# Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter
## 1 Introduction  
In the vehicle moving process, the accurate acquisition of motion state information is one of the key factors for the effective control of a vehicle active safety system [1; 2; 3]. However, the direct measurement of the vehicle motion states, including the longitudinal velocity, lateral velocity and sideslip angle, requires the help of some expensive and specialized sensors. For example, the measurement of vehicle sideslip angle can only be realized by means of a non-contact optical sensor (such as the S-Motion produced by Corrsys-Datron) [4]. Because of high cost and special installation requirements, it is difficult to apply this kind of sensors in production vehicles. On the other hand, vehicle motion states can also be determined via the soft-sensing theory which combines the linear/nonlinear filtering algorithms with the information obtained by ordinary vehicle sensors. In recent years, utilizing the soft-sensing theory including the Kalman filter (KF) to acquire vehicle motion states has become a research hotspot due to low cost and easy migratability. Many scholars have investigated the strategies for vehicle state estimation based on the KF and its improved algorithms [5; 6; 7; 8; 9; 10]. [PERSON] et al. [11] combined the extended Kalman filter (EKF) with a 2-degree-of-freedom (2-DOF) single-track vehicle model and the simplified Magic formula tire model to obtain the sideslip angle. Considering the influence of the vehicle parameters on estimation results,Wenzel et al. [12] designed a dual extended Kalman filter (DEKF) structure, where a parallel loop was constructed between the vehicle parameter and state estimators to acquire high-accuracy and simultaneous determination. Moreover, to alleviate the adverse impact caused by the truncation error existed in the EKF, a series of sigma-point Kalman filters were utilized for the acquirement of vehicle information, such as the unscented Kalman filter (UKF) [13; 14] and the cubature Kalman filter (CKF) [15; 16]. These filters draw nonlinear transformations into the framework of the KF and carry out the transformations on the sigma sampling points. On the premise that the variance and mean of these points are the same as those of the states to be estimated, the probability distributions of nonlinear functions can be approximately approached. An overview of the typical strategies for vehicle state estimation is summarized in Table 1, where the UKF is most widely used. This table also shows that the robust performance is rarely involved in the traditional studies about estimation strategy. It should be noted that in the application of the series of filters mentioned above, system noise source is usually considered as Gaussian white noise, where the fixed mean and variance are used to describe the statistical characteristics of system noise. However, for real system, the prior knowledge of the process and measurement noises cannot be acquired accurately. When the filters mentioned above are employed, their actual performance is inevitably discounted. As the consideration that the distribution of system noise generally satisfies the condition of being unknown but bounded, the theory of set-membership filter (SMF) was proposed by [PERSON] [17] in 1968. Under this assumption for noise distribution, the estimated result obtained from the SMF is a set containing the real states, rather than one point for the stochastic Kalman algorithms, such as the EKF, UKF, et al. Therefore, it can provide 100% confidence for the estimated states. [PERSON] [17] used an ellipsoid