## 1 Introduction
Reference [12] gave a virtual wideband (VWB) receiver architecture, discussed the applicability of the signal provided by the PNT service on the giant LEO constellation, and analyzed the Doppler upper limit level that could be tolerated, however, although VWB can tolerate harsh Doppler conditions and reduce the required processing bandwidth, LEO has a limited time to pass the zenith; long-term band occupation will waste band resources and cause unstable frequency sources, and the error increases with time. _Aiming at the difficulty of navigation and positioning caused by the lack of visible satellites_. References [13; 14] designed a low-order Kalman filter for the BeiDou double-star positioning system, and the successive approximation method was used to solve the problem of position lag in the BeiDou double-star system. The actual sports car verification was carried out, however, the low order filter will bring some problems, such as phase lag, low sensitivity and poor smoothing effect, which will affect the final navigation and positioning effect. To overcome the shortcomings of the BeiDou dual-satellite navigation and positioning system with active positioning, a new passive positioning algorithm was developed that was based on the data collected by two synchronous satellites, as well as the atomic clock and altimeter carried by the user and through a grid search which realized passive positioning [15]. However, this algorithm required the assistance of an atomic clock, which undoubtedly increases the size, power consumption and cost of the device. Reference [16] was given an elevation model of the BeiDou double-star navigation and positioning system and combined with two BeiDou satellites for positioning; the simulation results proved that the algorithm had the characteristics of fast convergence and accurate positioning. The algorithm can be regarded as a typical traditional double star + altimeter navigation and positioning algorithm; the application in some challenging environments will be restricted. Reference [17] proposed a dual-satellite positioning algorithm to improve the positioning performance under the satellite signal gap; the algorithm used the previously stored positioning data under complete navigation conditions to simplify the positioning equation, since the clock deviation lasts for a short time, so, the method requires only two visible satellites to obtain accurate positioning results without calculating clock deviations, but the premise of the algorithm is to store navigation data under complete conditions, and requires high clock stability, it is known that the algorithm will also be affected by navigation lag and face the problem of high equipment cost. Aiming at the problem that the GNSS receiver cannot achieve positioning due to insufficient available satellites under incomplete conditions, a navigation and positioning algorithm based on multi-epoch double-differential pseudorange observation (MDPO) is proposed, which uses multi-epoch observations to reduce the number of navigation satellites required [18]; obviously, this algorithm requires multiple observations to achieve positioning solution. The solution is aimed at the situation where global positioning system (GPS) signals may be accidentally shielded in complex and challenging environments such as obstructions, urban blocks and mountains. Reference [19] considered accessing the stronger signal transmitted from the LEO communication satellite system to assist GPS services; the algorithm used the Doppler frequency shift as the measurement value and developed a positioning algorithm based on Doppler measurement, but the algorithm can only be used as an enhanced system to assist GPS in navigation and positioning. Reference [20] proposed a comprehensive navigation algorithm based on a giant low earth orbit (LEO) constellation; the algorithm realizes alternate switching of ranging by introducing a real ranging value and a virtual ranging value.