High-Precision Positioning in Power Applications Using BDS PPP-RTK for Sparse Reference Station Areas

Abstract

To address the urgent demand for high-precision positioning in power industry operations within sparse reference station areas, this paper proposes a real-time kinematic positioning method integrating BeiDou multi-antenna Precise Point Positioning–Real-Time Kinematic (PPP-RTK) with inertial measurement unit (IMU) assistance. By combining the strengths of Precise Point Positioning (PPP) and Real-Time Kinematic (RTK) technologies, we establish a multi-antenna observation model based on State Space Representation (SSR), incorporating satellite-based augmentation signals and atmospheric correction information from sparse reference station networks. Lie group theory is employed to enhance the Extended Kalman Filter (EKF) for simultaneous estimation of position, attitude, and ambiguity parameters. The integration of IMU measurements significantly improves robustness against environmental interference in dynamic scenarios. Experimental results demonstrate average positioning errors of 3.12 cm, 3.71 cm, and 6.23 cm in the East, North, and Up (ENU) directions, respectively, with an average convergence time of 1.62 min. Compared with non-IMU-augmented single-antenna PPP-RTK solutions, the proposed method achieves accuracy improvements up to 59.6% while maintaining stability in signal-occluded environments. This approach provides centimeter-level real-time positioning support for critical power grid operations in remote areas such as desert and Gobi regions, including infrastructure inspection and precise tower assembly, thereby significantly improving the efficiency of intelligent grid operation and maintenance.

1. Introduction

In the power industry, remote areas and desert, Gobi, and barren (DGB) regions face urgent demands for high-precision positioning in transmission line inspection, fault location, and power infrastructure monitoring. These regions are characterized by complex terrain, harsh environments, and sparse distribution of reference stations, where traditional positioning technologies struggle to meet the precision requirements for electrical equipment management due to inadequate ground infrastructure coverage [1]. The BeiDou Navigation Satellite System (BDS) demonstrates unique advantages in such scenarios through its all-weather and continuous availability, coupled with extremely low installation and maintenance costs. Leveraging BDS’s wide-area coverage and satellite-based augmentation services, it delivers stable positioning signals without relying on dense ground reference networks. This capability fundamentally supports spatial coordinate calibration of power facilities, digital modeling of transmission corridors, and disaster emergency response. Moreover, it plays a vital role in ensuring grid security and advancing the intelligent development of new power systems [2,3,4].

However, traditional satellite positioning methods like pseudorange-based single-point positioning, which relies on satellite code signals, typically achieve meter-level accuracy. This precision falls short of meeting centimeter-level requirements for critical power industry applications such as structural deformation monitoring and precision tower installation. To address this limitation, researchers have developed advanced positioning solutions by integrating satellite carrier-phase observations with differential correction data. Two prominent centimeter-level positioning techniques have emerged: Real-Time Kinematic (RTK) and Precise Point Positioning (PPP) [5,6].

PPP-RTK technology combines the strengths of PPP and RTK by integrating BDS global satellite-based augmentation signals and regional correction data from sparse reference stations. This approach enables receivers to achieve near-instantaneous centimeter-level positioning while drastically shortening initialization time. It provides a high-precision positioning solution for power industry operations in remote areas without relying on dense ground stations, effectively overcoming the technical limitations of traditional methods [7,8,9]. The foundation of PPP-RTK lies in precise products generated by the BDS reference station network, encompassing satellite orbits, clock offsets, signal biases, and optionally, atmospheric delays. Following encoding in State Space Representation (SSR) format, these parameters are broadcast to users (refer to Figure 1). Contrasted with RTK, PPP-RTK demonstrates clear advantages regarding bandwidth consumption, service coverage, and user scalability: it demands significantly less bandwidth, enables wide-area coverage, and imposes no restrictions on user capacity. This combination of attributes makes PPP-RTK particularly well-suited to the positioning requirements inherent to power industry applications. Building on this foundation, this study focuses on developing Inertial Measurement Unit (IMU)-assisted BDS multi-antenna PPP-RTK technology to achieve real-time high-precision positioning in regions with sparse reference stations.

The multi-antenna PPP-RTK technology discussed in this paper originated from the Matrix-Aided Precise Point Positioning (A-PPP) concept proposed by Teunissen [10]. This technology requires a multi-antenna BDS receiver system and state-space representation correction data streams. By precisely measuring the relative positions of the antenna array within the carrier platform, it offers two key advantages: (1) it enables direct or global positioning information acquisition, and (2) it enhances the redundancy of BDS observations, thereby effectively shortening convergence time and improving resistance to signal reflection interference.

For dynamic equipment positioning applications such as power inspection drones or robots, the study proposes a tightly coupled integration of multi-antenna PPP-RTK with high-precision gyroscope-based IMU measurements of acceleration and angular velocity [11]. This inertial data is fused with PPP-RTK through an engineered recursive Bayesian estimator. The estimator employs an Extended Kalman Filter (EKF) to simultaneously process parameters spanning multiple mathematical domains: real-valued states (position, velocity, inertial biases), integer-valued ambiguities (BDS carrier phase measurements), and SO(3) manifold parameters [12,13,14,15]. To address the geometric constraints of the SO(3) manifold, an error-state EKF formulation based on Lie group theory is adopted, where the estimated states reside on the manifold while their perturbations lie in the corresponding tangent space. This approach avoids the singularity issues associated with Euler angles and enhances both the numerical stability and computational efficiency of attitude estimation. For integer ambiguity resolution, the standard three-step approach is implemented: float solution estimation, integer ambiguity fixing, and fixed solution validation.

The positioning solution proposed in this study achieves centimeter-level accuracy in quasi-real-time dynamic positioning, effectively addressing the high-precision localization requirements for power industry applications in areas with sparse reference stations. The primary contributions and innovations of this research include three key aspects:

(1)

We develop an SSR parameter-based corrected multi-antenna PPP-RTK observation model that enhances observation redundancy through a multi-antenna array and incorporates dynamic attitude and displacement information from the IMU. This integration improves ambiguity resolution efficiency and positioning convergence speed, particularly in sparse reference station environments.

(2)

We establish a unified mathematical framework that integrates multi-antenna PPP-RTK with IMU through an Extended Kalman Filter (EKF) implementation. The framework is capable of uniformly processing multiple types of states, including position, velocity, attitude, IMU biases, and ambiguities, achieving tight coupling integration of PPP-RTK and IMU. This integration improves the system’s robustness in dynamic and occluded environments.

(3)

Through comprehensive validation combining field experiments and simulation tests, we systematically demonstrate the algorithm’s effectiveness in positioning accuracy, convergence time, and positioning stability, especially under challenging conditions such as frequent signal obstructions and sparse reference stations in complex terrains.

The remaining sections of this paper are organized as follows: Section 2 details the BeiDou multi-antenna PPP-RTK observation model and introduces a Kalman filter-based ambiguity resolution method for high-precision positioning terminals in the multi-antenna PPP-RTK system. Section 3 elaborates on experimental configurations and results analysis. Finally, Section 4 concludes the study and outlines future research directions.

