## 2 Constraints of the Reconfigurable (Rc)PU Limb
### Bifurcated Motion Mode Identification of 3(Rc)PU PM  
According to the two kinematic configurations of the (Rc)PU limb, it can be concluded through linear independence analysis of the reconfigurable circular groove cross joint that the 3(Rc)PU parallel mechanism has four different motion branches under the initial configuration, namely 3T motion branches (3(Rc)\({}_{1}\)PU configuration), the 1R3T motion branch (2(Rc)\({}_{1}\)PU-(Rc)\({}_{2}\)PU configuration), the 2R3T motion branch ((Rc)\({}_{1}\)PU-2(Rc)\({}_{2}\)PU configuration), and the 3R3T motion branch (3(Rc)\({}_{2}\)PU configuration). Figure 4 shows four configurations of the 3(Rc)PU parallel mechanism. As shown in Figure 4a, when the three limbs of the 3(Rc)PU PM are all (Rc)\({}_{1}\)PU limbs, then the moving platform will be acted on by three constraint force vectors. The constraint screw system composed of these three constraint force vectors is as follows:  
\[ \begin{split} W=W_{1}\cup W_{2}\cup W_{3}=\left(\begin{array}{ c}s_{B1}\\ s_{0B1}\end{array}\right)\cup\left(\begin{array}{c}s_{B2}\\ s_{0B2}\end{array}\right)\cup\left(\begin{array}{c}s_{B3}\\ s_{0B3}\end{array}\right)\end{split}  \tag{11}\]  
Due to the reciprocity between the constraint screw and the motion screw, the screw system in 3T motion mode of the 3(Rc)PU PM can be obtained as shown:  
\[ \begin{split} S_{1}=\left(0,\ 0,\ 0;\ 1,\ 0,\ 0\right)^{\text{T}},S_{2}= \left(0,\ 0,\ 0;\ 0,\ 1,\ 0\right)^{\text{T}},S_{3}=\left(0,\ 0,\ 0;\ 0,\ 0,\ 1\right)^{\text{T}} \end{split}  \tag{12}\]  
Figure 4: Four configurations of the 3(Rc)PU PM: (**a**) 3(Rc)\({}_{1}\)PU configuration; (**b**) 2(Rc)\({}_{1}\)PU-(Rc)\({}_{2}\)PU configuration; (**c**) (Rc)\({}_{1}\)PU-2(Rc)\({}_{2}\)PU configuration; (**d**) 3(Rc)\({}_{2}\)PU configuration. The kinematic description of the moving platform in 3T motion mode can be obtained, as shown in Figure 4. At this time, the moving platform has three translation degrees of freedom along the \(X\) axis, \(Y\) axis, and \(Z\) axis. As shown in Figure 4, when the limb structure of the 3(Rc)PU PM is in the 2(Rc)\({}_{1}\)PU-(Rc)\({}_{2}\)PU configuration, the moving platform will be subject to the action of two constraint vectors. The constrained screw system composed of these two constraint vectors is as follows:  
\[ \begin{split}\mathbf{W}=\mathbf{W}_{1}\cup\mathbf{W}_{2}|\mathbf{W}_{1}\cup\mathbf{W} _{3}|\mathbf{W}_{2}\cup\mathbf{W}_{3}\ =\left(\begin{array}{c}\mathbf{s}_{\mathbf{B}1}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}1}\end{array}\right)\cup\left(\begin{array}{c}\mathbf{s}_{ \mathbf{B}2}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}2}\end{array}\right)\bigg{|}\left(\begin{array}{c}\mathbf{s} _{\mathbf{B}1}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}1}\end{array}\right)\cup\left(\begin{array}{c}\mathbf{s}_{ \mathbf{B}3}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}3}\end{array}\right)\bigg{|}\left(\begin{array}{c}\mathbf{s} _{\mathbf{B}2}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}2}\end{array}\right)\cup\left(\begin{array}{c}\mathbf{s}_{ \mathbf{B}3}\\ \mathbf{s}_{\mathbf{0}\mathbf{B}3}\end{array}\right)\end{split}  \tag{13}\]  
In Equation (13), the symbol \"\(\cup\)\" represents a union, and the symbol \" \(\cup\)\" represents the logical operator \"or\". The screw system of the 3(Rc)PU PM in the 2(Rc)\({}_{1}\)PU-(Rc)\({}_{2}\)PU configuration is obtained:  
\[ \begin{split}\mathbf{S}_{1}=(0,\ 0,\ 0;\ 1,\ 0,\ 0)^{\mathsf{T}},\mathbf{S}_{2}=(0, \ 0,\ 0;\ 0,\ 1,\ 0)^{\mathsf{T}},\mathbf{S}_{3}=(0,\ 0,\ 0;\ 0,\ 0,\ 1)^{\mathsf{T}},\\ \mathbf{S}_{4}=\left(\begin{array}{c}\frac{a_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+1 }},\ \frac{b_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+1}},\ \frac{1}{\sqrt{a_{1}^{2}+b_{1}^{2}+1}};\ 0,\ 0,\ 0 \end{array}\right)^{\mathsf{T}}\end{split}  \tag{14}\]  
where  
\[ \begin{split} a_{1}&=\frac{3r_{31}y_{-73}y_{-32}r_{21}+r_{ 23}-3r_{21}y_{31}+3r_{23}y_{1}}{3r_{21}x-r_{23}x-3r_{31}y+r_{13}y-3r_{11}r_{23} +3r_{31}y_{21}},\\ b_{1}&=\frac{-(3r_{31}-r_{32}x-3r_{31}x+r_{12}x-3r_{11}y_{ 31}+3r_{11}y_{21})}{3r_{21}x-r_{23}x-3r_{11}y+r_{13}y-3r_{11}r_{23}+3r_{11}y_{2 1}}\end{split}