A Meta-Reinforcement Learning Method for Enhancing GNSS Positioning by Integrating Spatiotemporal Correlations

2.1 ML-Based GNSS Modeling

Significant research has explored ML techniques for modeling GNSS signal reception, with notable progress in LOS/NLOS differentiation (Mohanty & Gao, 2024). Phan et al. (2013) employed support vector machines using satellite elevation angle and altitude to distinguish NLOS signals. Yozevitch et al. (2016) developed a robust decision tree model incorporating C/No, elevation angle, and pseudorange measurements for NLOS identification. R. Sun et al. (2019) proposed a classification framework integrating multiple complementary error-inducing features for LOS/NLOS discrimination, applying principal component analysis for dimensionality reduction followed by an adaptive neuro-fuzzy inference system for classification. L. Li et al. (2024) introduced a visually augmented random forest (RF) classifier with factor analysis-based error correction and enhanced NLOS detection by extracting diverse GNSS features, including pseudorange, carrier phase, C/No, and elevation angle, significantly improving positioning accuracy and reliability in complex urban environments.

To mitigate GNSS positioning errors, researchers have proposed various effective techniques. R. Sun et al. (2021) extended upon previous work by introducing a gradient-boosted decision tree algorithm that efficiently corrects multipath-induced NLOS pseudorange errors, improving positioning accuracy. Similarly, X. Zhang et al. (2024) developed an innovative LightGBM approach using multiple GNSS features, including the signal-to-noise ratio, elevation angle, pseudorange/carrier-phase consistency, code-minus-carrier values, and multipath combination observations, to detect and mitigate NLOS effects. W. Li et al. (2024) proposed a decision tree method for GNSS amplitude scintillation detection using geodetic receivers, extracting scintillation-relevant features and applying ML classification to enhance accuracy. P. Xu et al. (2024) modeled GNSS errors using a factor graph framework to enable robust probabilistic inference to identify and suppress corrupted signals, reducing the impacts of multipath and NLOS. In a related study, Özarpacı et al. (2024) emphasized the significance of tectonic structures and employed four ML algorithms, i.e., Gradient Boosting Machines (GBM), LightGBM, Random Forest (RF), and Extreme Gradient Boosting (XGBoost), to model velocity fields associated with different tectonic plates, enhancing predictive performance for geophysical applications. Despite promising results, ML-based methods rely on manual feature engineering and static modeling, hindering dynamic adaptation to complex multipath environments and thereby limiting GNSS positioning accuracy and generalization capability.

2.2 DL-Based GNSS Modeling

As a specialized subset of ML, DL has been increasingly applied to measurement-domain modeling for mitigating GNSS positioning errors. Several studies leverage DL techniques for error correction and performance enhancement. Kanhere et al. (2022) developed a DL network based on a fixed transformer architecture to estimate position corrections by processing LOS vectors and pseudorange residuals. Weng et al. (2024) proposed a neural network model called PrNet based on a multilayer perceptron for pseudorange bias correction that extracts features including signal strength, carrier phase, Doppler frequency, and transmission and reception times from raw GNSS measurements to train the model. Wu et al. (2024) introduced a transformer-based GNSS displacement estimation algorithm that learns nonlinear relationships between raw measurements and position changes, using a denoising autoencoder for preprocessing and adaptively fusing conventional single-point positioning (SPP) to improve positioning accuracy through dynamic fusion.

To optimize GNSS positioning in specific environments, various advanced approaches have been developed. Zeng et al. (2025) proposed a dual self-attention network model to separately process satellite observation features and spatiotemporal correlations, improving traditional elevation-angle-based positioning results by incorporating NLOS identification and SPP. Quan et al. (2018) presented a CNN architecture with sparse autoencoders to enhance GPS multipath detection, using GPS signal characteristics such as signal strength and phase delay. W. Sun et al. (2024) developed a hybrid DL model integrating CNNs, bidirectional gated recurrent units, and attention mechanisms for predicting navigation satellite clock deviations. The Beluga whale optimization algorithm is used to perform hyperparameter optimization, demonstrating superior predictive accuracy over standalone DL architectures. Despite the promise of DL approaches, their performance often degrades when faced with unseen or highly dynamic environmental conditions.