To address these challenges, there is an urgent need to implement carrier-phase measurement technologies. While methods such as Real-Time Kinematic (RTK) and Precise Point Positioning (PPP) offer promising pathways for establishing high-precision spatiotemporal service systems in complex terrains, several critical research gaps remain inadequately addressed-particularly in the context of power industry applications under sparse reference station networks and challenging signal conditions.

For instance, Reference [16] demonstrated that network RTK can support centimeter-level horizontal accuracy and sub-decimeter vertical precision in hydraulic engineering applications, while Reference [17] proposed that RTK/INS integration helps reduce positioning errors in obstructed environments. Nevertheless, both approaches still face limitations in maintaining performance across large areas with sparse infrastructure. Similarly, Reference [18] showed that multi-system PPP improves point cloud accuracy in airborne inspection, and Reference [19] indicated that it enables structural monitoring; however, its convergence time and reliability deteriorate under frequent signal blockages and in sparse reference station scenarios.

A more promising approach lies in PPP-RTK technology, which leverages SSR corrections for atmospheric and orbit errors, offering wider coverage and lower bandwidth requirements than RTK. Recent studies have validated its feasibility in power inspection contexts: Reference [20] implemented UAV positioning with BDS PPP-RTK in complex terrains, and Reference [21] developed multi-system PPP-RTK/VIO integration to mitigate signal loss in urban or underground settings. Furthermore, References [22,23] demonstrated that incorporating IMU through tightly or loosely coupled schemes improves short-term navigation and system robustness during BDS outages.

However, two persistent research gaps hinder their deployment in remote or infrastructure-sparse power operations: (1) Slow convergence of PPP-RTK in sparse reference networks, limiting its utility in quasi-real-time dynamic applications. (2) Positioning drift and reliability degradation during frequent GNSS signal interruptions, especially in rugged or occluded environments where IMU-only navigation accumulates error rapidly.

To systematically address these issues, this study introduces a tightly coupled multi-antenna PPP-RTK/IMU integration framework. The proposed model enhances observation redundancy through a multi-antenna configuration and incorporates SSR-aided atmospheric corrections, accelerating ambiguity resolution and convergence under sparse networks. Moreover, an error-state EKF based on Lie group theory is employed to unify the processing of positioning, attitude, and ambiguity parameters, improving numerical stability and extending positioning continuity during signal outages.

These methodological advances are rigorously validated through static and dynamic experiments in challenging settings, demonstrating robust centimeter-level accuracy and improved convergence-effectively bridging the gap between high-precision requirements and operational constraints in power industry applications.

The proposed PPP-RTK technique is a recursive motion estimation algorithm that integrates multi-antenna BDS system data with IMU measurements. As illustrated in Figure 2, this system integrates three core modules within a unified framework:

2. Methods

This section elaborates in detail on the PPP-RTK modeling approach for multi-antenna BDS terminals, the Error-State Kalman filtering process, and the ambiguity resolution procedure for generating observations.

2.1. PPP-RTK Modeling

This section elaborates in detail on the PPP-RTK modeling approach for multi-antenna BDS terminals, the Error-State Kalman filtering process, and the ambiguity resolution procedure for generating observations. In the PPP-RTK observation model, aside from satellite orbit errors, clock offsets, and hardware delays, atmospheric delays (ionospheric and tropospheric) and multipath effects are the primary error sources affecting positioning accuracy. This study utilizes atmospheric delay products provided by SSR to apply undifferenced corrections to the main antenna, while for the auxiliary antenna, most atmospheric errors are eliminated through double-difference observations. Regarding multipath effects, due to their strong environmental dependency, an elevation-dependent stochastic model is employed for mitigation, specifically by incorporating an elevation-dependent weighting strategy into the observation noise model.

BDS pseudorange and phase observation equations from Satellite $s$ to Main Antenna $n$:

$$\left\{\begin{array}{l}{P}_{n,1}^s={\rho }_n^s+c\left(t_n-d{t}^{s,P1}\right)+M^s\cdot T_n+I_n^s+\epsilon \left(P_{n,1}^s\right)\\ P_{n,2}^s={\rho }_n^s+c\left(t_n-d{t}^{s,P2}\right)+M^s\cdot T_n+g\cdot I_n^s+h_{n,2}+\epsilon \left(P_{n,2}^s\right)\\ L_{n,1}^s={\rho }_n^s+c\left(t_n-d{t}^{s,L1}\right)+M^s\cdot T_n-I_n^s+{\lambda }_1\cdot {\tilde{a}}_{n,1}^s+\epsilon \left(L_{n,1}^s\right)\\ L_{n,2}^s={\rho }_n^s+c\left(t_n-d{t}^{s,L2}\right)+M^s\cdot T_n-g\cdot I_n^s+{\lambda }_2\cdot {\tilde{a}}_{n,2}^s+\epsilon \left(L_{n,2}^s\right)\\ S_{n,T}^s=M^s\cdot T_n+\epsilon \left(S_{n,T}^s\right)\\ S_{n,I}^s=I_n^s+\epsilon \left(S_{n,I}^s\right)\end{array}\right.$$

(1)

(1)

$P_{n,1}^s$ and $P_{n,2}^s$ represent the pseudorange observations for satellite $s$ received by master antenna $n$ on frequency 1 and frequency 2, respectively.

(2)

$L_{n,1}^s$ and $L_{n,2}^s$ are the carrier phase observations for the corresponding frequencies.

(3)

The geometric distance $\rho n^s$ is computed from the satellite coordinates $\left(X^s,Y^s,Z^s\right)$ and the antenna position $\left(X_n,Y_n,Z_n\right)$:${\rho }_n^s=\sqrt{{\left(X^s-X_n\right)}^2+{\left(Y^s-Y_n\right)}^2+{\left(Z^s-Z_n\right)}^2}$.

(4)

$c$ is the speed of light in vacuum; $t_n$ is the receiver clock error.

(5)

$d{t}^{s,P1}$, $d{t}^{s,P2}$, $d{t}^{s,L1}$, $d{t}^{s,L2}$ denote the SSR-provided combined satellite clock error and signal bias terms for each observation type.

(6)

$Tn$ represents the zenith tropospheric delay at the receiver; $M^s$ is the mapping function [24,25] that projects the zenith delay $Tn$ to the satellite $s$ line-of-sight direction.

(7)

$I_n^s$ is the line-of-sight ionospheric delay; $g=f_1^2/f_2^2$ is the ionospheric delay scaling factor between frequency 1 ($f_1$) and frequency 2 ($f_2$).

(8)

$h_{n,2}$ is the receiver pseudorange hardware delay (bias) of frequency 2 relative to frequency 1.

(9)

${\lambda }_1$ and ${\lambda }_2$ are the wavelengths of the corresponding frequency signals.

(10)

$\tilde{a}$ denotes the float ambiguity parameter, which encompasses both the integer ambiguity and the receiver- and satellite-end phase delays/biases.

(11)

The SSR atmospheric correction model provides estimates $S_{n,T}^s$ and $S_{n,I}^s$, representing slant-path signal delay due to the troposphere and ionosphere, respectively.

