# Combined Identification of Vehicle Parameters and Road Surface Roughness Using Vehicle Responses
### Vehicle Modeling and Simulation of Road Roughness  
In this paper, the four-degree-of-freedom half-vehicle model shown in Figure 1 is used. The four-degree-of-freedom half-vehicle model is commonly used in vehicle dynamics simulation [21,23]. Compared with the quarter-vehicle model, which can only represent the vertical bouncing motion of the body and wheels, the half-vehicle model can consider the motion of the body pitching, and also allows to take into account the effect of the position of the center of gravity, and the different damping and stiffness of the front and rear axles. A four-degree-of-freedom semi-vehicle model is traveling at a uniform speed \(v\) on a road of length L with roughness. In Figure 1, \(y_{v}\), \(\theta_{v}\), \(y_{1}\), \(y_{2}\) are the vertical displacement of the center of gravity of the body, the angular displacement and the vertical displacement of the front and rear axles, respectively; \(m_{v}\), \(I_{v}\), \(m_{1}\), \(m_{2}\) are the center of gravity mass, the moment of inertia and the front and rear axles' mass, respectively; \(x_{1}(t)\) and \(x_{2}(t)\) denote the distances from the front and rear axles to the starting position of the left end, respectively; \(r(x_{1}(t))\) and \(r(x_{2}(t))\) are the roughness of the road surface at the contact points of the front and rear wheels, respectively. \(s\) denotes the distance between the front and rear axles; \(a_{1}\) and \(a_{2}\) are the proportionality coefficients of the distance from the front and rear axles to the center of gravity of the body, respectively, and satisfy \(a_{1}+a_{2}{=}1\). \(k_{s1}\), \(k_{s2}\) and \(c_{s1}\), \(c_{s2}\) are the suspension stiffness and suspension damping coefficients for the front and rear, respectively; \(k_{t1}\), \(k_{t2}\) are the stiffness coefficients of the front and rear axles, respectively. The equations of motion with the vehicle part above the tire spring as the isolator are as follows:  
\[\mathbf{M}_{v}\mathbf{\dot{X}}_{v}(t)+\mathbf{C}_{v}\mathbf{\dot{X}}_{v}(t)+\mathbf{K}_{v}\mathbf{X}_{v }(t)=\mathbf{\eta}_{v}^{u}\mathbf{\dot{\mu}}(t) \tag{1}\]  
where \(\mathbf{X}_{v}=\left\{y_{v}\quad\theta_{v}\quad y_{1}\quad y_{2}\right\}^{\text{T}}\) denotes the vehicle state vector; \(\mathbf{\eta}_{v}^{u}= \begin{bmatrix}0&0&1&0\\ 0&0&0&1\end{bmatrix} ^{\text{T}}\) is the matrix of effects of unknown forces. \(\mathbf{\dot{\mu}}^{u}(t)\) is the unknown force vector applied to the vehicle portion above the tire spring as an isolator, denoted as follows:  
\[\mathbf{\dot{f}}^{u}(t)=\left\{ \begin{array}{l}\mathbf{\dot{f}}_{1}^{u}(t)\\ \mathbf{\dot{f}}_{2}^{u}(t)\end{array} \right\}=\left\{ \begin{array}{l}-k_{t1}(y _{1}(t)-r(x_{1}(t)))\\ -k_{t2}(y_{2}(t)-r(x_{2}(t)))\end{array} \right\} \tag{2}\]  
\(\mathbf{M}_{v}\), \(\mathbf{C}_{v}\), \(\mathbf{K}_{v}\) are the mass matrix, damping matrix, and stiffness matrix of the study object, respectively, which are expressed in the following form:  
Figure 1: Four-degree-of-freedom half-vehicle model. \[\mathbf{M}_{v}= \begin{bmatrix}m_{v}&0&0&0\\ 0&I_{v}&0&0\\ 0&0&m_{1}&0\\ 0&0&0&m_{2}\end{bmatrix} ;\ \mathbf{C}_{v}= \begin{bmatrix}c_{s1}+c_{s2}&(-c_{s1}a_{1}+c_{s2}a_{ 2})s&-c_{s1}&-c_{s2}\\ (-c_{s1}a_{1}+c_{s2}a_{2})s&(c_{s1}a_{1}^{2}+c_{s2}a_{2}^{2})s^{2}&c_{s1}a_{1}s &-c_{s2}a_{2}s\\ -c_{s1}&c_{s1}a_{1}s&c_{s1}&0\\ -c_{s2}&-c_{s2}a_{2}s&0&c_{s2}\end{bmatrix} \]  
\[\mathbf{K}_{v}= \begin{bmatrix}k_{s1}+k_{s2}&(-k_{s1}a_{1}+k_{s2}a_{2})s&-k_{s1}&-k_ {s2}\\ (-k_{s1}a_{1}+k_{s2}a_{2})s&(k_{s1}a_{1}^{2}+k_{s2}a_{2}^{2})s^{2}&k_{s1}a_{1}s &-k_{s2}a_{2}s\\ -k_{s1}&k_{s1}a_{1}s&k_{s1}&0\\ -k_{s2}&-k_{s2}a_{2}s&0&k_{s2}\end{bmatrix} \]  
It is worth noting that, unlike previous analyses that were directly based on a semi-vehicle model [30], this study analyzes the vehicle portion above the tire stiffness as the object of study. The stiffness matrix \(K_{v}\) does not contain the tire stiffness \(k_{t1}\) and \(k_{t2}\).