Cascaded Ambiguity Resolution for Pseudolite System-Augmented GNSS PPP

Abstract

Global navigation satellite System (GNSS) precise point positioning (PPP) enables high-precision global positioning using a single receiver, yet its widespread application is hindered by long convergence times. In contrast, pseudolite system (PLS) transmitters are located relatively close to receivers, and the movement of receivers induces rapid spatial geometry changes, which greatly facilitate fast parameter convergence. Therefore, leveraging the fast-converging PLS to augment GNSS PPP presents a promising solution. This study proposes a tightly coupled PLS and GNSS observation-level integration model. A key factor influencing the augmentation effectiveness is the strategy of ambiguity resolution. In this work, we design a novel strategy of ambiguity resolution, in which the fast convergence property of PLS is taken into account, and the PLS ambiguities are picked out to be fixed independently. This strategy can resolve the PLS ambiguities, GNSS wide-lane (WL) ambiguities, and GNSS L1 ambiguities cascadingly. Further, the fixed ambiguities can be treated as constraints in the filtering process. The experimental results demonstrate that the proposed strategy substantially improves the ambiguity fixing rates, especially in short-duration augmentation.

1. Introduction

Mass-market applications such as autonomous vehicles and unmanned aerial vehicles are increasingly demanding fast, high-accuracy, and low-bandwidth location services [1]. Precise point positioning (PPP) is one of the mainstream global navigation satellite system (GNSS) positioning technologies that can achieve centimeter-level positioning accuracy globally with a single receiver [2]. However, slow geometry variations, atmospheric delays, and uncalibrated phase delays result in PPP usually taking tens of minutes to converge, which hinders its wide application.

Integer ambiguity resolution (AR) can effectively shorten the convergence time of GNSS PPP. This technique utilizes precise satellite orbit, clock, and code/phase bias provided by the server side to recover the integer nature of ambiguity, thus achieving integer ambiguity resolution [3,4,5]. Once integer ambiguities are fixed correctly, the positional parameters will converge instantaneously, which greatly reduces the convergence time of the PPP [6,7]. In addition, precise atmospheric delay corrections obtained from a regional network can also be applied to constrain the relevant parameters to further accelerate the ambiguity resolution and convergence [8,9,10].

Some scholars utilize other positioning systems such as the pseudolite system (PLS) to augment GNSS to accelerate its convergence. PLS transmits GNSS-like signals near the ground [11,12,13,14]. It can not only work independently but can also be used to augment GNSS. Due to PLS transmitters being close to receivers, moving receivers can produce quick spatial geometry variations, which is conducive to the rapid convergence of parameters [15,16]. Jiang et al. [17] proposed a Locata (a ground-based pseudolite system) augmented GPS PPP, which was referred to as Locata/GPS-PPP. The convergence time of the integrated Locata/GPS-PPP was only about 10 s. However, neither Locata nor GPS ambiguities were fixed to integers. If the ambiguity can be fixed to an integer, the convergence time is expected to be further reduced. In order to fix the integer ambiguity of PLS, our previous work proposed an on-the-fly method to estimate the PLS transmitter phase bias (TPB) and recover the integer nature of PLS ambiguities [18]. Furthermore, we presented a concept of PLS short-duration augmented GNSS PPP and realized the integer ambiguity resolution of PLS and GNSS [19].

However, in prior research, PLS has been typically treated as a subsystem of GNSS, with its ambiguities resolved jointly. This approach overlooks a critical characteristic: PLS ambiguities converge significantly faster than those of GNSS. Simultaneous ambiguity resolution may inadvertently degrade the PLS ambiguity fixing rate. Therefore, it is worthwhile to further investigate how to perform ambiguity resolution and how to utilize the fixed ambiguities under the PLS augmented GNSS PPP model. In this work, a strategy of ambiguity resolution has been proposed based on the characteristics of PLS and GNSS, and its performance has been analyzed.

The structure of this paper is organized as follows: Firstly, the observation models for GNSS and PLS are presented; a tightly coupled model for PLS-augmented GNSS PPP is then established. Based on this tightly coupled model, a strategy of ambiguity resolution has been proposed. Subsequently, real-time PPP experiments are conducted to evaluate the performance under different strategies of ambiguity resolution. Finally, conclusions are presented.

2. Observation Model

Here, the GNSS and PLS observation models are presented, respectively. The GNSS pseudorange observation (${\rho }_{r,j}^{\mathrm{G},s}$) and phase observation ($L_{r,j}^{\mathrm{G},s}$) are described as follows:

$$\left\{\begin{array}{l}{\rho }_{r,j}^{\mathrm{G},s}=R_r^{\mathrm{G},s}+\mathrm{c}\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},s}\right)+{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},s}+T_r^{\mathrm{G},s}+b_{r,j}^{\mathrm{G}}-b_j^{\mathrm{G},s}+e_{r,j}^{\mathrm{G},s}\\ L_{r,j}^{\mathrm{G},s}=R_r^{\mathrm{G},s}+\mathrm{c}\left(\delta t_r^{\mathrm{G}}-\delta t^{G,s}\right)-{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},s}+T_r^{\mathrm{G},s}+{\lambda }_j^{\mathrm{G}}\left(N_{r,j}^{\mathrm{G},s}+B_{r,j}^{\mathrm{G}}-B_j^{\mathrm{G},s}\right)+{\epsilon }_{r,j}^{\mathrm{G},s}\end{array}\right.$$