(12)

The random noise component affecting the relevant observable is designated as $\epsilon (\cdot )$.

The linearized form of the BDS observation equation from satellite $s$ to the master antenna can be expressed in matrix form as

$$\left[\begin{array}{c}{\tilde{P}}_{n,1}^s\\ {\tilde{P}}_{n,2}^s\\ {\tilde{L}}_{n,1}^s\\ {\tilde{L}}_{n,2}^s\\ S_{n,T}^s\\ S_{n,I}^s\end{array}\right]=\left[\begin{array}{ccccccc}{\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& 1& 0& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 1& M^s& g& 0& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& -1& {\lambda }_1& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& -g& 0& {\lambda }_2\\ 0& 0& 0& M^s& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}\delta {\mathbf{p}}_n\\ t_n\\ h_{n,2}\\ T_n\\ {\tilde{a}}_{n,1}^s\\ {\tilde{a}}_{n,2}^s\end{array}\right]$$

(2)

where

$$\left\{\begin{array}{l}{\tilde{P}}_{n,1}^s=P_{n,1}^s-{\rho }_n^s+c\cdot d{t}^{s,P1}\\ {\tilde{P}}_{n,2}^s=P_{n,2}^s-{\rho }_n^s+c\cdot d{t}^{s,P2}\\ {\tilde{L}}_{n,1}^s=L_{n,1}^s-{\rho }_n^s+c\cdot d{t}^{s,L1}\\ {\tilde{L}}_{n,2}^s=L_{n,2}^s-{\rho }_n^s+c\cdot d{t}^{s,L2}\end{array}\right.$$

(3)

$\delta {\mathbf{p}}_n=\left(\delta N_n,\delta E_n,\delta D_n\right)$ represents the position correction vector of the master antenna in the $n$-frame. In (1) and (2), the structure of the design matrix reflects the linear relationship between observations and state parameters, with its columns corresponding to position corrections, ambiguity parameters, atmospheric delays, etc.; the state vector contains parameters such as the main antenna position corrections, baseline vectors, and ambiguities. The observation noise is modeled as zero-mean Gaussian white noise, with its covariance matrix determined by the elevation-dependent model.

The Jacobian matrix ${\mathbf{J}}_n^s$ can be calculated as

$${\mathbf{J}}_n^s=\left[-{\displaystyle \frac{X^s-X_n}{{\rho }_n^s}},-{\displaystyle \frac{Y^s-Y_n}{{\rho }_n^s}},-{\displaystyle \frac{Z^s-Z_n}{{\rho }_n^s}}\right]$$

(4)

Let $C_{e~n}$ denote the rotation matrix from coordinate system $e$ to coordinate system $n$, with its specific form given by

$${\mathbf{C}}_{e~n}=\left[\begin{array}{ccc}-\sin \left(la\right)\cos \left(lo\right)& -\sin \left(lo\right)& -\cos \left(la\right)\cos \left(lo\right)\\ -\sin \left(la\right)\sin \left(lo\right)& \cos \left(lo\right)& -\cos \left(la\right)\sin \left(lo\right)\\ \cos \left(la\right)& 0& -\sin \left(la\right)\end{array}\right]$$

(5)

where $\left(lo,la\right)$ are the geodetic longitude and latitude of the antenna. Given the close proximity of the secondary antennas to the master antenna, the observation equation from satellite $s$ to auxiliary antenna $u$ can be linearized similarly to Equation (4):

$$\left[\begin{array}{c}{\tilde{P}}_{u,1}^s\\ {\tilde{P}}_{u,2}^s\\ {\tilde{L}}_{u,1}^s\\ {\tilde{L}}_{u,2}^s\end{array}\right]=\left[\begin{array}{ccccccc}{\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& 1& 0& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 1& M^s& g& 0& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& -1& {\lambda }_1& 0\\ {\mathbf{J}}_n^s{\mathbf{C}}_{e~n}& c& 0& M^s& -g& 0& {\lambda }_2\end{array}\right]\left[\begin{array}{c}\delta {\mathbf{p}}_u\\ t_u\\ h_{u,2}\\ T_u\\ {\tilde{a}}_{u,1}^s\\ {\tilde{a}}_{u,2}^s\end{array}\right]$$

(6)

Here, $\delta {\mathbf{p}}_u=\left(\delta N_u,\delta E_u,\delta D_u\right)$ is defined as the position correction vector of secondary antenna $u$ in the $n$-frame, and

$$\left\{\begin{array}{l}{\tilde{P}}_{u,1}^s=P_{u,1}^s-{\rho }_n^s+c\cdot d{t}^{s,P1}\\ {\tilde{P}}_{u,2}^s=P_{u,2}^s-{\rho }_n^s+c\cdot d{t}^{s,P2}\\ {\tilde{L}}_{u,1}^s=L_{u,1}^s-{\rho }_n^s+c\cdot d{t}^{s,L1}\\ {\tilde{L}}_{u,2}^s=L_{u,2}^s-{\rho }_n^s+c\cdot d{t}^{s,L2}\end{array}\right.$$

(7)

The dual-differenced BDS pseudorange and carrier phase observation equations between satellite pair $\left(s_1,s_2\right)$ and antenna pair $\left(n,u\right)$ can be expressed as

$$\left[\begin{array}{c}{\tilde{P}}_{nu,1}^{s_1{s}_2}\\ {\tilde{P}}_{nu,2}^{s_1{s}_2}\\ {\tilde{L}}_{nu,1}^{s_1{s}_2}\\ {\tilde{L}}_{nu,2}^{s_1{s}_2}\end{array}\right]=\left[\begin{array}{ccc}{\mathbf{J}}_n^{s_1{s}_2}{\mathbf{C}}_{e~n}& 0& 0\\ {\mathbf{J}}_n^{s_1{s}_2}{\mathbf{C}}_{e~n}& 0& 0\\ {\mathbf{J}}_n^{s_1{s}_2}{\mathbf{C}}_{e~n}& {\lambda }_1& 0\\ {\mathbf{J}}_n^{s_1{s}_2}{\mathbf{C}}_{e~n}& 0& {\lambda }_2\end{array}\right]\left[\begin{array}{c}{\mathbf{d}}_{nu}\\ a_{nu,1}^{s_1{s}_2}\\ a_{nu,2}^{s_1{s}_2}\end{array}\right]$$

(8)

In the above equations:

(1)

The vector ${\mathbf{d}}_{nu}=\delta p_u-\delta p_n$ defines the baseline vector from the master antenna to the secondary antenna in the $n$ coordinate frame.

(2)

The difference operator ${\mathbf{J}}_n^{s_1{s}_2}={\mathbf{J}}_n^{s_1}-{\mathbf{J}}_n^{s_2}$ represents the differenced form of the Jacobian matrix ${\mathbf{J}}_n^s$ for the satellite pair $(s_1,s_2)$.

