# Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Filter** & **Vehicle Model** & **Tire Model** & **Trait** & **Ref.** \\ \hline \multirow{6}{*}{EKF and its improvements} & 3-DOF dual-track & Longitudinal force observer and Magic formula & Simple estimation structure. Real-time tracking capability. Complex steps for obtaining tire force. & [6] \\  & 2-DOF single-track & \multirow{2}{*}{Linear} & Estimate vehicle mass simultaneously. Model is too simplified to get accurate estimation results. & [8] \\  & 2-DOF single-track & & & \\  & 2-DOF single-track & Simplified Magic formula & Weak universality in varied testing conditions. & [11] \\ \hline \multirow{3}{*}{DEKF} & 3-DOF dual-track & HRSI/ & Estimate other parameters simultaneously. & [7]/ \\  & 3-DOF dual-track & TMeasy and Magic formula & Issues of computational efficiency. & [12] \\ \hline \multirow{6}{*}{UKF and its improvements} & 3-DOF dual-track & Duggoff/ Magic formula/ Piecewise linear & Parameters in tire model rely on a large amount of experimental data. & [5]/ \\  & 2-DOF & Linear and Magic formula & Not affected by parameter uncertainties. Need to train a lot of data. & [13] \\ \hline \multirow{6}{*}{CKF and its improvements} & 3-DOF single-track & Linear & Simple state estimation structure. Small computational burden. Low robustness in high maneuvers. & [15] \\  & 3-DOF dual-track & Magic formula & High estimation accuracy. Poor real-time performance caused by adaptive calculation process. & [16] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of the strategies for vehicle state estimation. set to contain the real states, but did not offer an algorithm for finding the smallest ellipsoid among the ellipsoid clusters. [PERSON] and [PERSON] [18] deduced a minimal-volume ellipsoid method and a minimal-trace ellipsoid method to find the smallest ellipsoid set for optimization. Subsequently, [PERSON] and [PERSON] [19] proposed an extended set-membership filter (ESMF), similar to the EKF, through a linearization approach for nonlinear system. In the ESMF, the higher-order term errors produced by the linearization process are integrated into the system noise by means of interval analysis. The practicability of the ESMF has been verified in some fields [20; 21; 22]. For example, in the power dynamic system, guaranteed estimation was achieved with hard bounds containing real rotor angle and speed [20]. The estimated results of this case are robust because the system noise is always included in the assumed bounds. Additionally, for common onboard sensor, the boundary of its noise distribution can be determined offline through calibration, which motivates the application of the ESMF in the field of vehicle state estimation. In this study, a robust estimation strategy for vehicle motion states is proposed, which consists of an ESMF-based estimator for vehicle motion states and a KF-based scheme for tire force calculation. The remaining parts of this paper are organized as follows. A dual-track vehicle dynamic model and the design of the robust estimation strategy are discussed in detail in Section 2. The steps of the ESMF algorithm are presented in Section 3. To replace the complex semi-empirical tire models commonly used, Section 4 introduces the scheme of tire force calculation with simple structure. Then Section 5 is devoted to validating the effectiveness of the proposed robust estimation strategy. The last section is the conclusion of this paper.