2.3 RL-Based GNSS Modeling

RL has recently gained increasing attention in GNSS positioning research. As a specialized branch of ML, RL is capable of dynamically modeling complex and time-varying environments through continuous interaction and feedback. Building upon the dynamic learning capabilities of RL, several studies have explored its application to enhance the performance of GNSS positioning. Zhang and Masoud (2021) implemented an RL-based positioning model using the asynchronous advantage actor-critic (A3C) algorithm, enabling adaptive learning and environmental modeling under evolving GNSS conditions. Tang et al. (2024) developed an RL algorithm incorporating a non-convex logarithmic regularizer and utilizing dictionary learning to mitigate the instability commonly observed in deep neural networks (DNNs), effectively reducing state correlations and suppressing interference during learning. Zhao et al. (2024) proposed a sparse representation-based multi-view DRL-based model for GNSS positioning correction. This model employs attention-based multi-view fusion to integrate multiple observational perspectives and uses sparse representation to mitigate height-related observation interference, resulting in substantial improvements in positioning performance.

In summary, existing DL-based GNSS positioning models have demonstrated strong environmental feature learning capabilities. However, their adaptability to dynamic and rapidly changing environments remains limited. In contrast, RL-based models possess the ability to learn and adapt to environmental variations through continuous interaction. Nevertheless, RL-based approaches also face critical limitations when applied to new trajectories. These limitations include the need for extensive training time, limited capacity to transfer previously acquired knowledge for accelerated learning, and restricted generalization performance across diverse scenarios (Panda & Guo, 2025). To overcome these limitations inherent to single-task RL, meta-RL offers a promising direction. Meta-RL aims to train agents that can quickly adapt to new tasks by learning transferable knowledge from a distribution of related tasks, thus directly addressing the needs for fast adaptation and strong generalization in dynamic GNSS scenarios. To address these challenges, this study proposes a meta-RL GNSS positioning method by integrating spatiotemporal correlations, following a four-stage framework: (1) CNN-LSTM networks are used to extract spatiotemporal features from GNSS measurements; (2) a POMDP is employed to model interactions between the environment and the receiver; (3) a PPO-based DRL agent is designed to learn adaptive positioning strategies; and (4) the ProMP algorithm is applied to enable rapid adaptation to new environments while preserving robust generalization capability.

3.1 Feature Extraction

Multiple data sets are constructed by selecting various GNSS positioning environments including feature matrices of satellite receiver parameters and positioning information. Because GNSS positioning requires signals from at least four satellites, the four visible satellites with the highest elevation angles are selected from the receiver. For each satellite, seven features are extracted: pseudorange residual, three LOS vector components, GDOP, C/No, and elevation angle. The feature matrix is structured with a fixed size of 4 × 7 dimensions, corresponding to four satellites and seven features per satellite. Each column represents a satellite, while each row corresponds to a specific feature. The detailed features are described below.

1. Pseudorange Residual: The pseudorange residual serves as a critical metric in satellite navigation systems for assessing positioning accuracy. This term represents the discrepancy between the measured pseudorange value and its theoretically expected value, mathematically expressed as follows:

   δρm=ρm−(rm−br)1

   where ρ^m represents the pseudorange measurement of the m-th satellite, b_r denotes the clock bias of the receiver, and r^m represents the geometric distance from the m-th satellite to the receiver:

   rm=(x−xm)2+(y−ym)2+(z−zm)22

   Here, [x, y, z] denotes the initial estimated position of the receiver in the Earth-centered, Earth-fixed (ECEF) coordinate system, while [x^m, y^m, z^m] indicates the position of the m-th satellite in ECEF coordinate system.

2. LOS Vector: The LOS vector characterizes the direct geometric connection between the satellite and the receiver. Let [gxm,gym,gzm] denote the components of the unit LOS vector from the m-th satellite to the receiver. Then, in the ECEF coordinate system, the x, y, and z components are defined as follows:

   gxm=x−xm(x−xm)2+(y−ym)2+(z−zm)23

   gym=y−ym(x−xm)2+(y−ym)2+(z−zm)24

   gzm=z−zm(x−xm)2+(y−ym)2+(z−zm)25

3. GDOP: The GDOP is a parameter that quantifies how the geometric distribution of satellites influences the positioning accuracy. A smaller GDOP value corresponds to a more favorable satellite geometry and a higher positioning accuracy. Conversely, a larger GDOP value corresponds to a poorer satellite geometry and a lower positioning accuracy. Typically, an optimal GDOP value approaches one, indicating highly favorable satellite geometry for precise positioning. The GDOP is mathematically defined as follows:

   GDOP=(tr(GTG)−1)6

   G=[gx1gy1gz11gx2gy2gz21gx3gy3gz31gx4gy4gz41]7

   where G^T, G^–1, and tr(G) denote the transpose, inverse, and trace of a matrix G, respectively.

