## 3 Overall Jacobian Matrix
\tag{4}\]  
The physical significance of \(\hat{\mathbf{S}}_{\text{wc},1,i}^{2}\) and \(\hat{\mathbf{S}}_{\text{wc},2,i}^{2}\) are the constraint wrench around \(\mathbf{m}_{i}\) and \(\mathbf{q}_{i}\), \(\mathbf{m}_{i}=\mathbf{q}_{i}\times\mathbf{s}_{1,i}\), and \(\hat{\mathbf{S}}_{\text{wc},3,i}^{2}\) represents the constraint force along \(\mathbf{s}_{3,i}\). Figure 3: Two types of driving chain. (**a**) PRR-type driving chain; (**b**) RPR-type driving chain. \[\hat{\mathbf{\hat{s}}}_{\text{wc},1,i}^{2}=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{\mathbf{m}}_{i}\end{array} \right)\hat{\mathbf{\hat{s}}}_{\text{wc},2,i}^{2}= \left( \begin{array}{c}\mathbf{0}\\ \mathbf{\mathbf{q}}_{i}\end{array} \right)\hat{\mathbf{\hat{s}}}_{\text{wc},3,i}^{2}= \left( \begin{array}{c}\mathbf{\mathbf{s}}_{3,i}\\ \mathbf{\mathbf{b}}_{i}\times\mathbf{\mathbf{s}}_{3,i}\end{array} \right). \tag{5}\]  
The dot product of Formulas (1)-(3) can be represented as follows:  
\[\hat{\mathbf{\hat{s}}}_{\text{wc},k,i}^{\text{T}}\hat{\mathbf{\mathbf{S}}}_{\text{p}}=0. \tag{6}\]  
Reorganized matrix form is given as follows:  
\[\mathbf{J}_{c}\hat{\mathbf{\mathbf{S}}}_{\text{p}}=\mathbf{0}, \tag{7}\]  
where, \(\mathbf{J}_{c}^{1}\) and \(\mathbf{J}_{c}^{2}\) are the constraint sub-matrixes of overall Jacobian. \[\mathbf{J}_{c}^{1}=\left[ \begin{array}{ccc}\mathbf{0}&\mathbf{n}_{1}^{\text{T} }\\ \mathbf{0}&\mathbf{s}_{1,1}^{\text{T}}\\ \mathbf{\mathbf{s}}_{2,1}^{\text{T}}&[\mathbf{b}_{i}\times\mathbf{\mathbf{s}}_{3,1}]^{ \text{T}}\\ &\vdots\\ \mathbf{0}&\mathbf{n}_{i}^{\text{T}}\\ \mathbf{0}&\mathbf{s}_{1,i}^{\text{T}}\\ \mathbf{\mathbf{s}}_{2,i}^{\text{T}}&[\mathbf{b}_{i}\times\mathbf{\mathbf{s}}_{3,i}]^{ \text{T}}\\ \end{array} \right]_{(3\times i)\times 6},\mathbf{J}_{c}^{2}=\left[ \begin{array}{ccc} \mathbf{0}&\mathbf{m}_{1}^{\text{T}}\\ \mathbf{0}&\mathbf{q}_{1}^{\text{T}}\\ \mathbf{\mathbf{s}}_{3,1}^{\text{T}}&[\mathbf{b}_{i}\times\mathbf{\mathbf{s}}_{3,1}]^{ \text{T}}\\ &\vdots\\ \mathbf{0}&\mathbf{m}_{i}^{\text{T}}\\ \mathbf{0}&\mathbf{q}_{i}^{\text{T}}\\ \mathbf{\mathbf{s}}_{3,i}^{\text{T}}&[\mathbf{b}_{i}\times\mathbf{\mathbf{s}}_{3,i}]^{ \text{T}}\\ \end{array} \right]_{(3\times i)\times 6} \tag{8}\]  
When the input driving joint of each limb is locked, a reciprocal screw, which is reciprocated with other screws in a limb, \(\hat{\mathbf{\mathbf{s}}}_{\text{wa},1,i}\) can be expressed in Equation (9),  
\[\hat{\mathbf{\mathbf{s}}}_{\text{wa},1,i}^{1}=\left( \begin{array}{c}\mathbf{q}_{i} \\ \mathbf{r}_{i}\times\mathbf{q}_{i}\end{array} \right)\hat{\mathbf{\mathbf{s}}}_{ \text{wa},1,i}^{2}=\left( \begin{array}{c}\mathbf{q}_{i}\\ \mathbf{r}_{i}\times\mathbf{q}_{i}\end{array} \right) \tag{9}\]  
where, \(\mathbf{r}_{i}=\mathbf{b}_{i}\cdot\frac{[\mathbf{b}_{i}\times\mathbf{q}_{i}]} {[\mathbf{b}_{i}\times\mathbf{q}_{i}]}\), \(\mathbf{r}_{i}\) is a vertical vector from origin P of the local reference frame to vector \(\mathbf{q}_{i}\). Take dot product of Equations (1) and (9),  
\[\hat{\mathbf{\mathbf{s}}}_{\text{wa},1,i}^{\text{T}}\hat{\mathbf{\mathbf{S}}}_{\text{p }}=\dot{\mathbf{d}}_{i}. \tag{10}\]  
Arranging Equation (9) into matrix form,  
\[\mathbf{J}_{\mathbf{a}}^{1}=\left[ \begin{array}{cc}\mathbf{q}_{i}^{\text{T} }&(\mathbf{r}_{i}\times\mathbf{q}_{i})^{\text{T}}\\ \mathbf{q}_{i}^{\text{T}}\mathbf{s}_{1}&\mathbf{q}_{i}^{\text{T}}\mathbf{s}_{ 1}\end{array} \right]\times 6, \tag{11}\]  
Remarkably, the velocity overall Jacobian \(\mathbf{J}\) of the mechanism is expressed as  
\[\mathbf{J}=\left[ \begin{array}{c}\mathbf{J}_{\mathbf{a}}\\ \mathbf{J}_{\text{c}}\end{array} \right] \tag{12}\]  
By observing the overall Jacobian matrix, the first three rows of the matrix are dimensionless, and the last three rows are related to the position vector \(\mathbf{r}\).