## 4 Vehicle Mass Estimation  
The vehicle mass is one of the critical parameters for obtaining accurate vertical tire forces and road friction condition. Although many studies of mass estimation are carried out based on model-based approaches along with adaptive filters [14; 15; 16; 17; 18; 19], they owned the limitation that the vehicle is required to be driven to obtain the proper persistent excitation and it requires a considerable estimation time. Therefore, this paper utilizes another method to estimate the vehicle mass based on the static load distribution by monitoring the static suspension deflections of all wheels [23] for quicker and more accurate results. Suspension deflection is assumed to be directly measured through a potentiometer or an LVDT (linear variable differential transformer) which are both economical and practical to be used. The advantage of this method is that it can be performed when the vehicle is stopped or is subject to a low speed thus it does not need to wait until the vehicle is subject to proper acceleration including persistent excitation. Moreover, the loading and unloading onto a vehicle are conducted when the vehicle is stopped. According to the loading condition, the suspension deflection is also changed. This variation can provide us with the increase or decrease of mass over the entire loading and unloading process.  
The total mass of the vehicle is the sum of sprung mass \(m_{s}\) and un-sprung mass \(m_{u}\). The un-sprung mass is seldom adjusted during vehicle operations. Hence, \(m_{u}\) is considered as a known constant in (11)  
\[\hat{m}_{v}=m_{s}+m_{u} \tag{11}\]The variation in vehicle mass means the change of sprung mass due to the loading and unloading of either items or passengers (or both). Nevertheless, the sprung mass of the empty vehicle \(m_{s0}\) is given by the manufacturer. Thus,  
\[m_{s}=m_{s0}+\Delta m_{s} \tag{12}\]  
In order to acquire the sprung mass variation, a quarter-car model in Figure 3 provides,  
\[\Delta m_{s}=\Delta m_{s.FL}+\Delta m_{s.FR}+\Delta m_{s.RL}+\Delta m_{s.RR} \tag{13}\]  
The suspension springs are varied by the corresponding sprung mass on each side. Given a conventional suspension without level regulation, and assuming that it is operating within its linear range, an individual variation sprung mass, \(\Delta m_{s.i}\), is defined,  
\[\Delta m_{s.i}=\frac{F_{sz.i}}{g}=\frac{k_{si}\ \Delta s_{i}e_{i}}{g}\ \ for\ i= FL,FR,RL,RR \tag{14}\]  
Here, \(\Delta s_{i}\) is the spring deflection variation from the initial deflection of the empty vehicle, \(k_{i}\) is the spring stiffness of the suspension, and \(e_{i}\) is the coefficient of spring force projection to the vertical axis.  
Since this approach is valid in \"stop\" or \"low speed\" conditions with a less lateral motion of the vehicle, we set that the mass estimation is triggered at the conditions \(V_{x}\leq 10\ \mathrm{km/h}\) and \(SWA\leq 20^{\circ}\). The pseudo-code for the mass estimation is shown in the Algorithm 1.  
```
1:\(s_{i},\ s_{0},\ V_{x},\ m_{u},\ m_{s0},\ k_{i},\ g,\ e_{i}\)
2:Output \(\hat{m}_{s.k}\)
3:\(\mathbf{If}\ V_{x}<10\ \mathrm{km/h}\) and \(SWA\leq 20^{\circ}\) %SWA: Steering_Wheel_Angle
4:\(\hat{m}_{v}=\hat{k}e_{i}(s_{i}-s_{0})/g\)
5:\(\hat{m}_{v}=\mathrm{sum}(m_{i})+m_{u}+m_{s0}\)
6:\(\hat{m}_{v,k}=\alpha\ \hat{m}_{v,k-1}+(1-\alpha)\hat{m}_{v}\) % Low-pass filter to obtain the smooth value
7:\(\hat{m}_{v.k-1}=\hat{m}_{v.k}\)
8:End
```  
**Algorithm 1** Mass Estimation