4. C/No: C/No is a critical metric for discriminating between LOS and NLOS signals. This parameter evaluates the signal quality through the ratio of received carrier power to noise spectral density. An elevated C/No value indicates strong signal strength, minimal noise interference, and superior signal quality. Conversely, a diminished C/No value reflects weaker signal strength, increased noise levels, and degraded signal quality.

5. Elevation Angle: The elevation angle of a satellite is defined as the angular elevation between the position of the satellite and the receiver, specifically representing the vertical angle of the satellite relative to the receiving device. The elevation angle of the satellite serves as a reliable indicator of the quality of pseudorange measurement. An elevation angle of less than 15° typically correlates with significant NLOS errors owing to the increased probabilities of atmospheric propagation and obstruction. Conversely, an elevation angle of more than 30° typically exhibits mitigated multipath effects through an improved geometry of signal reception.

The four highest-elevation satellites are selected to meet the minimum requirement for 3D positioning and compensation while effectively mitigating severe multipath effects, NLOS reception, and atmospheric delays typically associated with low-elevation signals. The seven selected features are largely complementary, including signal quality, measurement consistency, and observation geometry. Furthermore, the complete set of ECEF LOS vectors is preserved to maintain the physical integrity of the 3D spatial geometry, playing a critical role in defining the topological constraints for position estimation.

3.2 CL-PPO Positioning Model

A POMDP is introduced to model the environmental interactions. Figure 2 illustrates the structural framework of the CL-PPO positioning model. This model is composed of two main components: the CNN-LSTM model and the PPO model, which are described in detail below.

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FIGURE 2

Structural framework for the CL-PPO positioning model

3.3 CL-MPPO Model

The CL-PPO model serves as the base learner within the meta-learning framework to stabilize training and improve sample efficiency. However, when encountering new GNSS positioning environments, CL-PPO typically requires extensive training time, cannot effectively leverage prior knowledge, and exhibits limited generalization. To address these limitations, the proposed method integrates the ProMP algorithm with the CL-PPO model and trains this model on diverse GNSS positioning trajectories to learn shared meta-parameters θ that capture cross-environment patterns, enabling effective generalization. Consequently, when initialized with these meta-parameters and presented with a new trajectory, the model rapidly adapts to the new trajectory distribution. The CL-MPPO model process consists of two stages: meta-training and meta-testing, as detailed in Algorithm 1. In the meta-training phase, multiple distinct GNSS positioning environments are randomly selected. The optimization process is organized into two hierarchical levels:

1. In the meta-training stage, the inner-loop optimization algorithm of the CL-MPPO is adopted. For each sampled trajectory T_i, the updated parameter set θi′ is obtained via a single-step gradient descent, expressed as follows:

   θi′=θ−α∇θL(Ti,θ)11

   where θ denotes the meta-parameters prior to adaptation and θi′ represents the trajectory-adapted parameters after one inner-loop update. α is the inner-loop learning rate, which controls the adaptation speed to each trajectory. The function L(Ti,θ) defines the trajectory-specific proximal loss for T_i, measuring how well the current policy parameters θ perform on the sampled trajectory, which is formally defined as follows:

   L(Ti,θ)=E[min(rt(θ)A^t,clip(rt(θ),1−ϵ,1+ϵ)A^t)]12

   rt(θ)=πθ(at∣st,xtobs )πθold (at∣st,xtobs )13

   where E denotes the expectation, representing the average over the distribution of the minimum between the importance sampling ratio and the clipped importance sampling ratio. r_t(θ) is the importance sampling ratio, which measures how the current policy differs from the old policy. A^t denotes the GAE at time step t, with discount factor γ and GAE parameter λ. ϵ is the PPO clipping coefficient, which constrains large policy updates and stabilizes training. The operator clip(⋅) limits the policy ratio to (1−ϵ,1+ϵ), enforcing a trust region that prevents destructive updates. πθ(at∣st,xtobs ) denotes the probability of selecting action a_t given the current state s_t and the GNSS satellite observation feature matrix xtobs, under the current policy parameters θ. Correspondingly, π_θ denotes the action probability distribution of the current policy, whereas πθold represents the probability distribution of the previous policy used to collect trajectories for PPO. This inner-loop optimization allows the policy to perform rapid adaptation to each trajectory T_i by leveraging both state observations and satellite features.