(3)

The symbols ${\tilde{P}}_{nu,1}^{s_1{s}_2}$, ${\tilde{P}}_{nu,2}^{s_1{s}_2}$ and ${\tilde{L}}_{nu,1}^{s_1{s}_2}$, ${\tilde{L}}_{nu,2}^{s_1{s}_2}$ represent the dual-differenced pseudorange observations and dual-differenced carrier phase observations, respectively, on the first frequency and second frequency between the satellite pair $(s_1,s_2)$ and the station pair $(n,u)$ (i.e., the master-secondary antenna pair).

(4)

The variables $a_{nu,1}^{s_1{s}_2}$ and $a_{nu,2}^{s_1{s}_2}$ represent the two distinct frequencies in the dual -differenced integer ambiguity.

It is noteworthy that, (1) the pseudorange bias $h_{u,2}$ and the receiver’s clock bias $t_u$ have effectively been eliminated; and (2) due to the short baseline length (typically only a few meters) between the secondary antenna and the master antenna, the tropospheric delay and ionospheric delay are significantly reduced in Equation (8) and are therefore negligible.

In Equations (1) and (8):

The expression on the left-hand side is defined as the observed minus computed vector. The first term on the right-hand side is defined as the design matrix. And the second term on the right-hand side is defined as the state matrix. If the Observed Minus Computed Vectors (OMCV), design matrices, and state matrix from Equations (1) and (8) are combined, they can be expressed as

$$\left[\begin{array}{c}{\mathbf{y}}_n\\ {\mathbf{y}}_{nu}\end{array}\right]=\left[\begin{array}{ccccc}{\mathbf{H}}_n^{\delta \mathbf{p}}& \mathbf{0}& {\mathbf{H}}_n^{\mathbf{o}}& {\mathbf{H}}_n^{\mathbf{a}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{H}}_{nu}^{{\mathbf{d}}_{nu}}& \mathbf{0}& \mathbf{0}& {\mathbf{H}}_{nu}^{\mathbf{a}}\end{array}\right]\left[\begin{array}{c}\delta {\mathbf{p}}_n\\ {\mathbf{d}}_{nu}\\ {\mathbf{o}}_n\\ {\tilde{\mathbf{a}}}_n\\ {\mathbf{a}}_{nu}\end{array}\right]$$

(9)

In the above equation:

(1)

The non-differenced OMCV of the master antenna is denoted by ${\mathbf{y}}_n$.

(2)

${\mathbf{y}}_{nu}$ represents the dual-differenced OMCV for the auxiliary antenna pair $nu,u\in \left(1,\cdots ,N\right)$.

(3)

The subscript of the design matrix $\mathbf{H}$ are used to identify whether it corresponds to the master antenna $n$ or the auxiliary antenna pair $nu$, while its superscript identifies the parameter type, including position $p$, baseline vector ${\mathbf{d}}_{nu}$, and ambiguity $a$.

(4)

$\mathbf{o}$ represents other parameters of the main antenna, including atmospheric delay and receiver bias, among others.

The above is an analysis of the observation equations for the BDS system in multi-antenna scenarios.

2.2. Error-State Kalman Filter

Consider a general state estimation vector composed of an unknown vector $\mathbf{u}$ and a directional vector $\mathbf{q}$, expressed in the form ${\mathbf{x}}^{\mathrm{T}}=\left[{\mathbf{u}}^{\mathrm{T}},{\mathbf{q}}^{\mathrm{T}}\right]$, where ${\left(\cdot \right)}^{\mathrm{T}}$ denotes vector/matrix transposition. The error state is described by $\delta {\mathbf{x}}^{\mathrm{T}}=\left[\delta {\mathbf{u}}^{\mathrm{T}},\delta {\mathbf{\theta }}^{\mathrm{T}}\right]$. Here, $\delta \mathbf{\theta }$ represents a rotation vector. The Euclidean space of $\delta \mathbf{\theta }$ is associated with its Lie algebra $\mathbf{e}\phi \in {\mathbb{S}}^3$ (where $\delta \mathbf{\theta }$ denotes the unit vector of rotation and $\delta \mathbf{\theta }$ represents the rotation angle), connected through the isomorphism ${\left(\cdot \right)}^{\wedge }:{ℝ}^3\mapsto {\mathbb{S}}^3$. The Lie algebra is then linked to the 3D unit spherical manifold ${\mathcal{S}}^3$ via the exponential map [26,27]. The overall process is illustrated as follows:

$$\begin{array}{l}\delta \mathbf{\theta }\in {ℝ}^3\stackrel{{\left(\cdot \right)}^{\wedge }}{{\displaystyle \mapsto }}\mathbf{e}\phi \in {\mathbb{S}}^3\stackrel{{\left(\cdot \right)}^{\wedge }}{{\displaystyle \mapsto }}\delta \mathbf{q}\in {\mathcal{S}}^3\\ {\left(\delta \mathbf{\theta }\right)}^{\wedge }:\left\{\begin{array}{c}\mathbf{e}={\displaystyle \frac{\delta \mathbf{\theta }}{{‖\delta \mathbf{\theta }‖}^2}}\\ \phi ={‖\delta \mathbf{\theta }‖}^2\end{array}\right.,\exp \left(\mathbf{e}\phi \right):\left[\begin{array}{c}\cos \left(\phi /2\right)\\ \mathbf{e}\sin \left(\phi /2\right)\end{array}\right]\end{array}$$

(10)

Hence, $\mathbf{q}\left\{\mathbf{\theta }\right\}$ corresponds to the mapping between Euclidean space and unit quaternions. With the widespread application of Lie group theory tools, the composition of the nominal state and error state can be expressed as

$$\mathbf{x}={\mathbf{x}}_0\otimes \delta \mathbf{x}=\left\{\begin{array}{c}{\mathbf{u}}_0+\delta \mathbf{u}\\ {\mathbf{q}}_0\oplus \delta \mathbf{q}\left\{\delta \mathrm{\theta }\right\}\end{array}\right.$$

(11)

Consequently, the extended Kalman filter adjusts the structure of the standard Kalman filter under the chosen nonlinear parameterization framework. This reformulates the general state vector as Equation (11) to preserve the unit-norm constraint on quaternions.

The system employs a multi-rate asynchronous filtering architecture to process sensor data with varying sampling rates. The IMU, serving as the primary motion sensor, continuously integrates at a high frequency of 200 Hz to independently predict the system’s state vector $\mathbf{x}$ and error covariance matrix $\mathbf{P}$. This process constitutes the prediction step of the filter. When BDS observations arrive at specific 1 Hz timestamps, the system executes the update step of the filter. To ensure temporal synchronization, an interpolation-extrapolation strategy is adopted: at the time of BDS observation, the IMU state is precisely interpolated using adjacent IMU data to align with the GNSS timestamp, thereby constructing the observation residual $\mathbf{y}$. This mechanism of “high-frequency IMU prediction and low-frequency BDS update” leverages the short-term high accuracy of the IMU to smooth dynamic trajectories, while effectively correcting the accumulated IMU errors through BDS observations, achieving complementary advantages of the sensors.

