Deep-Learning-Assisted Kalman Filter for GNSS/MEMS IMU Integration in GNSS-Denied Environments

2 METHODOLOGY

GNSS/INS integration can be performed loosely, tightly, or deeply (Teunissen & Montenbruck, 2017). In this study, we apply a loose integration, where the GNSS and INS operate independently and their outputs are fused via an MBKF. Although tightly coupled integration generally yields better performance than loosely coupled integration, the core challenges of the MBKF such as model linearization, incomplete error modeling, and the assumption of white Gaussian noise are common to both loosely and tightly coupled architectures. The primary objective of this study is to address these shared limitations by introducing a DNN. Therefore, while we focus on loose integration in this work, the proposed DLKF framework is equally applicable to tightly coupled GNSS/INS integration schemes. This feature opens opportunities for future work in generalizing the DLKF approach across various integration architectures. In loose integration, as illustrated in Figure 1, the high-rate INS performs strapdown computation using acceleration and angular rate measurements from the IMU, providing INS solutions that include position, velocity, and attitude (PVA). The low-frequency GNSS provides more accurate position and velocity (PV) solutions, which are used as measurements for the MBKF. When the low-frequency GNSS solutions are available, the propagated errors from the strapdown computation are corrected by the MBKF. After the correction, the estimated error states of the MBKF are reset to zero. The INS uses the IMU error estimates from the previous MBKF update to correct the strapdown navigation solution until the next GNSS solution arrives. It should be noted that the loosely coupled integration requires a minimum of four visible satellites to generate valid GNSS PV solutions, a condition that may not always be satisfied in urban environments. Thus, an extension to a tightly coupled scheme would be beneficial but is beyond the scope of this work, owing to the increased DNN complexity that arises when pseudoranges are considered as features instead of PV.

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FIGURE 1

Loosely coupled GNSS/INS integration framework

In a 15-state loosely coupled system, the state vector of the MBKF is typically structured as follows:

x=[δp,δv,δΨ,ϵ,ζ]T1

where δp,δν, and δΨ represent the estimated PVA errors and ϵ and ζ denote the accelerometer and gyroscope biases, respectively. The MBKF comprises two primary steps: system propagation and measurement update. During the system propagation step, considering the GNSS solution time interval Δτ, the state vector and its covariance P(15 × 15) at the current time t are predicted using the state transition matrix F(15 × 15):

xt∣t−Δτ=Fxt−Δτ∣t−Δτ2

Pt∣t−Δτ=FPt−Δτ∣t−ΔτFT+Q3

where Q(15 × 15) is the process noise covariance matrix. The state transition matrix F is derived by linearizing the nonlinear error propagation equations in the strapdown computation. During the measurement update step, the predicted state vector and its covariance matrix are updated using the following equations:

K=Pt∣t−ΔτHT[HPt∣t−ΔτHT+R]−14

xt∣t=xt∣t−Δτ+K(yt−Hxt∣t−Δτ)5

Pt∣t=[I−KH]Pt∣t−Δτ6

where K is the KG, calculated based on the measurement matrix H(6 × 15), the measurement covariance matrix R(6 × 6), and the predicted covariance matrix Pt∣t−Δτ. The KG determines the weight assigned to the predicted states and the measurements. The predicted states are updated by applying the KG to the measurement innovation, which is the difference between the actual measurements y and the predicted measurements Hx. The H matrix projects the predicted states from the state space to the measurement space. Once the states are updated, the estimated PVA errors and IMU biases are fed back into the INS, and the integration system continuously iterates under normal operating conditions.

When the GNSS signal is obstructed, the system transitions to an INS-only mode, using the most recent integrated navigation solutions as the initial value for strapdown computation. The accuracy of the INS solutions is significantly influenced by the quality of the IMU measurements. In particular, MEMS IMUs are prone to various errors. Beyond laboratory-calibrated errors and high-order errors, all types of accelerometers and gyroscopes have a constant bias, scale factor error, misalignment error, and random noise as follows:

α=ϵa+(I+Ma)αˇ+ζa7

ω=ϵg+(I+Mg)ωˇ+ζg8

where α and ω denote the actual accelerometer and gyroscope measurements, respectively, and αˇ and ωˇ represent their corresponding true values. The variables ϵa and ϵg are constant biases, ζa and ζg denote the random noise components, and the matrices Ma and Mg represent the scale factor and misalignment error of the accelerometer and gyroscope, respectively.

