## 5 Kinematic Performance Analysis  
The essential function of a parallel mechanism is to output motion and resist external load. The motion/force transmission and constraint characteristics reflect the essential characteristics of parallel mechanisms. The 3(Rc)PU parallel mechanism has four kinds of motion branches, and it is necessary to study the motion/force transmission performance of the parallel mechanism in different motion modes. By selecting the P pair on each limb and the outer axis on the Rc joint as the driving joint, the input actuation screw of the mechanism can be obtained by solving the inverse kinematics of the PM. The input actuation screw system of the mechanism can be obtained using Equations (21) and (22):\[\mathbf{T}=\left\{ \begin{array}{l}\mathbf{S}_{A1}=(0,\,0,\,0;\,m_{1},\,n_{1},\,d_{1})^{ \mathsf{T}}\\ \mathbf{S}_{A2}=(0,\,0,\,0;\,m_{2},\,n_{2},\,d_{2})^{\mathsf{T}}\\ \mathbf{S}_{A3}=(0,\,0,\,0;\,m_{3},\,n_{3},\,d_{3})^{\mathsf{T}}\\ \mathbf{S}_{A4}=(1,\,0,\,0;\,0,\,R,\,0)^{\mathsf{T}}\\ \mathbf{S}_{A5}=\left(0,\,0,\,1;\,0,-\sqrt{3}R/2,\,0\right)^{\mathsf{T}}\\ \mathbf{S}_{A6}=\left(0,\,0,\,1;\,0,\,\sqrt{3}R/2,\,0\right)^{\mathsf{T}}\\ \end{array} \right. \tag{26}\]  
Since the transmission wrench screw exerted by each limb on the moving platform must pass through the joint center point of each limb, the U joint center of the (Rc)PU limb is selected as the characteristic point of the transmission wrench screw of each limb, and the screw system of the mechanism's transmission force on the moving platform can be obtained as  
\[\mathbf{T}=\left\{ \begin{array}{l}\mathbf{S}_{T1}=(m_{1},n_{1},d_{1};-\text{Rn}_{1},Rm_{1},0)^{\mathsf{T}}\\ \mathbf{S}_{T2}=\left(m_{2},n_{2},d_{2};Rn_{2}/2,-\sqrt{3}Rd_{2}/2-Rm_{2}/2,\sqrt{ 3}Rn_{2}/2\right)^{\mathsf{T}}\\ \mathbf{S}_{T3}=\left(m_{3},n_{3},d_{3};Rn_{3}/2,\sqrt{3}Rd_{3}/2-Rm_{3}/2,-\sqrt{ 3}Rn_{3}/2\right)^{\mathsf{T}}\\ \mathbf{S}_{T4}=\left(0,-d_{1},n_{1}^{2}n_{1}^{2}+d_{1}^{2}l_{1}+d_{1}R_{-}m_{1}n_ {1}l_{1},-m_{1}d_{1}l_{1}\right)^{\mathsf{T}}\\ \mathbf{S}_{T5}=\left(-n_{2},m_{2},0;-m_{2}d_{2}l_{2}+Rm_{2}/2,-n_{2}d_{2}l_{2}+Rn _{2}/2,m_{2}^{2}l_{2}+\sqrt{3}Rm_{2}/2+n_{2}^{2}l_{2}\right)^{\mathsf{T}}\\ \mathbf{S}_{T6}=\left(n_{3},-m_{3},0;\,m_{3}d_{3}l_{3}-Rm_{3}/2,n_{3}d_{3}l_{3}-Rn _{3}/2,-m_{3}^{2}l_{3}+\sqrt{3}Rm_{3}/2-n_{3}^{2}l_{3}\right)^{\mathsf{T}}\\ \end{array} \right. \tag{27}\]  
Equation (27) is the set of transmission wrench screws obtained by locking the actuated joints on each limb. Among them, \(\mathbf{S}_{T1}\), \(\mathbf{S}_{T2}\), and \(\mathbf{S}_{T3}\) are the transmission wrench screws corresponding to the prismatic input actuated joints on limbs I, II, and III, and \(\mathbf{S}_{T4}\), \(\mathbf{S}_{T5}\), and \(\mathbf{S}_{T6}\) are the transmission wrench screws corresponding to the input actuated joints on the outer axis of limbs I, II, and III. The input transmission performance index of the 3(Rc)PU PM can be obtained from Equations (26) and (27):  
\[\psi_{i}=\frac{\left|\mathbf{S}_{Ai}\circ\mathbf{S}_{Ti}\right|}{\left|\mathbf{S}_{Ai} \circ\mathbf{S}_{Ti}\right|_{\max}},\,(i=1,\,2,\,\cdots,\,n) \tag{28}\]  
In Equation (28), the value of \(n\) is related to the motion mode of the 3(Rc)PU parallel mechanism and is the mechanism's number of input actuated joints in each motion mode. When the parallel mechanism is in 3T motion mode, the relation between the output twist screw and the constrained screw corresponding to limb I is as follows:  
\[\left\{ \begin{array}{l}\mathbf{S}_{O1}\circ\mathbf{S}_{Ti}=0,(i=1,\,2)\\ \mathbf{S}_{O1}\circ W_{j}=0,(j=1,\,2,\,3)\\ \end{array} \right. \tag{29}\]  
The output twist screw corresponding to limb I of the parallel mechanism in 3T motion mode is  
\[\mathbf{S}_{O1}=(0,\,0,\,0;-n_{2}d_{1}+n_{1}d_{2},\,m_{2}d_{1}-m_{1}d_{2},\,m_{1}n _{2}-m_{2}n_{1}) \tag{30}\]  
Similarly, the other two output twist screws under the 3T motion branch and the output twist screw under other motion modes can be obtained.