As indicated by the previously defined observation model, the state matrix comprises elements such as position coordinates, receiver bias, relevant atmospheric delays, and carrier-phase ambiguities. However, within a multi-sensor architecture integrating BDS and IMU, the complete state matrix X must incorporate IMU-specific error states and kinematic parameters, including gyroscope calibration biases, the system’s velocity vector, and accelerometer calibration biases. To enhance conceptual clarity, these diverse parameters are partitioned into four distinct subsets, as detailed below:

$$\begin{array}{l}\mathbf{x}={\left[\begin{array}{cccc}{\mathbf{x}}_k^T& {\mathbf{x}}_{\mathrm{IMU}}^T& {\mathbf{x}}_{\mathrm{BDS}}^T& {\mathbf{x}}_{\mathrm{a}}^T\end{array}\right]}^T\\ {\mathbf{x}}_k={\left[\begin{array}{ccc}\delta {\mathbf{p}}^T& \delta {\mathbf{v}}^T& \delta {\mathbf{\theta }}^T\end{array}\right]}^T\\ {\mathbf{x}}_{\mathrm{IMU}}={\left[\begin{array}{cc}{\mathbf{b}}_a^T& {\mathbf{b}}_{\omega }^T\end{array}\right]}^T\\ {\mathbf{x}}_{\mathrm{BDS}}={\left[\begin{array}{ccc}{\mathbf{t}}_n^T& T_n& {\mathbf{i}}^T\end{array}\right]}^T\\ {\mathbf{x}}_{\mathrm{a}}={\left[\begin{array}{cccc}{\tilde{\mathbf{a}}}_n^T& {\mathbf{a}}_{1n}^T& \dots & {\mathbf{a}}_{Un}^T\end{array}\right]}^T\end{array}$$

(12)

where

(1)

${\mathbf{x}}_k$ contains the error states of relevant kinematic parameters such as position $\mathbf{p}$, velocity $\mathbf{v}$, and attitude $\mathbf{\theta }$.

(2)

${\mathbf{x}}_{\mathrm{IMU}}$ contains IMU parameters: accelerometer bias (${\mathbf{b}}_a$) and gyroscope bias (${\mathbf{b}}_{\omega }$).

(3)

${\mathbf{x}}_{\mathrm{BDS}}$ contains BDS-related parameters: receiver clock offset (${\mathbf{t}}_n$), tropospheric delay ($T_n$), and satellite-receiver path ionospheric delay ($\mathbf{i}$).

(4)

${\mathbf{x}}_{\mathrm{a}}$ contains ambiguity parameters: master antenna ambiguity (${\mathbf{a}}_n$) and secondary antenna-master antenna double-difference ambiguities $\left({\mathbf{a}}_{1n},\cdots ,{\mathbf{a}}_{Un}\right)$.

The subset ${\mathbf{x}}_{\mathrm{IMU}}$ of the state vector $\mathbf{x}$ includes the accelerometer bias ${\mathbf{b}}_a$ and the gyroscope bias ${\mathbf{b}}_{\omega }$. These biases are modeled as random walk processes. The parameters of their process noise covariance matrix are primarily configured based on the typical noise density (Accelerometer: $0.06 \mathrm{m}/{\mathrm{s}}^2/\sqrt{\mathrm{Hz}}$, Gyroscope: $0.0016 \mathrm{rad}/\mathrm{s}/\sqrt{\mathrm{Hz}}$) and bias instability (Accelerometer: $0.003 \mathrm{m}/{\mathrm{s}}^2$, Gyroscope: $0.0004 \mathrm{rad}/\mathrm{s}$) provided in the ADIS16505-2BMLZ datasheet. By estimating these biases online within the filter and compensating for them in the raw IMU measurements, the system effectively mitigates the impact of low-frequency bias drift—inherent to MEMS sensors—on navigation accuracy.

Leveraging IMU-derived specific force and angular rate data, the INS mechanization process computes estimates of an object’s attitude, velocity, and position. The state transition matrix for state vectors ${\mathbf{x}}_k$ and ${\mathbf{x}}_{\mathrm{IMU}}$ is derived from the INS relative state dynamics model as

$$\left\{\begin{array}{l}\delta \dot{\mathbf{p}}=-{\mathsf{\omega }}_{en}^n\times \delta \mathbf{p}+\delta \mathbf{v}\hfill \\ \delta \dot{\mathbf{v}}=-\left({\mathsf{\omega }}_{en}^n+2{\mathsf{\omega }}_{ie}^n\right)\times \delta \mathbf{v}+{\mathbf{C}}_b^n{\mathbf{f}}^b\times \delta \mathbf{\theta }+\delta {\mathbf{g}}^n+{\mathbf{C}}_b^n\delta {\mathbf{f}}^b\hfill \\ \delta \dot{\mathbf{\theta }}=-\left({\mathsf{\omega }}_{en}^n+{\mathsf{\omega }}_{ie}^n\right)\times \delta \mathbf{\theta }-{\mathbf{C}}_b^n\delta {\mathsf{\omega }}_{ib}^b\hfill \end{array}\right.$$

(13)

where

(1)

$\delta \dot{\mathbf{\theta }}$, $\delta \dot{\mathbf{p}}$, and $\delta \dot{\mathbf{v}}$ are the derivatives of Euler angle error, position, and velocity, respectively.

(2)

${\omega }_{en}^n$ is the projection of the angular rate of the navigation frame ($n$) relative to the Earth frame ($e$) onto the $n$-frame.

(3)

${\omega }_{ie}^n$ is the projection of the angular rate of the Earth frame ($e$) relative to the inertial frame ($i$) onto the $n$-frame.

(4)

$\delta {\mathbf{g}}^n$ denotes the gravity correction vector.

(5)

${\mathbf{f}}^b$ and ${\omega }_{ib}^b$ are the specific force measured by the accelerometers and the angular rate measured by the gyroscopes, respectively, in the body frame ($b$).

(6)

${\mathbf{C}}_b^n$ is the attitude direction cosine matrix transforming from the $b$ -frame to the $n$ -frame.

(7)

$\delta {\mathbf{f}}^b$ and $\delta {\omega }_{ib}^b$ represent the systematic errors of the accelerometers and gyroscopes, respectively, in the $b$ -frame.

During the correction phase, for a multi-antenna platform, the BeiDou Observation Residual Vector at time $t$ is constructed as

$${\mathbf{y}}_t={\mathbf{H}}_t{\mathbf{x}}_{t\left|t-1\right.}+{\mathbf{W}}_t$$

(14)

where ${\mathbf{x}}_{t\left|t-1\right.}$ is the state matrix predicted by the IMU. The observation noise ${\mathbf{W}}_t$ can be represented by additive white Gaussian noise with zero mean, and construct its covariance matrix ${\mathbf{R}}_t$ using the noise $\epsilon$, while considering the cross-correlation characteristics between the master and auxiliary antennas.