(1)

where indices G, s, r, and j refer to the GNSS, satellite, receiver, and frequency band, respectively. $R_r^{\mathrm{G},s}$ is the geometric distance between the phase center of the receiver antenna and the satellite antenna. $\mathrm{c}$ is the speed of light in vacuum. $\delta t_r^{\mathrm{G}}$ and $\delta t^{\mathrm{G},s}$ denote the clock offsets of the GNSS receiver and satellite, respectively. $I_{r,1}^{\mathrm{G},s}$ represents the first-order slant ionospheric delay on the first frequency, and the slant ionospheric delay on the j-th frequency can be obtained by a multiplier factor ${\gamma }_j^{\mathrm{G}}={\left[{\lambda }_j^{\mathrm{G}}/{\lambda }_1^{\mathrm{G}}\right]}^2$ that depends on the wavelength ${\lambda }_j^{\mathrm{G}}$. $T_r^{\mathrm{G},s}$ is the slant tropospheric delay. $b_{r,j}^{\mathrm{G}}$ and $b_j^{\mathrm{G},s}$ are the code biases of the GNSS receiver and satellite, respectively. $B_{r,j}^{\mathrm{G}}$ and $B_j^{\mathrm{G},s}$ are the phase biases of the GNSS receiver and satellite, respectively. $N_{r,j}^{\mathrm{G},s}$ denotes the GNSS integer ambiguity. $e_{r,j}^{\mathrm{G},s}$ and ${\epsilon }_{r,j}^{\mathrm{G},s}$ are the sum of measurement noise and multipath error of code and phase observations, respectively.

In GNSS PPP, the errors caused by the Sagnac effect, relativistic effect, polar tide, ocean tide, solid earth tide, phase center offset, phase center variations, and phase wind-up are corrected by models [20]. The tropospheric delays consist of dry and wet components. The dry component can be accurately modeled, while the wet component is difficult to eliminate by models, and thus the zenith wet delay is usually estimated with mapping functions. The satellite orbit, clock, and code/phase bias are corrected by precise products generated by analysis centers.

Compared with GNSS observations, the PLS observations have relatively fewer errors. The tropospheric delay can be ignored or corrected by a simplified model [21,22] if the signal propagation distance is short. Furthermore, due to PLS signals propagating near the surface of the Earth, PLS observations are not affected by ionospheric delays. The coordinates of transmitters are accurately measured in advance. For synchronized PLS, the clock offsets of transmitters are equivalent and can be absorbed by the clock offset of the receiver. The transmitter code/phase biases can be estimated by the service end and broadcast to the user [18]. Therefore, the PLS observations equation can be simplified from Equation (1) as follows:

$$\left\{\begin{array}{l}{\rho }_{r,j}^{\mathrm{P},s}=R_r^{\mathrm{P},s}+c\delta t_r^{\mathrm{P}}+b_{r,j}^{\mathrm{P}}+e_{r,j}^{\mathrm{P},s}\\ L_{r,j}^{\mathrm{P},s}=R_r^{\mathrm{P},s}+c\delta t_r^{\mathrm{P}}+{\lambda }_j^{\mathrm{P}}\left(N_{r,j}^{\mathrm{P},s}+B_{r,j}^{\mathrm{P}}\right)+{\epsilon }_{r,j}^{\mathrm{P},s}\end{array}\right.$$

(2)

where the superscript P denotes PLS. ${\rho }_{r,j}^{\mathrm{P},s}$ and $L_{r,j}^{\mathrm{P},s}$ are the PLS pseudorange and phase observations, respectively. $\delta t_r^{\mathrm{P}}$ is the clock offset of the PLS receiver. It should be noted that $\delta t_r^{\mathrm{P}}$ absorbs the clock offsets of the transmitters so that it drifts differently from that of the GNSS receiver. This should be taken care of in the setting of the clock offset parameter within the integration model.

3. PLS-Augmented GNSS PPP Model

In this work, we implement PLS-augmented GNSS PPP based on a tightly coupled (TC) model [19]. The diagram of the GNSS/PLS TC model is shown in Figure 1. The input data include GNSS observations, State Space representation (SSR) of GNSS corrections, PLS observations, and PLS ephemeris. SSR corrections include satellite orbit, clock, and code/phase biases, which are used for GNSS real-time PPP. PLS ephemeris contains the position, attitude, and code/phase biases of the pseudolite [18]. The attitude of the pseudolite antenna is primarily used to calculate the wind-up correction, and the phase bias is mainly employed to recover the integer nature of the PLS ambiguity. In this study, dual-frequency GNSS observations and single-frequency PLS observations are integrated with an Extended Kalman Filter (EKF). The undifferenced and uncombined PPP model is very convenient for system and frequency expansion [23]. Therefore, it is adopted in the GNSS/PLS TC model.

