## 2 The Proposed VA-RAIM
### Vision Model with Calibration
#### 2.2.1 Vision Model  
Inspired by the principle of GPS position model [18], the landmarks are regarded as pseudo-satellites to obtain the similar navigation measurements and a vision pseudorange is defined as the estimation of the distance between the user and a landmark. However, different from GPS pseudoranges, the vision pseudoranges are calculated but directly measured. As shown in Figure 3, at discrete time \(k\), denote the vision pseudorange vector \(\mathbf{m}_{(k)}=[m_{1},\,m_{2},\,\ldots\,,\,^{m_{N_{V(k)}}}]^{T}\in\,\textbf{R}^{ N_{V(k)}}\), which is composed of \(N_{V(k)}\) vision pseudoranges. By applying the cosine rule, the vision pseudorange vector is the solution of the over-determined equation set composed by \(N_{C(k)}\)=\(N_{V(k)}\)(\(N_{V(k)}-1\))/2 equations as:  
\[m_{i(k)}^{2}+m_{j(k)}^{2}-2m_{i(k)}m_{j(k)}c_{ij(k)}-d_{ij}^{2}=0,\underset{i( k)\
eq j(k)}{\forall}i(k),j(k)\in 1,2,\,...\,,N_{V(k)} \tag{1}\]  
where \(m_{i(k)}\in\mathbf{R}^{+}\) is the vision pseudorange of the \(i^{th}\) landmark \(\mathbf{p}_{i}\in\mathbf{R}^{3}\) at time \(k\). \(c_{ij(k)}\) = cos\(\theta_{ij(k)}\in[-1,\,1]\), in which \(\theta_{ij(k)}\in[0,\,\pi]\) is the angle between the two lines-of-sight of landmarks \(\mathbf{p}_{i}\) and \(\mathbf{p}_{j}\in\mathbf{R}^{3}\). As for the other terms in Equation (1), \(d_{ij}=\|\mathbf{p}_{i}-\mathbf{p}_{j}\|\in\mathbf{R}^{+}\) is the nominal distance between landmarks \(\mathbf{p}_{i}\) and \(\mathbf{p}_{j}\), which is time-invariant. Figure 2: The framework of VA-RAIM. (A) Error Analysis  
As shown in Equation (1), the vision pseudorange vector is decided by \(d\stackrel{{ 2}}{{{}_{\mathit{ij}}}}\) and \(c_{ij(k)}\). In this section, we take into account \(N_{V(k)}=3\) to simplify the following analysis, while our approach can also be easily extended to any number of landmarks. The relation between the vision pseudorange error \(\delta\mathbf{m}_{(k)}=[m_{1(k)}\), \(m_{2(k)}\), \(m_{3(k)}]^{T}\) and the error of these parameters \(\delta\mathbf{d}=[\delta d\stackrel{{ 2}}{{{}_{\mathit{12}}}}\), \(\delta d\stackrel{{ 2}}{{{}_{\mathit{23}}}}\), \(\delta d\stackrel{{ 2}}{{{}_{\mathit{31}}}}]^{T}\) and \(\delta\mathbf{c}_{(k)}=[\delta c_{12(k)}\), \(\delta c_{23(k)}\), \(\delta c_{31(k)}]^{T}\) can be obtained according to the differential parameters of Equation (1):  
\[\mathbf{A}_{(k)}\delta\mathbf{m}_{(k)}=\delta\mathbf{c}_{(k)}+\mathbf{B}_{(k)} \delta\mathbf{d} \tag{2}\]  
where \(\mathbf{A}_{(k)}= \begin{bmatrix}\dfrac{1}{m_{2(k)}}-\dfrac{c_{12(k)}}{m_{1(k) }}&\dfrac{1}{m_{1(k)}}-\dfrac{c_{12(k)}}{m_{2(k)}}&0\\ 0&\dfrac{1}{m_{3(k)}}-\dfrac{c_{23(k)}}{m_{2(k)}}&\dfrac{1}{m_{2(k)}}-\dfrac{c _{23(k)}}{m_{3}}\\ \dfrac{1}{m_{3(k)}}-\dfrac{c_{3(k)}}{m_{1(k)}}&0&\dfrac{1}{m_{1(k)}}-\dfrac{c _{31(k)}}{m_{3(k)}}\end{bmatrix} \) and \(\mathbf{B}_{(k)}=\dfrac{1}{2m_{(k)}m_{2(k)}m_{3(k)}} \begin{bmatrix}m_{3(k)} &0&0\\ 0&m_{(k)}&0\\ 0&0&m_{2(k)}\end{bmatrix} \). Then we can have:  
\[\delta\mathbf{m}_{(k)}=\mathbf{A}_{(k)}^{-1}\delta\mathbf{c}_{(k)}+\mathbf{A}_{( k)}^{-1}\mathbf{B}_{(k)}\delta\mathbf{d} \tag{3}\]  
In Equation (3), since \(\delta\mathbf{d}\) is fixed with a set of given lengths, it generates the time-invariant error of the vision system. \(\delta\mathbf{c}_{(k)}\) is decided by the detection and matching results at time \(k\), which generates the random time-variant error. The properties of \(\delta\mathbf{d}\) and \(\delta\mathbf{c}_{(k)}\) are analyzed as follows. If the distance of each pair of landmarks is accurately obtained, we have \(\delta\mathbf{d}=\mathbf{0}\) and the second part on the right of Equation (3) disappears. However, it is very challenging to measure the distance directly. In practice, we can measure the position of a landmark up to centimeter-level accuracy with the utilization of real-time kinematic (RTK), and then calculate the distance.