# A Novel Longitudinal Speed Estimator for Four-Wheel Slip in Snowy Conditions
## 3 Results and Discussion
### Experimental Result and Discussion  
The vehicle is accelerated at full throttle from 0 km/h to 30 km/h, holding the steering wheel in the centre. Then, we obtain the proposed controller input variables from the test car's in-vehicle network. These include the rotation speed of the four wheels collected by the wheel encoders, as shown in Figure 8. The longitudinal acceleration obtained from the acceleration sensor is shown in Figure 9.  
Left side wheel speeds are much slower than right side speeds after 1.7 s, but their speed change is smoother, which means the left two wheels slip later and more gently than the right wheels. This is because the left two tires are located on the asphalt road, while right tires contact the snowy road. The left rear wheel rotates smoothly before 2.8 s without dramatic change, indicating no slip on this wheel until 2.8 s. While the left front wheel has no slip before 2.1 s, the other two wheels slip at 1.7 s.  
The longitudinal acceleration signal not only provides a reference for the integrated speed value but also verifies the slip condition determined by the speed sensors. For example, the longitudinal acceleration changed dramatically due to the loss of traction force at 1.7 s when the vehicle slipped.  
Although the wheel rotation speed input variables originate from different encoders, there is nearly no wheel slip from 0 to approximately 1.7 s, two wheels slip from 1.7 s to 2.1 s, three wheels slip from 2.1 to 2.8 s and four wheels slip from 2.8 s to 3.2 s. These cases are divided into four parts for further analysis: no wheel slip, two-wheel slip, three-wheel slip and four-wheel slip. The calculated calibrated sensor-based wheel speeds used as the output speeds from the controller are shown in Figure 10.  
Figure 8: Wheel rotation speed controller input variables.  
Figure 9: Longitudinal acceleration controller input variable.  
To compare and validate the estimated vehicle speed, we collect the longitudinal vehicle speed from the in-vehicle network and define the signal as Ref.Vehicle Speed. Meanwhile, we obtain the estimated longitudinal vehicle speed though the estimator and define it as Est.Vehicle Speed. The comparison between the controller-estimated vehicle speed and the reference vehicle speed from the production vehicle is shown in Figure 11. The smooth curves show that the controller-estimated longitudinal speed is similar to the reference vehicle speed.  
To verify the performance of the proposed controller, the error rate is calculated from the estimated speed and the reference speed, as shown in Figure 12.  
\[\delta=\frac{\
u_{est}-\
u_{ref}}{\
u_{ref}}\cdot 100\%\]  
in which \(\delta:error\ rate\), \(\
u_{est}\): estimated longitudinal speed, and \(\
u_{ref}\): reference vehicle speed.  
Because this estimator is mainly designed to address high slip levels, we focus on vehicle speeds from 1.5 s to 3.2 s to study the error rate with obvious slip (Figure 12). The largest absolute error rate is below 9%. There are four experimental conditions categorized by the number of slipping wheels, as shown in Table 3 below.  
Figure 11: Controller-estimated speed and reference vehicle speed from the production vehicle.  
Figure 12: Estimated vehicle speed error rate.  
Figure 10: Controller output of wheel speed.  
The proposed estimator focuses on the high slip level condition, as supported by the error rates shown in Table 3. The largest error rate is approximately 9% in the second condition (two slipping wheels). The third condition (three slipping wheels) benefits most, with an error rate below 3.5%. The last condition is slightly higher but still below 4.321%.