According to observation Equation (9), the design matrix ${\mathbf{H}}_t$ and the OMCV ${\mathbf{y}}_t$ are constructed as follows

$$\left\{\begin{array}{l}{\mathbf{y}}_t={\left[\begin{array}{cccc}{\mathbf{y}}_n^T& {\mathbf{y}}_{1n}^T& \dots & {\mathbf{y}}_{un}^T\end{array}\right]}^T\hfill \\ {\mathbf{H}}_t=\left[\begin{array}{cccc}{\mathbf{H}}_k& {\mathbf{H}}_{\mathrm{IMU}}& {\mathbf{H}}_{\mathrm{BDS}}& {\mathbf{H}}_{\mathrm{a}}\end{array}\right]\hfill \end{array}\right.$$

(15)

where

$$\begin{array}{l}{\mathbf{H}}_k=\left[\begin{array}{ccc}{\mathbf{H}}_n^{\delta \mathbf{p}}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{H}}_{1n}^{\delta \mathbf{\theta }}\\ ⋮& ⋮& ⋮\\ \mathbf{0}& \mathbf{0}& {\mathbf{H}}_{Um}^{\delta \mathbf{\theta }}\end{array}\right],{\mathbf{H}}_{\mathrm{IMU}}=\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\\ ⋮& ⋮\\ \mathbf{0}& \mathbf{0}\end{array}\right]\\ {\mathbf{H}}_{\mathrm{BDS}}=\left[\begin{array}{c}{\mathbf{H}}_n^{\mathbf{o}}\\ \mathbf{0}\\ ⋮\\ \mathbf{0}\end{array}\right],{\mathbf{H}}_{\mathbf{a}}=\left[\begin{array}{cccc}{\mathbf{H}}_n^{\mathbf{a}}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathbf{H}}_{1n}^{\mathbf{a}}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \ddots & \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& {\mathbf{H}}_{Un}^{\mathbf{a}}\end{array}\right]\end{array}$$

(16)

The calibration procedure refines the inertial measurement unit (IMU) derived state predictions by employing BDS observational data. Applying the Kalman filter algorithm yields adjustments to both the state matrix itself and its covariance matrix as

$$\left\{\begin{array}{c}{\widehat{\mathbf{x}}}_{t\left|t\right.}={\mathbf{K}}_t\left({\mathbf{y}}_t-{\mathbf{H}}_t{\mathbf{x}}_{t\left|t-1\right.}\right)\\ {\mathbf{P}}_{t\left|t\right.}=\left(\mathbf{I}-{\mathbf{K}}_t{\mathbf{H}}_t\right){\mathbf{P}}_{t\left|t-1\right.}\end{array}\right.$$

(17)

Among them, ${\mathbf{K}}_t$ is a matrix used to represent the Kalman gain. The state matrix update is then performed using the following equation:

$${\mathbf{x}}_{t\left|t\right.}={\mathbf{x}}_{t\left|t-1\right.}\otimes {\widehat{\mathbf{x}}}_{t\left|t\right.}$$

(18)

In Equation (18), ${\mathbf{x}}_{t\left|t\right.}$ denotes the float solution state vector prior to ambiguity resolution; ${\mathbf{x}}_{t\left|t-1\right.}$ represents the state vector predicted by the IMU; ${\widehat{\mathbf{x}}}_{t\left|t\right.}$ indicates the state vector obtained after calibrating the IMU-predicted state using BDS measurement data.

2.3. Ambiguity Resolution Method

Traditional ambiguity resolution methods primarily target ambiguity resolution scenarios without aiding, using single-frequency signals, and single-epoch processing. Under these conditions, the mathematical model often lacks sufficient strength, leading to integer ambiguity resolution difficulty that is highly dependent on the strength of the underlying model [28]. Therefore, to enhance the solving capability, attitude determination applications typically require prior analysis and integration of certain a priori information, including the properties of the rotation matrix and baseline length data, among others.

This research focuses on multi-epoch BDS ambiguity resolution aided by an IMU. It is evident that introducing IMU aiding, increasing the number of tracked satellites, accumulating more observation epochs, and utilizing multiple frequencies will significantly enhance the strength of the BDS mathematical model. Furthermore, during the modeling stage, the baseline length constraint is inherently satisfied by incorporating coordinate differences into the observation model via quaternion representation; simultaneously, the linearization process of the quaternion eliminates quadratic constraint terms.

Once the ambiguity is successfully fixed as the integer vector ${\overline{\mathbf{x}}}_{\overline{\mathrm{a}}}$, we can derive the ambiguity-fixed solutions as

$${\overline{\mathbf{x}}}_k={\mathbf{x}}_k\otimes {\mathbf{P}}_{\overline{\mathrm{a}},k}{\mathbf{P}}_{\overline{\mathrm{a}},\overline{\mathrm{a}}}^{-1}\left({\overline{\mathbf{x}}}_{\overline{\mathrm{a}}}-{\widehat{\mathbf{x}}}_{\overline{\mathrm{a}}}\right)$$

(19)

In Equation (19), ${\overline{\mathbf{x}}}_k$ represents the fixed solution kinematic parameter vector following ambiguity resolution; ${\mathbf{x}}_k$ denotes the kinematic parameters; ${\mathbf{P}}_{\overline{\mathrm{a}},k}$ and ${\mathbf{P}}_{\overline{\mathrm{a}},\overline{\mathrm{a}}}^{-1}$ are sub-blocks of the variance-covariance matrix for the state vectors used in ambiguity resolution, characterizing the degree of correlation (covariance) between ambiguities and other state parameters in the float solution; $\left({\overline{\mathbf{x}}}_{\overline{\mathrm{a}}}-{\widehat{\mathbf{x}}}_{\overline{\mathrm{a}}}\right)$ signifies the difference between the fixed and float solutions of the ambiguity vector after converting double-difference ambiguities to single-difference ambiguities.

The server-side generates precise atmospheric products using geodetic BDS data from regional networks and broadcasts them to user-end devices. At the user end, raw BDS observations are acquired through a multi-antenna platform, and the observation model is calibrated using IMU-predicted states. Subsequently, BDS multi-antenna PPP-RTK parameter estimation is performed by integrating precise satellite orbits, clock offsets, OSB, and atmospheric correction products.

It is worth mentioning that the BDS/IMU integrated positioning system designed in this paper adopts a tightly coupled architecture. Unlike loosely coupled systems, which treat GNSS and IMU as independent navigation systems and perform fusion at the position/velocity level, this scheme integrates measurements at the observation level. As shown in Equations (14)–(16), the raw BDS pseudorange and carrier phase observations (after SSR correction) are directly fed into the extended Kalman filter together with the IMU-predicted vehicle states. The core advantage of this tightly coupled approach lies in its ability to constrain the error states of the IMU using residual BDS observation information even during short periods when the number of visible satellites is fewer than four, significantly enhancing the system’s robustness and continuous operational capability in signal-obstructed environments.

3. Results and Discussion

To evaluate the positioning performance of PPP-RTK technology in areas with sparse reference stations, this study conducted experiments at the Beijing Campus of North China Electric Power University. A custom-built multi-antenna BDS PPP-RTK receiver was employed as the testing device, with detailed hardware configurations presented in

Table 1. Receiver Hardware Functional Module Configuration.

