## 1 Introduction
### Multi-Factor Test Design
#### 2.4.2 Evaluation Method  
* [PERSON]'s Correlation Coefficient  
As mentioned above, LA, SA, LTR, and RA were generally utilized to evaluate the lateral stability of the vehicle. However, the results determined from these four indicators may show diverse states of stability. So, the [PERSON]'s correlation analysis was conducted to examine the consistency in evaluating the sideslip stability and roll-over stability of the vehicle. The calculation of [PERSON]'s correlation \(r_{xy}\) was expressed as:  
\[r_{xy}=\frac{\text{cov}(x,y)}{\sqrt{\text{var}(x)}\cdot\sqrt{\text{var}(y)}} \tag{3}\]  
where \(x\) and \(y\) denote any two of LA, SA, LTR, and RA. \(\text{cov}(x,y)\) is the covariance of \(x\) and \(y\). \(\text{var}(x)\) and \(\text{var}(y)\) are the variances of \(x\) and \(y\), respectively. * Grey Relational Analysis (GRA)  
Grey relational analysis (GRA) is generally used to evaluate the correlation between factors in multi-factor systems. In this section, grey relational coefficient (GRC) and grey  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \(\mathbf{V_{\text{c}}}\) & **WHR** & \(\mathbf{R}\) & \(\mathbf{e}\) & \(\mathbf{\delta_{\  
relational grade (GRG) are adopted to characterize the correlation between the testing factors and targeted indicators LA, SA, LTR, and RA. The basic process for grey relational analysis mainly includes the following four steps [50]. (1) Confirmation of reference sequence and comparison sequence  
As expressed in the Equations below, four indicators, i.e., LA, SA, LTR, and RA, were taken as reference sequences, respectively. The comparison sequence was composed of above seven factors. \[\Phi_{0}(i)=\{\ \Phi_{0}(1),\Phi_{0}(2),\Phi_{0}(3),\ldots,\Phi_{0}(i)\} \tag{4}\]  
\[X_{k}(i)=\{X_{1}(1),X_{2}(2),X_{3}(3),\ldots,X_{k}(i)\} \tag{5}\]  
where \(\Phi_{0}(i)\) denotes targeted output (LA, SA, LTR, and RA) at case \(i\) in Table 2 (\(i=1,2,3,\ldots\), 32), and \(X_{k}(i)\) denotes inputting factors (\(V_{c}\), WHR, \(R\), \(e\), \(\delta_{\rm hw}\), \(V_{\rm hw}\), and \(f\)), \(k=1,2,3,4,5,6,7\). (2) Normalization of the original test data  
The units and orders of magnitude for seven factors are different, so they need to be compared in the same range. Therefore, the original test data were processed by normalization. The calculation processes were expressed as Equations (6) and (7). \[Y_{0}(i)=\frac{\Phi_{0}(i)-\Phi_{0}(\min)}{\Phi_{0}(\max)-\Phi_{0}(\min)} \tag{6}\]  
\[X^{\prime}{}_{k}(i)=\frac{X_{k}(i)-X_{k}(\min)}{X_{k}(\max)-X_{k}(\min)} \tag{7}\]  
where \(Y_{0}(i)\) denotes normalized value of LA, SA, LTR, and RA at case \(i\), and \(X^{\prime}{}_{k}(i)\) denotes normalized value at case \(i\) of experimental factor \(k\). \(\Phi_{0}(\min)\) and \(\Phi_{0}(\max)\) denote the maximum and minimum value of targeted output. \(X_{k}(\min)\) and \(X_{k}(\max)\) denote the maximum and minimum value of inputting factor \(k\). (3) Calculation of the proximity  
The proximity matrix can be obtained through calculation absolute value of the difference of \(Y_{0}(i)\) and \(X^{\prime}{}_{k}(i)\), as expressed in Equation (8). \[\delta_{0,k}(i)=\left|Y_{0}(i)-X^{\prime}{}_{k}(i)\right| \tag{8}\]  
where \(\delta_{0,k}(i)\) denotes the absolute value of the difference of \(Y_{0}(i)\) and \(X^{\prime}{}_{k}(i)\). (4) Calculation of GRC and GRG  
GRC is calculated as Equation (9). \[\xi_{i}(k)=\frac{\delta_{0,k}(\min)+\rho\delta_{0,k}(\max)}{\delta_{0,k}(i)+ \rho\delta_{0,k}(\max)} \tag{9}\]  
And GRG is express as:  
\[\zeta_{k}=\frac{\sum\limits_{i=1}^{m}\xi_{i}(k)}{m} \tag{10}\]  
where \(\xi_{i}(k)\) denotes grey relational coefficient (GRC), \(\delta_{0,k}(\min)\) and \(\delta_{0,k}(\max)\) denote minimum and maximum of the absolute value of \(\delta_{0,k}(i)\), respectively. \(\rho\) represents distinguishing coefficient, which usually takes 0.5. \(\zeta_{k}\) denotes grey relational grade (GRG), and \(m\) is the number of experiment case (equal to 32).