# A Multibody Model Calibration Method for Vehicle Misuse Testing
## 3 Linear Parameter Calibration Using the Least Squares Method
The force difference between the two front wheels is  
\[\Delta f_{\rm F}=f(\xi_{\rm F},\eta_{\rm L})-f(\xi_{\rm F},\eta_{\rm R})=k_{\rm f }(a\xi_{\rm F}+b\eta_{\rm L}+c)-k_{\rm f}(a\xi_{\rm F}+b\eta_{\rm R}+c)=k_{\rm F} b(\eta_{\rm L}-\eta_{\rm R}), \tag{3}\]  
and the force difference between the two rear wheels is  
\[\Delta f_{\rm R}=f(x_{\rm R},\eta_{\rm L})-f(x_{\rm R},\eta_{\rm R})=k_{\rm r }(ax_{\rm R}+b\eta_{\rm L}+c)-k_{\rm R}(ax_{\rm R}+b\eta_{\rm R}+c)=k_{\rm R} b(\eta_{\rm L}-\eta_{\rm R}). \tag{4}\]  
Formulas (3) and (4) illustrate that both \(\Delta f_{\rm F}\) and \(\Delta f_{\rm R}\) are either zero or non-zero simultaneously. However, the recorded average forces of the rear wheels are the same, whereas those of the front wheels are not. This discrepancy reflects the difference between the rigid chassis model and the actual vehicle chassis, due to flexible deformation of the chassis. By applying the least squares method, the descent height parameters \(a\), \(b\), and \(c\) can be calibrated. However, in practice, the vehicle's mass \(m\) and centroid position \((x,y)\) are used instead of the descent height parameters. On the one hand, \(c\) is directly influenced by \(m\), while \(a\) and \(b\) are influenced by \((x,y)\); on the other hand, utilizing \(m\) and \((x,y)\) can circumvent the impact of the suspension stiffness on the results. The actual vehicle experiment data (test data) are taken as the target, and we start with the simulation results for the initial parameters. The parameter sensitivity is computed using the finite difference method, with significant variations \(\Delta m=100\) kg, \(\Delta x=100\) mm, and \(\Delta y=100\)mm to suppress numerical errors in the Adams/Car simulation. As depicted in Table 3, the sensitivity matrix \(A\) with respect to the parameters \([m,x,y]^{\rm T}\) can be computed according to the quotient between the column's differences and the variation as  
\[A=\left[f_{1}-f_{0}\ \ \ f_{2}-f_{0}\ \ \ f_{3}-f_{0}\right]/100. \tag{5}\]  
Then, with the response error  
\[\mathbf{e}=f_{0}-f_{\rm test}, \tag{6}\]  
the predicted parameters are computed as  
\[[m_{\rm c},x_{\rm c},y_{\rm c}]^{\rm T}=[m_{0},x_{0},y_{0}]^{\rm T}+(A^{\rm T }A)^{-1}A^{\rm T}\mathbf{e}. \tag{7}\]  
With the calibrated parameters \([m_{\rm c},x_{\rm c},y_{\rm c}]^{\rm T}\), the Adams/Car simulation results \(f_{\rm cali}\) are very close to the actual data from the vehicle experiment (test data), as shown in Figure 5 and Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Wheel Position** & **Tested** & **Initial** & **Initial** & **Initial** & **Initial** & **Predicted Using** & **Caliibrated** \\  & **(Test Data)** & & **+\(\Delta m\)** & **+\(\Delta x\)** & **+\(\Delta y\)** & **Least Squares** & **Calibrated** \\ \hline  & \(f_{\rm test}\) & \(f_{0}\) & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{\rm pred}\) & \(f_{\rm cali}\) \\ \hline Left Front (LF) & 7.03 & 6.50 & 6.74 & 6.22 & 5.84 & 6.95 & 6.93 \\ Right Front (RF) & 6.52 & 6.50 & 6.74 & 6.22 & 7.15 & 6.59 & 6.49 \\ Left Rear (LR) & 8.00 & 6.88 & 7.11 & 7.16 & 6.47 & 8.12 & 8.20 \\ Right Rear (RR) & 8.00 & 6.88 & 7.11 & 7.16 & 7.30 & 7.89 & 7.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calibration of the vehicle’s mass and centroid position using the least squares method: four-wheel load data for the target, the initial model, and the variant models; that predicted using the least squares method; and that for the simulated model (\(\Delta m=100\) kg, \(\Delta x=\Delta y=100\) mm); the value unit is kN.