System Function	Subfunction	Description
Cortex-M7	Main chip	Core: Cortex-M7; Max frequency: 400 MHz (algorithm required); RAM: 1 MB; ROM: 2 MB
STM32H743VIT6	Microprocessor	–
Positioning Function	BDS Module	BDS Module:UM482
Gyroscope & Accelerometer	IMU Module	Model: ADIS16505-2BMLZ; Manufacturer: ADI (Melville, MA, USA)
Network Interface	Network PHY	10/100 M Ethernet PHY chip supporting RMII interface

. The data processing strategy is specified in

Table 2. Data Processing Strategies for BDS Multi-Antenna PPP-RTK Scheme.

Parameter Type	Processing Strategy
BDS Signal	BDS (L-band B11 frequency) pseudo-range and carrier-phase signals
Observation Noise	Elevation-dependent model: ${\left({\displaystyle \frac{\sigma }{\sin \left(Elev\right)}}\right)}^2$, where $\sigma =0.6 \mathrm{m}$, pseudo-range-to-phase noise ratio:100:1
Sampling Frequency	1 Hz
Cutoff Elevation Angle	15°
Ambiguity Resolution	BDS ambiguities resolved via LAMBDA method; ratio test threshold: 99.5%
Satellite Orbit, Clock Offsets, and Signal Biases	Corrected using SSR atmospheric products received from the BDS system
Atmospheric Delay	Suppressed via double-difference for secondary antennas; residual ionospheric/tropospheric delays estimated for the main antenna using SSR atmospheric products

. The multi-antenna BDS receiver used in the experiment features the STM32H743VIT6 as the main control chip, which is equipped with a Cortex-M7 core running at a maximum frequency of 400 MHz, along with 1 MB of RAM and 2 MB of ROM. The GNSS module employs the UM482, capable of receiving BDS L-band B11 frequency signals. The IMU module is the ADIS16505-2BMLZ from Analog Devices Inc. (Wilmington, MA, USA), with a sampling rate of 200 Hz. A quadrifilar helical antenna is utilized, offering a gain of better than 3 dBi. The receiver’s firmware version is V2.1.6, and the data sampling rate is set at 1 Hz. All equipment was calibrated in the laboratory to ensure time synchronization and data consistency.

The experiment continuously collected 6.5 h of observation data from 9:00 to 15:30 GMT on 25 April 2025. During the first three hours (9:00–12:00), static measurements were performed in open terrain, followed by dynamic testing (12:00–15:30) where the receiver underwent uniform motion through multiple shadowed areas. The GNSS receiver operated at a 1 Hz sampling rate while the inertial measurement unit (IMU) collected data at 200 Hz. During this measurement period, the BDS system received SSR correction information. As illustrated in Figure 3, these SSR products were generated through collaborative processing from three sparsely distributed reference stations around the campus. These stations collected multi-antenna BDS observation data over 24 h, with regional atmospheric products being resolved through PPP technology.

While the reference network itself with ~50–100 km baselines (Figure 3) satisfies the spatial sparsity criterion for PPP-RTK, the current validation is constrained by the single test location and limited duration. These factors should be considered when assessing the method’s readiness for operational deployment in all target environments.

3.1. Positioning Accuracy Analysis

In Figure 3, the RTK reference station combined with the RTKLib software (version 2.4.2) forms the corresponding RTK reference trajectory. And

Table 3. Data Processing Strategies for RTK Scheme.

Parameter Type	Processing Strategy
BDS Signal	BDS (L-band B11 frequency) pseudo-range and carrier-phase signals
Observation Noise	Elevation-dependent model: $a^2+{\left(b/\sin \left(Elev\right)\right)}^2$, where $a=0.002\mathrm{m},b=0.002\mathrm{m}$ pseudo-range-to-phase noise ratio: 100:1
Sampling Frequency	1 Hz
Cutoff Elevation Angle	15°
Ambiguity Resolution	BDS ambiguities resolved via LAMBDA method; ratio test threshold:2
Satellite Orbit, Clock Offsets, and Signal Biases	Corrected using Wuhan University (WUM) BDS precise orbit/clock products; signal biases calibrated via BDS OSB products
Atmospheric Delay	Suppressed via double-difference observations and ignored

illustrates the data processing strategies in RTKLib.

To validate the performance enhancement of the proposed PPP-RTK algorithm, this study comparatively analyzed two additional positioning techniques:

(a)

An IMU-aided single-antenna PPP-RTK positioning solution;

(b)

A conventional RTK positioning solution.

As illustrated in Figure 4, which presents the positioning errors in East, North, and Up (ENU) directions for three positioning schemes when the receiver is fixed at a reference station in a sparse open area, the proposed scheme (red) demonstrates significant improvements over scheme (a) (blue) and scheme (b) (green) across six 30-min observation intervals. The results show that, except for isolated intervals, the proposed scheme achieves markedly enhanced positioning accuracy and superior stability compared to scheme (a).

Table 4. Positioning Errors in ENU Directions for Different Schemes.

Positioning Scheme	E-Direction Error	N-Direction Error	U-Direction Error
Proposed PPP-RTK	3.12 ± 0.45 cm	3.71 ± 0.52 cm	6.23 ± 0.89 cm
Scheme (a)	7.74 ± 1.21 cm	8.21 ± 1.35 cm	11.31 ± 1.67 cm
Scheme (b)	37.26 ± 4.82 cm	35.19 ± 5.13 cm	42.53 ± 6.24 cm

reveals the average ENU errors of scheme (a) as 7.74 cm, 8.21 cm, and 11.31 cm, respectively, while scheme (b) exhibits decimeter-level errors of 37.26 cm, 35.19 cm, and 42.53 cm. This degradation in scheme (b) stems from increased residual spatially correlated errors due to the extended baseline distances between sparse reference stations and the receiver. In contrast, the proposed scheme achieves substantially lower average errors of 3.12 cm, 3.71 cm, and 6.23 cm. Comparative analysis indicates that scheme (a) outperforms scheme (b) with positioning accuracy improvements of 79.2%, 76.6%, and 73.4% in the ENU directions, demonstrating the effectiveness of PPP-RTK integration over standalone RTK in sparse reference station environments. Furthermore, the proposed PPP-RTK scheme achieves additional accuracy enhancements of 59.6%, 54.8%, and 44.9% compared to scheme (a) in respective directions. This superior performance can be attributed to the developed multi-antenna PPP-RTK mathematical model, which effectively improves ambiguity resolution success rates and reduces average positioning errors through enhanced error mitigation.

On the other hand, to validate the performance improvement of the integrated IMU unit in positioning stability within the proposed scheme, experimental tests were conducted by repeatedly moving the receiver into signal-obstructed shadowed areas, as indicated by the dashed segments in Figure 5. Notably, most significant positioning errors occurred shortly after exiting these shadowed zones. The results demonstrate that the proposed scheme effectively suppresses the pronounced positioning deviations observed in schemes (a) and (b) under such conditions.

This enhancement stems from the synergistic integration of IMU with PPP-RTK technology, which substantially improves positioning stability through several key error suppression mechanisms. When satellite signals are obstructed, the IMU compensates for positioning gaps by continuously estimating motion through real-time measurements of acceleration and angular velocity, leveraging the short-term high-precision characteristics of inertial navigation. Concurrently, the PPP-RTK framework provides high-accuracy correction parameters to mitigate IMU error accumulation.