For a low-cost MEMS IMU, accurate error estimation by the filter is crucial to prevent rapid accumulation of PVA errors over time. For example, an accelerometer bias of 0.1 m/s² can result in a position error of 180 m within 60 s. In a 15-state integration system, the filter only estimates biases for the accelerometer and gyroscope, without fully addressing the complex error characteristics in the MEMS IMU system. As a model-based algorithm, the MBKF relies on predefined system models and noise statistics to estimate accumulated errors in the INS solutions. The process noise in the MBKF is typically configured based on the Allan variance plot, which provides a detailed characterization of the IMU noise components (El-Sheimy et al., 2007). However, owing to the high nonlinearity of the integration system, the accuracy of the MBKF is constrained by system model deficiencies, noise statistics, and noise model assumptions.

To overcome these limitations, we propose a DLKF that integrates a DNN into the GNSS/INS integration system, as illustrated in Figure 2. Instead of manually defining the matrices that describe system dynamics and noise, the DNN utilizes the MBKF states and GNSS solutions as input features (S. Li, Mikhaylov, Pany, & Mikhaylov, 2023). The DNN learns the system and measurement noise statistics through its extensive network of neurons and estimates a more accurate KG. Furthermore, the IMU errors are learned directly from the high-rate raw IMU measurements and the INS solutions through the DNN, rather than being estimated solely by the MBKF. Consequently, the filter employs the KG provided by the DNN to correct the accumulated errors in the strapdown system. The overall structure of the DLKF remains consistent with the MBKF, performing Kalman filter updates whenever new GNSS measurements arrive. Thus, the DNN serves as a supplement to learn the components that cannot be accurately estimated through conventional model-based methods. The primary difference between the DLKF and the MBKF is the treatment of IMU errors. In a conventional MBKF, each type of IMU error, such as bias, scale factor, misalignment, and g-dependent terms, is typically modeled within the state vector; however, this approach results in increasing complexity and a nonlinear state transition matrix F. In this study, we use a 15-state MBKF, where only the accelerometer and gyroscope biases are included in the state vector, which avoids excessive model complexity while still offering a reasonable level of performance in conventional settings. However, the MBKF inherently neglects many error sources present in MEMS IMUs. In contrast, the DLKF estimates the complete IMU error directly through a neural network, which learns to map raw IMU measurements and INS solutions to IMU error estimates without requiring explicit modeling of each error component. As a result, the state vector in the DLKF is reduced to nine elements, representing only the PVA errors. It is worth noting that, throughout this paper, the term “IMU error” refers to the complete IMU error profile. Given the sensor-dependent nature of IMU error characteristics, the objective of this work is to present a framework adaptable to different IMU configurations through retraining, rather than a single model intended to generalize across all IMU types.

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FIGURE 2

High-level architecture of the DLKF

As shown in Figure 3, the proposed DNN combines a CNN and an LSTM network. Periodic IMU measurements and INS solutions are fed into the CNN module, which comprises two one-dimensional convolutional layers. The first convolutional layer includes 32 hidden channels, while the second layer reduces the output to a single channel. The CNN outputs are concatenated with four components to form the input to the LSTM network. The four components include the successive MBKF posterior state difference (xt∣t−xt−Δτ∣t−Δτ), predicted and posterior state difference (xt∣t−xt∣t−Δτ), current and previous GNSS solution difference (yt−yt−Δτ), and innovation between GNSS and INS PV solutions (S. Li, Mikhaylov, Pany, & Mikhaylov, 2023). The LSTM has a hidden size of 256 and consists of four stacked recurrent layers, followed by two fully connected layers with sizes of 512 and 256, respectively. LeakyReLU activation functions are employed between each layer to introduce nonlinearity (Maas et al., 2013).

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FIGURE 3

Architecture of the proposed DNN module

The CNN input is an M × 15 matrix, formed by stacking sequential high-rate samples of strapdown INS and IMU data. Each row contains 3D PVA estimates from the INS solution, along with raw IMU measurements. M corresponds to the number of time steps within the selected input window. In this work, the IMU sampling rate is 50 Hz; thus, a 1-s window yields M = 50. The LSTM input is a 1 × 30 matrix.

The DNN produces two final outputs: the IMU errors and the KG, each from separate output layers. The network architecture was selected empirically by evaluating multiple configurations, with the number of hidden units chosen as powers of two. We observed that increasing the number of layers or hidden units did not lead to a significant accuracy improvement, whereas smaller networks resulted in degraded performance. The final configuration was chosen to balance estimation accuracy and computational efficiency.

Although CNNs are traditionally used for spatial analysis, such as image recognition, their ability to extract features across multiple dimensions makes them well-suited for this application because the input features, including IMU measurements and INS solutions, exhibit complex interdependencies. Additionally, by sliding convolutional filters over time windows, the CNN can capture temporal information and identify temporal features such as motion dynamics.

