## 4 Result Verification
To further evaluate the accuracy characteristics of the CFF-SIR algorithm in the identification results of the vehicle longitudinal tire force estimation model, this study uses statistical comparison methods to quantitatively analyze the performance indicators of the proposed algorithm. As shown in Table 1, by establishing a dual-index evaluation system of mean error and mean square error, under the J-turn simulation condition, the average deviation of longitudinal tire force estimation obtained by the C-SIR algorithm is 1.8566, and the mean square error of the fluctuation amplitude indicator is 0.9604. In contrast, the improved CFF-SIR algorithm exhibits significant advantages, with the average deviation of its estimates reduced to 0.9337 and the mean square error reduced to 0.4377. In addition, when considering noise, the average estimation error and mean square error under the C-SIR algorithm are 2.2607 and 1.8662, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 0.9886 and 0.6987, respectively. When considering interference, the average estimation error and mean square error under the C-SIR algorithm are 2.4039 and 1.4306, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 0.9218 and 0.6331,  
Figure 6: Estimation results of longitudinal tire force under J-turn condition. (**a**) Without noise and interference effects, (**b**) with noise effects, (**c**) with interference effects. respectively. The results indicate that the improved algorithm achieves performance improvements in both model estimation accuracy and result stability, reflecting the application value of the improved CFF-SIR algorithm in identifying estimation models under complex operating conditions. In order to further reflect the longitudinal tire force estimation effect of the proposed model identification method under more complex vehicle operating conditions, simulation tests were conducted under variable speed sinusoidal steering conditions. The steering angle and vehicle speed of the vehicle under variable speed sinusoidal steering conditions are shown in Figure 7. The longitudinal tire force estimation results obtained under different settings are shown in Figure 8.  
As shown in Figure 8, under the variable speed sine-steering condition, the longitudinal tire force of the vehicle changes more dramatically, and there is a scenario of sudden changes in longitudinal tire force during the switching between constant speed and acceleration conditions of the vehicle. In this case, when noise and interference effects are not considered, the longitudinal tire force estimation results under the CFF-SIR algorithm are closer to the measured values. When there is noise influence, there is a certain degree of fluctuation in the longitudinal tire force estimation values under both algorithms, but compared to the C-SIR algorithm, the results under the C-SIR algorithm have more obvious deviations and fluctuations, while the CFF-SIR algorithm is relatively less affected by noise. Similarly, when there is interference, there is a significant estimation amplitude deviation in the longitudinal force under the C-SIR algorithm, while the estimation value under the CFF-SIR algorithm is relatively small. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Condition** & **Method** & **Mean Error** & **Mean Square Error** \\ \hline \multirow{3}{*}{Without noise or interference} & CFF-SIR & 0.9337 & 0.4377 \\  & C-SIR & 1.8566 & 0.9604 \\ \multirow{3}{*}{With noise} & CFF-SIR & 0.9886 & 0.6987 \\  & C-SIR & 2.2607 & 1.8662 \\ \multirow{3}{*}{With interference} & CFF-SIR & 0.9218 & 0.6331 \\  & C-SIR & 2.4039 & 1.4306 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean error and mean square error of estimation results in J-turn maneuver. Figure 7: Vehicle state under sinusoidal steering condition. (**a**) Steering wheel steering angle, (**b**) vehicle speed.