## 1 Introduction  
Low earth orbit (LEO) satellites are usually equipped with a dual-frequency or single-frequency global navigation satellite system (GNSS) receiver for computing orbit. The GNSS-based orbit determination has been widely used for various LEO satellites, such as TOPEX/POSEIDON [1, 2, 3], CHAMP [4], Jason-1 [5], GRACE [6], Swarm [7], Sentinel [8], FengYun-3C [9], and so on. Precise orbit determination (POD) at centimeter-level accuracy can be achieved with dual-frequency observations by reduced dynamic filtering that exploits the unique geometric strength and considers the pseudo-stochastic orbit modeling techniques [10] or zero-difference kinematic precise point positioning (PPP) [11, 12, 13, 14, 15]. Some researchers have also worked toward the integrated adjustment of CHAMP, GRACE, and GPS data [16], enhancing the performance of POD with LEO onboard multiple GNSS data [17, 18] and improving the robustness of the solution by piece-wisely linearly modeling the onboard GPS receiver oscillators [19]. Dual-frequency POD can obtain higher orbit accuracy than a single-frequency method through eliminating the first-order ionosphere effects, but needs an expensive geodetic GNSS receiver and complex computation algorithm. A small LEO satellite platform that has been popular in commercial aerospace applications controls costs strictly. Determining satellite orbit with a low-cost single-frequency receiver will be the primary option for future small LEO satellites since it not only saves cost, but also meets the accuracy requirement for real-time orbit [20]. Some low-cost single-frequency receivers have successfully implemented the real-time orbit determination of the LEO satellite for high dynamics applications, such as the miniature Phoenix-XNS GPS receiver [21; 22], the single-frequency GPS receiver carried by the SJ-9A LEO satellite [23]. The methods for the reduced dynamic orbit generation or the kinematic orbit determination (KOD) with single-frequency data can achieve orbit accuracies of one meter or the sub-meter level, respectively, depending on the noise of code observations and the correction of ionospheric error [24; 25]. The KOD can be applied for both free-flying and maneuvering spacecraft since it is free of any assumptions on the motion of LEO satellites. Therefore, the single-frequency KOD method is considered suitable for low-cost space applications demanding meter-level orbit precision. However, there are still some challenges for LEO orbit determination in processing the impact of satellite attitude since some LEO satellites use the sun-pointing attitude mode rather than the earth-pointing attitude mode to obtain enough energy from the solar panel. The sun-pointing satellite attitude mode has a significant impact on computing elevation since the onboard GNSS antenna is fixed to the LEO satellite, and its motion around the earth produces a changing discrepancy that the antenna boresight direction is not in line with the up direction of the local Cartesian coordinate system, where the origin is the onboard GNSS receiver, and the up direction always points towards outer space. The rotation of the GNSS antenna radiation pattern changes the noise characteristics and the true elevation cannot be computed because there is no rotation angle information of the antenna plane for most low-cost LEO satellites. The negative elevations occur relative to the local horizontal plane, such as the CANX-2 mission [26], BIRD mission [27], PROBA-2 mission [28], etc. For these small satellite missions, a C/N0-based weighting strategy is more suitable than the elevation-dependent weighting function. Moreover, some low-cost onboard GNSS receivers equipped with low-stability oscillators cause the fast changes of the receiver bias. The receiver clock is frequently adjusted to maintain a controllable apparent clock bias. As a result, the receiver clock steering event occurs frequently, which is a big disadvantage for the single-frequency cycle slip detection.