However, the temporal information captured by the CNN input features spans only a few seconds. To address the limitations of CNNs in capturing long-term dependencies, we further employ an LSTM network. Both the MBKF and LSTM are designed to predict future states based on historical data, making them ideal for time-series analysis. The MBKF assumes a Markov process, indicating that the future state depends only on the current state. While the LSTM network does not strictly follow a Markov process in the traditional sense, it can retain and leverage information over a long sequence through a large memory cell. Consequently, the CNN-LSTM combination is particularly effective for enhancing filter quality and modeling a complex, time-dependent integration system whose process noise is highly correlated in time.

3 TRAINING STRATEGY

The proposed DLKF framework is implemented using PyTorch (Paszke et al., 2019). Because the DNN incorporates an LSTM network, the backpropagation-through-time (BPTT) algorithm is employed for training (Werbos, 1990). The learning targets are the KG and IMU errors; however, directly training the DNN on these quantities is challenging because of the difficulty in obtaining their ground-truth values. To address this challenge, we adopt an end-to-end approach where the reference PVA serves as the learning target. Through gradient propagation, the model can learn the KG and IMU errors indirectly, as they are embedded in the optimization process. The reference PVA information is used solely during the training phase of the DLKF, to supervise the DNN in learning the KG and IMU errors. Once the training is complete, in real-world operation and testing, the DLKF functions as an MBKF driven only by IMU and GNSS measurements. The trained DNN model is intended to generalize across trajectories under similar IMU characteristics. Training a DNN in this end-to-end manner for GNSS/ INS integration faces additional challenges, such as gradient instability in long sequences and the need to maintain accuracy during GNSS outages. To overcome these challenges, we introduce specialized training strategies to improve the stability of the training process, allowing for longer sequence training and enhancing position accuracy during GNSS outages.

1. Dropout layers. We have implemented several modifications to the DNN structure and training process compared with our previous work (S. Li, Mikhaylov, Mikhaylov, & Pany, 2023). Notably, dropout layers, which are generally known to prevent overfitting, as discussed by Srivastava et al. (2014), have been integrated between fully connected layers. We observed that using a high dropout rate (0.5) greatly improves the numerical stability of the training process and the position accuracy for our specific problem. The regularization effect of the dropout layers makes the model less sensitive to outliers in the training data, which aids in maintaining the stability of predictions and enhancing generalization across unseen data. Without dropout layers, these outliers can cause large gradients, which leads to substantial changes in the model's weights, causing the training process to diverge. Applying dropout layers helps to avoid this issue by reducing the impact of outliers on the model's weights, making the training process more stable and the model more robust.

2. Weighted loss function. The DNN is trained in a supervised manner, and the total loss is computed with respect to the ground-truth trajectory. The losses for each component are weighted, and the total loss function is then given by L:

   L=cpLp+cvLv+chLh9

   where Lp,Lv, and Lh are the losses for PVA, respectively, and Cp,Cv, and Ch are the corresponding weights. Attitude is denoted by h in this work to avoid confusion with a, which denotes acceleration. The individual loss is computed by using the smooth L1 loss defined by Girshick (2015), because this loss is more robust to outliers and causes less divergence during training. Because the PVA errors have different units (meters, meters per second, and radians), the scales differ significantly. To address this scale difference, the loss weights are chosen proportionally to the losses of the DNN when trained without weights. Selecting suitable weights for the loss terms in a model is crucial to ensure that each term is given the appropriate level of importance, regardless of the differences in their scales. Without proper weighting, the model prioritizes optimizing one particular quantity at the expense of the others, leading to sub-optimal performance and inaccurate results. Furthermore, the weights allow some quantities to be prioritized over others. This feature allows the model's behavior to be fine-tuned and adapted to the specific requirements of a task. The weights in the loss function are dynamically adjusted during training. During the early stages, position errors are relatively large and dominate the loss. Therefore, larger weights are assigned to the velocity and attitude terms to keep the PVA components on comparable scales. As training progresses and the filter converges, the weights are reduced accordingly. A normalization-based approach was also considered. However, because of the large variation in PVA error magnitudes across different training phases and during GNSS outage scenarios, an adaptive weighting strategy was found to be more suitable, as it better reflects the varying error characteristics across different training phases.

3. Truncated trajectories. Directly applying the BPTT algorithm to long trajectories can lead to instability and high computational cost. To mitigate these effects, we truncate the trajectories into fixed-length sub-trajectories, each spanning 1 or 2 min. This approach not only helps prevent the vanishing gradient problem, which is common when training DNN on long sequences, but also ensures that the model retains critical information over time. Additionally, using fixed-length sub-trajectories facilitates batch processing, further enhancing the efficiency and scalability of the training process.

