## 2 The Proposed VA-RAIM
### Vision Model with Calibration
#### 2.2.1 Vision Model
Specifically, the position difference is calculated as \(d\stackrel{{ 2}}{{{}_{\mathit{ij}}}}=||\mathbf{p}_{i}-\mathbf{p}_{j}||^{2}\), and the error of the position difference is obtained as \(\delta d\stackrel{{ 2}}{{{}_{\mathit{ij}}}}=2(\mathbf{p}_{i}-\mathbf{p}_{j})^{T}( \delta\mathbf{p}_{i}-\delta\mathbf{p}_{j})\). Suppose the landmark position errors in each degree of freedom obey \(N(0,\sigma_{s}^{2})\) and independent from each other, \(\delta d_{\mathit{ij}}^{2}\) obey Gaussian distribution as with zero mean and variance \(\delta d_{\mathit{ij}}^{2}\sigma_{s}^{2}\), which is proportional to the \(d_{\mathit{ij}}^{2}\). Figure 3: The model of vision pseudoranges. With the transformation projective from object to image space [19], the parameter \(c_{ij(k)}\) is calculated by  
\[c_{ij(k)}=\left\langle\frac{\textbf{p}_{i(k)}^{c}}{\left|\textbf{p}_{i(k)}^{c} \right|}\frac{\textbf{p}_{j(k)}^{c}}{\left|\textbf{p}_{j(k)}^{c}\right|}\right\rangle \tag{4}\]  
where \(<,>\) is the vector inner product, \(\textbf{p}_{i(k)}^{c}\in\textbf{R}^{2}\) and \(\textbf{p}_{j(k)}^{c}\in\textbf{R}^{2}\) is the \(i^{th}\) and the \(j^{th}\) landmarks in the camera frame, as shown in Figure 3. The landmarks in the camera frame are obtained through the detection and matching algorithm, and the detection error \(\delta\textbf{p}_{i(k)}^{c}\) is assumed to follow the independent normal distribution with zero mean and covariance matrix \(\Sigma_{\textbf{p}}\in\textbf{R}^{2\times 2}\). Linearizing Equation (4) yields:  
\[\delta c_{ij(k)}=\left\langle\delta\frac{\textbf{p}_{i(k)}^{c}}{\left|\textbf{ p}_{i(k)}^{c}\right|}\frac{\textbf{p}_{j(k)}^{c}}{\left|\textbf{p}_{j(k)}^{c} \right|}\right\rangle+\left\langle\frac{\textbf{p}_{i(k)}^{c}}{\left|\textbf{ p}_{i(k)}^{c}\right|}\cdot\delta\frac{\textbf{p}_{j(k)}^{c}}{\left|\textbf{p}_{j(k)}^{c} \right|}\right\rangle \tag{5}\]  
As proofed in Appendix, \(\delta c_{ij(k)}\) can be obtained as a linear combination of \(\delta\textbf{p}_{i(k)}^{c}\) and \(\delta\textbf{p}_{j(k)}^{c}\), _i.e._,  
\[\delta c_{ij(k)}=\textbf{\mu}_{j(k)}^{T}\delta\textbf{p}_{i(k)}^{c}+\textbf{ \mu}_{j(k)}^{T}\delta\textbf{p}_{j(k)}^{c} \tag{6}\]  
which follows a normal distribution with zero mean and variance \(\textbf{\mu}_{j(k)}^{T}\Sigma_{\textbf{p}}\textbf{\mu}_{ij(k)}+\textbf{\mu}_{ j(k)}^{T}\Sigma_{\textbf{p}}\textbf{\mu}_{ji(k)}\). Thus, we can obtain that \(\delta\textbf{c}_{(k)}\) follows a normal distribution with zero mean and covariance matrix \(\Sigma_{\textbf{c}(k)}\in\textbf{R}^{N_{\text{c}(k)},N_{\text{c}(k)}}\). (B) Fault Analysis  
For the RAIM algorithm, a GPS fault is defined as a pseudorange bias deviating from its nominal behavior [2]. The probabilities of simultaneous satellite faults with different number of visible satellite are plotted in Figure 4. Since the scenario of this paper focuses on the approach and landing phase that is a low GPS visibility (often less than 8), the probability of multi-fault simultaneity is less than the integrity requirement \(10^{-7}\) and can be ignored in applications. Thus, the threat of GPS is a fault in at most one visible satellite in this paper. However, if the visible satellites are sufficient that multi-fault cannot be ignored, we can detect the faults by using standalone satellite observations [9]. Figure 4: Probabilities of simultaneous satellite faults. When used for aiding navigation, vision system may also suffer from faults. With the introduction of vision pseudorange, the fault of vision system ultimately causes a vision pseudorange bias similar to GPS pseudorange bias. As shown in Equation (3), the pseudorange fault is decided by the fault of \(\delta\mathbf{d}\) and \(\delta\mathbf{c}_{(k)}\). In real applications, there are various types of threats that might cause a fault on \(\delta\mathbf{d}\) or \(\delta\mathbf{c}_{(k)}\). In this paper, two typical examples are considered as follows, _i.e._, fault in the landmark location and fault in the feature detection result during image processing.