# Driving Safety Improved with Control of Magnetorheological Dampers in Vehicle Suspension
## Vehicle Model and Its Coupling with Road and Environment
### Road Profile  
Random road profile exhibits an integral character according to the norm ISO 8608 [41], i.e., low frequency components of the road profile present greater amplitudes and vice versa. Thus, the spatial power spectral density (PSD) of an ideal road profile, denoted as \(S_{r}(\omega_{p})\), can be described with respect to the spatial angular frequency \(\omega_{p}\) as follows:  
\[S_{r}(\omega_{p})=\Theta_{0}\cdot\left(\frac{\omega_{p0}}{\omega_{p}}\right)^{ 2}, \tag{1}\]  
where \(\Theta_{0}\) is a reference value of \(S_{r}(\omega_{p})\) that is dedicated to a selected road class at an angular frequency \(\omega_{p0}=1\). Here, class F is used in order to reflect off-road conditions; thus, \(S_{r}(\omega_{p0})=\Theta_{0}=1024\times 10^{-6}\) was assumed. The value of \(\Theta_{0}\) was set as the geometric mean of the range for class F. Figure 2b presents the resultant characteristics of \(S_{r}\) evaluated with respect to the spatial frequency \(f_{p}\) as well as the simulated road profile.  
The generation of road profile was realized by filtering a Gaussian white noise \(w_{r}(x)\) dependent on the distance coordinate \(x\) while using an integrating digital discrete-distance filter of transfer  
Figure 2: Standard road profile of class F: (**a**) with respect to the longitudinal coordinate, (**b**) spatial power spectral density (PSD) with respect to the spatial frequency.  
function \(H(z_{p}^{-1})\). A continuous-distance filter \(H(s_{p})\) equivalent to \(H(z_{p}^{-1})\) was defined in order to more clearly discuss its parameters and relate them to the continuous-distance spatial PSD \(S_{r}\):  
\[H(s_{p})=\omega_{p0}\cdot\sqrt{\Theta_{0}}\cdot\frac{a_{H}}{1+s_{p}\cdot a_{H}}, \tag{2}\]  
where \(a_{H}\) = 6.6 m is a cutoff wavelength (an equivalent of time constant in the time domain, noticeable for low frequencies in Figure 2b), which is much greater than the wheelbase of the vehicle equal to 1.29 m. Consequently, Figure 2a shows a diagram of a sample road profile \(z_{r}(x_{r})\) with respect to distance.  
The filtering of white noise can only be accomplished in the discrete domain, since white noise cannot be realized in reality. Besides, the resolution of the generated road profile was assumed as \(f_{ps}\) = 1000 points per meter. Thus, a discrete realization of white noise \(w_{r}(x)\) was used with variance \(\sigma_{r}^{2}=2\pi\frac{f_{ps}}{2}\) in order to obtain a unit level of the input random signal in spatial PSD \(S_{r}(\omega_{p})\), defined with respect to \(\omega_{p}=2\pi f_{p}\).  
Higher spatial frequencies of the generated road profile can be neglected in further analysis due to the much greater dimension of wheels. Furthermore, the road profile needs to be differentiated in order to obtain \(\frac{dz_{r}(x_{r})}{dx_{r}}\) indicating an inclination of road in a given point, which is required by the vehicle simulator. Thus, the bandwidth of the road profile \(z_{r}\) was limited to the cutoff spatial frequency that was equal to 200 cycles per meter (see Figure 2b) while using an additional discrete domain finite impulse response filter (FIR).