## 3 Overall Jacobian Matrix  
By using the closed vector method, achieving the spatial stiffness of a planar parallel mechanism is no longer satisfied. In this section, screw theory is used to calculate the overall Jacobian of the mechanism by analyzing the form of the active and constraint screws of each driving chain, which is the foundation of building the spatial stiffness model. We can obtain the joint instantaneous screws according to geometric relation of driving chains as shown in Figure 3.  
The instantaneous velocity of MP can be represented as \(\hat{\mathbf{S}}_{\text{p}}=\left[ \begin{array}{cc}\mathbf{v}^{\mathsf{T}}& \boldsymbol{\omega}^{\mathsf{T}}\end{array} \right]^{\mathsf{T}}\); \(\mathbf{v}\) and \(\boldsymbol{\omega}\) indicate the vector of translational velocity and vector of rotational velocity, respectively; the instantaneous screw of the MP can be represented through three linear un-correlation joint instantaneous screws. \[\hat{\mathbf{S}}_{\text{p}}=\dot{\mathbf{d}}_{1i}\hat{\mathbf{S}}_{\text{ta, 1},i}+\dot{\mathbf{0}}_{2}\hat{\mathbf{S}}_{\text{ta,2},i}+\dot{\mathbf{0}}_{3 }\hat{\mathbf{S}}_{3,\text{4}n,i}, \tag{1}\]  
where \(\hat{\mathbf{S}}_{\text{ta,}k,i}\) denotes the \(k\)th joint instantaneous screw of the \(i\)th limb, \(k=1\)-\(3\), \(i\) indicates the driving chain number, \(\dot{\mathbf{0}}_{ki}\) is the \(k\)th joint instantaneous rotational velocity of the \(i\)th limb, \(\dot{\mathbf{d}}_{ki}\) represents the \(k\)th joint instantaneous translational velocity of the \(i\)th limb, \(\hat{\mathbf{S}}_{\text{ta}}\) is expressed in the Plucker ray coordinate, \(\hat{\mathbf{\xi}}^{1}\) is the screw of each joint for \(\underline{\text{PRR-type driving chain}}\), \(\hat{\mathbf{\xi}}^{2}\) is the screw of each joint for \(\underline{\text{PRR-type driving chain}}\), \(\mathbf{0}\) represents a \(3\times 1\) zero vector, \(\mathbf{s}_{1,i}\) is a unit vector along the P joint, \(\mathbf{q}_{i}\) is a unit vector along the \(\overline{BC}\), \(\mathbf{L}_{0,i}\) is the length of BC, \(\mathbf{s}_{2,i}\) and \(\mathbf{s}_{3,i}\) are the unit vectors along the axis of R joints, \(\mathbf{b}_{i}\) is the vector along \(\overline{PB}\), \"\(\times\)\" denotes cross-product. \[\hat{\mathbf{S}}_{\text{ta,1},i}^{1}=\left( \begin{array}{c}\mathbf{s}_{1,i} \\ \mathbf{0}\end{array} \right),\hat{\mathbf{S}}_{\text{ta,2},i}^{1}=\left( \begin{array}{c}(\mathbf{b}_{i}+\mathbf{q}_{i}\mathbf{L}_{0,i})\times \mathbf{s}_{2,i}\\ \mathbf{s}_{2,i}\end{array} \right)\hat{\mathbf{S}}_{\text{ta,3},i}^{1}=\left( \begin{array}{c}\mathbf{b}_{i}\times\mathbf{s}_{3,i}\\ \mathbf{s}_{3,i}\end{array} \right), \tag{2}\]  
\[\hat{\mathbf{S}}_{\text{ta,1},i}^{2}=\left( \begin{array}{c}(\mathbf{b}_{i}+ \mathbf{q}_{i}\mathbf{L}_{0,i})\times\mathbf{s}_{2,i}\\ \mathbf{s}_{2,i}\end{array} \right)\hat{\mathbf{S}}_{\text{ta,2},i}^{2}=\left( \begin{array}{c}\mathbf{q}_{i}\\ \mathbf{0}\end{array} \right)\hat{\mathbf{S}}_{\text{ta,3},i}^{2}=\left( \begin{array}{c}\mathbf{b}_{i}\times\mathbf{s}_{3,i}\\ \mathbf{s}_{3,i}\end{array} \right), \tag{3}\]  
where \(\hat{\mathbf{S}}_{\text{wc},k,i}\) denotes the \(k\)th reciprocal basis screw of the \(i\)th limb; \(\mathbf{n}\) is a unit vector perpendicular to the P joint; the physical significance of \(\hat{\mathbf{S}}_{\text{wc},1,i}^{1}\) and \(\hat{\mathbf{S}}_{\text{wc},2,i}^{1}\) are constraint wrench around \(\mathbf{n}\) and \(\mathbf{s}_{1,i}\), respectively; and \(\hat{\mathbf{S}}_{\text{wc},3,i}^{1}\) represents the constraint force along \(\mathbf{s}_{3,i}\). \[\hat{\mathbf{S}}_{\text{wc},1,i}^{1}=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{n}_{i}\end{array} \right)\hat{\mathbf{S}}_{\text{wc},2,i}^{1}=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{s}_{1,i}\end{array} \right)\hat{\mathbf{S}}_{\text{wc},3,i}^{1}=\left( \begin{array}{c}\mathbf{s}_{3,i}\\ \mathbf{b}_{1}\times\mathbf{s}_{3,i}\end{array} \right).