This paper presents a Fast Covariance Intersection-based Adaptive Extended Kalman Filter (FCI-AEKF) framework for PPP-RTK and 5G integrated positioning. The framework addresses key challenges in GNSS-5G fusion, including limited use of PPP-RTK's high precision, slow error convergence from simple motion models, and distancedependent 5G measurement noise. It enables multi-rate fusion of PPP-RTK and 5G data, combining high-rate 5G updates with PPP-RTK corrections. A dynamic motion model switching strategy is proposed to adaptively select between constant velocity (CV) and constant acceleration (CA) models based on real-time vehicle dynamics. Additionally, a distance-based noise model is introduced to adjust measurement noise covariance, enhancing robustness under varying conditions. Experimental results on open-source GNSS data from The Hong Kong Polytechnic University demonstrate that the proposed method outperforms existing GNSS-5G fusion approaches in accuracy, robustness, and resistance to interference.
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To address these gaps, this paper introduces an adaptive model-switching strategy that dynamically selects appropriate motion models based on vehicle dynamics. In addition, a distance-aware noise model and a dynamic fusion weighting mechanism are designed to enable real-time covariance adaptation, enhancing robustness and positioning accuracy under diverse signal conditions.
Fig. 1 illustrates our proposed hybrid positioning system integrating PPP-RTK, 5G PRS, and CORS corrections for urban scenarios.
To this end, we propose a Fast Covariance Intersection-based Adaptive Extended Kalman Filter (FCI-AEKF) framework, with key contributions: • First integration of PPP-RTK and 5G in a multi-rate fusion framework, leveraging precise corrections for enhanced accuracy over SPP-based methods. • An adaptive motion model switching strategy to improve tracking under varying dynamics. • A distance-based noise model for real-time adjustment of measurement noise covariance, improving robustness. • A dynamic localization mode-switching strategy to maintain positioning continuity under degraded GNSS/5G conditions.
2.1. PPP-RTK Measurement Model The linearized observation equations for pseudorange and carrier phase of the th satellite on frequency are: Δ = ⋅ + ⋅ + + − + , +
ΔΦ = ⋅ + + − + + Here, and represent the direction cosine and position correction, respectively. is the speed of light, is the receiver clock bias, and are mapping functions, and , denote tropospheric and ionospheric delays. is carrier phase ambiguity, and , are Gaussian noise terms. Satellite clock and relativistic efects are assumed corrected [ 7].
in (
The pseudorange and carrier phase diferences are further expressed as: Δ = ΔΦ = − √( − ,−1 )2 + ( − ,−1 )2 + ( − ,−1 )2 − √( − ,−1 )2 + ( − ,−1 )2 + ( − ,−1 )2 where ( , , )and ( ,−1 , ,−1
, ,−1 )are satellite and receiver positions.
The PPP-RTK state vector is then:
−
= (
,
)
(
As signal acquisition is not the focus of this work, technical details are omitted. When the 5G receiver detects signals at time , the 5G measurement model is: Z5 [] = [ , , ] = h5 (s ) +n5 [] s = [X , V , A , , ]
T
azimuth and elevation, respectively. n5 [] represents measurement noise.
Here,
= − 1 is the time-diference between base stations and 1. The angles and denote where X , V , and A are the 3D position, velocity, and acceleration, respectively; and denote
h5 (s ) =⎢ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎢arctan ( − 1 +
arctan ( Δ ) Δ Δ √(Δ )2 + (Δ )2 ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ )⎥ where is the distance to base station : = √(Δ )2 + (Δ )2 + (Δ )2
Given low-cost 5G oscillators, clock drift relative to GNSS must be compensated. The clock ofset evolves as:
= −1 + Δ = ⋅ −1 + where is set to 1, and ∼ (0, 2)is white Gaussian noise. 3. Proposed Approach We propose a GNSS-5G hybrid positioning framework leveraging multi-rate fusion, designed to enhance positioning accuracy and robustness. The framework integrates the following core components: • Dynamic Motion Model Switching: An adaptive strategy that switches between constant velocity (CV) and constant acceleration (CA) models based on real-time vehicle dynamics, ensuring accurate state prediction in varying motion conditions. • Distance-Dependent Noise Modeling: A noise adaptation mechanism that adjusts observation noise covariance in the Kalman filter according to the varying distances between the user and 5G base stations, improving the reliability of 5G-based measurements. • Fast Covariance Intersection (FCI) Fusion: An eficient fusion algorithm that combines GNSS and 5G positioning estimates without requiring iterative optimization, ensuring robust performance even under partial observability.