2. After inner-loop updates have been completed for all sampled trajectories, the CL-MPPO algorithm proceeds to the outer-loop meta-update, which refines the meta-parameters θ. To prevent overfitting to heterogeneous trajectory distributions and enhance cross-trajectory stability, a ProMP-inspired proximal constraint is introduced, ensuring that the updated meta-parameters do not deviate excessively from the previous values. Formally, the meta-update can be expressed as the following constrained optimization problem:

   minθ∑TiL(Ti,θi′), s.t. ‖θ−θold ‖2≤δprox14

   where δ_prox controls the maximum allowed deviation and θold denotes the meta-parameters before the update. The meta-loss for a specific trajectory T_i is defined as follows:

   Lmeta (Ti,θi′)=Eτ∼πθi′[L(Ti,θi′)]15

   where τ denotes trajectories sampled under the trajectory-adapted policy πθi′ and L(Ti,θi′) evaluates the effectiveness of the adapted parameters in trajectory T_i. This loss evaluates how well the meta-parameters support fast adaptation to each trajectory and serves as the optimization objective for the meta-update. In practical implementation, the proximal constraint is enforced via gradient clipping to stabilize the meta-update. The resulting update rule is as follows:

   θ←θ−β⋅clip(∇θ∑TiLmeta (Ti,θi′),−δclip,δclip )16

   where β is the meta-learning rate controlling the magnitude of the metaparameter update. δ_clip is the gradient clipping threshold, which enforces the proximal constraint by limiting the update magnitude and stabilizing meta-training. ∇θΣTiLmeta (Ti,θi′) computes the gradient of the aggregated meta-loss across all sampled trajectories with respect to the original meta-parameters θ. This design effectively improves cross-trajectory generalization by stabilizing the outer-loop updates and preventing overfitting to individual trajectories. Through multiple iterations of the inner-loop update (Equation (11)) and the outer-loop update (Equation (16)), the meta-parameters θ gradually converge. The final converged parameters are denoted as θ*, which are used as initialization for the subsequent metatesting phase.

3. In the meta-testing phase, the learned meta-parameters θ* serve as the initialization for adapting to previously unseen trajectories. Given a new trajectory T_new, only the inner-loop adaptation (Equation (11)) is performed to rapidly obtain trajectory-specific parameters θnew ′ without any outerloop update. Starting from θ*, the model performs a few gradient-based fine-tuning steps to quickly and effectively adapt to the new environment, thereby learning the MPPO decision model. The adapted parameters θnew ′ define a Gaussian policy that generates position-correction actions:

   θnew′=θ∗−α∇θL(Tnew ,θ∗)17

   πθnew′(at∣st,xtobs )=N(at∣μθnew ′(st,xtobs ,at−1,rt−1),σθnew′(st,xtobs ,at−1,rt−1))18

   where the mean μθnew′ and standard deviation σθnew′ are jointly parameterized by neural networks, conditioned on the current state s_t, the GNSS satellite observation feature matrix xtobs, and the historical action-reward pair (at−1,rt−1).N(⋅∣μ,σ) represents a Gaussian distribution with mean μ and standard deviation σ. This meta-testing procedure demonstrates the framework’s ability to leverage prior knowledge from meta-training and to achieve rapid, effective adaptation in previously unseen GNSS positioning environments.

