# BeiDou Augmented Navigation from Low Earth Orbit Satellites
## 2 Advantages of Navigation with LEO Constellation  
### Positioning Error  
Like GNSS systems, the users will also use the trilateration to compute their position with navigation service provided by LEO constellation. Hence, statistically, the point positioning accuracy can be described by the product of the user equivalent range error (UERE), which combines all measurement and modeling errors of an individual pseudo-range, and the dilution of precision (DOP), which maps such errors to the position uncertainty for a given number and geometric distribution of observed navigation satellites [20]. The relation can be expressed as following. \[\sigma_{3D}=PDOP\cdot\sigma_{UERE} \tag{1}\]  
where \(\sigma_{3D}\) is the user 3D root-mean-square (RMS) positioning error, PDOP is the position DOP, and \(\sigma_{UERE}\) is the UERE. The DOP factors are derived from the terms associated with the user satellite geometry. Therefore, only geometry maps range errors into user space. This means that DOP is the geometry factor of the visual satellites related to their distribution. Whether the satellites are distributed more homogeneously, it will be helpful to the position estimation for the observation information gathered from more directions. Usually, the DOP value is resolved into horizontal (HDOP) and vertical dilution of precision (VDOP), these give us the relationship between UERE and errors in the vertical and horizontal directions. Thanks to the fast motion and numerous satellites, the DOP will be significantly improved by LEO constellation. The simulated results will be presented in Section 4.1. The UERE is further divided into a signal-in-space [user] range error (SIS URE or SISRE) and a user equipment error (UEE). While the UEE term comprises all receiver-related contributions such as noise, multipath and uncorrected atmospheric errors, the SISRE describes the statistical uncertainty of the modeled pseudo-range due to errors in the broadcast orbit and clock information. It is driven by the space segment characteristics (e.g., clock stability and predictability of orbital motion), as well as the control segment capabilities (orbit and clock determination performance, distribution of monitoring stations and upload capacity). The orientation of the line-of-sight vector impacts the orbit errors \(\Delta r=(\Delta r_{R},\Delta r_{A},\Delta r_{C})\) in the radial (\(\Delta r_{R}\)), along-track (\(\Delta r_{A}\)) and cross-track (\(\Delta r_{C}\)) directions on the line-of-sight range, and hence the user location. The average contribution over all points of the Earth within the visibility cone of the satellite is considered normally for a statistical description. The orbit-only contribution to the SISRE can then be described as a weighted average:  
\[SISRE(orb)=RMS\Big{[}w_{R}^{2}\cdot\Delta r_{R}^{2}+w_{A,C}^{2}\cdot\Big{(} \Delta r_{A}^{2}+\Delta r_{C}^{2}\Big{)}\Big{]}=\sqrt{w_{R}^{2}\cdot R^{2}+w_{ A,C}^{2}\cdot(A^{2}+C^{2})} \tag{2}\]  
of the RMS errors \(R=RMS(\Delta r_{R})\), \(A=RMS(\Delta r_{A})\) and \(C=RMS(\Delta r_{C})\) scale like text along the three axes. The altitude of the GNSS satellite determines the weight factors \(w_{R}\) and \(w_{A,C}\)[21]. For MEO, the \(w_{R}\) is closed to 1, and it indicates that the radial orbits largely project onto the user. However, for LEO orbits, the weight factors become almost equally important to the SISRE [3], as their cones should be wider to provide larger converge. The combined orbit and clock SISRE is obtained from the expression:  
\[SISRE=\sqrt{(w_{R}\cdot\Delta r_{R}-\Delta cdt)^{2}+w_{A,C}\cdot(A^{2}+C^{2})} \tag{3}\]  
the \(\Delta cdt\) represents the error of the broadcast clock offset. It takes into consideration that the modeled pseudo-range is derived from the difference of the radial orbit error and the clock offset. The analysis performed by [PERSON] et al. [21] indicates that the SISRE for GPS, GLONASS, Galileo, QZSS, and BeiDou regional system (BeiDou-2) are 0.71 m, 1.97 m, 1.64 m, 0.57 m, and 1.46 m, respectively.