In the EKF, the state equation and measurement equation at the k-th epoch are as follows:

$$\left\{\begin{array}{l}{x}_k=x_{k-1}+w_{k-1}\\ z_k=h(x_k)+v_k\end{array}\right.$$

(3)

where the unknown state vector $x_k={\left[{r_r}^T,\mathrm{c}\delta t_r^{\mathrm{G}},\mathrm{c}\delta t_r^{\mathrm{P}},{\left(I_{r,1}^{\mathrm{G}}\right)}^T,T_{r,w}^{\mathrm{G}},{\left(N_{r,1}^{\mathrm{G}}\right)}^T,{\left(N_{r,2}^{\mathrm{G}}\right)}^T,{\left(N_{r,1}^{\mathrm{P}}\right)}^T\right]}_k^T$. $r_r$ represents the receiver position; $I_{r,1}={\left[I_{r,1}^{\mathrm{G},1},I_{r,1}^{\mathrm{G},2},\dots I_{r,1}^{\mathrm{G},m}\right]}^T$ is the slant ionosphere delay, $N_{r,1}^{\mathrm{G}}={\left[N_{r,1}^{\mathrm{G},1},N_{r,1}^{\mathrm{G},2},\dots ,N_{r,1}^{\mathrm{G},m}\right]}^T$, $N_{r,2}^{\mathrm{G}}={\left[N_{r,2}^{\mathrm{G},1},N_{r,2}^{\mathrm{G},2},\dots ,N_{r,2}^{\mathrm{G},m}\right]}^T$, and $N_{r,1}^{\mathrm{P}}={\left[N_{r,1}^{\mathrm{P},1},N_{r,1}^{\mathrm{P},2},\dots ,N_{r,1}^{\mathrm{P},n}\right]}^T$ are the GNSS first frequency band, GNSS second frequency band, and PLS first frequency band ambiguity vectors, respectively. The measurement model vector reads $h(x_k)={\left[{\left(h_{\rho ,1}^{\mathrm{G}}\right)}^T,{\left(h_{L,1}^{\mathrm{G}}\right)}^T,{\left(h_{\rho ,2}^{\mathrm{G}}\right)}^T,{\left(h_{L,2}^{\mathrm{G}}\right)}^T,{\left(h_{\rho ,1}^{\mathrm{P}}\right)}^T,{\left(h_{L,1}^{\mathrm{P}}\right)}^T\right]}^T$, with

$$h_{\rho ,j}^G=\left[\begin{array}{c}{R}_r^{\mathrm{G},1}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},1}\right)+{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},1}+T_r^{\mathrm{G},1}+b_{r,j}^{\mathrm{G}}-b_j^{\mathrm{G},1}\\ R_r^{\mathrm{G},2}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},2}\right)+{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},2}+T_r^{\mathrm{G},2}+b_{r,j}^{\mathrm{G}}-b_j^{\mathrm{G},2}\\ ⋮\\ R_r^{\mathrm{G},m}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},m}\right)+{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},m}+T_r^{\mathrm{G},m}+b_{r,j}^{\mathrm{G}}-b_j^{\mathrm{G},m}\end{array}\right],$$

(4)

$$h_{L,j}^G=\left[\begin{array}{c}{R}_r^{\mathrm{G},1}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},1}\right)-{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},1}+T_r^{\mathrm{G},1}+{\lambda }_j^{\mathrm{G}}\left(B_{r,j}^{\mathrm{G}}-B_j^{\mathrm{G},1}+N_{r,j}^{\mathrm{G},1}\right)\\ R_r^{\mathrm{G},2}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},2}\right)-{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},2}+T_r^{\mathrm{G},2}+{\lambda }_j^{\mathrm{G}}\left(B_{r,j}^{\mathrm{G}}-B_j^{\mathrm{G},2}+N_{r,j}^{\mathrm{G},2}\right)\\ ⋮\\ R_r^{\mathrm{G},m}+c\left(\delta t_r^{\mathrm{G}}-\delta t^{\mathrm{G},m}\right)-{\gamma }_j^{\mathrm{G}}{I}_{r,1}^{\mathrm{G},m}+T_r^{\mathrm{G},m}+{\lambda }_j^{\mathrm{G}}\left(B_{r,j}^{\mathrm{G}}-B_j^{\mathrm{G},m}+N_{r,j}^{\mathrm{G},m}\right)\end{array}\right],$$

(5)

and

$$h_{\rho ,1}^{\mathrm{P}}=\left[\begin{array}{c}{R}_r^{\mathrm{P},1}+c\delta t_r^{\mathrm{P}}+b_{r,1}^{\mathrm{P}}\\ R_r^{\mathrm{P},2}+c\delta t_r^{\mathrm{P}}+b_{r,1}^{\mathrm{P}}\\ ⋮\\ R_r^{\mathrm{P},n}+c\delta t_r^{\mathrm{P}}+b_{r,1}^{\mathrm{P}}\end{array}\right],h_{L,1}^{\mathrm{P}}=\left[\begin{array}{c}{R}_r^{\mathrm{P},1}+c\delta t_r^{\mathrm{P}}+{\lambda }_1^{\mathrm{P}}\left(N_{r,1}^{\mathrm{P},1}+B_{r,1}^{\mathrm{P}}\right)\\ R_r^{\mathrm{P},2}+c\delta t_r^{\mathrm{P}}+{\lambda }_1^{\mathrm{P}}\left(N_{r,1}^{\mathrm{P},1}+B_{r,1}^{\mathrm{P}}\right)\\ ⋮\\ R_r^{\mathrm{P},n}+c\delta t_r^{\mathrm{P}}+{\lambda }_1^{\mathrm{P}}\left(N_{r,1}^{\mathrm{P},1}+B_{r,1}^{\mathrm{P}}\right)\end{array}\right].$$

