# A Multibody Model Calibration Method for Vehicle Misuse Testing
## 3 Linear Parameter Calibration Using the Least Squares Method  
In this section, a linear parameter calibration process utilizing the least squares method is presented for vehicle misuse testing. Certain vehicle parameters, such as its mass and centroid position, exhibit a nearly linear relationship with their corresponding responses. This process can also be applied to nonlinear parameters, provided they are sufficiently close to the actual vehicle values. The calibration process for the vehicle's mass and centroid position is described in this section as an example. The vehicle's mass and centroid position must be calibrated for misuse tests. Generally, the design department can only supply an Adams/Car (or any other vehicle simulation software) model with a half-load counterweight, initially set up for durability analyses. However, misuse tests typically require a full-load counterweight configuration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Resposes** & **Description** & **Measurement Channels** & **Corresponding Model Parameters** \\ \hline \(v\) & Driving speed & GPS sensor & Driving speed \(v\) \\ \(f_{\mathrm{z}}\) & The vertical wheel’s & Six-component force meter & The vehicle’s total mass \(m\), the vehicle’s moment of inertia \(J\), the centroid position \((x,y)\) \\ \(\tilde{z}_{\mathrm{w}}\) & The wheel’s center acceleration & Acceleration sensor & The shock absorber’s stiffness \(k_{\mathrm{s}}\), damping \(c_{\mathrm{s}}\), preload \(f_{\mathrm{s}}\), the wheel’s center mass \(m_{\mathrm{w}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Vehicle responses, measurement channels, and corresponding model parameters. To calibrate the vehicle's mass and centroid position using road test data, the wheel loads are initially calculated from the experimental responses on a flat road segment, as shown in Figure 4a. Prior to encountering the obstacle, the vehicle must maintain a specified speed on the flat road segment. The six-component force meters record the forces acting on the wheel centers, including the vertical forces. Then, the averages of the vertical forces are considered as the loads acting on each wheel--for instance, \(\overline{f}_{\text{RF}}\) on the right front wheel, \(\overline{f}_{\text{LF}}\) on the left front wheel, \(\overline{f}_{\text{RR}}\) on the right rear wheel, and \(\overline{f}_{\text{LR}}\) on the left rear wheel. For the target vehicle discussed in this paper, the wheel loads are 6.5 kN, 7 kN, 8 kN, and 8 kN, respectively, as indicated in Figure 4b. In fact, these values indicate a discrepancy between the rigid chassis model and the actual vehicle chassis. To explain this, it is hypothesized that the rigid chassis's descent height \(\Delta h\), caused by the loads, exhibits a bilinear relationship with respect to the chassis's local longitudinal coordinates \(\xi\) and lateral coordinates \(\eta\) as follows:  
\[\Delta h=a\xi+b\eta+c. \tag{1}\]  
wherein \(a\) and \(b\) represent the longitude slope and the lateral slope, respectively, and \(c\) is the average descent height of the four wheels. In the vehicle examined in this paper, the air springs of the front wheels differ from those of the rear wheels both in structure and stiffness. Therefore, let \(k_{\text{F}}\) and \(k_{\text{R}}\) denote the suspension stiffness of the front and rear wheels, respectively. The reaction force function can be given through Lagrange interpolation as  
\[f(\xi,\eta)=[k_{\text{R}}+\frac{\xi-\xi_{\text{R}}}{\xi_{\text{F}}-\xi_{\text {R}}}(k_{\text{F}}-k_{\text{R}})](a\xi+b\eta+c), \tag{2}\]  
Figure 4: Calibration of the vehicle’s mass and centroid position through the wheel load responses on a flat road segment: (**a**) computation of the average wheel loads from the responses on the flat road segment and (**b**) theoretical support model of the rigid chassis. wherein \(\xi_{\rm F}\) and \(\xi_{\rm R}\) represent the local longitudinal coordinates of the front and rear wheels, respectively.