## 3 Mechanical Error Analysis
### The Evaluation Indices  
From the stochastic viewpoint, the worst-case deviation of the output link of the lower sub-mechanism will be [27; 28]  
\[\begin{array}{ll}\left(\Delta L_{\text{wor}}\right)_{L}&=\ |\Delta L_{1L}|+| \Delta L_{2L}|+|\Delta L_{3}|+|\Delta L_{4a}|+|\Delta L_{4b}|+|\Delta L_{4c}|+ |\Delta L_{5}|+|\Delta L_{eL}|\end{array} \tag{38}\]  
in which, \((\Delta L_{\text{wor}})_{L}\) represents the worst combinations of the output deviation caused by each link-length error (i.e., dimensional errors or tolerances). The maximum expected deviation of the output link of the lower sub-mechanism will be [27; 28]  
\[\begin{array}{ll}\left(\Delta L_{\text{rss}}\right)_{L}&=\ \sqrt{\Delta L_{1L}^{2}+\Delta L_{2L}^{2}+ \Delta L_{3}^{2}+\Delta L_{4a}^{2}+\Delta L_{4b}^{2}+\Delta L_{4c}^{2}+\Delta L _{5}^{2}+\Delta L_{eL}^{2}}\end{array} \tag{39}\]  
in which, \((\Delta L_{\text{rss}})_{L}\) is obtained by the widely used root-sum-of-squares (RSS) approach [27; 28]. Likewise, the worst-case deviation of the output link of the upper sub-mechanism will be [27; 28]  
\[\begin{array}{ll}\left(\Delta L_{\text{wor}}\right)_{U}&=\ |\Delta L_{1U}|+| \Delta L_{2U}|+|\Delta L_{6}|+|\Delta L_{7a}|+|\Delta L_{7b}|+|\Delta L_{7c}|+ |\Delta L_{8}|+|\Delta L_{eU}|\end{array} \tag{40}\]  
Then, the maximum expected deviation of the output link of the upper sub-mechanism will be [27; 28]  
\[\begin{array}{ll}\left(\Delta L_{\text{rss}}\right)_{U}&=\ \sqrt{\Delta L_{1U}^{2}+\Delta L_{2U}^{2}+ \Delta L_{6}^{2}+\Delta L_{7a}^{2}+\Delta L_{7b}^{2}+\Delta L_{7c}^{2}+\Delta L _{8}^{2}+\Delta L_{eU}^{2}}\end{array} \tag{41}\]  
Equations (38)-(41) are based on widely used stochastic approaches to obtain four evaluated tolerance ranges \(\pm(\Delta L_{\text{wor}})_{L}\), \(\pm(\Delta L_{\text{rss}})_{L}\), \(\pm(\Delta L_{\text{wor}})_{U}\), and \(\pm(\Delta L_{\text{rss}})_{U}\) of the output link of the lower or upper sub-mechanism, which may be caused by the specified tolerance ranges of the geometric parameters of all link members.  
In order to evaluate an extreme condition, when the dimensional errors of all geometric parameters in the lower sub-mechanism are the lower deviations of their specified bilateral tolerance ranges, and those in the upper sub-mechanism are the upper deviations of their specified bilateral tolerance ranges, or vice versa, two evaluation indices can be established as  
\[\delta L_{\text{wor}}\;=\;\left|\pm(\Delta L_{\text{wor}})_{U}-\mp(\Delta L_{ \text{wor}})_{L}\right|\;=\;\left(\Delta L_{\text{wor}}\right)_{U}+\left( \Delta L_{\text{wor}}\right)_{L} \tag{42}\]  
\[\delta L_{\text{rss}}\;=\;\left|\pm(\Delta L_{\text{rss}})_{U}-\mp(\Delta L_{ \text{rss}})_{L}\right|\;=\;\left(\Delta L_{\text{rss}}\right)_{U}+\left( \Delta L_{\text{rss}}\right)_{L} \tag{43}\]  
which are quantitative indices for evaluating the asymmetry between the lower and upper sub-mechanisms. The extreme condition indicates the extent of inconsistency between the horizontal positions of joints F and C of the output link (as shown in Figure 5) that may further lead to excessive wear, large deformations, and even fatigues of link members.