## 2 Methods
### State-Augmented EKF Design  
In this study, the position and velocity error states of GNSS positioning are estimated using EKF. Conventionally, the EKF design presents only the system error at a particular time. Whereas, a delta phase or delta position represents the integrated velocity over time [5]. To sufficiently use such delta measurement for estimating the change in the receiver position, the conventional system model should be modified and extended to include the estimated position and receiver clock bias from the last epoch as follows [5; 37]:  
\[\delta\mathbf{x}=\left[ \begin{array}{cccc}\delta\mathbf{r}_{u}^{e}(t_{k})& \delta\mathbf{v}_{u}^{e}(t_{k})&\delta\mathbf{r}_{u}^{e}(t_{k-1})&\delta b_{u} (t_{k})&\delta b_{d}(t_{k})&\delta b_{u}(t_{k-1})\end{array} \right]_{1\times 1 2}^{T} \tag{1}\]  
Figure 1: GNSS/Odometer integration scheme with WLS-based robust regression. \[\mathbf{\Phi}_{p_{1}\overline{v}p_{0}t}=\left[ \begin{array}{ccc} \mathbf{Q}_{p_{1}\overline{v}}&0_{6\times 3}&0_{6\times 3}\\ \mathbf{I}&0_{3\times 3}\end{array} \right]_{3\times 6} \left.0_{3\times 3}&0_{3\times 3}\\ 0_{3\times 6}&0_{3\times 3}&\left[ \begin{array}{cc}\mathbf{Q}_{t}&0 \\ 1&0\end{array} \right]_{1\times 2}&0\end{array}\right]_{3\times 3}\right]_{12\times 12} \tag{2}\] \[\mathbf{Q}_{p_{1}\overline{v}p_{0}t}=\left[ \begin{array}{ccc} \mathbf{Q}_{p_{1}\overline{v}}&0_{6\times 3}&0_{6\times 3}\\ 0_{3\times 6}&0_{3\times 3}&0_{3\times 3}\\ 0_{3\times 6}&0_{3\times 3}&\left[\begin{array}{cc}\mathbf{Q}_{t}&0_{1 \times 1}\\ 0_{1\times 2}&0_{1\times 1}\end{array}\right]_{3\times 3}\end{array} \right]_{12 \times 12} \tag{3}\]  
with  
\[\mathbf{\Phi}_{p_{1}\overline{v}}=\left[ \begin{array}{cccccc}1&0&0&\Delta t &0&0\\ 0&1&0&0&\Delta t&0\\ 0&0&1&0&0&\Delta t\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]_{6\times 6}\,\ \mathbf{I}=\left[ \begin{array}{ ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right]_{3\times 3}\]  
\[\mathbf{Q}_{p_{1}\overline{v}}=\left[ \begin{array}{cccccc}q_{p}\Delta t+ \frac{1}{3}q_{v}\Delta t^{3}&0&0&\frac{1}{2}q_{v}\Delta t^{2}&0&0\\ 0&q_{p}\Delta t+\frac{1}{3}q_{v}\Delta t^{3}&0&0&\frac{1}{2}q_{v}\Delta t^{2} &0\\ 0&0&q_{p}\Delta t+\frac{1}{3}q_{v}\Delta t^{3}&0&0&\frac{1}{2}q_{v}\Delta t^{2} \\ \frac{1}{2}q_{v}\Delta t^{2}&0&0&q_{v}\Delta t&0\\ 0&0&\frac{1}{2}q_{v}\Delta t^{2}&0&0&q_{v}\Delta t&0\\ \end{array} \right]_{6\times 6}\]  
where \(\delta\mathbf{\hat{x}}\) is the error state vector including the position errors \(\delta\mathbf{r}_{u}^{e}\), velocity error \(\delta\mathbf{v}_{u}^{e}\), receiver clock bias \(\delta b_{u}\), and receiver clock drift \(\delta b_{d}\); \(t_{k}\) and \(t_{k-1}\) denote the corresponding epochs of the error states; each of \(\delta\mathbf{r}_{u}^{e}\) and \(\delta\mathbf{v}_{u}^{e}\) is a \(3\times 1\) vector, where the superscript \(e\) denotes a vector in the e-frame (earth-centered-earth-fixed coordinate system); \(\mathbf{\Phi}_{p_{1}\overline{v}}\) and \(\mathbf{Q}_{p_{1}\overline{v}}\) denote the transition matrix (\(t_{k}\) to \(t_{k+1}\) epoch) and the corresponding process noise matrix under assumed constant velocity conditions, for the current time; \(q_{p}\) and \(q_{v}\) are the spectral densities of the above process noise sequences; \(\mathbf{\Phi}_{t}\) and \(\mathbf{Q}_{t}\) denote the transition matrix and process noise matrix for the receiver clock state; \(\Delta t\) is the time interval between adjacent epochs; \(\mathbf{\Phi}_{p_{1}\overline{v}p_{0}t}\) and \(\mathbf{Q}_{p_{1}\overline{v}p_{0}t}\) are the augmented models from \(\mathbf{\Phi}_{p_{1}\overline{v}}\) and \(\mathbf{Q}_{p_{1}\overline{v}}\) to incorporate with the delta-measurement update; \(\mathbf{I}\) is the identity matrix, corresponding to each component of the current position error state. The subscript \(p_{1}\), \(v\), and \(p_{0}\) denotes that the matrix contains the components corresponding to the current position, velocity, and previous position error state.