4.1 Experimental Data Set

This paper utilizes an open data set published by Google in 2022 (Fu et al., 2022), which contains both raw GNSS measurements and ground-truth positioning information. The data set comprises the position information collected by identical mobile GNSS devices across diverse public environments in the San Francisco Bay and Los Angeles areas. It encompasses open spaces, mountainous forests, urban streets, and densely built metropolitan regions, thereby covering a wide spectrum of GNSS positioning scenarios. The data set contains both open-sky and light urban traces. Light urban roads, which are commonly used as an initial benchmark in GNSS learning studies, introduce realistic NLOS and multipath effects through signal obstruction and reflection. This feature, along with the data set’s public availability and established benchmark status, makes this data set suitable for evaluating learning-based correction methods and enables direct comparison with existing approaches. The data set comprises 29 samples, each containing an average of approximately 1500 data points. In this study, we selected driving trajectories collected with a Xiaomi 8 smartphone across different regions of the United States in 2021. After the exclusion of trajectories with missing key observations, the remaining trajectories are split into a training set and a non-overlapping test set, with five trajectories randomly selected from each trial for training. Each data point includes raw GNSS measurements across GPS, Galileo, GLONASS, and BeiDou systems (L1/L5 bands), specifically C/No ratios, elevation angles, satellite transmission times, and signal arrival times. Calculated parameters include pseudorange residuals, LOS vectors, GDOP, C/No ratios, elevation angles, and derived GNSS positioning solutions. Table 1 presents the parameters received by the GPS satellite receiver from different satellites at a given time. Specifically, satellite-x, satellite-y, and satellite-z represent the 3D coordinates of each satellite in the ECEF coordinate system. The data set provides GNSS positioning results with weighted least squares, which serve as the baseline solution and the input to all correction methods in our experiments.

4.2 Data Preprocessing

In this paper, the ground-truth positions of the data set are expressed in longitude–latitude–altitude coordinates, whereas the measured positions are represented in the ECEF coordinate system. For consistent observation of data characteristics, the north–east–down (NED) coordinate system is uniformly adopted as the reference framework. In the NED coordinate system, the positive directions are defined as north, east, and downward. The north direction points towards the Earth’s north pole, the east direction points towards the eastern longitude, and the downward direction points towards the Earth’s surface. These directions are denoted as X, Y, and Z, respectively. To ensure consistency in spatial representation and to facilitate the analysis of data characteristics, this paper uniformly adopts the NED coordinate system as the reference frame throughout all positioning calculations and evaluations. The data set is partitioned into training and testing sets. To mitigate the effects of scaling during network training, min-max normalization is applied for data preprocessing, which is computed as follows:

S=x−xminxmax−xmin19

where x represents the original data, S represents the normalized data, and x_max and x_min are the maximum and minimum values in the data, respectively.

4.3 Evaluation Metrics

Three standard errors, including the mean error (ME), mean absolute error (MAE), and root mean square error (RMSE), were selected as evaluation metrics. The ME is computed as follows:

ME=1N∑i=1N(‖Pigt−Picorr‖)220

where Pigt and Picorr denote the true and corrected coordinates, respectively. The MAE and RMSE for the three directions, X, Y, and Z, are calculated as shown below. For simplicity, only the calculation for the X direction is presented as an example; the definitions for the Y and Z directions are the same as that for the X direction:

MAE=1N∑i=1N|Pxigt−Pxicorr|21

RMSE=1N∑i=1N(Pxigt−Pxicorr)222

where Pxigt and Pxicorr denote the true and corrected coordinates in the X direction, respectively.

An official evaluation metric, the score for the Decimeter Challenge (SDC), is also used as an evaluation metric. This term is defined as the average of the 50th and 95th percentiles of the horizontal positioning error sequence (Fu et al., 2022). Unlike the directional errors mentioned above, the SDC only evaluates the horizontal positioning accuracy and does not consider the vertical direction:

SDC=P50+P95223

where P₅₀ and P₉₅ denote the 50th and 95th percentiles of the error sequence {e_i} Here, ei=(ΔNi)2+(ΔEi)2, where ΔN_i and ΔE_i are the north and east differences between the estimated and ground-truth positions, respectively.

