0&0&1\end{array} \right]\]  
An example is shown in Figure 5. The arm can achieve a six-DoF motion with double segments. The translation motion of the base is \(\gamma\), and the rolling motion is \(\psi\). The proximal segment has two joints and can simultaneously bend with \(\theta_{1}\) and \(\theta_{2}\) in orthogonal directions. The distal segment has four joints and can bend with \(\theta_{3}\) and \(\theta_{4}\) in orthogonal directions. Each set of two distal joints is coupled. It is assumed that the coupled joints equally share a bending angle. The thickness between joint bases is \(l_{t}\), and the lengths of the medial linkage and end-effector are \(l_{m}\) and \(l_{e}\), respectively. The transformation matrices for the translation motion and rolling motion are Equation (2) and Equation (3), respectively. The matrix \(\mathbf{G}\) in Equation (4) just moves the frames by the length of the components. The transformation matrix from the inertial frame to the end-effector's frame can be obtained from the product of the matrices, as shown in Equation (5). In this work, the product of the exponential was used to derive the forward kinematics, and the configuration parameters are arranged in Table 1. As a result, the position and orientation of the end-effector can be calculated, as shown in Equation (6), where the subscripts of the matrix represent the number of rows and columns. Figure 4: Diagram of a single-joint section. Figure 5: Articulated joints robotic arm with double segments. The orientation is expressed as \(z\)-\(y\)-\(x\) Euler angles for the roll, pitch, and yaw. This is the exact forward kinematic model of the articulated robotic arm with a joint structure. Equations (2)-(6) are  
\[\mathbf{Tr}(\gamma)=\left[ \begin{array}{cccc}1&0&0&\gamma\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right], \tag{2}\]  
\[\mathbf{Ro}(\psi)=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\psi&-\sin\psi&0\\ 0&\sin\psi&\cos\psi&0\\ 0&0&0&1\end{array} \right], \tag{3}\]  
\[\mathbf{G}(l)=\left[ \begin{array}{cccc}1&0&0&l\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}