Furthermore, we incorporate a multi-rate mode switching mechanism that dynamically selects the optimal positioning strategy based on the availability and quality of GNSS and 5G signals. This design ensures consistent, reliable performance across complex urban environments. 3.1. Dynamic Motion Model Switching We propose a Dynamic Motion Model Switching (DMMS) framework that adaptively switches between CV and CA models by analyzing velocity and acceleration. The selection rule is:
Model Selection = ⎪ ⎩ ⎧CV Model, ⎨⎪CA Model, |V ⋅ A | ‖V ‖2 + |V ⋅ A | ‖V ‖2 + < ≥
s = A ⋅ s−1 + B + W where A is selected as either ACV or ACA, depending on the switching rule, and W is Gaussian process noise.
The CV model uses a transition matrix ACV that accounts for position, velocity, and clock drift: ACV = ⎢⎢⎡⎢I300×3 ⎣ 0 Δ ⋅ I3×3
I3×3
0 Δ −1 ⋅ I3×3 0 0 0 0 0
⎤ 0 ⎥ 0 ⎥⎥ , A ⎦
A = [
1 Δ
0
Here, A models clock bias as in (
where Q , U, and L are standard formulations capturing process noise.
QCV = [ 0
Q 0 0 U 0 0
L ACA = [0
I Δ ⋅ I 0 Δ22 ⋅ I Δ ⋅ I ] 3.2. Dynamic Noise Covariance Modeling
This simplified formulation retains key equations and modeling insights while reducing redundancy. nents, each modeled as independent zero-mean Gaussian variables: Observation noise is a critical factor afecting positioning accuracy. Traditional EKF methods typically assume fixed noise covariance, which neglects variations caused by changes in sensor-target geometry and environmental conditions. To address this, we incorporate a distance-dependent noise model based on the Cramér–Rao Lower Bound (CRLB), improving the robustness of 5G-based localization.
At time step , the 5G measurement noise vector n5 [] comprises TDOA and DOA noise compo ,1 ∼ (0, ,21 ), , ∼ (0, ,2 ), , ∼ (0, ,2 ) The CRLBs for TOA and DOA are given by [9, 10, 11, 12]: 2 = 2 = 4 2 ⋅ ⋅ 3 ⋅ SNR , ,2 =
2 8 2 ⋅ ⋅ ⋅
SNR 16 2 ⋅ ⋅ 10 /10 (18) (19) (20) (21) (22) (23) (24) (25)
2 0 R 0 R = [ 0
R 0 0 with the process noise covariance matrix QCA defined as: = −1 + , = 1 ∶ = −1 + where denotes the GNSS epoch index (starting from 0), and =
5 represents the ratio of 5G and DOA: where = 1 is the reference distance.
By combining these relationships, we obtain the distance-dependent noise variances for TDOA and 2 = 2 (
) , 2 = 2 (
) , 2 = 2 (
)
This distance-aware model dynamically adjusts measurement noise according to the relative geometry between the user and 5G base stations, enhancing localization reliability and adaptability under varying signal conditions. 3.3. FCI-AEKF-based filtering process 5G systems typically rely on GPS as the primary time synchronization source, with synchronization errors generally within tens of nanoseconds [15]. In real vehicular environments, timing deviations may be negligible, and thus, this work assumes perfect synchronization for simplicity. In our fusion framework, the GNSS and 5G measurements are processed in two stages, each at diferent sampling rates.
