# Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter
## 2 Robust Estimation Strategy for Vehicle Motion States
### Design of Robust Estimation Strategy  
The proposed strategy is achieved based on the combination of the established vehicle dynamics and estimation algorithms. We choose the robust ESMF algorithm based on the minimal-trace ellipsoid method, which makes it adaptively select the most reasonable parameters to optimize the estimation sets. In addition, considering the problem of numerical stability, the process of interval analysis for bounding truncation error is replaced by appropriately increasing the boundary values of process and measurement noises.  
On the basis of Equation (1), the state space of the nonlinear vehicle state estimation can be expressed as:  
\[\dot{\mathbf{x}} =m(\mathbf{x},\mathbf{u})+\mathbf{w} \tag{3}\] \[\mathbf{z} =n(\mathbf{x})+\mathbf{v}\]  
where \(m()\) and \(n()\) are the process and measurement functions, \(\mathbf{x}\) is the state vector composed of the longitudinal velocity, lateral velocity, yaw rate, longitudinal acceleration, lateral acceleration and yaw moment, i.e., \(\mathbf{x}=[v_{x}\;\;v_{y}\;\;r\;\;a_{x}\;\;a_{y}\;\;M_{z}]^{\mathrm{T}}\), \(\mathbf{z}\) is the measurement vector consisting of the yaw rate, longitudinal acceleration and lateral acceleration, that is, \(\mathbf{z}=[r\;\;a_{x}\;\;a_{y}]^{\mathrm{T}}\), and \(\mathbf{w}\) and \(\mathbf{v}\) are the process and measurement noises, respectively, \(\mathbf{u}\) is the input of the state estimation system composed by the steering wheel angle and the longitudinal and lateral force of each tire.  
The signals in the measurement vector can be directly obtained through the standard sensors equipped on vehicle. The components of the state vector are the vehicle states that need to be estimated, where \(a_{x}\), \(a_{y}\) and \(r\) are extended as estimated states to make full use of the measurement information.  
To implement a linear/nonlinear estimation algorithm, the continuous system expressed in Equation (3) must be discretized according to the sampling time \(\triangle T\), and the discrete system can be noted as:  
\[\begin{bmatrix}v_{x}\\ v_{y}\\ r\\ a_{x}\\ a_{y}\\ M_{z}\end{bmatrix}_{k+1} =\begin{bmatrix}v_{y}r+a_{x}\\ -v_{x}r+a_{y}\\ \frac{M_{z}}{l_{z}}\\ 0\\ 0\\ 0\end{bmatrix}_{k}\cdot\Delta T+\begin{bmatrix}v_{x}\\ v_{y}\\ r\\ a_{x}\\ a_{y}\\ M_{z}\end{bmatrix}_{k}+\mathbf{w}_{k} \tag{4a}\] \[\mathbf{z}_{k+1} =\begin{bmatrix}0&0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&1&0\end{bmatrix}\cdot\mathbf{x}_{k+1}+\mathbf{v}_{k+1} \tag{4b}\]where the subscripts \(k\) and \(k+1\) denote the \(k\)th and \((k+1)\)th sampling instants.  
Apply the steps of the ESMF algorithm in Section 3 to the state space Equation (4), and substitute the result of the tire force calculation in Section 4 into Equation (4). Subsequently, the robust estimation strategy for vehicle motion states can be constructed, which can be found in Figure 2.  
The vehicle sideslip angle \(\beta\) is approximately calculated through the estimated longitudinal velocity and lateral velocity:  
\[\hat{\beta}\approx\arctan\left(\frac{\partial_{y}}{\partial_{x}}\right) \tag{5}\]