# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 2 The Navigation Filter  
The navigation filter implements a tight coupling between IMU, the Iridium Doppler shift and the magnetometer (Figure 1). Here, we take advantage of the complementarity between the inertial navigation that has a good performance in the short term but diverges with time, and the Doppler measurements from Iridium satellites that are very noisy in the short term but should be relatively stable at the long term. This algorithm is different from the usual GNSS/IMU tight coupling in three aspects: (1) it uses only LEO SV Doppler shift measurements, not pseudoranges; (2) the measurements from satellites in view over the same period of time come at different instances as the corresponding bursts do, and thus the innovation at each time update contains usually a single Doppler shift measurement, in addition to the Earth's magnetic field measurements; and (3) the use of magnetometers for a direct attitude observation, especially when dealing with a spinning vehicle.  
We define the local frame (\(O\), \(x\), \(y\), \(z\)) such that the \(x\)-\(y\) plane is tangent to the Earth ellipsoid at point \(O\), with (\(Ox\)) towards a given direction (the geographic north for instance), (\(Oz\)) is the vertical downward direction and (\(Oy\)) completes the direct coordinate system. The body frame (\(O_{b}\), \(x_{b}\), \(y_{b}\), \(z_{b}\)) is centred on the vehicle's center of mass, with (\(O_{b}x_{b}\)) being the vehicle longitudinal axis oriented towards the movement direction, and the \(y_{b}\)-\(z_{b}\) plane being perpendicular to (\(O_{b}x_{b}\)). Both frames are visible in Figure 2. The attitude angles (roll: \(\phi\), pitch: \(\theta\), yaw: \(\psi\)) are defined as the angles that allow the transformation between the body frame and the local frame. The three-axis accelerometer, the three-axis gyroscope and the three-axis magnetometer are supposed to be aligned with the body frame axes.  
The L1 band (1.626 GHz) antenna is supposed to be located at \(O_{b}\). These assumptions are intended to simplify the filter model.  
The navigation algorithm implements an extended Kalman filter (EKF). The state vector is formed by the position error \(\delta\mathbf{r}=\hat{\mathbf{r}}-\mathbf{r}\) and the velocity error \(\delta\mathbf{v}=\hat{\mathbf{v}}-\mathbf{v}\), both in the navigation frame, the attitude error \(\delta\mathbf{e}=\hat{\mathbf{e}}-\mathbf{e}\), the accelerometer biases error \(\delta\mathbf{b_{a}}=\hat{\mathbf{b}}_{a}-\mathbf{b_{a}}\) and the gyroscope biases error \(\delta\mathbf{b_{g}}=\hat{\mathbf{b}}_{g}-\mathbf{b_{g}}\), both in the body frame, the receiver clock bias error \(\delta(\delta t)=\hat{\delta}t-\delta t\) and the receiver clock drift error \(\delta\Big{(}\hat{\delta t}\Big{)}=\widehat{\delta}t-\hat{\delta}t\) converted in m and in m/s, respectively:  
\[\delta\mathbf{x}=\Big{[}\delta\mathbf{r},\delta\mathbf{v},\delta\mathbf{e},\delta\mathbf{b_{a}}, \delta\mathbf{b_{g}},\delta(c\delta t),\delta\Big{(}c\hat{\delta t}\Big{)}\Big{]} ^{T} \tag{1}\]  
\(c\) is the signal propagation speed.  
Figure 1: The principle of the INS/SOP tight coupling.  
Figure 2: The local frame (\(O,x,y,z\)) and the body frame (\(O_{b}x_{b}y_{b},z_{b}\) ).