## 2 Problem Description and Formulation (INS Motion Model)
### INS/GNSS Motion Model  
Tracking the position of a vehicle is usually done relative to a reference. The INS's measurements usually provided in the body (sensors) frame would need to be transformed into the navigation frame for tracking purposes [21]. In this study, we adopt the North-East-Down convention in defining the navigation frame. The transformation matrix \(R^{nb}\) from the body frame to navigation frame is as shown in Equation \((1)\)  
\[R^{nb}=\left[\begin{array}{ccc}cos\theta cos\Psi&-cos\theta\ sin\Psi+sin \phi\ sin\theta\ cos\Psi&sin\theta\ sin\theta\ cos\Psi\\ cos\theta\ sin\Psi&cos\phi\ cos\Psi+sin\theta\ sin\theta\ sin\Psi&-sin\phi\ cos \Psi+cos\phi\ sin\theta\ sin\Psi\\ -sin\theta&sin\phi cos\theta&cos\phi cos\theta\end{array}\right]. \tag{1}\]  
where \(\phi\) is the roll, \(\theta\) is the pitch and \(\Psi\) is the yaw. However, as our study is limited to the two-dimensional tracking of vehicles, \(\phi\) and \(\theta\) are thus considered to be zero, thus the rotation matrix \(R^{nb}\) becomes:  
\[R^{nb}=\left[\begin{array}{ccc}cos\Psi&-sin\Psi&0\\ sin\Psi&cos\Psi&0\\ 0&0&1\end{array}\right] \tag{2}\]  
The gyroscope measures the rate of change of attitude (angular velocity) in yaw, roll and pitch with respect to the inertial frame as expressed in the body frame [1; 21]. Giving initial orientation information \(\Psi_{0}\), the attitude rate \(\omega^{b}\) can be integrated to provide continuous orientation \(\Psi_{INS}\) information in the absence of the GNSS signal.  
\[\Psi_{INS}=\Psi_{0}+\int_{t-1}^{t}\omega^{b} \tag{3}\]  
The accelerometer measures the specific force (In the vehicle tracking application, the centrifugal acceleration is considered absorbed in the local gravity sector and the centrifugal acceleration considered negligible due to its small magnitude.) \(f^{b}\) on the sensor in the body frame and is as expressed in Equation (4); where \(g^{n}\) represents the gravity vector, \(R^{bn}\) is the rotation matrix from the navigation frame to the body frame and \(a^{n}\) denotes the linear acceleration of the sensor expressed in the navigation frame.  
\[f^{b}=R^{bn}(a^{n}-g^{n}) \tag{4}\]  
However, the accelerometer measurements at each time \(t\) are usually corrupted by a bias \(\delta^{b}_{INS}\) and noise \(\epsilon^{b}_{a}\), and is thus represented by \(F^{b}_{INS}\), as shown in Equation \((5)\).  
\[F^{b}_{INS}=f^{b}_{INS}+\delta^{b}_{INS}+\epsilon^{b}_{a} \tag{5}\]  
\[\left[\begin{array}{c}a_{INS}^{n},\,\text{\emph{North}}\\ a_{INS}^{n},\,\text{\emph{North}}\\ v_{IN}^{n},\,\text{\emph{North}}\\ v_{IN}^{n},\,\text{\emph{North}}\\ v_{IN}^{n},\,\text{\emph{North}}\\ x_{INS}^{n},\,\text{\emph{North}}\\ x_{INS}^{n},\,\text{\emph{East}}\\ x_{INS}^{n},\,\text{\emph{East}}\\ \end{array}\right]=\left[\begin{array}{c}a_{INS}^{b}\,.\,cos\textbf{\Psi}_{INS} \\ a_{INS}^{l}\,.\,sin\textbf{\Psi}_{INS}\\ v_{INS}^{l}\,.\,cos\textbf{\Psi}_{INS}\\ v_{INS}^{l}\,.\,sin\textbf{\Psi}_{INS}\\ x_{INS}^{l}\,.\,cos\textbf{\Psi}_{INS}\\ x_{INS}^{l}\,.\,cos\textbf{\Psi}_{INS}\\ x_{INS}^{l}\,.\,sin\textbf{\Psi}_{INS}\\ x_{INS}^{l}\,.\,sin\textbf{\Psi}_{INS}\\ x_{INS}^{l}\,.\,sin\textbf{\Psi}_{INS}\\ \end{array}\right] \tag{19}\]  
The vehicle's true displacement \(x_{GNSS}^{b}\) is estimated as the distance between two points on the surface of the earth specified in longitude and latitude. The accuracy of \(x_{GNSS}^{b}\) is limited to the precision and accuracy of the GNSS, which is defined as \(\pm 3\) according to Reference [22]. There is the possibility to improve the accuracy of \(x_{GPS}^{b}\) using approaches such as in Reference [3]. \(x_{GNSS}^{b}\) is determined using the Vincenty's Inverse and applied according to Reference [1] using the Python implementation in Reference [23].