## 2 Methods
### Measurement Model for Odometer Observation with Vehicular-Motion Constraint  
The odometer-derived velocity indicates that unless a vehicle is off the ground or slides on the ground, the velocity of the vehicle in the plane perpendicular to the forward direction is nearly zero [44; 45]. Because an odometer provides the forward speed of a vehicle, the measurement is expressed as follows:  
\[\mathbf{v}^{v}=\left[ \begin{array}{ccccc}v&0&0\\ \end{array} \right]^{T} \tag{15}\]  
where \(\mathbf{v}^{v}\) denotes the velocity measurement in the v-frame, and the lateral and vertical speeds are presumed to be zero. Because (15) need not be obtained in the e-frame, it cannot be used directly in the state estimation. In this study, based on the fact that \(v\) is the observation as the resultant speed of \(x\), \(y\), and \(z\) components for the velocity state in (1), the corresponding measurement model can be formulated as,  
\[\mathbf{H}_{\alpha do}=\left[ \begin{array}{cccccccc}0_{1\times 3}&(v_{3d})^{- \frac{1}{2}}v_{x}^{e}&(v_{3d})^{-\frac{1}{2}}v_{y}^{e}&(v_{3d})^{-\frac{1}{2}}v_{ z}^{e}&0_{1\times 3}&0_{1\times 1}&0_{1\times 1}&0_{1\times 1}\end{array} \right]_{1 \times 12} \tag{16}\]  
\[\delta\mathbf{z}_{\alpha do}=\left[v-v_{3d}\right]_{1\times 1}=\mathbf{H}_{ \alpha do}\delta\mathbf{\hat{x}}+\eta_{\alpha do} \tag{17}\]  
\[\mathbf{R}_{\alpha do}=\left[\sigma_{\alpha do}^{2}\right]_{1\times 1} \tag{18}\]  
with  
\[v_{3d}=\left(v_{x}^{e}\right)^{2}+\left(v_{y}^{e}\right)^{2}+\left(v_{z}^{e} \right)^{2}\]  
where \(\mathbf{H}_{\alpha do}\) is the design matrix for the odometer; \(\eta_{\alpha do}\) comprises the measurement noise and abnormal sensing such as sliding on the ground; \(v_{x}^{e}\), \(v_{y}^{e}\), \(v_{z}^{e}\) are the initial GNSS velocity at the current epoch in the three directions; \(v_{3d}\) is the initial resultant speed. The determination of \(\sigma_{\alpha do}^{2}\) can be given by a pre-calibrated process (e.g., comparison between the resultant speed of the reference system and the OBD II odometer). It is worth mentioning that this model only constrains the combined velocity for an estimation that is non-directional, and could face the local minimum problem during state estimation. Due to the lateral velocity of a land vehicle is nearly zero under normal conditions, it can act as an additional aiding source to enhance such state estimation. The relationship between the vehicle frame and the local-level frame (l-frame) in the horizontal direction is depicted in Figure 2, and the corresponding equation given by,  
\[\mathbf{H}_{lateral}=\left[ \begin{array}{cccccccc}0_{1\times 3}&A(t_{k-1})&B(t_ {k-1})&C(t_{k-1})&0_{1\times 3}&0_{1\times 1}&0_{1\times 1}&0_{1\times 1} \end{array} \right]_{1\times 12} \tag{19}\]  
\[\delta\mathbf{z}_{lateral}=\left[0-v_{lateral}\right]_{1\times 1}=\mathbf{H}_{lateral }\delta\mathbf{x}+\eta_{lateral} \tag{20}\]  
\[\mathbf{R}_{lateral}=\left[\sigma_{lateral}^{2}\right]_{1\times 1} \tag{21}\]  
with  
\[v_{lateral}=A(t_{k-1})\times v_{x}^{e}(t_{k})+B(t_{k-1})\times v_{y}^{e}(t_{k} )+C(t_{k-1})\times v_{z}^{e}(t_{k}),\]  
\[A=-cos(-\theta)\times sin(\Lambda)+sin(-\theta)\times sin(\varphi)\times cos( \Lambda),\]  
\[B=cos(-\theta)\times cos(\Lambda)+sin(-\theta)\times sin(\varphi)\times sin( \Lambda),\ C=-sin(-\theta)\times\cos(\varphi)\]  
where \(v_{lateral}\) is the lateral velocity derived from the velocity state estimation; \(\eta_{lateral}\) is the combination of velocity estimation error and heading error; \(\mathbf{H}_{lateral}\) is the design matrix of the lateral constraint based on the transformation of the e-frame, the l-frame, and vehicle frame; \(\theta\), \(\varphi\), and \(\Lambda\) denote the vehicular heading, latitude, and longitude angles at \(t_{k-1}\) epoch. The determination of \(\sigma_{lateral}^{2}\) is based on the uncertainty of the estimated heading, which is an empirical coefficient in this study.