## 2 Road Excitation Model
### 3-D Road Roughness  
#### 2.2.1 Straight Driving Condition  
One dimensional (1-D) autocorrelation function of the road roughness PSD based on white noise can be acquired using inverse Fourier transform [13]. 2-D auto-correlation function can be obtained by expanding the 1-D auto-correlation function and using isotropy of road roughness, as follows.  
\[G_{q}(n_{1},n_{2})=\frac{4\pi\alpha\rho^{2}}{\left[\alpha^{2}+4\pi^{2}({n_{1}}^ {2}+{n_{2}}^{2})\right]^{3/2}} \tag{6}\]  
where, \(n_{1}\) is spatial frequency with \(X\)-axis in \(m^{-1}\), \(n_{2}\) is spatial frequency with \(Y\)-axis in \(m^{-1}\) and \(G_{q}(n_{1},n_{2})\) is PSD of 2-D road roughness with spatial frequency in \(m^{3}\).  
According to [13], Fourier transform and PSD of 2-D road roughness can be expressed as Equations (7) and (8).  
\[H(f_{x},f_{y})=\lim_{l_{x},l_{y}\rightarrow\infty}\int_{-l_{x}}^{l_{x}}\int_{ l_{y}}^{l_{y}}h(x,y)e^{-j2\pi(xf_{x}+yf_{y})}dxdy \tag{7}\]  
\[G(f_{x},f_{y})=\lim_{l_{x},l_{y}\rightarrow\infty}\frac{1}{4l_{x}l_{y}}\big{|} H(f_{x},f_{y})\big{|}^{2} \tag{8}\]  
where, \(f_{x}\) and \(f_{y}\) are spatial frequencies with \(X\)-axis and \(Y\)-axis in \(m^{-1}\) respectively, \(l_{x}\) and \(l_{y}\) are surface road lengths; \(n_{2}\) is spatial frequency with \(Y\)-axis in \(m^{-1}\), \(h(x,y)\) is 3-D road roughness in \(m\), \(G(f_{x},f_{y})\) is PSD of 2-D road roughness for spatial frequency in \(m^{3}\), and \(H(f_{x},f_{y})\) is Fourier transform of \(h(x,y)\).  
Due to boundary of the actual road [10], Equations (7) and (8) can be expressed as Equations (9) and (10).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Road Level** & **a/(m\({}^{-1}\))** & **\(\rho\)/m** & \(G_{q}(n_{0})\) \\ \hline \(A\) & & 0.0153 & 16 \\ \(B\) & & 0.0306 & 64 \\ C & & 0.0611 & 256 \\ \(D\) & & 0.1222 & 1024 \\ \(E\) & 0.0011 & 0.2444 & 4096 \\ \(F\) & & 0.4888 & 16,384 \\ G & & 0.9776 & 65,536 \\ \(H\) & & 1.9552 & 26,2144 \\ \hline \end{tabular}
\end{table}
Table 1: Estimated results of road parameters.  
Figure 1: Results of parameters estimation.  
\[H(f_{p},f_{q})=\Delta x\Delta y\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}h(m\Delta x,n\Delta y )e^{-j2\pi(\frac{p}{M}m+\frac{q}{N}n)} \tag{9}\]  
\[G(f_{p},f_{q})=\frac{\Delta x\Delta y}{MN}\big{|}F(f_{p},f_{q})\big{|}^{2} \tag{10}\]  
In addition,  
\[f_{p}=\frac{p}{M\Delta x};\ p=0,1,2,\cdots,M-1 \tag{11a}\] \[f_{q}=\frac{q}{N\Delta x};\ q=0,1,2,\cdots,N-1 \tag{11b}\]  
where, \(M\) and \(N\) are sampling numbers, \(\Delta x\) and \(\Delta y\) are sample intervals, \(f_{p}\) and \(f_{q}\) are discrete spatial frequencies in \(m^{-1}\) and \(F(f_{p},f_{q})\) is Fourier transform of \(h(m\Delta x,n\Delta y)\).  
With assumption of stationary random process, road defined in Equation (12) can be computed according to Equation (10) as follows.  
\[F(f_{p},f_{q})=\big{|}F(f_{p},f_{q})\big{|}e^{\varphi(f_{p},f_{q})} \tag{12}\]  
where \(\phi(f,f)\) is phase angle of \(F(f_{p},f_{q})\).  
Based on properties of 2-D Fourier transform, function \(H(p,q)\), i.e., the Fourier transform of \(h\) (\(m\), \(n\)), may be satisfied by Equation (13).  
\[H(1,N+1-n)=conj[H(1,n)]; \tag{13a}\] \[H(M+1-n,1)=conj[H(m,1)];\] (13b) \[H(M/2,N+1-n)=conj[H(M/2,n)];\] (13c) \[H(M+1-m,N/2)=conj[H(m,N/2)];\] (13d) \[H(M+1-m,N+1-n)=conj[H(m,n)]; \tag{13e}\]  
where \(conj\) represents the conjugate; \(m=1\), \(2\),.., \(M/2-1\); \(n=1\), \(2\),..., \(N/2-1\).  
Based on the above analysis, 3-D road roughness can be expressed by Equation (14).  
\[h(m,n)=\frac{1}{MN}\sum_{p=1}^{M-1}\sum_{q=1}^{N-1}F(f_{p},f_{q})e^{j2\pi( \frac{p}{M}m+\frac{q}{N}n)} \tag{14}\]  
Following the analysis presented above, 3-D road roughness for level ISO-\(A\) and ISO-C cases are obtained. For instance, analysis is carried out considering level ISO-\(A\) excitation (at a velocity of 80 km/h). Results are shown in Figure 2.  
Figure 2: 3-D road roughness for Level ISO-\(A\) (at a velocity of 80 km/h).