## 2 Characteristic Analysis of the Iridium System
### The Satellite Level
#### 2.1.2 The Orbital Configuration  
The orbital configuration of satellites directly influences the linearity of the observation matrix in positioning solutions. Assume that in a LEO SOP Doppler positioning, a total of \(N_{e}\) epochs are received; then the observation matrix \(\mathbf{G}_{doppler}\) can be defined as:  
\[\mathbf{G}_{Doppler}=\begin{bmatrix}\dot{\boldsymbol{\dot{\rho}}}_{x}^{1}/ \boldsymbol{\gamma}&\dot{\boldsymbol{\dot{\rho}}}_{y}^{1}/\boldsymbol{\gamma}& \dot{\boldsymbol{\dot{\rho}}}_{z}^{1}/\boldsymbol{\gamma}&1\\ \dot{\boldsymbol{\dot{\rho}}}_{x}^{2}/\boldsymbol{\gamma}&\dot{\boldsymbol{ \dot{\rho}}}_{y}^{2}/\boldsymbol{\gamma}&\dot{\boldsymbol{\dot{\rho}}}_{z}^{2}/ \boldsymbol{\gamma}&1\\ \dot{\boldsymbol{\dot{\rho}}}_{x}^{3}/\boldsymbol{\gamma}&\dot{\boldsymbol{ \dot{\rho}}}_{y}^{3}/\boldsymbol{\gamma}&\dot{\boldsymbol{\dot{\rho}}}_{z}^{3}/ \boldsymbol{\gamma}&1\\ \vdots&\vdots&\vdots&\vdots\\ \dot{\boldsymbol{\dot{\rho}}}_{x}^{N_{e}}/\boldsymbol{\gamma}&\dot{\boldsymbol {\dot{\rho}}}_{y}^{N_{e}}/\boldsymbol{\gamma}&\dot{\boldsymbol{\dot{\rho}}}_{z}^ {N_{e}}/\boldsymbol{\gamma}&1\end{bmatrix} \tag{2}\]  
where \(\dot{\boldsymbol{\dot{\rho}}}_{x}^{i}\), \(\dot{\boldsymbol{\dot{\rho}}}_{y}^{i}\), and \(\dot{\boldsymbol{\dot{\rho}}}_{z}^{i}\) denote the rates of change of the pseudo-range direction vector of the \(i\)-th epoch in the \(x\), \(y\), and \(z\) directions, respectively (which can be derived using the  
Figure 1: The orbital errors predicted using the Simplified General Perturbations Version 4 (SGP4) model. (**a**) Three-dimensional position errors; (**b**) three-dimensional velocity errors.  
SPG4 model). \(\gamma\) denotes the maximum rate of change of the pseudo-range direction vector, which is defined as:  
\[\gamma=\left\|\hat{\mathbf{\rho}}\right\|_{max}=\left[\frac{1}{1-\left(\text{Re}/a_{ orb}\right)}\right]\sqrt{\frac{\mathbf{\mu}}{a_{orb}^{3}}} \tag{3}\]  
where \(a_{orb}\) denotes the semi-major axis of the satellite orbit, Re donates the radius of the Earth, and \(\mathbf{\mu}\) denotes the gravitational constant, with a value of 398,600 \(\times\) 10\({}^{9}\) m\({}^{3}\)/s\({}^{2}\).  
The dimensionless Doppler positioning cofactor array \(\mathbf{Q}_{Doppler}\) from \(\gamma\) can be defined as:  
\[\mathbf{Q}_{Doppler}=\left(\mathbf{G}_{doppler}^{\text{T}}\cdot\mathbf{G}_{ doppler}\right)^{-1}=\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\ g_{21}&g_{22}&g_{23}&g_{24}\\ g_{31}&g_{32}&g_{33}&g_{34}\\ g_{41}&g_{42}&g_{43}&g_{44}\end{pmatrix} \tag{4}\]  
where the subscripts 1, 2, and 3 of \(g\) respectively represent the position estimates in the \(x\), \(y\), and \(z\) directions, and 4 represents the clock bias estimation. \(g_{11}\) denotes the cofactor of the position estimation in the \(x\) direction, \(g_{12}\) denotes the cofactor between the position estimation in the \(x\) direction and \(y\) direction, and so forth. The dimensionless analysis described above is termed generalized Doppler analysis, where the generalized Doppler geometry dilution of precision (GDOP) and its position dilution of precision (PDOP) can be expressed as [33]:  
\[GDOP=\sqrt{g_{11}+g_{22}+g_{33}+g_{44}} \tag{5}\]  
\[PDOP=\sqrt{g_{11}+g_{22}+g_{33}} \tag{6}\]  
The corresponding positioning accuracy can be expressed in terms of the PDOP as [34]:  
\[\sigma_{r}=PDOP\cdot\left(\mathbf{\lambda}\cdot\sigma_{D}/\gamma\right) \tag{7}\]  
where \(\sigma_{r}\) denotes the positioning error, \(\sigma_{D}\) denotes the observable error of Doppler frequency, and \(\lambda\) denotes the wavelength of the carrier.  
Therefore, unlike pseudo-range positioning, generalized Doppler GDOP requires not only diverse geometric distributions of satellites but also diverse velocity directions of satellites; that is, it has higher requirements for the diversity of orbits. More diverse Doppler projection directions can reduce the generalized Doppler GDOP and improve the Doppler positioning accuracy.