4.4 Experimental Parameter Setting

In the meta-training phase, trajectories are randomly sampled from a predefined list of GNSS positioning environment data sets, where the total number of trajectories corresponds to the length of the data set list. In this study, five distinct GNSS positioning environments are selected. Each sampled trajectory is then individually trained using the meta-RL algorithm MPPO, with 200 training episodes per trajectory. The training process is organized in an inner-outer loop structure. In the inner loop, the policy and value networks of CL-PPO share the same learning rate, which is set to 0.0005, and the clipping range for the surrogate loss is set to 0.2. In the outer loop, instead of a fixed learning rate, adaptive parameters are used to regulate the global parameter update magnitude. Once all trajectories have been trained, the meta-training process is completed, yielding a set of globally optimal model parameters, denoted as θ*. These parameters are saved as the weights of the meta-policy and meta-value networks for subsequent meta-testing. During the meta-testing phase, a GNSS positioning environment that has not been encountered during meta-training is randomly selected for evaluation. The globally optimal parameters θ* obtained from meta-training are used to initialize the CL-MPPO model, which is then fine-tuned with an adaptively adjusted learning rate to enhance adaptation to the new environment. The parameter configurations for CNN-based feature extraction and LSTM-based temporal processing are listed in Table 2, while training and optimization hyperparameters for meta-training and meta-testing are summarized in Table 3. These parameter configurations were selected to ensure efficient convergence in GNSS positioning environments.

4.5 Comparative Analysis of Ablation Experiments

4.5.1 CL-PPO Ablation Experiment

To evaluate the performance of the proposed CL-PPO positioning model, the model was decomposed into three constituent models: PPO, CNN-PPO, and LSTM-PPO. These baseline variants were individually implemented and compared against the full CL-PPO model introduced in this study. Figure 3(a) illustrates the comparative performance of all models in terms of the ME. The corresponding ME values for each model are summarized in Table 4, with the results achieved by the proposed method indicated in bold font. Figure 3(b) presents the comparative MAE across the X, Y, and Z directions for all evaluated models, while Table 5 provides a detailed summary of the directional error statistics for each model, including the maximum error, minimum error, MAE, and RMSE along all three coordinate axes. The experimental results reveal two key findings. First, all PPO-based models significantly outperform the raw GNSS measurements, as illustrated in Figure 3(a) and Figure 3(b). Second, the proposed CL-PPO architecture achieves superior performance across all evaluation metrics compared with the baseline models, as evidenced by the numerical results in Table 4 and Table 5. Compared with CNN-PPO, which leverages only spatial features, CL-PPO integrates LSTM-based temporal modeling and achieves consistent improvements. Specifically, CL-PPO reduces the ME by 8.73%, with the most substantial gains observed along the Z axis, where the MAE decreases by 21.48% and the RMSE improves by 16.15%. In the X and Y directions, the MAE is reduced by 4.94% and 8.59%, respectively. In comparison to LSTM-PPO, which focuses solely on temporal dependencies, the incorporation of CNN-based spatial feature extraction in CL-PPO leads to more pronounced benefits. The ME is reduced by 20.74%, whereas the 3D error metrics exhibit improvements ranging from 16.01% to 24.87%, with the highest enhancement being a 24.87% reduction in the MAE along the Z axis. These results demonstrate the effectiveness of integrating spatial and temporal features within the CL-PPO framework to enhance GNSS positioning accuracy.

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FIGURE 3

Ablation study of CL-MPPO for GNSS positioning: (a) Euclidean position error; (b) X, Y, and Z direction errors

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TABLE 4 Overall ME Metrics from Ablation Study on Model Performance

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TABLE 5 Directional Error Metrics from Ablation Study on Model Performance

4.5.2 Comparison of CL-MPPO and CL-PPO

In this experimental setup, both the CL-PPO and CL-MPPO models are trained over 1,000 episodes within a specified GNSS positioning environment. Model performance is evaluated using four key metrics: convergence rate, reward value, positioning accuracy, and generalization capability. The primary results concerning reward dynamics and convergence speed are illustrated in Figure 4 and summarized in Table 6. These results demonstrate that the CL-MPPO model effectively leverages prior knowledge to achieve convergence within approximately 200 training episodes in specified GNSS environments, attaining a stabilized reward value of –3.73. This rapid convergence stems from the core advantage of the meta-RL framework, whereby meta-trained parameters serve as an effective prior to enable rapid adaptation with minimal fine-tuning, a key requirement for practical applications. In contrast, the CL-PPO model, which begins with randomly initialized parameters, undergoes a prolonged exploration phase and requires approximately 850 training episodes to converge, ultimately reaching a final reward value of only –3.93. The observed learning curves further confirm that CL-MPPO converges substantially faster than CL-PPO. Quantitative analysis indicates that CL-MPPO achieves a 4.25-fold improvement in convergence speed and a 5.09% increase in reward value compared with CL-PPO.

