# Combined Identification of Vehicle Parameters and Road Surface Roughness Using Vehicle Responses
### Identification of Unknown Vehicle Physical Parameters, State, and Unknown Force  
Kalman filtering (KF) is a method of taking the initial state estimate and observations of a system, minimizing the estimation error covariance, and continually updating the estimate of the state of the system by recursive computation [32]. For traditional KF, it is required that the external excitation of the system and information about the system model are available, and it is clear that traditional KF is not applicable when the vehicle model parameters are unknown and the roughness is unknown. Some scholars have proposed an extended Kalman filter with unknown inputs (EKF-UI) on the basis of a traditional KF, which realizes the simultaneous identification of structural state, structural parameters, and unknown excitation through structural responses. In this section, the EKF-UI is used \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Roughness Class**} & \multirow{2}{*}{**Description**} & \multicolumn{4}{c}{\(G(n_{0})(\mathbf{10^{-6}m^{3}})\)} \\ \cline{3-5}  & & **Lower Limit** & **Geometric Mean** & **Upper Limit** \\ \hline A & Very good & - & 16 & 32 \\ B & Good & 32 & 64 & 128 \\ C & Average & 128 & 256 & 512 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficient values for different roughness classes according to ISO-8608. for the identification of unknown physical parameters, states, and unknown forces of the vehicle [33; 34]. Wheel mass is assumed to be known in advance because it can be easily estimated. It can be seen that the unknown force Equation (2), where the axle stiffness \(k_{t}\) is coupled to the roughness \(r(x(t))\), cannot be recognized directly, so the remaining vehicle parameters are first expanded into the state vector. The vehicle system state vector is defined as \(\mathbf{Z}(t)=\left\{\mathbf{X}_{v}^{T}(t)\quad\dot{\mathbf{X}}_{v}^{T}(t)\quad\mathbf{\theta} \right\}^{T}\), \(\mathbf{\theta}=\left\{m_{v}\quad I_{v}\quad k_{s1}\quad k_{s2}\quad c_{s1}\quad c _{s2}\quad a_{1}s\quad a_{2}s\right\}\). The system state equation containing the unknown parameters (1) is rewritten in the form of a state space; the first derivative with respect to time is expressed as Equation (4). After linearization and discretization, it is obtained as Equation (5). \[\dot{\mathbf{Z}}(t)=\left\{-\mathbf{M}_{v}^{-1}\Big{[}(\mathbf{C}_{v})_{\mathbf{\theta}}\dot{ \mathbf{X}}_{v}(t)+(\mathbf{K}_{v})_{\mathbf{\theta}}\mathbf{X}_{v}(t)-\mathbf{\eta}_{v}^{u}\mathbf{f} ^{u}(t)\Big{]}\right\}+\mathbf{w}(t)=\mathbf{g}(\mathbf{Z}(t),\mathbf{f}^{u}(t))+\mathbf{w}(t) \tag{4}\]  
\[\mathbf{g}(\mathbf{Z},\mathbf{f}^{u})=\mathbf{g}\Big{(}\dot{\mathbf{Z}}_{k|k},\dot{\mathbf{f}}_{k|k}^{ u}\Big{)}+\mathbf{G}_{k|k}\Big{(}\mathbf{Z}_{k}-\dot{\mathbf{Z}}_{k|k}\Big{)}+\mathbf{B}_{k|k}^{ u}\Big{(}\mathbf{f}^{u}-\dot{\mathbf{f}}_{k|k}^{u}\Big{)} \tag{5}\]  
where \(\mathbf{w}(t)\) is mean-zero process noise vector, \(\mathbf{G}_{k|k}=\frac{\partial\mathbf{g}(\mathbf{Z},\mathbf{f}^{u})}{\partial\mathbf{Z}}|_{\mathbf{Z} _{k}=\dot{\mathbf{Z}}_{k|k}\vec{\mathbf{f}}=\vec{\mathbf{f}}_{k|k}^{u}}\). \(\mathbf{B}_{k|k}^{u}=\frac{\partial\mathbf{g}(\mathbf{Z},\mathbf{f}^{u})}{\partial\mathbf{f}}|_\mathbf{{Z}_{k}=\dot{\mathbf{Z}}_{k|k}\vec{\mathbf{f}}^{u}-\vec{\mathbf{f}}_{k|k}^{u}}\). \(\dot{\mathbf{\cdot}}\) is denoted as the estimated value of the parameter. This method solves the vertical and cornering displacements and accelerations of the body, and the vertical displacements and accelerations of the front and rear axes. Therefore, the non-linearized discrete observation equation can be expressed as Equation (6):  
\[\mathbf{y}_{k+1}=\left\{ \begin{array}{c}\mathbf{X}_{v}\\ \ddot{\mathbf{X}}_{v}\end{array}