## 2 Methods
### Robust Regression with Odometer Aiding  
WLS estimation is more susceptible to outliers compared with KF-based estimation. Robust regression, based on WLS residuals analysis, can adjust the weight measurements probably affected by multipath/NLOS reception whose corresponding residuals are large [18; 19]. Such a process is a generalization of the maximum likelihood estimator (M-estimator), and is applied in this study. Considering that the state estimation for WLS is an iterative process (until the convergence is reached), the weightings depending on the residuals are adaptively reweighted in each iteration. The weighting function of this robust regression in each iteration can be expressed as follows [18; 19]:  
\[\left(w^{i}\right)_{j+1}=\left\{ \begin{array}{cc}\left(w^{i}\right)_{j},& \frac{\left|\left(r^{i}\right)_{j}\right|}{\delta_{0}}\leq\alpha\\ \left(w^{i}\right)_{j}\left|\frac{\left(r^{i}\right)_{j}}{\delta_{0}}\right|& \frac{\left|\left(r^{i}\right)_{j}\right|}{\delta_{0}}>\alpha\end{array} \right. \tag{27}\]  
where \(\left(w^{i}\right)_{j}\) denotes the diagonal element of the weighting matrix corresponding to the \(i^{\text{th}}\) satellite measurement at the \(j^{\text{th}}\) iteration; \(\delta_{0}\) is the a posteriori scale factor given by median absolute deviation due to the spread of the residuals; \(\alpha\) is the tuning constant and is set to 1.345 based on [46]; \(\alpha/\left|\left(r^{i}\right)_{j}/\delta_{0}\right|\) is the weighting scale. Using (27), the residuals are considered as an approximation of the measurement errors, and the results until WLS convergence are implemented to adaptively adjust the measurement covariance matrix used in EKF estimation [19]. A potential limitation of this method emerges if abnormal measurements are in majority, as mentioned previously. To improve the performance of robust regression under the emergence of multiple faulty measurements, this study proposes the use of accumulated distance based on the odometer information as a positional-displacement constraint to the WLS filter during the robust regression operation. When the vehicle does not change its heading orientation significantly, the moving distance from the wheel is similar to the difference between the corresponding position states of the adjacent epochs. Considering the error accumulation, due to the non-linear acceleration and the driving path challenges this assumption, such distance information is used only as an aided constraint for obtaining a weighting scale in (27), rather than as a direct measurement update in EKF estimation. The wheel information from the OBD II interface is transformed into instantaneous velocity; therefore, the accumulated distance can be calculated as follows,  
\[\Delta d_{3d}=\frac{1}{\xi}\sum\
olimits_{ \begin{subarray}{c}\kappa=1 \end{subarray} }^{\xi}v_{\kappa} \tag{28}\]  
where \(\Delta d_{3d}\), \(\xi\), and \(v_{\kappa}\) indicate the averaged displacement between consecutive GNSS epochs, odometer rating between GNSS consecutive epochs, and instantaneous velocity measurement through the OBD II interface. Similar to (16)-(18), this measurement model of position constraint for the WLS filter can be formulated as,  
\[\delta\mathbf{\hat{x}}_{WLS}=\left[ \begin{array}{cccccc}\delta\mathbf{r}_{u} ^{e}(t_{k})&\delta b_{u}(t_{k})&\delta\mathbf{r}_{u}^{e}(t_{k-1})&b(t_{k-1}) \end{array} \right]_{1\times 8}^{T} \tag{29}\]  
\[\mathbf{H}_{WLS}=\left[ \begin{array}{cccccc}r_{3d}{}^{-\frac{1}{2}}\Delta r _{x}^{e}&r_{3d}{}^{-\frac{1}{2}}\Delta r_{y}^{e}&r_{3d}{}^{-\frac{1}{2}} \Delta r_{z}^{e}&0&-r_{3d}{}^{-\frac{1}{2}}\Delta r_{x}^{e}&-r_{3d}{}^{-\frac{1 }{2}}\Delta r_{y}^{e}&r_{3d}{}^{-\frac{1}{2}}\Delta r_{z}^{e}&0\end{array}