# Credible Navigation Algorithm for GNSS Attack Detection Using Auxiliary Sensor System
## 2 Credible Kalman Filter Model
### DDE/INS System Module  
The auxiliary sensor system needs to independently maintain the output position of the mobile terminal in the navigation coordinate system in order to verify the credibility of the GNSS positioning result. The auxiliary sensor system used in this paper is the DDE/INS auxiliary sensor system. The working principle of the DDE/INS system is shown in Figure 2.  
Given the initial position, the DDE/INS system will calculate the position and heading of the mobile terminal through the DDE odometer. Before deriving the DDE/INS system model, we should derive the DDE system module first. The two wheels of the DDE odometer are independently controlled to realize the movement and steering control of the chassis [28]. By collecting the encoded data of the two wheels in a DDE unit time slot \(\Delta t^{\rm D}\), the relative displacement in the navigation coordinate system can be solved as \((\Delta e^{\rm D},\Delta n^{\rm D},\Delta yaw^{\rm D})\). The moving distance of the left and right wheels in \(\Delta t\) is denoted as \((s^{\rm l}_{k},s^{\rm f}_{k})\). Then, in the navigation coordinate system, the calculation model of \((s^{\rm l}_{k^{\prime}},s^{\rm f}_{k})\) is  
\[ \begin{split} s^{\rm l}_{k}&=o^{\rm l}_{k-1,k}\cdot 2 \pi\cdot W_{\rm r}\cdot L_{\rm l}\\ s^{\rm f}_{k}&=o^{\rm r}_{k-1,k}\cdot 2\pi\cdot W_{\rm r} \cdot L_{\rm r}\end{split}  \tag{2}\]  
Figure 1: Credible Kalman filter model. where \((o^{1}_{k-1,k},o^{\rm r}_{k-1,k})\) is the increment of the left and right wheels in \(\Delta t^{\rm D}\), \((L_{\rm t},L_{\rm r})\) is the maximum count per rotation of the left or right wheel, \(W_{\rm r}\) is the radius of the wheel, and \(L_{\rm w}\) is the distance between the two wheels. Therefore, the navigation coordinate \((e^{\rm D}_{k},n^{\rm D}_{k})\) and yaw angle \(yaw^{\rm D}_{k}\) at time \(k\) in the DDE system model is  
\[ \begin{split} e^{\rm D}_{k}&=e^{\rm D}_{k-1}+\frac {1}{2}\cdot(s^{1}_{k}+s^{\rm r}_{k})\cdot\sin(yaw^{\rm D}_{k-1}+\frac{(s^{1}_{ k}-s^{\rm r}_{k})}{L_{\rm w}})\\ n^{\rm D}_{k}&=n^{\rm D}_{k-1}+\frac{1}{2}\cdot(s^{1 }_{k}+s^{\rm r}_{k})\cdot\cos(yaw^{\rm D}_{k-1}+\frac{(s^{1}_{k}-s^{\rm r}_{k} )}{L_{\rm w}})\\ yaw^{\rm D}_{k}&=yaw^{\rm D}_{k-1}+\frac{(s^{1}_{k}-s^ {\rm r}_{k})}{L_{\rm w}}\end{split}  \tag{3}\]  
However, the DDE system model is greatly affected if the road surface is not flat, which causes a large error in the mileage calculation. In order to improve the navigation performance of the auxiliary sensor system and the credibility of the ACNA algorithm, the DDE system model is optimized by tightly coupling the yaw angle \(yaw^{\rm I}\) and the height \(\Delta u^{\rm I}\) calculated by INS, as shown in the following formula:  
\[ \begin{split} e^{\rm D}_{k}&=e^{\rm D}_{k-1}+\frac {1}{2}\cdot(s^{1}_{k}+s^{\rm r}_{k})\cdot\sin(yaw^{\rm I}_{k})\\ n^{\rm D}_{k}&=n^{\rm D}_{k-1}+\frac{1}{2}\cdot(s^{1 }_{k}+s^{\rm r}_{k})\cdot\cos(yaw^{\rm I}_{k})\\ u^{\rm D}_{k}&=u^{\rm D}_{k-1}+\Delta u^{\rm I}_{k} \end{split}  \tag{4}\]  
where \((e^{\rm D}_{k},n^{\rm D}_{k},u^{\rm D}_{k})\), \(yaw^{\rm I}_{k}\) and \(\Delta u^{\rm I}_{k}\) are the navigation coordinate, the yaw angle and the height at time \(k\) in the DDE/INS system. In the navigation coordinate system, the velocity components \((v_{x},v_{y},v_{z})\) of the mobile terminal in the east and north directions at time \(k\) is  
\[ \begin{split} v_{x,k}&=\frac{1}{2\Delta t^{\rm D}} \cdot(s^{1}_{k}+s^{\rm r}_{k})\cdot\sin(yaw^{\rm I}_{k})\\ v_{y,k}&=\frac{1}{2\Delta t^{\rm D}}\cdot(s^{1}_{k}+s ^{\rm r}_{k})\cdot\cos(yaw^{\rm I}_{k})\\ v_{z,k}&=\frac{1}{\Delta t^{\rm D}}\cdot\Delta u^{ \rm I}_{k}\end{split}  \tag{5}\]  
Figure 2: Working principle of DDE/INS system. Therefore, the state parameter \(\mathbf{x}^{\text{DT}}\) of the DDE/INS system model is  
\[\mathbf{x}^{\text{DI}}=\left[ \begin{array}{c}e_{k}^{\text{DI}}\\ n_{k}^{\text{DI}}\\ u_{k}^{\text{DI}}\end{array} \right]=\left[