## 3 Kinematic Analysis of CDHRM
### Cable-Joint Kinematics  
The length of each cable consists of the length inside the tubular structure and the length between adjacent joints. The cable length inside the tubular structure is equal to the length of the tubular structure, which is a measurable constant. The length between adjacent joints is derived with regard to the joint angles, which can be calculated as follows:  
Figure 3: Geometric model of the universal joint.  
Figure 2: Multilevel mapping relationships of the manipulator.  
Suppose that the manipulator has \(N\) joints, \(H\) represents the length of each tubular structure, \(R\) represents the radius of each tubular structure, \(D\) represents the distance from the tubular structure to the center of universal joint. To establish the relationship between the length between adjacent joints and the joint angles, the transformation between the adjacent joint frames \(I_{i}\) and \(J_{i}\) can be derived as (1) and (2), which are shown at the top of next page, where if \(i\) (\(i\) =1,2,...,\(N\)) is an odd number, the transformation is (1) otherwise the transformation is (2).  
\[\begin{split}\int\limits^{I}T=& transl\left(D,0,0 \right)\,trotz\left(\alpha_{i}\right)\,troty\left(\beta_{i}\right)\,transl \left(D,0,0\right)\\ =&\left[\begin{array}{ccc}cos\left(\alpha_{i} \right)\,cos\left(\beta_{i}\right)&-sin\left(\alpha_{i}\right)&sin\left( \beta_{i}\right)\,cos\left(\alpha_{i}\right)&D+Dcos\left(\alpha_{i}\right)\, cos\left(\beta_{i}\right)\\ cos\left(\beta_{i}\right)\,sin\left(\alpha_{i}\right)&cos\left(\alpha_{i} \right)&sin\left(\beta_{i}\right)\,sin\left(\alpha_{i}\right)&Dcos\left( \beta_{i}\right)\,sin\left(\alpha_{i}\right)\\ -sin\left(\beta_{i}\right)&0&cos\left(\beta_{i}\right)&-Dsin\left(\beta_{i} \right)\\ 0&0&0&1\end{array}\right]\end{split} \tag{1}\]  
\[\begin{split}\int\limits^{I}T=& transl\left(D,0,0\right)\,troty\left(\beta_{i}\right)\,trotz\left(\alpha_{i}\right)\,transl \left(D,0,0\right)\\ =&\left[\begin{array}{ccc}cos\left(\beta_{i}\right)cos\left( \alpha_{i}\right)&-cos\left(\beta_{i}\right)sin\left(\alpha_{i}\right)&sin \left(\beta_{i}\right)&D+Dcos\left(\beta_{i}\right)cos\left(\alpha_{i}\right) \\ sin\left(\alpha_{i}\right)&cos\left(\alpha_{i}\right)&0&Dsin\left(\alpha_{i} \right)\\ -sin\left(\beta_{i}\right)cos\left(\alpha_{i}\right)&sin\left(\beta_{i} \right)sin\left(\alpha_{i}\right)&cos\left(\beta_{i}\right)&-Dsin\left(\beta_{ i}\right)cos\left(\alpha_{i}\right)\\ 0&0&0&1\end{array}\right]\end{split} \tag{2}\]  
where \(transl()\) represents the translation function, \(troty()\) and \(trotz()\) represent the rotation function refer to \(y\)-axis and \(z\)-axis, \(\alpha_{i}\) and \(\beta_{i}\) represent counterclockwise rotation angle of the \(i\)-th joint around the \(z\)-axis and \(y\)-axis.  
The counterclockwise mounting angle \(\phi_{j,k}\) of the \(j\)-th (\(j\) = 1,2,3) cable actuating the \(k\)-th (\(k\) = 1,2,...,\(N\)) joint can be derived in (3), The \(p_{i,j,k}\) is the mounting position of the \(j\) th cable actuating the \(k\) th joint at the \(I_{i}\) frame, which can be derived in (4).  
\[\phi_{j,k}=k\times\frac{360}{M}+(j-1)\times 120 \tag{3}\]  
\[p_{i,j,k}=\left[0,R\times\cos(\phi_{j,k}),-R\times\sin(\phi_{j,k})\right]^{T} \tag{4}\]  
where \(M\) represents the number of the holes at the end plate.  
Define \(l_{ijk}\) as the length of the \(j\) th cable actuating the \(k\) th joint between the frames \(I_{i}\) and \(J_{i}\), \(L_{jk}\) as the total length of the \(j\) th cable actuating the \(k\) th joint, which can be derived as follows:  
\[l_{i,j,k}=\left|\right|_{I}^{I}T\times p_{i,j,k}-p_{i,j,k}| \tag{5}\]  
\[L_{jk}=\sum_{i=1}^{k}(l_{ijk}+H) \tag{6}\]  
Therefore, the geometric relationship between cable length (\(L_{jk}\)) and joint angle \(\alpha_{i}\) and \(\beta_{i}\) can be derived with regard to (1)-(6).