## 5 Parameter Optimization
### Response Surface Analysis
#### 5.4.2 Regression Model Analysis  
The quadratic regression equation of the five parameters and the volume ratio of the posture adjustment mechanism were developed by Design-Expert as follows:  
\[ \begin{array}{l}\Re=-4.2009-0.002039\mathrm{X}_{9}+0.001488\mathrm{X}_{0}+3.47535\mathrm{X}_{5}+0.001569\mathrm{X}_{3}-0.001287\mathrm{X}_{2}\\ -(1.3333\mathrm{3E}-07)\mathrm{X}_{9}\mathrm{X}_{0}+0.000937\mathrm{X}_{9} \mathrm{X}_{5}+(8\mathrm{E}-08)\mathrm{X}_{9}\mathrm{X}_{3}+(4.66666\mathrm{F} \mathrm{E}-07)\mathrm{X}_{9}\mathrm{X}_{2}\\ -0.00076\mathrm{X}_{0}\mathrm{X}_{5}-(1.28889\mathrm{E}-07)\mathrm{X}_{0} \mathrm{X}_{3}+(4.44444\mathrm{E}-07)\mathrm{X}_{0}\mathrm{X}_{2}-0.000835\mathrm{ X}_{5}\mathrm{X}_{3}\\ +0.000648\mathrm{X}_{5}\mathrm{X}_{2}+(8.88889\mathrm{E}-08)\mathrm{X}_{3} \mathrm{X}_{2}+(4.06667\mathrm{E}-07)\mathrm{X}_{9}{}^{2}-(4.22222\mathrm{E}-08 )\mathrm{X}_{0}{}^{2}\\ -0.602754\mathrm{X}_{5}{}^{2}-(6.14815\mathrm{E}-08)\mathrm{X}_{3}{}^{2}+(6.2963 0\mathrm{E}-08)\mathrm{X}_{2}{}^{2}\end{array}  \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Mechanical Parameters** & **Low Level** & **Middle Level** & **High Level** \\  & **(\(-\)1)** & **(0)** & **(1)** \\ \hline \(X_{9}\) & Minimum drive trajectory of the second driving pair of limb \(l_{2\mathrm{min}}\) & \(-225\) & \(-200\) & \(-175\) \\ \(X_{0}\) & Width of moving platform \(m\) & 775 & 850 & 925 \\ \(X_{5}\) & Angle between \(P_{3}\) and \(P_{2}\) axis \(\theta\) & \(15\pi/32\) & \(\pi/2\) & \(17\pi/32\) \\ \(X_{3}\) & Length of fixed platform \(N\) & 1425 & 1500 & 1575 \\ \(X_{2}\) & Width of fixed platform \(M\) & 875 & 950 & 1025 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Factors and levels for Box–Behnken test. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No. ** & \(\mathbf{X_{9}}\) & \(\mathbf{X_{0}}\) & \(\mathbf{X_{5}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{X_{2}}\) & \(\Re\) \\ \hline \(1\) & \(-150\) & 1000 & 9\(\pi\)/16 & 1650 & 800 & 6.1715 \(\times\) 10\({}^{-4}\) \\ \(2\) & \(  
The ANOVA of this quadratic regression model is shown in Table 5, and based on the F-value and \(p\)-value, the effect of the parameters on the volume ratio of the posture adjustment mechanism is in the order as follows:  
\[ \begin{array}{l}X_{5}>X_{2}>X_{9}>X_{0}>{X_{5}}^{2}>X_{5}X_{3}>X_{0}X_{5}>X_{5}X _{2}>X_{3}>X_{9}X_{5}>\\ X_{9}X_{3}>X_{0}X_{3}>{X_{2}}^{2}>X_{3}{}^{2}>X_{3}X_{2}>{X_{9}}^{2}>X_{0}^{2}>X_ {9}X_{0}>X_{9}X_{3}>X_{0}X_{2}\end{array}  \tag{17}\]  
The parameter with \(p\) < 0.05 has a significant effect on the posture adjustment mechanism volume ratio. This regression model with \(p\) < 0.05 shows the validity of the regression model, indicating that the relationship between the posture adjustment mechanism volume ratio and the resulting regression equation is significant. CV = 8.03%, so the reliability of this test is fine; Adeq Precision = 19.7373, so the accuracy of this regression model is fine. The correlation coefficient of the regression model is a measure of the degree of model fit and is calculated as following equation. Its expression can be described as an equation:  
\[R^{2}=SSR/SST \tag{18}\]  
where \(SSR\) is the sum of squares of the difference between the predicted and actual values of all observations in the regression model; \(SST\) is the sum of squares of the difference between all observations and their mean values. The results of the correlation coefficient of the regression model established in this paper are shown in Table 5, \(R^{2}\) = 0.9436 and \(R^{2}_{\rm adj}\) = 0.8985, which indicate the validity of the above regression model, and therefore, the reliability of the obtained regression equation is high. The quadratic regression model was optimized by removing the terms with insignificant effects. The optimized regression equation is:  
\[\begin{array}{l}\Re=-3.6877-0.00028X_{9}+0.001254X_{0}+3.2892X_{5}+0.001343X_ {3}-0.001124X_{2}\\ -0.00076X_{0}X_{5}-0.000835X_{5}X_{3}+0.000648X_{5}X_{2}-0.603143X_{5}