###### Abstract
The Lunar GNSS proposed by these previous studies was based on time of arrival (TOA), i.e., pseudorange measurement, and it requires that at least four satellites be visible all the time, which inevitably requires a large number of satellites and, consequently, a large cost, but it only requires users to carry a passive ranging receiver. In an attempt to further reduce the costs of the system, some researchers have investigated the reduction of the number of satellites. Navigation technologies that use fewer than two satellites with a passive user receiver have been discussed extensively in the field of Earth GNSS applications [9]. One of these studies used angle of arrival (AOA) data and reduced the number of navigation satellites down to one [10]. However, this algorithm provides low position accuracy because a very small error in the AOA measurement results in a large error in the user position; i.e., a 1-degree error in the AOA for a distance of a few hundred km between the satellite and the user results in an error of a few kilometers in the position on the lunar surface. Another algorithm uses time difference of arrival (TDOA or single-differenced pseudorange) and/or frequency difference of arrival (FDOA, or single-differenced Doppler) to reduce the number of navigation satellites to a minimum of two [11]. Moreover, one study successfully showed that single-differenced Doppler using a static reference station, known as Law of Cosines (LOC), can provide a high-accuracy position with as little as one satellite on the lunar surface at the specified condition with several tens of minutes observation [12]. Furthermore, the authors of reference [12] also proposed an algorithm that uses a combination of range and Doppler measurements with a static reference station, known as Joint Doppler and Ranging (JDR), and achieved 3D positioning with as little as one satellite [13]. Basically, these previous studies [7; 8; 9; 10; 11; 12; 13] are based on the assumption that both the satellite and the user system or either the satellite system or the user system can provide a stable clock and/or a stable frequency without an offset (bias), and the estimation algorithms do not have to account for those errors. Such approaches also require the satellites and/or the user to carry a highly stable clock source, such as an atomic clock, in order to maintain a sufficiently small clock bias and frequency bias between bias estimations by ground segments (i.e., ground stations on the earth); otherwise the accuracy of the user position deteriorates immediately. In addition, the target of our study comprises a micro-sized satellite and rover systems whose power generation capability is limited by size and consequently, not compatible with the deep space atomic clock (DSAC). In this case, the best current clock technology that is compatible with the micro-sized satellite is the Chip Scale Atomic Clock (CSAC). As reported in [14], while CSAC can suppress the frequency instability of the clock down to about 1 ppb for 24 h, CSAC incurs several tens to hundreds of meters of error in pseudorange observation after 24 h, which further increases over time. As a result, using CSAC inevitably requires pseudorange-based navigation systems to conduct frequent estimations of the satellite and/or user clock bias using earth ground stations, which is very challenging in Lunar GNSS because of the limitation of the availability and number of earth ground stations that are capable of Earth-Moon distance communication. Another approach to eliminating the biases of the satellite and user clocks at the same time is to use two-way ranging between the user and the satellite [15]. However, this requires an active ranging operation between the satellite segment and the user segment (the user must send a radio signal to the satellite, and the satellite has to send the received radio signals back to each user separately), and this consequently imposes an extra burden and cost on the user segment as well as on the satellite side.