(6)

The partial derivatives matrix of $h(x_k)$ reads

$$\begin{array}{cc}\hfill H(x_k)& ={\displaystyle \frac{\partial h(x_k)}{\partial x_k}}{|}_{x_k={\widehat{x}}_k}\hfill \\ & =\left[\begin{array}{cccccccc}-g_r^{\mathrm{G}}& E_m& 0& I_m& M_T& 0& 0& 0\\ -g_r^{\mathrm{G}}& E_m& 0& -I_m& M_T& {\lambda }_1^{\mathrm{G}}{I}_m& 0& 0\\ -g_r^{\mathrm{G}}& E_m& 0& {\gamma }_2^{\mathrm{G}}{I}_m& M_T& 0& 0& 0\\ -g_r^{\mathrm{G}}& E_m& 0& -{\gamma }_2^{\mathrm{G}}{I}_m& M_T& 0& {\lambda }_1^{\mathrm{G}}{I}_m& 0\\ -g_r^{\mathrm{P}}& 0& E_n& 0& 0& 0& 0& 0\\ -g_r^{\mathrm{P}}& 0& E_n& 0& 0& 0& 0& {\lambda }_1^{\mathrm{P}}{I}_n\end{array}\right]\hfill \end{array}$$

(7)

where $g_r^{\mathrm{G}}$ and $g_r^{\mathrm{P}}$ are the line-of-sight unit vector from the receiver to GNSS satellite and pseudolite, respectively; $E_m={{\left[\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right]}_{1\times m}}^T$; $I_m$ is a m-dimensional identity matrix; $M_T={\left[\begin{array}{cccc}{m}_r^{G,1}& m_r^{G,2}& \cdots & m_r^{G,m}\end{array}\right]}^T$ is the mapping matrix of zenith wet delay. The covariance matrixes of process noise $w_k$ read

$$Q=\left[\begin{array}{cccccccc}{\infty }_r& & & & & & & \\ & {\infty }_{\delta t}^G& & & & & & \\ & & {\infty }_{\delta t}^{\mathrm{P}}& & & & & \\ & & & Q_I^{\mathrm{G}}& & & & \\ & & & & Q_T^{\mathrm{G}}& & & \\ & & & & & 0_{m\times m}& & \\ & & & & & & 0_{m\times m}& \\ & & & & & & & 0_{n\times n}\end{array}\right]$$

(8)

Adequately large process noises ($\infty$ = 10⁶) of receiver position and receiver clock offset are added to the variance at every epoch. $Q_I^{\mathrm{G}}$ and $Q_T^{\mathrm{G}}$ is the process noise covariance slant ionosphere and zenith wet troposphere terms, respectively [24]. The covariance matrixes of measurement error $v_k$ read

$$R_k=\left[\begin{array}{cccc}{R}_{\rho ,k}^{\mathrm{G}}& & & \\ & R_{L,k}^{\mathrm{G}}& & \\ & & R_{\rho ,k}^{\mathrm{P}}& \\ & & & R_{L,k}^{\mathrm{P}}\end{array}\right]$$

(9)

where $R_{\rho ,k}^{\mathrm{G}}=\mathrm{diag}\left({({\sigma }_{\rho ,k}^{\mathrm{G},1})}^2,{({\sigma }_{\rho ,k}^{\mathrm{G},2})}^2,\cdots {({\sigma }_{\rho ,k}^{\mathrm{G},m})}^2\right)$, $R_{L,k}^{\mathrm{G}}=\mathrm{diag}\left({({\sigma }_{L,k}^{\mathrm{G},1})}^2,{({\sigma }_{L,k}^{\mathrm{G},2})}^2,\cdots {({\sigma }_{L,k}^{\mathrm{G},m})}^2\right)$, $R_{\rho ,k}^{\mathrm{P}}=\mathrm{diag}\left({({\sigma }_{\rho ,k}^{\mathrm{P},1})}^2,{({\sigma }_{\rho ,k}^{\mathrm{P},2})}^2,\cdots {({\sigma }_{\rho ,k}^{\mathrm{P},n})}^2\right)$, and $R_{L,k}^{\mathrm{P}}=\mathrm{diag}\left({({\sigma }_{L,k}^{\mathrm{P},1})}^2,{({\sigma }_{L,k}^{\mathrm{P},2})}^2,\cdots {({\sigma }_{L,k}^{\mathrm{P},n})}^2\right)$. ${\sigma }_{\rho ,k}^{\mathrm{G},s}$ and ${\sigma }_{L,k}^{\mathrm{G},s}$ are the standard deviations of GNSS pseudorange and phase measurement errors, respectively. ${\sigma }_{\rho ,k}^{\mathrm{P},s}$ and ${\sigma }_{L,k}^{\mathrm{P},s}$ are the standard deviations of PLS pseudorange and phase measurement errors, respectively. The setting of the standard deviations of GNSS measurement errors employed an elevation-dependent model [24]. However, for PLS measurements, their standard deviations are primarily related to environmental factors. Based on empirical values, the standard deviations of PLS pseudorange and phase measurement errors are set to constants of 2 m and 0.01 m, respectively.

