# Combined Identification of Vehicle Parameters and Road Surface Roughness Using Vehicle Responses
### Identification of Tire Stiffness and Road Roughness  
This section proposes a method for identifying tire stiffness and road roughness based on the characteristics of the front and rear axles of a vehicle driving over the same road area using the identified unknown forces and vehicle states. The identified unknown force, as shown in Equation (2), contains unknown roughness \(r(x_{1}(t))\), \(r(x_{2}(t))\) through which the front and rear axles pass and unknown stiffness \(k_{t1},k_{t2}\) of the front and rear axles, rewriting Equation (2) as follows:  
\[\begin{cases}f_{2}^{u}(t)=-k_{t1}(y_{1}(t)-r(x_{1}(t)))\\ f_{2}^{u}(t+\frac{z}{v})=-k_{t2}\big{(}y_{2}(t+\frac{z}{v})-r(x_{2}(t+\frac{z }{v}))\big{)}\end{cases}t=\bigg{\{}0,\ldots,\frac{\text{L}-2s}{v}\bigg{\}} \tag{14}\]  
In the proposed method, the vehicle start and end positions are described in Figure 2. The starting position is where the vehicle rear wheel just enters left end of road, and the end position is where the vehicle front wheel is about to leave right end of the road. When the front and rear axles of the vehicle are driven at a uniform speed \(v\), in the common part of the road where both the front and rear axles of the vehicle pass through (blue dotted line in Figure 2), the rear axle contact point roughness is the same as that of the front axle contact point roughness and the time lag \(s/v\), which is considered as \(r(x_{1}(t))=r(x_{2}(t+s/v))\), Equation (15) is given by Equation (14):  
\[y_{1}(t)-y_{2}(t+s/v)=\frac{f_{2}^{u}(t+s/v)}{k_{t2}}-\frac{f_{1}^{u}(t)}{k_{t 1}} \tag{15}\]  
Provided that the unknown forces and vehicle states are identified accurately, the exact \(k_{t1},k_{t2}\) makes Equation (15) hold. The objective function defined by Equation (16) is as follows:  
\[erf(k_{t1},k_{t2})=\sum_{i=1}^{\left[(\text{L}-2s)/v\right]/\delta t}\Bigg{(}y _{1,i}-y_{2,i+\left\lfloor s/v/\delta t\right\rfloor}-(\frac{f_{2,i+\left\lfloor s /v/\delta t\right\rfloor}^{u}}{k_{t2}}-\frac{f_{1,i}^{u}}{k_{t1}})\Bigg{)} \tag{16}\]  
where L\(-2s\) is the length of the front and rear axles of the vehicle driving over the same road area, and their sampling interval is \(\delta t\). The operator [] denotes the rounding operation. Combined with the identified unknown force \(f_{u}(t)\) and axle vertical displacement \(y(t)\), Equation (16) can identify the front and rear axle stiffness \(k_{t1}\), \(k_{t2}\) by nonlinear least squares. Then, the roughness of the front and rear axles passing through can be obtained by Equation (17).  
\[\left\{\begin{array}{c}r(x_{1}(t))\\ r(x_{2}(t))\end{array}\right\}=\left\{\begin{array}{c}y_{1}(t)+f_{1}^{1}(t)/ k_{t1}\\ y_{2}(t)+f_{2}^{2}(t)/k_{t2}\end{array}\right\} \tag{17}\]  
For the whole section of road roughness, the start and end positions of the vehicle are shown in Figure 2. The roughness of the left end of the road for the portion driven by the rear axle only is identified by the rear axle, the roughness of the right end of the road for the portion driven by the front axle only is identified by the front axle, and the portion driven by the common axle is determined by the average of the roughness values identified by each axle, i.e.,  
The flowchart of the method is shown in Figure 3: