## 3 Comprehensive Evaluation of Configuration
### Mechanism Performance Indicators
The dynamic sensitivity of the whole mechanism configuration is represented by matrix \(D\). Any element \(d_{i}\) in matrix \(D\) is the minimum number of joints between each joint \(i\) and the frame. The sensitivity matrix for the configuration of Figure 6 is \(D= \begin{bmatrix}0&1&2&1&0&2&1&0&1&0\end{bmatrix} \). The dynamic sensitivity can be expressed in terms of the maximum value of the element \(d_{max}\) of matrix \(D\). The calculated dynamic sensitivity index \(d_{max}\) for the configuration of Figure 6 is 2. Similarly, the dynamic sensitivity indices for all of the configurations in Figure 4 are presented in Figure 7c. Figure 7: Values of various indicators for different configurations: (**a**) compactness values; (**b**) stiffness values; (**c**) sensitivity values; (**d**) _KIC_ values; (**e**) transmission efficiency of the joint. **(4) Motion transferability. ** In mechanism design, it is essential to coordinate component parameters and perform kinematic analysis. For single-loop mechanisms, fewer components lead to simpler kinematic analysis. However, in multi-loop mechanisms, the number of co-loop components can vary due to different combinations of components and joint connections. A greater number of co-loop components increases the correlation between loops, generally making the kinematic analysis more complex. Since the specific dimensions of the components are not considered at the configuration selection stage, it is not possible to quantitatively analyze the positions, velocities, and accelerations of the mechanism's components. To derive the structural characteristics of the mechanism, the topological characteristic matrix is first established. This matrix should capture information about the loops that constitute the mechanism, the types of components and joints within the loops, their connections, and their combination relationships. It is defined mathematically as shown in Equation (5). \[LLJM= \begin{bmatrix}L_{11}/F&L_{12}/F&\dots&L_{1m}/F\\ L_{21}/F&L_{22}/F&\dots&L_{2m}/F\\ \dots&\dots&\dots&\dots\\ L_{11}/F&L_{12}/F&\dots&L_{lm}/F\end{bmatrix}  \tag{5}\]  
where \(L_{ij}\) is the type of no. j component of the no. i loop, \(L_{ij}=\pm 2\) for the two-joint component, and \(L_{ij}=\pm 3\) for the three-parameter member. When \(L_{ij}\) is counted for the first time in the loop, the \"+\" sign is taken; otherwise, the \"\(-\)\" sign is taken. \(F\) is the number of degrees of freedom of the mechanism; the number of rows of the \(LLJM\) matrix, \(l\), is the number of independent loops of the mechanism; and the number of columns of the \(LLJM\) matrix, \(m\), is the maximum value of the number of components in each independent loop. Each loop is filled with the corresponding element of the \(LLJM\) matrix in the order of frame, input point, and output point. In Figure 6, the mechanism has an \(l\) of 3 and an \(m\) of 6, the frame is a four-joint component, the input slider is a two-joint component, and all three output components are two-joint components, so the \(LLJM\) is as follows:  
\[LLJM= \begin{bmatrix}4&2&3&2&0&0\\ -4&-2&-3&-2&-3&2\\ -4&-2&-3&-2&3&2\end{bmatrix} \]  
According to the characteristics of \(LLJM\), \(KIC\) is used to express the structural characteristics of the mechanism, which can be represented by Equation (6). \[KIC=\sum k_{ij}\quad \begin{cases}k_{ij}=1&\text{When }L_{ij}\
eq 0\\ k_{ij}=0&\text{When}L_{ij}=0\end{cases} \quad(i=1,2,\cdots,l;\ j=1,2,\cdots,m) \tag{6}\]  
Informed by the analysis above, a higher number of co-loop components correlates with a greater \(KIC\), indicating increased complexity in motion analysis. Conversely, efficient motion and power transmission along the mechanism's loop are preferred, with shorter transmission paths being advantageous. For the mechanism configuration depicted in Figure 6, the \(KIC\) is 16. Figure 7 illustrates the \(KIC\) for each configuration in Figure 5.  
**(5) Transmission efficiency of joint. ** During the transfer of motion and power, joints inevitably incur power loss.