4. Strategies of Ambiguity Resolution

In the undifferenced and uncombined PPP model, ambiguities are affected by both initial phase biases and clock offsets, thus lacking the integer nature. Specifically, satellite-dependent initial phase biases and precise clock offsets are typically provided by the International GNSS Service (IGS) analysis center, which can be removed from the observations. Nevertheless, the raw zero-difference ambiguity parameters estimated within the filter are still affected by receiver-dependent initial phase biases and clock offsets. Before fixing the ambiguities using the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) algorithm [25], the receiver clock offsets and initial phase biases can be eliminated through the between-satellites single difference. Here, the first satellite is assumed as the reference satellite, thus defining the single-difference transformation matrix as follows:

$$D=\left[\begin{array}{ccccc}-1& 1& & & \\ -1& & 1& & \\ ⋮& & & \ddots & \\ -1& & & & 1\end{array}\right]$$

(10)

Single-difference ambiguities are obtained through the single-difference transformation matrix as follows:

$$\Delta \widehat{N}=D\widehat{N}=\left[\begin{array}{ccccc}-1& 1& & & \\ -1& & 1& & \\ ⋮& & & \ddots & \\ -1& & & & 1\end{array}\right]\left[\begin{array}{c}{\widehat{N}}^1\\ {\widehat{N}}^2\\ ⋮\\ {\widehat{N}}^s\end{array}\right]$$

(11)

The float single-difference ambiguity vector $\Delta \widehat{N}$ is processed by the LAMBDA algorithm for integer ambiguity resolution. Upon successful resolution, the fixed single-difference ambiguity vector $\Delta \stackrel{⌣}{N}$ is obtained. The $\Delta \stackrel{⌣}{N}$ can be used as a pseudo-observation to constrain other parameters to obtain fixed solutions.

This constraint can be distinguished between temporary constraint and tight constraint, which correspond to the modes termed “Continuous” (Figure 2a) and “Fix and Hold” (Figure 2b) in the open-source software RTKLIB [24]. In “Continuous” mode, upon successful ambiguity resolution, the fixed integer ambiguity vector constrains a temporarily copied filter, yielding a fixed solution. Crucially, the integer ambiguity vector is not fed back to the original filter, whereas, in “Fix and Hold” mode, the fixed integer ambiguity vector is fed back to the original filter, constraining its ambiguity parameters to integers to obtain the fixed solution.

The fundamental distinction between the two modes is as follows: in the “Continuous” mode, the fixed ambiguities influence only the current epoch, while in the “Fix and Hold” mode, they persist to constrain subsequent epochs. Notably, “Fix and Hold” mode carries higher operational risk. Should an incorrect ambiguity vector be fed back into the original filter, it propagates systematic errors, causing deviations in positioning solutions for the following epochs.

In the GNSS/PLS TC model, PLS can be regarded as a subsystem of GNSS, in which PLS observations and GNSS observations are processed together. However, the ambiguity convergence rate of PLS is significantly faster than that of GNSS. Ignoring this characteristic and jointly fixing their ambiguities simultaneously may compromise the optimal fixation rate, thereby compromising augmentation effectiveness. Therefore, it can be considered to prioritize fixing the PLS ambiguities, and the fixed PLS ambiguity vector can subsequently be employed as a virtual observation to constrain the remaining parameters. This strategy reduces the correlation between GNSS ambiguities and other parameters (e.g., atmospheric delays), facilitating the subsequent GNSS ambiguity resolution.

Furthermore, GNSS dual-frequency data enables the formation of wide-lane (WL) ambiguity. Owing to the fact that the WL ambiguities have substantially longer wavelength (~86 cm for GPS WL) compared to fundamental-frequency ambiguities (~19 cm for GPS L1), WL ambiguities can be fixed rapidly and reliably. The WL ambiguity can be constructed by the following transformation:

$${\widehat{\mathit{N}}}_{\mathrm{WL}}^{\mathrm{G}}=\left[1\hspace{1em}-1\right]\left[\begin{array}{c}{\widehat{N}}_{\mathrm{L}1}^{\mathrm{G}}\\ {\widehat{N}}_{\mathrm{L}2}^{\mathrm{G}}\end{array}\right]$$

(12)

where ${\widehat{N}}_{\mathrm{L}1}^{\mathrm{G}}$ and ${\widehat{N}}_{\mathrm{L}2}^{\mathrm{G}}$ represent the estimated GNSS ambiguities of the first and second frequency bands, respectively. Once WL ambiguities are fixed, they can be used as a pseudo-observation to constrain the fundamental-frequency (L1/L2) ambiguities, reducing parameter correlations and accelerating their resolution.

