## Appendix A Analysis of the Rank of the Jacobian Matrix of the 3 DJM  
The transpose of the Jacobian matrix \(\mathbf{J}\in\mathbb{R}^{6\times 3}\) is denoted as \(\mathbf{S}\in\mathbb{R}^{3\times 6}\), which is represented as below  
\[\mathbf{S}=\mathbf{J}^{T}=-(\mathbf{a}_{1}\times\mathbf{u}_{1},\mathbf{a}_{2}\times\mathbf{u}_{2},\cdots,\mathbf{a}_{6}\times\mathbf{u}_{6})^{T}. \tag{12}\]  
Assuming the column rank of \(\mathbf{S}\) satisfies  
\[\mathbf{\mathrm{rank}}(S)<3, \tag{13}\]  
the mixed product of any three column vectors of the matrix \(\mathbf{S}\) yields  
\[\mathbf{S}_{i}\cdot(\mathbf{S}_{j}\times\mathbf{S}_{k})=\left|\begin{array}{ccc}s_{i1}&s _{j1}&s_{k1}\\ s_{i2}&s_{j2}&s_{k2}\\ s_{i3}&s_{j3}&s_{k3}\end{array}\right|\equiv 0, \tag{14}\]  
where \(i,j,k=1,2,\ldots,6\).  
For the 3 DJM of the CHRA, the mixed product of any three column vectors of the matrix \(\mathbf{S}\) can be written as  
\[\begin{split}&\mathbf{S}_{i}\cdot(\mathbf{S}_{j}\times\mathbf{S}_{k})\\ =&(\mathbf{a}_{i}\times\mathbf{u}_{i})\cdot[(\mathbf{a}_{j}\times\mathbf{u}_{j}) \times(\mathbf{a}_{k}\times\mathbf{u}_{k})]\\ =&\frac{[\mathbf{a}_{i}\times(\mathbf{b}_{i}-\mathbf{a}_{i})]\cdot\{[\mathbf{a}_ {j}\times(\mathbf{b}_{j}-\mathbf{a}_{j})]\times[\mathbf{a}_{k}\times(\mathbf{b}_{k}-\mathbf{a}_{k} )]\}}{c_{i}c_{j}c_{k}}\\ =&\frac{(\mathbf{a}_{i}\times\mathbf{b}_{i})\cdot[(\mathbf{a}_{j}\times\mathbf{b}_{j}) \times(\mathbf{a}_{k}\times\mathbf{b}_{k})]}{c_{i}c_{j}c_{k}}\\ \equiv& 0\quad(i,j,k=1,2,\cdots,6).\end{split} \tag{15}\]  
In the design of the 3 DJM, \(e_{A}\approx 0\) and \(e_{B}\approx 0\), we have \(\mathbf{a}_{1}\approx\mathbf{a}_{2},\mathbf{a}_{3}\approx\mathbf{a}_{4},\mathbf{a}_{5}\approx\bm {a}_{6}\), and \(\mathbf{b}_{1}\approx\mathbf{b}_{6},\mathbf{b}_{2}\approx\mathbf{b}_{3},\mathbf{b}_{4}\approx\bm {b}_{5}\). Then, the following mixed product can be represented as  
\[\begin{split}&(\mathbf{a}_{1}\times\mathbf{b}_{1})\cdot[(\mathbf{a}_{2} \times\mathbf{b}_{2})\times(\mathbf{a}_{3}\times\mathbf{b}_{3})]\\ \approx&(\mathbf{a}_{1}\times\mathbf{b}_{1})\cdot[(\mathbf{a}_{1 }\times\mathbf{b}_{3})\times(\mathbf{a}_{3}\times\mathbf{b}_{3})]\\ =&(\mathbf{a}_{1}\times\mathbf{b}_{1})\cdot(\mu\mathbf{b}_{3}) \\ \equiv& 0,\\ &(\mathbf{a}_{4}\times\mathbf{b}_{4})\cdot[(\mathbf{a}_{2}\times\mathbf{b}_{2}) \times(\mathbf{a}_{3}\times\mathbf{b}_{3})]\\ \approx&(\mathbf{a}_{3}\times\mathbf{b}_{5})\cdot[(\mathbf{a}_{1 }\times\mathbf{b}_{3})\times(\mathbf{a}_{3}\times\mathbf{b}_{3})]\\ =&(\mathbf{a}_{3}\times\mathbf{b}_{5})\cdot(\mu\mathbf{b}_{3}) \\ \equiv& 0,\end{split} \tag{16}\]  
where \(\mu\in\mathbb{R}\) and \(\mu\
eq 0\).  
Equation (14) shows that vectors \(\mathbf{b}_{3}\), \(\mathbf{a}_{1}\) and \(\mathbf{b}_{1}\) are in the same plane, (16) shows that vectors \(\mathbf{b}_{3}\), \(\mathbf{a}_{3}\) and \(\mathbf{b}_{5}\) are in the same plane, and (17) shows that vectors \(\mathbf{b}_{3}\), \(\mathbf{a}_{5}\) and \(\mathbf{b}_{5}\) are in the same plane.  
In summary, we have that vectors \(\mathbf{a}_{1}\), \(\mathbf{a}_{3}\), \(\mathbf{a}_{5}\), \(\mathbf{b}_{1}\), \(\mathbf{b}_{3}\) and \(\mathbf{b}_{5}\) are in the same plane, which is impossible for the 3 DJM. The assumption (A2) does not hold, hence  
\[\mathbf{\mathrm{rank}}(S)=3,\] (A8)  
i.e.,  
\[\mathbf{\mathrm{rank}}(J)=3.\] (A9)