## 3 Comprehensive Evaluation of Configuration
### Mechanism Performance Indicators  
The components and joints are numbered for easy analysis, as shown in Figure 6 for mechanism configuration (a). Figure 6: Numbering of components and joints of configuration (a). In Figure 6, the Roman numerals (I, II, III, etc.) represent the joint identifiers of the mechanism, while the Arabic numerals (I, 2, 3, etc.) denote the component identifiers of the mechanism. **(1) Compactness. ** In the design of mechanism configurations, the lengths and dimensions of the links (links are primary components of a planar mechanism, serving as a key part of its structure) are not considered. When evaluating the compactness of a link mechanism, a smaller number of links and joints indicates better compactness. If two configurations have the same number of links and joints, compactness is further assessed by determining the minimum number of joints separating two connected links. From the definition of compactness, the space matrix \(L\) of mechanism configuration (a) of Figure 6 is obtained. \[L= \begin{matrix}\text{Link 1}\\ \text{Link 2}\\ \text{Link 3}\\ \text{Link 4}\\ \text{Link 5}\\ \text{Link 6}\\ \text{Link 7}\\ \text{Link 8}\\ \end{matrix}\begin{bmatrix}0&1&2&3&2&1&1&1\\ 1&0&1&2&3&2&2&2\\ 2&1&0&1&2&3&1&3\\ 3&2&1&0&1&2&2&2\\ 2&3&2&1&0&1&3&1\\ 1&2&3&2&1&0&2&2\\ 1&2&1&2&3&2&0&2\\ 1&2&3&2&1&2&2&0\end{bmatrix} \]  
The elements of the spatial matrix \(L\) represent the minimum number of joints between a selected link and any other link, with diagonal elements being zero. A lower sum of the matrix components indicates a more compact mechanism. Using this method, the compactness values of all the mechanism configurations in Figure 5 are calculated by analogy, with the results presented in Figure 7a. **(2) Stiffness. ** The actual stiffness of each link in the mechanisms primarily depends on the size, elasticity, and support of the links. For the configurations in Figure 5, the dimensions of the links are not considered. The links are treated as elastic bodies, with the number of joints for each link equal to the number of support points. The stiffness of the mechanism is then represented using the stiffness matrix (Equation (3)). \[K= \begin{bmatrix}k_{1}&k_{2}&\cdots&k_{i}\end{bmatrix}  \tag{3}\]  
where \(i\) denotes the number of independent loops (loops refer to closed kinematic chains in a planar linkage mechanism, consisting of the frame-driving components-driven components-output components and returning to the frame) of the mechanism, and the elements in the matrix can be calculated according to the Equation (4). \[k_{i}=\frac{1}{\
icefrac{{1}}{{c_{1}}}+\
icefrac{{1}}{{c_{2}}}+\ldots+ \
icefrac{{1}}{{c_{ij}}}} \tag{4}\]  
where \(c_{i}\), \(c_{j}\), and \(c_{s}\) are the number of joints from the input to the output component 1, 2, \(\ldots\), \(j\) in the independent loop. As shown in Figure 6, \(k_{1}\) represents the stiffness value of the loop 2-3-7. Component 2 has two kinematic pairs, component 3 has three kinematic pairs, and component 7 has two kinematic pairs, resulting in \(k_{i}=\frac{1}{\
icefrac{{1}}{{2}}+\
icefrac{{1}}{{3}}+\
icefrac{{1}}{{2}}} =3/4\). The rigidity matrix of the mechanism configuration of Figure 6 can be obtained as \(K= \begin{bmatrix}3/4&6/13&6/13\end{bmatrix} \). In the stiffness matrix \(K\), each element is summed to yield the overall stiffness value of the mechanism configuration, with the configuration in Figure 6 having a stiffness value of 1.637. Figure 7b shows the calculated stiffness values for all the mechanism configurations depicted in Figure 5.  
**(3) Dynamic sensitivity. ** The dynamic performance of the mechanism can be evaluated through dynamic sensitivity. The dynamic sensitivity of each configuration, without considering the size and mass of the links, is determined by the minimum number of joints separating the mechanism from the frame. A lower sensitivity indicates better dynamic performance, while a higher sensitivity suggests poorer performance.