Combining the aforementioned ambiguity resolution sequence and the constrained mode after the ambiguity is fixed, this paper proposes a new strategy of ambiguity resolution, as illustrated in Figure 3. In the traditional strategy, the PLS is treated as a subsystem of GNSS, and its ambiguities are fixed together. However, in the proposed strategy, the PLS ambiguities are fixed first, and the fixed PLS integer ambiguity vector is used to update the temporary filter; subsequently, the GNSS WL ambiguities are fixed, and the temporary filter is updated with the fixed WL ambiguity vector; finally, the GNSS L1 (first frequency band) ambiguities are fixed. Updating the temporary filter only affects the current epoch, meaning it solely constrains the remaining parameters at that epoch. To propagate the influence of fixed ambiguities across all subsequent epochs, these fixed ambiguities should be fed back into the original filter. This feedback mechanism is implemented through the “Fix and Hold” mode, which operates as an optional strategy in the processing workflow.

It should be noted that in PPP mode, attempting to fix the full set of ambiguities is highly challenging due to high-dimensional correlations and residual atmospheric errors. Consequently, partial ambiguity resolution (PAR), which only fixes a suitable subset of the ambiguities, is typically employed [26,27,28,29]. Several kinds of strategies have been proposed to select ambiguity subsets, such as the elevation order strategy, the Signal-to-Noise Ratio (SNR) order strategy. Pseudolites typically have low elevation angles; using the elevation order strategy is likely to exclude them. As for the SNR order strategy, pseudolites generally exhibit significantly higher SNR than GNSS satellites. Once a new pseudolite appears, if it cannot be fixed, it will block the fixing of subsequent GNSS satellites. Furthermore, considering the fast geometric variations between pseudolites and moving users, the correlation among the PLS ambiguity parameters decreases fast, and theoretically, PLS ambiguities will be fixed faster than GNSS ambiguities. Therefore, selecting a subset of ambiguities based on the ordering of ambiguity correlation ensures that the PLS ambiguities can be effectively retained during the subset selection process, thereby improving the fixing rate.

Figure 4 shows the flowchart of the PAR procedure adopted in this study. The core of this method involves sorting the decorrelated ambiguity variances in ascending order and sequentially removing ambiguities with the largest variances until the remaining subset satisfies the minimum predefined success rate. First, the PAR process starts with the decorrelation of the ambiguities [25]. Then, the full set of ambiguity resolution success rates is computed as $P_{\mathrm{s}}$ [30]. If $P_{\mathrm{s}}$ > 99%, a normal full AR process is performed; otherwise, PAR is performed. It should be noted that the diagonal elements of $D_{\mathrm{z}}$ have been sorted in ascending order during the decorrelation process. Therefore, we sequentially remove the ambiguities with the largest covariances from the first to the last until the success rate exceeds 99%. If the number of selected ambiguities is more than 4, the AR search is performed. In this PAR, a fixed solution is obtained only if both the success rate and the ratio-test [31] are satisfied.

In summary, the proposed strategy employs a PAR algorithm based on decorrelated ambiguity variances and leverages the rapid convergence advantage of PLS. The PLS ambiguities are first sequentially resolved, followed by the GNSS WL ambiguities, and finally the GNSS L1 ambiguities. Furthermore, during the ambiguity fixing process, the already fixed ambiguities are utilized to constrain the remaining ambiguities, thereby enhancing the success rate of fixing the remaining ambiguities.

5. Experiment and Analysis

5.1. Overview of the Experiment

To evaluate the performance of different ambiguity resolution strategies on real-time kinematic PPP augmented by PLS, we conducted a two-hour kinematic experiment on a building rooftop. Due to frequency regulation standards, the PLS transmits signals in the 2.4 GHz band, which is license-free for industrial, scientific, and medical (ISM). Therefore, GNSS and PLS work in different frequency bands, meaning that separate antennas need to be used to receive GNSS and PLS signals, respectively. The experimental environment and platform are illustrated in Figure 5. The rover was equipped with one PLS receiver and two GNSS receivers. The PLS receiver and its antenna are custom developed, while the model of the GNSS receivers is Septentrio Mosaic X5, with two low-cost quadrifilar helix GNSS antennas. The PLS receiving antenna was located in the middle of the two GNSS antennas. This dual-GNSS-antenna configuration enabled lever-arm correction, thereby correcting measurements from the main GNSS antenna to the PLS antenna phase center [19]. Additionally, the average coordinates of two GNSS antennas, derived from moving-RTK solutions, served as the reference trajectory for assessing positioning results during the experiment.

The coordinates of the GNSS RTK base station were determined through long-term static GNSS PPP processing, while the coordinates of the 6 PLs were obtained from long-term static GNSS RTK and total station measurements. Figure 6 presents the planimetric view of the layout of the 6 PLs and the rover trajectory during the experiment. Within this figure, the PLs are denoted by yellow circles. The two-hour walking trajectory (about 1 min to walk one circuit) is indicated by the blue line.

Figure 7 illustrates the variation in the number of tracked GNSS satellites. The black, red, blue, and green lines represent the number of satellites of GPS, Galileo, GLONASS, and BDS, respectively, while the purple line denotes the total number of all satellites. The number of satellites changes frequently, due to the fact that some of the satellites’ signals are blocked when the user approaches the wall. At certain moments, the total number of GNSS satellites drops to approximately 20, primarily caused by the loss of BDS observations output from the receiver. Overall, the total number of GNSS satellites is about 35 throughout the experiment.

