# A Novel Longitudinal Speed Estimator for Four-Wheel Slip in Snowy Conditions
## 2 Materials and Methods
### Longitudinal Speed Estimator of Four-Wheel Drive Slipping Vehicle on Split Roads
#### 2.1.2 Vehicle Speed Estimator Based on Wheel Speed  
The main task of this estimator is to estimate the vehicle speed by a fuzzy controller, based on a minimum wheel speed and four wheels' slip rate. Although the estimated result is not accurate in high slip conditions, it is meaningful as a reference value. Obviously, the estimated vehicle speed based on the wheel speed encoder is a very important part. Fuzzy control logic is used to guarantee its accuracy. The key to estimating longitudinal vehicle speed accurately is the tire slip ratio. In other words, lower tire slip levels yield higher confidence levels. In the vehicle estimate, speed logic is based on wheel speed encoders and the minimum wheel speed is selected as a basic reference; then, the fuzzy logic algorithm is used to consider the effect of each slip condition. The fuzzy logic algorithm determines the confidence level based on the number of wheels that slip and each wheel's slip level. The output is a weighting coefficient to represent the influence at the minimum wheel speed to accurately estimate the vehicle's longitudinal speed. The wheel rotation speed input signal is directly from the wheel rotational speed encoders; it is sent to and optimized by the vehicle's control unit. Signal filtering is used to avoid zero shift. Any wheel can have the lowest speed and the lowest slip rate. The lowest wheel speed is that with the lowest slip rate among the four wheels at a certain time. The objective is to calculate the longitudinal vehicle speed based on the lowest slip rate of the four wheels  
\[\mathrm{V}_{\mathrm{min}}=\mathrm{min}\big{\{}\mathrm{V}_{i,j}\big{\}},\ 1\leq i,j\leq 2\]  
There are three driving condition cases from the analysis. The first case is no wheel slip, in which the vehicle speeds calculated from each wheel speed encoder are equal to each other when the car is driving straight. \[\mathrm{V}_{i,j}=\mathrm{real\ longitudinal\ vehicle\ speed}\]  
The second case is at least one wheel slips, but the number of slipping wheels is three or fewer. Assuming the calculated vehicle speed based on the non-slipping wheel is \(\mathrm{V}_{m,m}\), then  
\[\mathrm{V}_{i,j}\geq\mathrm{V}_{m,m}=\mathrm{V}_{real}\]  
in which, \(m\
eq i,j,1\leq m,i,j\leq 2\), \(\mathrm{V}_{real}\) is the real longitudinal vehicle speed. The third case is when all four wheels' slip. \[\mathrm{V}_{i,j}\geq\mathrm{V}_{m,m}\geq\mathrm{V}_{real}\]  
in which, \(m\
eq i,j,1\leq m,i,j\leq 2\)  
Besides the minimum wheel speed, the wheel slip rate is the most important input variable in the fuzzy control logic,  
\[\lambda_{i,j}=\frac{\left|\min\left(V_{i,j}\right)-V_{i,j}\right|}{\min\left(V _{i,j}\right)}\]  
in which, \(1\leq i,j\leq 2\). To make the first-layer calculation strategy smooth, slip rates are divided into low, mid and large slip in the fuzzy control rule. The strategy is to estimate vehicle speed on not only a high friction asphalt road but also a special (snowy) road, where the slip friction between the tire and the road is approximately 0.1 and the maximum friction is approximately 0.2. The low slip category only considers snowy road driving conditions; the slip friction between the tire and the snow surface is between 0.1 and 0.175. Mid slip considers asphalt road and snow road driving, in which the friction coefficient is between 0.1 and 0.25. The large slip condition considers friction coefficients above 0.2 [21] (Figure 2). In addition to the fuzzy logic input variables, the output variables are also important. The output variables are defined as small, mid and large rates in the fuzzy control logic, as shown in Figure 3. When a high slip rate occurs, the actual vehicle speed could be smaller than the calculated value for wheel speed, so this output variable coefficient is the inverse of the fuzzy control output variable. Finally, we obtain the fuzzy logic algorithm. Fuzzy controller input variables, \(\lambda_{i,j}\) are defined as the tire slip ratios, in which \(i=1,2\) indicate the vehicle's front and rear.