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FIGURE 4

Training reward dynamics and convergence speed: CL-PPO versus CL-MPPO

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TABLE 6 Comparison of Reward Values and Convergence Counts Between CL-PPO and CL-MPPO

In terms of positioning accuracy, Figure 5(a) and Figure 5(b) compare the Euclidean distance errors and the MAE along the X, Y, and Z axes for both models. Visual inspection clearly shows that the CL-MPPO model consistently delivers higher positioning precision than the CL-PPO model across all spatial dimensions. To objectively assess positioning accuracy, a comprehensive comparison of error metrics between the baseline CL-PPO model and the CL-MPPO model was conducted. The results are systematically summarized in Table 7 and Table 8. As shown in Table 7, in terms of Euclidean distance error metrics, CL-MPPO outperforms the baseline across all three evaluation criteria: maximum, minimum, and average error. Specifically, the model achieves an ME of 3.1505 m, reflecting an 18.91% reduction compared with the CL-PPO model. For the 3D positioning accuracy, the CL-MPPO model also demonstrates significant improvements in MAE over CL-PPO, as shown in Table 8. The reductions are quantified as 30.49% in the X direction, 10.75% in the Y direction, and 13.62% in the Z direction. Corresponding improvements are observed in the RMSE, with reductions of 27.43%,7.99%, and 12.58% in the X, Y, and Z directions, respectively. These results provide strong empirical evidence about the superior overall performance of the CL-MPPO model. In conclusion, the CL-MPPO positioning model achieves markedly higher positioning accuracy than the baseline CL-PPO model across all evaluated dimensions.

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FIGURE 5

Positioning accuracy comparison between CL-PPO and CL-MPPO: (a) Euclidean position error (m); (b) X, Y, and Z absolute errors

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TABLE 7 Comparison of ME Evaluation Metrics Between CL-PPO and CL-MPPO

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TABLE 8 Comparison of X/Y/Z Direction Error Evaluation Metrics Between CL-PPO and CL-MPPO

Finally, with respect to generalization capability, the trained models were evaluated through meta-testing in three previously unseen environments. The corresponding model variants are denoted as CL-PPO1 and CL-MPPO1 for Environment 1, CL-PPO2 and CL-MPPO2 for Environment 2, and CL-PPO3 and CL-MPPO3 for Environment 3. The evaluation results for these unseen environments are shown in Figure 6 and summarized in Table 9. The experimental results summarized in Table 9 demonstrate that the CL-MPPO model consistently outperforms the baseline CL-PPO model across all three testing environments. The proposed model exhibits notable improvements in key performance metrics, including convergence rate, reward value, positioning accuracy, and generalization capability. In particular, CL-MPPO achieves faster training convergence and higher cumulative rewards, while maintaining stable performance across diverse and previously unseen environments. Moreover, the model demonstrates superior control over 3D positioning errors and exhibits strong adaptability to unseen GNSS scenarios. These results collectively confirm the enhanced generalization and adaptation capabilities of the CL-MPPO framework, validating its effectiveness as a next-generation positioning algorithm.

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FIGURE 6

Comparison of training reward values among different environments

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TABLE 9 Comparison of Reward Values and Convergence Counts Between CL-PPO and CL-MPPO

