## 1 Introduction
[7] showed that the rates of cycle slip on the Samsung Galaxy S8 and Huawei Honor V8 are greater than 50% and 90% after the duty cycle is turned on. Fortunately, it is possible to switch off the duty cycle mode after Android P (\"Pie\" = version 9), and [5; 6; 16] indicated the duty cycling is disabled in the Nexus 9 tablet; even [PERSON] et al. [9] showed that the observed carrier phase measurements of the Huawei P30 were continuous without the need to manually stop duty cycling, which seems to be disabled beforehand. Signal-to-noise ratio (SNR) refers to the ratio between the measured signal intensity and noise intensity at the same time and place in the circuit. Carrier-to-noise power density ratio (C/N\({}_{0}\)) is usually used to describe signal noise level in a GNSS receiver, and the C/N\({}_{0}\) value is SNR in 1 Hz width with the unit of dB-Hz. The C/N\({}_{0}\) is commonly affected by atmosphere, multipath, signal receiving antenna, internal circuit, etc. However, the majority of studies showed that the mean C/N\({}_{0}\) value of smartphones is generally lower than that of the geodetic receiver, and the C/N\({}_{0}\) difference between the geodetic receiver and smartphones even exceeds 10 dB-Hz [8; 17]. [PERSON] et al. [17] also indicated that the C/N\({}_{0}\) values of the Samsung Galaxy S3 sometimes suddenly and drastically decreased as the elevation increased. In addition, there are still some special properties such as gradual accumulation of phase errors [4] in GNSS carrier phase observations of smartphones. [PERSON] et al. [10] found the pseudorange and carrier phase observations (in meters) of the geodetic receiver are consistent, but not with the Xiaomi Mi 8 and Huawei Honor 9. This means the difference between pseudorange observations and carrier phase observations (in meters) are not fixed on smartphones. Various drifts of GNSS carrier phase observations, such as the code-phase difference ranged from \(-\)50 to 50 m for GPS satellites and \(-\)100 to 100 m for GLONASS satellites in Nexus 9 tablet, were also shown in [16]. Moreover, the positioning performance of smartphones is also limited by the poor multipath suppression capability. [PERSON] et al. [18] conducted the PPP experiment using the Huawei Mate20X in low and high multipath environments, even in the static PPP, the RMS of vertical errors in a high multipath environment is over ten times as in a low multipath environment within 60 min. In addition, the positioning accuracy of smartphones with an external high-grade antenna under high multipath can reach that of a smartphone with its own antenna under low multipath. Unlike the GNSS data observed by the geodetic receiver, the GNSS observations of a current smartphone are always affected by a variety of adverse factors mentioned above due to the hardware performance limitation. In order to reduce the adverse effects on positioning accuracy of multi-GNSS PPP, we first analyze in detail in Section 2 the GNSS signal characteristics, including the duty cycle, carrier-to-noise ratio, carrier phase cycle slip, gradual accumulation of phase errors, and pseudorange residual of smartphones. After that, in Section 3, we establish a multi-GNSS PPP mathematical model for smartphones with a stochastic model that is composed of GPS, BDS, and GLONASS step function variance depending on carrier-to-noise ratio, and the robust Kalman filter is used for parameter estimation. In Section 4, we show the multi-GNSS static and kinematic experiments and results, and the experimental results demonstrate the proposed PPP mathematical model provides higher positioning accuracy and accelerates the convergence of static positioning, compared with conventional PPP method depending on elevation. Lastly, we draw conclusions and add remarks in Section 5.