5.2. Analysis of the Experimental Results

In this experiment, we analyzed the GNSS/PLS ambiguity fixing rate and positioning accuracy. The ambiguity fixing rate was defined as the ratio of the number of epochs with successful ambiguity resolution to the total number of epochs. The experiment duration spanned 2 h, subdivided into 8 sequential segments, each lasting 15 min (900 s), with a reset procedure applied at the start of each segment. To enhance visual clarity in illustrating the positioning errors and the number of fixed ambiguities over time, only the data from the first 15 min segment (900 s) is presented graphically. However, for the statistical evaluation of the ambiguity fixing rate and root mean square error (RMSE) of positioning, data from all eight segments (the full 2 h) was utilized.

In the following experiments, we analyze both the traditional and the proposed strategies of ambiguity resolution. Based on the constraint mode after ambiguity is fixed, the traditional strategy is categorized into “Traditional (C)” and “Traditional (H)”, while the proposed strategy is similarly categorized into “Proposed (C)” and “Proposed (H)”. In “Traditional (C)” and “Proposed (C)”, the fixed ambiguities are not fed back to the original filter; that is, “Continuous” mode is adopted. Conversely, in “Traditional (H)” and “Proposed (H)”, the fixed ambiguities are fed back to the original filter, corresponding to the “Fix and Hold” mode. Additionally, based on the duration of PLS-augmented GNSS, we divided the experiments into long-duration augmentation and short-duration augmentation.

5.2.1. Long-Duration Augmentation

Figure 8 presents the number of fixed ambiguities with different strategies of ambiguity resolution, including the number of PLS ambiguities and GNSS L1 (first frequency band) ambiguities. First, we will focus on the PLS ambiguity resolution performance. It can be seen that PLS ambiguities are fixed in about 4 s for all strategies. Note that in the traditional strategy, the PLS and GNSS ambiguities are fixed together without deliberate separation, but it achieves fast resolution comparable to the proposed strategy. The traditional strategy achieved fast ambiguity resolution, primarily due to the PAR algorithm, which ranks ambiguity parameters based on their variance after decorrelation. The rapid geometric variations result in PLS ambiguity variances decreasing faster than that of GNSS ambiguities. Consequently, the PAR algorithm first removes GNSS ambiguities with large variances and preserves the PLS ambiguities with smaller variances to be fixed. This process inherently performs a selection mechanism functionally similar to the explicit PLS ambiguity selection employed in the proposed strategy. However, after approximately 160 s, the PLS ambiguities in “Traditional (C)” and “Traditional (H)” fail to be fixed. After careful analysis, it was found that after 160 s, the variances of GNSS ambiguities also decrease. Partial GNSS ambiguities satisfied the success rate threshold yet failed the ratio-test threshold, resulting in the failure of ambiguity resolution. Since the PLS and GNSS ambiguities are fixed together in this strategy, the overall failure consequently prevents the resolution of the PLS ambiguities. As the GNSS ambiguities continued to converge and passed the “ratio-test” threshold, PLS ambiguities were therefore fixed again, as shown in “Traditional (H)”. There is no obvious difference between the “Proposed (C)” and “Proposed (H)” in terms of PLS ambiguity resolution. In a word, the slow convergence of GNSS ambiguities may compromise the fixing rate of PLS ambiguities when they are resolved together. Therefore, it is advisable to resolve them separately.

Regarding GNSS L1 ambiguity resolution, the proposed strategy achieves a higher fixing rate compared to the traditional strategy in both “Continuous” and “Fix and Hold” modes. Notably, the fixing rate of the “Proposed (H)” strategy significantly exceeds that of other strategies, demonstrating the effectiveness of feeding fixed ambiguities back into the original filter for enhancing the ambiguity fixing rate.

Figure 9 presents the statistical results of the ambiguity fixing rates under different strategies of ambiguity resolution. The performance is consistent with the previous analysis.

Figure 10 depicts the positioning errors under these strategies. The statistical results of the positioning errors under strategies Traditional (C), Traditional (H), Proposed (C), and Proposed (H) are 0.021, 0.020, 0.019, and 0.020 cm, respectively. It can be seen that the positioning errors obtained by these strategies show no significant differences, with positioning errors consistently around 0.02 m. This consistency arises primarily because the PLS converges rapidly and stays converged throughout, which plays a major role in positioning.

5.2.2. Short-Duration Augmentation

On the one hand, it is difficult to see the effect of these strategies of ambiguity resolution on the positioning accuracy because the PLS augments GNSS throughout the whole period. On the other hand, due to the limited coverage area of PLS, sometimes GNSS can only be augmented for a short period of time. Therefore, we specifically investigated the performance of the short-duration (15 s) PLS augmentation to GNSS.

Figure 11 depicts the positioning errors under these strategies of ambiguity resolution with short-duration PLS augmentation. It can be seen that the positioning error in the “Proposed (H)” strategy remains consistently convergent, while other strategies exhibit a little divergence tendency. The statistical results of the positioning errors under strategies Traditional (C), Traditional (H), Proposed (C), and Proposed (H) are 0.111, 0.059, 0.097, and 0.048 cm, respectively. It can be seen that the positioning accuracy in the “Proposed (C)” strategy is higher than that achieved by the “Traditional (C)” strategy. When applying the “Fix and Hold” mode, the positioning accuracy is further improved, and the “Proposed (H)” mode still achieves higher positioning accuracy than the “Traditional (H)”. This demonstrates that regardless of whether the “Continuous” or “Fix and Hold” mode is employed, the proposed strategy of ambiguity resolution yields consistently higher positioning accuracy. However, the positioning accuracy of “Traditional (H)” is better than that of Proposed (C)”. This indicates that feeding back the fixed PLS integer ambiguities to the original filter is crucial in short-duration augmentation scenarios.

