# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 5 Optimization Design  
According to the kinematic analysis and performance analysis, the motion/force transmission performance indexes of the mechanism under the two modes of 2T1R and 1T2R are related to the dimension parameters \(l_{11},H,l_{12}\). Therefore, the dimension parameters \(l_{11}\), \(H,l_{12}\) are taken as the design variables in the optimization design. Based on the existing motion/force transmission performance and literature, the optimization objective functions are selected as the global transfer index (GTI) of the mechanism under the 2T1R motion mode and the 1T2R motion mode, as shown in Equation (53):  
\[ \begin{array}{c}\mathrm{GTI}=\dfrac{\sum\limits_{i=1}^{W_{i}}\Gamma}{W_{i}} \end{array}  \tag{53}\]  
where \(W_{i}\) is the number of points that meet the constraint conditions. The following equations can be obtained by dimensionless parameter optimization:  
\[\left\{ \begin{array}{l}D=\frac{l_{11}+H_{11}+l_{12}}{3}\\ r_{1}=\frac{l_{11}}{D},r_{2}=\frac{H_{11}}{D},r_{3}=\frac{l_{12}}{D}\end{array} \right. \tag{54}\]  
where \(D\) is the standardization factor, and the average of the dimension parameters. \(r_{1}\), \(r_{2}\), \(r_{3}\) represent the dimensionless values of \(l_{11}\), \(H_{11}\), \(l_{12}\). According to the actual size requirements of the mechanism, scale-free parameters also need to meet certain geometric conditions to limit the variation range of parameters, as shown in Equation (55):  
\[\left\{ \begin{array}{l}r_{1}>r_{2}\\ 0<r_{1},r_{2},r_{3}<3\end{array} \right. \tag{55}\]  
According to the optimization constraints of \(r_{1}\), \(r_{2}\), and \(r_{3}\), the optimal design space in three-dimensional space can be drawn, as shown in Figure 9.  
Figure 8: The motion/force transmission performance indexes of the mechanism (1T2R): (**a**) LTI in the workspace; (**b**) LTI of the mechanism (\(\alpha=0\)); and (**c**) LTI of the mechanism (\(\beta=0\)). The shaded region represented by \(\triangle DEF\) in the figure contains all possible size variable values. The shadow region can be mapped onto two-dimensional space by selecting a reasonable two-dimensional coordinate system in three-dimensional space. In two-dimensional space, the relationship between \(r_{1}\), \(r_{2}\), \(r_{3}\), and \(s\) and \(t\) can be represented by the following Equation (56):  
\[\left\{ \begin{array}{l}s=r_{2}\\ t=\frac{3-r_{3}+r_{1}}{\sqrt{3}}\end{array} \right. \tag{56}\]  
Based on the definition of GTI index under two motion modes and the given optimization design space, a corresponding performance map of the mechanism under the motion mode can be generated to describe the relationship between dimensionless dimensional variables and performance indexes, as shown in Figure 10.  
It is assumed that the optimization objective should meet GTI \(\geq\) 0.8325 in both 1T2R and 2T1R motion modes; the intersection region of the graphs meeting the performance requirements can be taken to obtain the optimal design space, as shown in Figure 11.  
In order to obtain better kinematic performance, parameters in the region with excellent performance can be selected as the mechanism parameters, such as \(t\) = 2.5, \(s\) = 0.2. Determination of the proportional coefficient \(D\) = 150 mm, and \(l_{11}\) = 375 mm, \(H_{11}\) = 30 mm,  
Figure 11: The intersection region of the graphs. Figure 10: GTI index of the mechanism under two motion modes: (**a**) GTI (2T1R) and (**b**) GTI (1T2R). Figure 9: Design space for mechanism parameters: (**a**) three-dimensional space diagram and (**b**) two-dimensional space diagram. \(l_{12}\) = 45 mm can be obtained. Tables 6 and 7 show the optimization results of the structural parameters and GTI, respectively. From Tables 6 and 7, it can be seen that \(l_{11}\) has increased while \(H_{11}\) and \(l_{12}\) have decreased. By calculation, GTI increased by 5.4% in 2T1R mode and 24.5% in 1T2R mode.