The core of the performance gain lies in how this fusion architecture suppresses critical error sources. First, the integration of high-frequency IMU data provides continuous constraints on the platform’s motion, effectively “bridging” gaps in carrier-phase tracking during signal outages. This, combined with the numerically stable attitude representation offered by the Lie group-based EKF, drastically strengthens the kinematic model. A stronger model reduces the uncertainty of the float ambiguity estimates, which is the primary mechanism for the accelerated re-convergence and ambiguity re-fixing observed upon signal recovery. Second, the stable attitude estimation from the Lie group EKF refines the known geometric constraints between the multiple antennas. This reduces the coupling between attitude errors and position/ambiguity estimates, contributing directly to the overall accuracy and robustness. Third, in the sparse network conditions of this experiment, the multi-antenna double-difference observations inherently cancel out spatially correlated atmospheric delays. The improved system stability afforded by the tight coupling allows these double-difference residuals to be estimated more reliably, further suppressing their impact on positioning accuracy.

In summary, this bidirectional cooperation does not merely combine sensors; it creates a synergistic loop where the IMU and Lie group EKF provide a stable dynamic backbone. This empowers the multi-antenna PPP-RTK algorithm to resolve its key parameters more efficiently and reliably after signal interruptions, effectively reducing positioning drift, extending availability during outages, and accelerating convergence upon signal recovery. As evidenced by the visual results in Figure 6 and error analysis, the proposed scheme achieves remarkable improvements in both positioning accuracy and stability, particularly excelling in BDS signal-constrained environments. The attained positioning accuracy (horizontal errors below 5 cm) adequately fulfills the requirements for power grid operations in remote areas and reference station-sparse regions with complex terrain, demonstrating substantial practical value for critical infrastructure applications.

3.2. Convergence Time Analysis

Many application scenarios require the system to achieve rapid convergence within minutes. Therefore, besides positioning accuracy, convergence time is also considered another critical metric for evaluating the performance of a positioning scheme. This paper defines convergence time as the duration starting when the horizontal positioning error first falls below 0.05 m and remains continuously below this threshold for a sustained period.

As shown in Figure 6, repositioning tests were conducted every 30 min between 9:00 and 12:00, with convergence times recorded. In regions with sparse reference stations, the spatial correlation of atmospheric errors (ionospheric, tropospheric) significantly decreases. This leads to increased double-differenced residuals in conventional RTK solutions, substantially prolonging convergence times to over 20 min, and sometimes even preventing ambiguity resolution. Therefore, Solution (b) is excluded from comparison. In contrast, the proposed PPP-RTK technique, especially when utilizing multiple antennas, typically achieves centimeter-level positioning by resolving ambiguities within 2 min. Out of six tests, convergence time exceeded 2 min only once; the remaining five instances were under 2 min, yielding an average convergence time of 1.62 min. These convergence results can meet the demand for rapid, high-precision positioning in most power industry applications within reference-sparse regions. The observed variations in convergence time are likely attributable to differences in observation quality during the initial solution stage and the inherent challenge of achieving the required positioning accuracy under sparse base station conditions. Overall, compared to Solution (a), the proposed PPP-RTK solution achieves an average convergence time reduction of 65.7%.

Meanwhile, to further investigate the convergence time performance of PPP-RTK technology under signal obstruction, each period following a receiver’s passage through shadowed areas was treated as a convergence sample.

Table 5. Comparison of Convergence Times Among Different Schemes Under Signal Occlusion Conditions.

Average Convergence Time (Minutes)	Sample Size
	$0<\mathit{t}\le 1$	$1<\mathit{t}\le 2$	$2<\mathit{t}\le 4$	$\mathit{t}>4$
Proposed PPP-RTK	7	2	0	0
Scheme (a)	0	1	2	6

presents the average convergence times of the proposed scheme and Scheme (a). Convergence times were categorized into four intervals (in minutes): t ϵ [0, 1], t ϵ [1, 2], t ϵ (2, 4] and t > 4.

For Scheme (a), among 9 samples: 1 sample converged within t ϵ [1, 2], with a convergence time of 1.89 min; 2 samples converged within t ϵ (2, 4], with an average convergence time of 3.64 min; 6 samples fell into t > 4, with an average convergence time of 6.28 min.

For the proposed PPP-RTK scheme: 7 samples converged within t ϵ [0, 1], averaging 0.67 min; 2 samples converged within t ϵ [1, 2], averaging 1.12 min.

In summary, the proposed PPP-RTK scheme demonstrates rapid and stable convergence after passing through shadowed or obstructed areas. Compared with Scheme (a), the proposed PPP-RTK scheme achieves an 84.3% reduction in average positioning convergence time following signal obstruction. This sufficiently validates the superiority of the proposed method. While the proposed method demonstrates promising performance under the semi-sparse conditions of this study, it is recognized that the single-location experiment and limited duration may not fully capture its long-term robustness in vast, challenging terrains like deserts and Gobi regions. The findings primarily validate the algorithmic efficacy and integration framework. Future work will prioritize long-term, large-scale field trials across diverse geographic and climatic conditions, particularly in the remote western parts of China with truly sparse reference networks, to comprehensively verify the system’s operational reliability for power grid applications.

4. Conclusions

To address the pressing need for high-precision positioning in power grid operations within areas with sparse reference stations, this paper proposed a method integrating BeiDou multi-antenna PPP-RTK with an IMU. The approach combines the advantages of PPP and RTK, leveraging BeiDou SBAS signals and regional atmospheric corrections from sparse reference stations, while utilizing the multi-antenna redundancy and IMU dynamic compensation capability. An EKF-based estimation model was developed, incorporating quaternion-based attitude representation to enhance computational efficiency and interference resistance. Experimental results demonstrated that the method achieves centimeter-level positioning accuracy—with average errors of 3.12 cm, 3.71 cm, and 6.23 cm in the East, North, and Up directions, respectively—and reduces convergence time to an average of 1.62 min. Compared to single-antenna PPP-RTK without IMU assistance, the proposed method improves accuracy by up to 59.6% and maintains stable performance under signal occlusion.

Despite these promising results, this study has certain limitations. The experimental validation was conducted under sparse network conditions at a single geographic location over a limited duration, which may not fully represent the performance in more varied or extreme operational environments such as vast desert or Gobi regions. Additionally, the current system relies on a tactical-grade IMU, and its adaptability to low-cost MEMS-based IMUs has not been assessed. Future work will focus on addressing these limitations through several directions. First, we will explore the integration of low-cost MEMS IMUs to enhance the system’s economic feasibility for large-scale power industry deployment. Second, we aim to collaborate with infrastructure providers to investigate regional enhancement network construction, improving the availability and reliability of SSR corrections in truly sparse regions. Third, dynamic baseline recalibration techniques will be developed to maintain multi-antenna geometric constraints under long-term operational conditions. Finally, long-term reliability tests in extreme environments and optimization of multi-source data fusion algorithms will be carried out to extend the method’s applicability in scenarios such as automated power line inspection and infrastructure health monitoring.