From the previous experiment, it can be seen that applying “Fix and Hold” mode to fixed ambiguities does not yield significant improvements in terms of positioning accuracy under long-duration augmentation scenarios. However, in short-duration augmentation scenarios, “Fix and Hold” mode provides considerable benefit. This is because in the case of long-duration augmentation, the PLS ambiguity parameters typically converge near integer values. As a result, whether they are fixed or not yields no significant difference in performance. However, in the case of short-duration augmentation, the PLS ambiguity parameters have not yet converged near integer values. Therefore, imposing additional integer constraints (“Fix and Hold” mode) leads to a significant augmentation improvement.

The effectiveness of ambiguity resolution under PLS short-duration augmentation to GNSS is further evaluated in Figure 12 and Figure 13. In the short-duration augmentation, PLS is available only for the initial 15 s and becomes unavailable thereafter. Therefore, the number of fixed PLS ambiguities dropped after 15 s. Here, the PLS ambiguity fixing rate is defined as the ratio of fixed epochs to the total number of available epochs. As can be seen, the PLS ambiguity fixing rate exhibits very little difference across these strategies. As we analyzed earlier, the PAR algorithm can automatically select PLS ambiguities for priority fixing. The previous experiments confirmed that GNSS ambiguities primarily impair the PLS ambiguity fixing rate after 100 s. However, in this experiment, PLS was unavailable after 100 s, thus its fixing rate was hardly affected by GNSS. Consequently, the PLS ambiguity fixing rates show comparable levels across all strategies under short-duration augmentation.

In contrast, significant differences exist in the GNSS L1 ambiguity fixing rates. In the “Proposed (C)” strategy, the L1 ambiguity fixing rate is approximately 30%, while in the “Traditional (C)” strategy, it is only approximately 3%. Furthermore, under the “Fix and Hold” mode, the “Proposed (H)” strategy achieves a remarkable fixing rate of approximately 97%, vastly exceeding the 15% attained by “Traditional (H)”. These results demonstrate that the cascaded ambiguity resolution approach and the “Fix and Hold” mode both greatly improve the GNSS ambiguity fixing rate.

Fixed PLS integer ambiguities adopt the “Fix and Hold” mode to provide a significant improvement in short-duration augmentation scenarios. However, the wavelength of PLS ambiguity (approximately 12 cm) is really short. Consequently, incorrectly fixed PLS ambiguities may introduce substantial deviations into positioning solutions. Figure 14 shows a positioning experiment, in which the duration of PLS-augmented GNSS is still 15 s. At approximately the 4th s, the PLS ambiguities were initially erroneously fixed in the “Proposed (H)” strategy and fed back into the original filter, contaminating the state parameters and resulting in significant positioning deviations. In contrast, the “Proposed (C)” strategy—which avoids feedback of wrong PLS ambiguities to the original filter—maintains relatively better positioning accuracy. As previously analyzed, in the long-duration augmentation, the “Fix and Hold” mode offers limited benefits yet carries substantial risks. Therefore, it is not recommended to apply the “Fix and Hold” mode to PLS. In the case of short-duration augmentation, feedback of PLS ambiguities to the original filter yields significant benefits but requires cautious validation mechanisms.

6. Conclusions

Based on the PLS-augmented GNSS PPP model, this study proposes a new strategy of ambiguity resolution. Within this proposed strategy, by leveraging the rapid convergence characteristics of PLS ambiguities, the PLS ambiguities and GNSS ambiguities are fixed cascadingly. Experiments were conducted to analyze the ambiguity fixing rate and positioning accuracy under different strategies of ambiguity resolution in both long-duration and short-duration PLS-augmented GNSS scenarios.

In the long-duration augmentation, if the rapid convergence characteristic of PLS ambiguities is not considered and they are fixed together with GNSS ambiguities, the PLS ambiguity fixing rate is only 16.9%. However, in the proposed strategy, the PLS ambiguity fixing rate can reach 99.5%. Furthermore, employing the proposed strategy also substantially enhanced the GNSS ambiguity fixing rate.

For short-duration PLS augmentation, the proposed strategy can also improve the GNSS ambiguity fixing rate and positioning accuracy. Additionally, during the short-duration augmentation, where PLS ambiguities have not yet converged sufficiently close to integer values, feeding back the fixed PLS integer ambiguities to the original filter effectively introduces additional integer constraints. This process significantly boosts positioning accuracy and the GNSS ambiguity fixing rate. However, the feedback of fixed PLS ambiguities to the original filter requires extreme caution, since the ambiguity with short wavelength is more prone to being incorrectly fixed, resulting in systematic deviations in the positioning solutions.

Given the rapid convergence characteristics of Low Earth Orbit (LEO) satellites, this study thus provides valuable reference for ambiguity resolution in LEO-augmented GNSS.