4.6 Comparative Experiments and Analysis

To further validate the effectiveness of the proposed positioning scheme, a comparative analysis was conducted using raw GNSS measurements, the Kalman filter (KF), the A3C algorithm (Quan et al., 2018; Wu et al., 2024; E. Zhang & Masoud, 2021), LSTM-based positioning (Yu et al., 2023), the MPPO model (Ye et al., 2025), and the proposed CL-MPPO model. The evaluation focuses on the Euclidean distance ME, as well as the MAE and RMSE along the X, Y, and Z coordinate axes. Figure 7(a) and Figure 7(b) present comparative results for Euclidean distance errors and directional absolute errors across the four positioning models. The proposed CL-MPPO model achieves substantial reductions in positioning error and significantly enhanced positioning accuracy relative to raw GNSS measurements. When compared with the other models, the raw GNSS baseline suffers from large positioning errors with high error fluctuation. The KF only delivers a tiny drop in positioning error relative to raw GNSS measurements, leading to very limited performance improvement. The A3C algorithm shows limited suitability for this trajectory. This result is primarily due to its design for complex, multi-threaded decision-making environments, which reduces the effectiveness of A3C in single-agent GNSS positioning applications. Although both the LSTM and MPPO models demonstrate improvements over conventional methods, their performance remains inferior to that of the CL-MPPO model. The LSTM-based approach lacks the adaptive decision-making capability of RL models such as PPO, which limits its effectiveness. Although the MPPO model incorporates RL, it does not explicitly model both spatial and temporal correlations in the input data. By contrast, the CL-MPPO model enhances positioning accuracy by integrating CNNs for spatial feature extraction and LSTM networks for capturing temporal dependencies in time-series data, while preserving the policy optimization strengths of the MPPO framework. As a result, the CL-MPPO model consistently outperforms the baseline approaches, demonstrating superior positioning performance.

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FIGURE 7

Comparison of positioning errors among different models: (a) Euclidean distance errors; (b) X, Y, and Z direction errors

To provide a more objective evaluation, the positioning errors of the original GNSS system, KF, A3C, MPPO, LSTM, and the proposed CL-MPPO model are systematically compared and summarized in Table 10 and Table 11. The bolded values in these tables highlight the performance metrics achieved by the proposed method. As shown in Table 10, the CL-MPPO model delivers the best positioning performance across all key Euclidean distance error metrics, including maximum, minimum, and mean values. Notably, the model achieves an ME of only 3.1505 m, representing a substantial 72.34% improvement compared with the original GNSS measurements. For reference, the winner of the Google Smartphone Decimeter Challenge 2022, denoted as GS-SOTA, achieved superior accuracy on the full test data set, reportedly reaching an SDC metric of 1.382 m. This performance is

typically attained through multi-sensor fusion, integrating data from inertial measurement units (IMUs), barometers, and other smartphone sensors. Nevertheless, our CL-MPPO model attains its significant positioning improvement using only GNSS measurements by effectively exploiting spatiotemporal correlations. The axial error statistics in Table 11 further confirm the model’s superiority in 3D positioning. The CL-MPPO model achieves an MAE of 1.3319 m in the X direction, 1.8203 m in the Y direction, and 1.6087 m in the Z direction, corresponding to reductions of 72.42%, 79.60%, and 66.55%, respectively, relative to the original GNSS system. With respect to the RMSE, the model exhibits excellent performance, yielding values of 1.7600 m in the X direction, 2.6110 m in the Y direction, and 2.1734 m in the Z direction, corresponding to reductions of 65.61%, 71.67%, and 57.67%, respectively, compared with the baseline GNSS system. In conclusion, the proposed CL-MPPO positioning model consistently outperforms the original GNSS system as well as the A3C, LSTM, and MPPO models across all evaluated metrics, confirming its effectiveness in improving position accuracy.

To visually compare the performance of the six different positioning models, Figure 8 displays the corrected trajectories of all methods against the ground truth and visualizes the 100-s trajectory within them. CL-MPPO adheres most closely to the ground-truth path in both the global view and the zoomed-in section, consistent with its low RMSE and MAE values. Compared with the other positioning models, CL-MPPO exhibits fewer off-track deviations and smaller local errors along the trajectory. This visual alignment supports the quantitative improvements, highlighting the effectiveness of CL-MPPO in correcting GNSS trajectories.

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FIGURE 8

Trajectory comparison of GNSS and learning-based positioning models

The single-epoch inference time is another key metric for assessing practical utility, defined as the time required to process one set of GNSS measurements and output one positioning correction. Tests were conducted on a standard Intel Core i5-12450H central processing unit platform and averaged over 3,000 independent trials. As shown in Table 12, the proposed CL-MPPO model achieves a mean inference time of 1.08 ms and a 99th-percentile latency of 1.99 ms. This result

represents an 11.5% reduction in computational cost compared with the MPPO baseline, while maintaining positioning accuracy. These results demonstrate that the inference overhead of CL-MPPO is satisfied while still meeting the requirement of real-time GNSS positioning.