## 4 Simulation Results and Analysis
### Results Analysis  
Figure 5 shows the tracking performance of the targeted driving process. The CarSim vehicle model equipped with VSC is able to accomplish the designed double lane change maneuver with acceptable tracking accuracy. Using the torque and rotational speed information directly acquired from CarSim, the longitudinal force of each tire can be calculated according to Equation (2). These longitudinal forces combined with the hand wheel steering angle signal comprise the control input, shown in Figure 6. Additionally, the measurements from virtual sensors under DLC test are depicted in Figure 7.  
UKF and MHE are implemented as two separate methods in order to directly compare the estimation performance and robustness. Figures 8-11 show the estimated results and errors of the longitudinal velocity, lateral velocity, yaw rate, and lateral tire forces, respectively, using these two estimation methods,. Figure 8 shows that both MHE and UKF can estimate the longitudinal speed satisfactorily. However, MHE shows a faster convergence and slightly higher accuracy compared with UKF. Figure 5: Tracking results on the desired trajectory. Figure 8: Estimated results and errors of longitudinal velocity using UKF and MHE. Figure 6: Control input signals of the double lane change (DLC) maneuver. Figure 7: Measured signals with Gaussian noises. During the lane changing interval around 4 s where the hand wheel input and lateral acceleration vary greatly, it can be seen from Figure 9 that UKF and MHE perform differently; near 4 s the estimated error of UKF is relatively larger and more fluctuating than that of MHE. Regarding the yaw rate estimation, the overall performance of both algorithms is satisfactory, as seen in Figure 10. Similar to the situation of lateral speed estimation, for the lateral tire force estimation in Figure 11, MHE also shows better performance than UKF under the strong nonlinearity caused by the large lateral acceleration. Table 2 summarizes the quantitative analysis for both UKF and MHE algorithms, including ME (maximum estimation) error, RMS (root mean square) error, and improvement percentage. Except that the longitudinal speed ME error of MHE is a bit larger than that of UKF, MHE outperforms UKF in all other cases. The ME errors of estimated states drop significantly when MHE is applied. Meanwhile, compared with UKF, the RMS errors of these seven vehicle states are improved at least 10% by MHE. For the longitudinal speed, the improvement even reaches 62%. Figure 10: Estimated results and errors of yaw rate using UKF and MHE. Figure 9: Estimated results and errors of lateral velocity using UKF and MHE. It is noteworthy that in real applications, measurement noise or other reasons may occasionally result in a wrong initial state value, which affects the accuracy of the subsequent estimation. In order to further evaluate the robustness of these two estimation algorithms, simulations with incorrect initial longitudinal speed estimates of 120 km/h and 80 km/h are conducted, the results of which are illustrated in Figures 12 and 13. Compared to UKF, MHE has the advantage of solving this kind of constrained state estimation problem. In this study, the constraint is set as Equation (40) in Section 3.2. In Figure 12a,c,e are the estimation with the initial value of 120 km/h and Figure 12b,d,f with the initial value of 80 km/h. Due to Equation (40), the estimated longitudinal speed by MHE approach converges to the reference value rapidly when the initial estimate equals 120 km/h. Even under an initial estimate of 80 km/h, MHE still converges faster than UKF. The other vehicle states, such as lateral speed, yaw rate, and lateral tire forces in Figure 13, are not affected too much by the wrong initial longitudinal velocity estimate. In Table 3 the ME error and RMS error of each state variable are listed.