In the first stage of FCI-AEKF, the time index of 5G measurements is defined as:
3.3.1. First Stage of FCI-AEKF
In the first stage, the FCI-AEKF predicts states using 5G TDOA and DOA measurements based on (
P5− [] =AP5+ [ − 1]AT + Q5 [] K5 [] =P5− []H5⊤ [](H5 []P5− []H5⊤ [] +R[])−1 s5+ [] =s5− [] +K5 [](Z5 [] −h5 (s5− []))
P5+ [] =(I − K5 []H5 [])P5− [] where H5 []is the Jacobian matrix of h5 . 3.3.2. Second Step of FCI-AEKF The second stage uses s5+ []as the reference position X and solves the PPP-RTK state via weighted least squares:
sppp-rtk[] = (HsatRs−a1tHsat) −1
HsatRs−a1tysat[] Pppp-rtk[] = (HsatRs−a1tHsat) −1 P[] = ( 5 [] P5− 1[] + sat[] Ps−a1t[] ) −1 (26) (27) (28) (29) (30) (31) (32) (33) (34) (35)
5 [] = ‖P5− 1[] + Ps−a1t[]‖ − ‖ Ps−a1t[]‖ + ‖ P5− 1[]‖ 2‖P5− 1[] + Ps−a1t[]‖
, sat[] = 1 − 5 [] ŝ[] = 5 P[] P5− 1[] s5 [] + satP[] Ps−a1t[] ssat[]
Here, ssat[] = X + X . The fused estimate ŝ[] is updated at the satellite sampling rate sat. 3.4. Positioning Mode Switching Strategy To enhance multi-rate GNSS/5G positioning, we propose a dynamic mode switching strategy that leverages high-rate 5G TDOA/DOA measurements and adapts to variations in GNSS and 5G signal conditions. The system operates in three modes according to signal availability: (i) GNSS/5G fusion mode, where 5G predictions are fused with GNSS updates using FCI-AEKF; (ii) 5G-only mode, activated when fewer than six GNSS satellites are visible; and (iii) PPP-RTK mode, used when fewer than two line-of-sight (LOS) 5G base stations are available. This adaptive mechanism ensures robust and continuous positioning across diverse and challenging environments, maintaining high accuracy even under degraded signal conditions.
The multi-rate switchover scheme for GNSS and 5G hybrid positioning is detailed in Algorithm 1. Algorithm 1 Multi-rate switchover algorithm for GNSS/5G hybrid positioning
Output: Fused positioning state ̂+ [] 1: for each GNSS epoch = 1 to do 2: Compute RSRP of all base stations 3.5. Results and Discussion 3.5.1. Experiment Setup To evaluate the proposed FCI-AEKF algorithm, a vehicle-based field experiment was conducted around the Haidian campus of Beijing Normal University. A GNSS-5G receiver was mounted on a car, and GNSS measurements were collected at 1 Hz during driving. Distances and angles from simulated 5G base stations to the user equipment (UE) were computed based on ground-truth positions. Synthetic noise was then added to emulate realistic 5G measurement errors. Hybrid positioning was performed by fusing actual GNSS observations with simulated 5G data. 3.5.2. Performance Comparison 3.6. Conclusions This paper presents the FCI-AEKF framework for GNSS/5G hybrid positioning. By fusing high-rate 5G and GNSS data, the algorithm efectively improves positioning accuracy while maintaining low computational complexity. It also integrates an adaptive motion model switching mechanism and a distance-based noise model to enhance robustness.
Field experiments confirm that FCI-AEKF outperforms conventional GNSS-5G methods, providing lower errors and higher reliability, particularly in challenging environments with limited GNSS visibility.
Overall, the proposed FCI-AEKF demonstrates strong potential for improving positioning performance and system stability in complex real-world scenarios, ofering a promising solution for precise GNSS/5G integration.
This work was supported by the National Key Research and Development Program of China underGrant No.2022YFB3904700.
Declaration on Generative AI The authors declare that AI tool was used to assist in improving the language fluency of this paper. All contents, ideas, and conclusions are the authors’ own, and the AI tool did not contribute to the scientific results or analysis.