## 2 Constraints of the Reconfigurable (Rc)PU Limb
From the reciprocity between the kinematic screw system and the constrained screw system, the constrained screw of the (Rc)\({}_{1}\)PU limb is  
\[{}^{1}\mathbb{S}^{r}=(0,\;0,\;0;\;0,\;\cos\alpha_{1},-\sin\alpha_{1})^{ \mathrm{T}} \tag{2}\]  
It can be seen from the constrained screw system of the (Rc)\({}_{1}\)PU limb that the limb provides a constraint couple acting on the moving platform, and the constraint couple restricts the rotational motion of the U pair center of the (Rc)\({}_{1}\)PU limb in the direction of the prismatic pair axis. The (Rc)\({}_{1}\)PU limb has five degrees of freedom. Figure 2b is the local coordinate system and screw distribution diagram of the (Rc)\({}_{2}\)PU limb. The (Rc)\({}_{2}\)PU screw's system of motion can be expressed as follows:  
Figure 2: Two configurations of (Rc)PU limb: (**a**) (Rc)\({}_{1}\)PU limb; (**b**) (Rc)\({}_{2}\)PU limb. The center of the circumscribed circle of \(\Delta A_{1}A_{2}A_{3}\) is \(O_{n}\), and its radius is \(R\). A fixed coordinate system \(O_{n}\)-\(XYZ\) of the 3(Rc)PU PM is connected to point \(O_{n}\), with the \(X\) axis on plane \(A_{1}A_{2}A_{3}\) and parallel to the outer axis of the Rc joint at point \(A_{1}\). The \(Z\) axis is on plane \(A_{1}A_{2}A_{3}\) and is parallel to the outer axis of the Rc joint at point \(A_{2}\), and the direction of the \(Y\) axis conforms to the right-hand rule. The center of the circumscribed circle of \(\Delta B_{1}B_{2}B_{3}\) is \(O_{u}\) and its radius is \(r\); the moving coordinate system \(O_{u}\)-\(xyz\) of the 3(Rc)PU PM is established at point \(O_{u}\), with the \(x\), \(y\), and \(z\) axes in the same direction as the \(X\), \(Y\), and \(Z\) axes, respectively, in the fixed coordinate system \(O_{n}\)-\(XYZ\). Using \(\mathbf{d}=[x,\,y,\,z]^{\mathrm{T}}\) to represent the shift operator of the position vector \({}^{u}\mathbf{p}\) to the position vector \({}^{u}\mathbf{p}\), and using the rotation matrix \({}_{n}^{u}\mathbf{R}\) to represent the shift operator of the position vector \({}^{u}\mathbf{p}\) to the position vector \({}^{u}\mathbf{p}\), let the matrix \({}_{n}^{u}\mathbf{R}\) be  
\[{}_{n}^{u}\mathbf{R}= \begin{bmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{bmatrix}  \tag{4}\]  
In vector space, the description of a point between different coordinate systems can be regarded as the result of an affine mapping of rotation and displacement. The position vector \({}^{u}\mathbf{p}\) of any point in the moving coordinate system \(O_{u}\)-\(xyz\) and the position vector \({}^{u}\mathbf{p}\) of that point in the fixed coordinate system \(O_{n}\)-\(XYZ\) can be described using the following equation:  
\[{}^{u}\mathbf{p}={}_{n}^{u}\mathbf{R}^{u}\mathbf{p}+\mathbf{d} \tag{5}\]  
As shown in Figure 3, the position vectors of points \(B_{1}\), \(B_{2}\), and \(B_{3}\) in the moving coordinate system \(O_{u}\)-\(xyz\) are as follows:  
\[{}^{u}\mathbf{p}_{B1}=(0,0,r)^{\mathrm{T}},\,\,{}^{u}\mathbf{p}_{B2}=\left(\sqrt{3}r/2,0,-r/2\right)^{\mathrm{T}},\,\,{}^{u}\mathbf{p}_{B3}=\left(-\sqrt{3}r/2,0,-r/2 \right)^{\mathrm{T}}. \tag{6}\]  
Then, the position vectors of points \(B_{1}\), \(B_{2}\), and \(B_{3}\) in the fixed coordinate system \(O_{n}\)-\(XYZ\) can be obtained from Equation (5) as follows:  
\[\left\{ \begin{array}{l}{}^{n}\mathbf{p}_{B1}=\left[rr_{13}+x,\,\,rr_{23}+y,\,\, rr_{33}+z\,\right]^{\mathrm{T}}\\ {}^{n}\mathbf{p}_{B2}=\left[\frac{\sqrt{3}}{2}rr_{11}-\frac{1}{2}rr_{13}+x,\,\, \frac{\sqrt{3}}{2}rr_{21}-\frac{1}{2}rr_{23}+y,\,\,\frac{\sqrt{3}}{2}rr_{31}- \frac{1}{2}rr_{33}+z\right]^{\mathrm{T}}\\ {}^{n}\mathbf{p}_{B3}=\left[-\frac{\sqrt{3}}{2}rr_{11}-\frac{1}{2}rr_{13}+x,-\frac {\sqrt{3}}{2}rr_{21}-\frac{1}{2}rr_{23}+y,-\frac{\sqrt{3}}{2}rr_{31}-\frac{1}{ 2}rr_{33}+z\right]^{\mathrm{T}}\end{array} \right.