## Appendix A  
The deducing process of general kinematics Jacobian matrix \(J_{m}\) given by (11), (14) and (17) are shown as the following, which refers to Figure 1:  
The centroid position vector of the link i in Arm-\(j\) is  
\[\mathbf{r}_{i}^{j}=\mathbf{r}_{0}+\mathbf{b}_{0}^{j}+\sum_{k=1}^{i-1}\left(\mathbf{a}_{k}^{j}+ \mathbf{b}_{k}^{j}\right)+\mathbf{a}_{i}^{j}, \tag{18}\]  
The centroid position vector of the end-effector of Arm-\(j\) is  
\[\mathbf{r}_{e}^{j}=\mathbf{r}_{0}+\mathbf{b}_{0}^{j}+\sum_{k=1}^{i-1}\left(\mathbf{a}_{k}^{j}+ \mathbf{b}_{k}^{j}\right), \tag{19}\]  
The mass center angular velocity of link 1 in Arm-\(j\) is  
\[\mathbf{\omega}_{i}^{j}=\mathbf{\omega}_{0}+\mathbf{\kappa}_{1}^{j}\dot{\mathbf{\theta}}_{1}^ {j}, \tag{20}\]  
where \(\mathbf{\kappa}_{1}^{j}=\left[\begin{array}{cc}0&0&1\end{array}\right]^{T}\) is the rotation axis represented in a fixed coordinate system.  
The mass center line velocity of link 1 in Arm-\(j\) is  
\[\begin{split}&\mathbf{v}_{i}^{j}=\mathbf{v}_{0}+\mathbf{\omega}_{0}^{j}\times \mathbf{b}_{0}^{j}+\mathbf{\omega}_{1}^{j}\times\mathbf{a}_{1}^{j}\\ &=\mathbf{v}_{0}+\mathbf{\omega}_{0}^{j}\times(\mathbf{r}_{1}^{j}-\mathbf{r}_{0})+[\mathbf{ \kappa}_{1}^{j}\times(\mathbf{r}_{1}^{j}-\mathbf{p}_{1}^{j})]\dot{\mathbf{\theta}}_{1}^{j} \end{split}\, \tag{21}\]  
According to (20), the mass center angular velocity of the link \(i\) in Arm-\(j\) is  
\[\mathbf{\omega}_{i}^{j}=\mathbf{\omega}_{0}+\sum_{k=1}^{i}\kappa_{k}^{j}\dot{\mathbf{ \theta}}_{k}^{j}, \tag{22}\]  
According to (21), the mass center line velocity of the link \(i\) in Arm-\(j\) is  
\[\mathbf{v}_{i}^{j}=\mathbf{v}_{0}+\mathbf{\omega}_{0}^{j}\times(\mathbf{r}_{i}^{j}-\mathbf{r}_{0}) +\sum_{k=1}^{i}\left[\mathbf{\kappa}_{k}^{j}\times(\mathbf{r}_{k}^{j}-\mathbf{p}_{k}^{j}) \right]\dot{\mathbf{\theta}}_{k}^{j}, \tag{23}\]  
According to (22), the angular velocity of the end-effector in Arm-\(j\) is  
\[\mathbf{\omega}_{e}^{j}=\mathbf{\omega}_{0}+\sum_{k=1}^{6}\kappa_{k}^{j}\dot{\mathbf{ \theta}}_{k}^{j}, \tag{24}\]According to (A6), the line velocity of the end-effector in Arm-\(j\) is  
\[\mathbf{v}_{e}^{j}=\mathbf{v}_{0}+\mathbf{\omega}_{0}^{j}\times(r_{e}^{j}-\mathbf{r}_{0})+\sum_{k= 1}^{6}\left[\kappa_{k}^{j}\times(r_{k}^{j}-\mathbf{p}_{k}^{j})\right]\hat{\mathbf{\theta}} _{k}^{j},\] (A8)  
From the above-mentioned (A7) and (A8), the general kinematics formula of the end-effector in Arm-\(j\) based on Jacobian matrix can be deduced  
\[\mathbf{V}_{e}^{j}=\mathbf{j}_{b}^{j}\mathbf{V}_{0}+\mathbf{j}_{m}^{j}\mathbf{\theta}^{k},\] (A9)  
where the matrixes are given by (A10)-(A12)  
\[\mathbf{j}_{b}^{j}=\left[\begin{array}{cc}\mathbf{E}_{3}&-\mathbf{\tilde{r}}_{0\mathbf{e}}^{ j}\\ 0&\mathbf{E}_{3}\end{array}\right]=\left[\begin{array}{c}\mathbf{j}_{0v}^{j}\\ \mathbf{j}_{0\omega}^{j}\end{array}\right],\] (A10)  
\[\mathbf{j}_{m}^{j}=\left[\begin{array}{ccc}\mathbf{\kappa}_{1}^{j}\times(r_{e}^{j}- \mathbf{p}_{1}^{j})&\cdots&\mathbf{\kappa}_{6}^{j}\times(\mathbf{r}_{e}^{j}-\mathbf{p}_{6}^{j} )\\ \mathbf{\kappa}_{1}^{j}&\cdots&\mathbf{\kappa}_{6}^{j}\end{array}\right]=\left[ \begin{array}{c}\mathbf{J}_{m\omega}^{j}\\ \mathbf{J}_{m\omega}^{j}\end{array}\right],\] (A11)  
\[V=\left[\begin{array}{c}\mathbf{v}\\ \mathbf{\omega}\end{array}\right],\] (A12)  
in which \(\mathbf{\tilde{r}}_{0\mathbf{e}}^{j}\) is the antisymmetric matrix of the vector \(\mathbf{r}=[x,y,z]^{T}\).