# IPCB: Intelligent Pseudolite Constellation Based on High-Altitude Balloons
## 3 Performance Indicator of IPCB Geometry Configuration  
Assuming that an IPCB composed of \(n_{p}\) IPBs is deployed above the service area initially and that \(n_{u}\) users are selected as samples to assess IPCB geometry performance. Then, the pseudo-range equation from the \(j\)-th IPB to the \(i\)-th user at time \(t\) can be expressed by Equation (1) [31; 32; 33]. \[\rho(i,j,t)=\sqrt{\left(x(j,t)-x_{u}(i,t)\right)^{2}+\left(y(j,t)-y_{u}(i,t) \right)^{2}+\left(z(j,t)-z_{u}(i,t)\right)^{2}}+ct_{u}(i,t) \tag{1}\]  
In Equation (1), \(\rho(i,j,t)\) represents the pseudo-range from the \(j\)-th IPB to the \(i\)-th user at time \(t\); (\(x(j,t)\), \(y(j,t)\), \(z(j,t)\)) represents the position of the \(j\)-th IPB at time \(t\); (\(x_{u}(i,t)\), \(y_{u}(i,t)\), \(z_{u}(i,t)\)) represents the position of the \(i\)-th user at time \(t\); \(c\) represents the speed of light; \(t_{u}(i,t)\) represents the clock difference of the \(i\)-th user at time \(t\). Equation (1) can be rewritten as Equation (2) after the first-order Taylor expansion. \[\Delta\rho(i,j,t)=a_{x}(i,j,t)\Delta x_{u}(i,t)+a_{y}(i,j,t)\Delta y_{u}(i,t)+ a_{z}(i,j,t)\Delta z_{u}(i,t)-c\Delta t_{u}(i,t) \tag{2}\]  
In Equation (2), '\(\Delta\)' represents the difference between the Taylor expansion point and its neighborhood, (\(a_{x}(i,j,t)\), \(a_{y}(i,j,t)\), \(a_{z}(i,j,t)\)) represents the direction cosine from the \(i\)-th user to the \(j\)-th IPB at time \(t\), which can be calculated by Equation (3). \[ \begin{array}{l}a_{x}(i,j,t)=\frac{x(j,t)-x_{u}(i,t)}{\sqrt{\left(x(j,t)-x_{ u}(i,t)\right)^{2}+\left(y(j,t)-y_{u}(i,t)\right)^{2}+\left(z(j,t)-z_{u}(i,t) \right)^{2}}}\\ a_{y}(i,j,t)=\frac{y(j,t)-y_{u}(i,t)}{\sqrt{\left(x(j,t)-x_{u}(i,t)\right)^{2}+ \left(y(j,t)-y_{u}(i,t)\right)^{2}+\left(z(j,t)-z_{u}(i,t)\right)^{2}}}\\ a_{z}(i,j,t)=\frac{z(j,t)-z_{u}(i,t)}{\sqrt{\left(x(j,t)-x_{u}(i,t)\right)^{2 }+\left(y(j,t)-y_{u}(i,t)\right)^{2}+\left(z(j,t)-z_{u}(i,t)\right)^{2}}}\end{array}  \tag{3}\]  
Equation (2) can be expanded from the \(j\)-th IPB to all the IPBs in the IPCB, which can be simplified as  
\[\mathbf{\Delta\rho}(i,t)=\mathbf{H}(i,t)\mathbf{\Delta x}(i,t) \tag{4}\]  
In Equation (4), \(\mathbf{\Delta\rho}(i,t)\) is a vector representing pseudo-range measurement error, \(\mathbf{\Delta x}(i,t)\) is a vector representing positioning error, and \(\mathbf{H}(i,t)\) is an observation matrix. Equation (4) can be rewritten as Equation (5) by the least square method. \[\mathbf{\Delta x}(i,t)=\left(\mathbf{H}(i,t)^{\mathrm{T}}\mathbf{H}(i,t) \right)^{-1}\mathbf{H}(i,t)^{\mathrm{T}}\mathbf{\Delta\rho}(i,t) \tag{5}\]  
Figure 2: Illustration of IPCB in regional positioning. If the pseudo-range noises of different IPBs are linearly independent, with an average of 0 and a variance of \(\sigma^{2}\), then the covariance of \(\Delta\mathbf{x}(i,t)\) can be expressed by Equation (6) [33]. \[\text{cov}(\Delta\mathbf{x}(i,t))=\sigma^{2}(\mathbf{H}(i,t)^{\text{T}}\mathbf{ H}(i,t))^{-1} \tag{6}\]  
From Equation (6), it can be concluded that \((\mathbf{H}(i,t)^{\text{T}}\mathbf{H}(i,t))^{-1}\) reveals the magnification from a user pseudo-range measurement error to its positioning error. Given the same user pseudo-range measurement error, the smaller \((\mathbf{H}(i,t)^{\text{T}}\mathbf{H}(i,t))^{-1}\) is, the smaller the positioning error is. Consequently, the square root of the trace of \((\mathbf{H}(i,t)^{\text{T}}\mathbf{H}(i,t))^{-1}\) is usually treated as an important indicator to assess the influence of a constellation geometry configuration on its positioning error, named GDOP (geometric dilution of precision), which can be expressed by Equation (7) [33; 34; 35; 36]. \[GDOP(i,t)=\sqrt{tr(\mathbf{H}(i,t)^{\text{T}}\mathbf{H}(i,t))^{-1}} \tag{7}\]  
It is obvious that the value of \(GDOP(i,t)\) will fluctuate due to IPCBs' dynamic movements. In particular, some IPBs in the constellation may leave approved airspace as time goes on.