# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 4 Performance Analysis  
### Workspace of Mechanism in Two Motion Modes  
The workspace of the mechanism is the collection of all points relative to the static coordinate system that a given reference point of it can reach. The size and shape of the workspace represent the range of motion of the end effector of the mechanism. Select the center point \(O_{1}\) of the moving platform as the reference point and discretize the workspace points using MATLAB 2021a. The angular variation range of the revolute pair is \([-60^{\circ},60^{\circ}]\)[32]. \(\alpha\) and \(\beta\) are only related to the rotation angle of the driving revolute pair, so the search range is the same as that of the rotation angle of the revolute pair. Then, determine whether each point satisfies the constraint conditions to obtain the reachable workspace of the mechanism in two motion modes, as shown in Figure 6.  
In the 2T1R motion mode, the color represents the value of \(\alpha\), and in the 1T2R motion mode, the color represents the value of \(\beta\). From Figure 6, it can be seen that the reconfigurable mechanism has a regular and large workspace shape in 2T1R mode, with a continuous and complete internal structure; in 1T2R mode, the internal structure of the workspace is continuous, and the movement distance along the \(X\)-axis tends to increase with the absolute value of \(\beta\).  
The input transmission performance represents the efficiency of the power transmitted from the driving joints to the serial chains, while the output transmission performance represents the efficiency of the power transmitted from the serial chains to the moving platform. Considering the input transmission performance and output transmission performance simultaneously, the local transmission index (LTI) is defined as  
\[\lambda_{i}=\frac{|\mathbf{\xi}_{\text{TI}}\circ\mathbf{\xi}_{\text{II}}|}{|\mathbf{\xi}_ {\text{TI}}\circ\mathbf{\xi}_{\text{II}}|_{\max}} \tag{40}\]  
\[\eta_{i}=\frac{|\mathbf{\xi}_{\text{OI}}\circ\mathbf{\xi}_{\text{TI}}|}{|\mathbf{\xi}_{ \text{OI}}\circ\mathbf{\xi}_{\text{TI}}|_{\max}} \tag{41}\]  
\[\Gamma=\min\{\lambda_{i},\eta_{i}\} \tag{42}\]  
Figure 6: The 3D workspace of the mechanism in two motion modes: (**a**) workspace in 2T1R mode and (**b**) workspace in 1T2R mode.  
where \(\lambda_{i}\) is the input transmission performance index of serial chain \(i\), \(\eta_{i}\) is the output transmission performance index of serial chain \(i\), \(\Gamma\) is the calculation formula of LTI, \(\mathfrak{F}_{\rm Ii}\) and \(\mathfrak{F}_{\rm Oi}\) are the input and output motion screws of serial chain \(i\), respectively, and \(\mathfrak{F}_{\rm Ti}\) is the transmission force screw of serial chain \(i\).  
The input and output transmission performance index represent the efficiency of energy transmission from the drive input to the serial chain and from the serial chain to the output, respectively. According to the definition, the values of \(\lambda_{i}\), \(\eta_{i}\), and \(\Gamma\) are all between 0 and 1, which are dimensionless indicators independent of the coordinate system. The closer the value of \(\Gamma\) is to 1, the better the motion/force transmission performance of the mechanism. Conversely, the closer the value of \(\Gamma\) is to 0, the closer the mechanism is to singularity.