4. Decreased GNSS solution frequency. The accuracy of INS solutions is highly dependent on the quality of IMU measurements. Consequently, accurate estimations of IMU errors are essential. One possible strategy is to add a loss term of the ground-truth IMU errors. However, this approach is not feasible in practice, as the IMU errors are complicated and ground-truth IMU errors are not available; therefore, we have chosen a different approach. During training, DLKF updates are applied every 2–5 s per sub-trajectory, despite GNSS solutions being available every second. This approach increases the number of time steps for which the system operates without DLKF updates, relying solely on the IMU error estimation rather than KG updates. Because strapdown computation solutions are strongly dependent on IMU error estimation, this approach increases the importance of IMU error estimation in the loss function; as a result, more weight is placed on the IMU error estimation as the DNN is optimized.

5. Simulated GNSS outages. To improve DLKF performance during GNSS outages, we manually simulate fixed GNSS outage intervals for each subtrajectory in the training phase. This simulation is achieved by selecting a random 10-s to 30-s interval within the sub-trajectory, during which only strapdown computation is performed. By introducing such outages during training, the DNN learns to handle situations in which GNSS signals are completely unavailable. This approach also ensures that the model is capable of recovering from GNSS outages and continuing to provide accurate trajectory estimates.

6. Warm-up period. Inaccuracies can lead to divergent predictions, resulting in high loss values or even NaN (not a number) values, which could disrupt the entire training process. To address this challenge, we introduce a warmup period at the start of training, where the model is trained on shorter subtrajectories. Specifically, during the first 20 training epochs, the model is trained on 20-s trajectories, with the trajectory length linearly increasing to the full sub-trajectory by the 40th epoch. Furthermore, GNSS outages are not simulated during the warm-up period to prevent prediction divergence. This strategy stabilizes the loss values and promotes model convergence.

The experimental parameters used in the training process are listed in Table 1. The loss function is highly sensitive to IMU errors. When the network is trained from scratch, an initial IMU error estimation that is close to random can lead to significantly large errors due to rapid error accumulation through strapdown computation. Therefore, a relatively large learning rate should initially be selected to allow the DNN to rapidly explore the solution space. The cosine annealing scheduler is designed to have an initial slow decrease, which allows the DNN to explore the space broadly and reduces the likelihood that the model will become trapped in a local minimum. Moreover, the rapid mid-stage convergence and the slow decrease in the learning rate during the final stage can help the DNN to efficiently converge to promising regions and enable fine-tuning. Additionally, the restart feature of cosine annealing can further avoid local minima (Z. Li & Arora, 2019). All remaining hyperparameters were determined through a grid search. The proposed DNN was trained using an NVIDIA GeForce RTX 2080 Ti 12-GB graphics processing unit. For an 8-h data set, the training took approximately 2.5 h to complete 500 epochs with a batch size of 50.

4 DATA COLLECTION

The proposed algorithm was first evaluated using a simulated data set, followed by testing on a real data set. The details of the data collection process for both data sets are presented in the following subsections.

4.1 Simulated Data Set

Raw GNSS measurements and vehicle motion were simulated using the Spirent GSS7000 GNSS simulator. The simulated data set spans a duration of 5 h, with the vehicle PVA recorded at 100 Hz as the ground truth. The raw GNSS measurements include GPS L1 pseudorange, Doppler measurements, and ephemeris data. The standalone PV solutions were calculated using weighted least squares. Because no additional measurement errors were simulated, the accuracy of these solutions is at the centimeter level. The vehicle motion was manually designed with a variety of maneuvers, including acceleration, straight driving, turning, and figure-eight motion. Based on the analysis of longitudinal and lateral accelerations in typical daily driving reported by Hugemann and Nickel (2003) and Yusof et al. (2016), the vehicle speed was simulated in the range of 0–80 km/h, with longitudinal accelerations ranging from 0 to 2 m/s². The turn radius was simulated over a range of 40–150 m, with lateral accelerations ranging from 0.5 to 5 m/s². Specific forces and angular rates were calculated considering Earth's rotation and gravity, and different types of IMU errors were simulated for various evaluation and analysis purposes, including sensor biases, scale factor and misalignment effects, and additive noise terms as defined in Equations (7) and (8), with an IMU sampling rate of 100 Hz. The simulation was implemented using an in-house Python package developed based on the work by Groves (2013) and the associated MATLAB code book. Following the training strategy described in Section 3, the entire trajectory was divided into several sub-trajectories, which were then shuffled and split into training and testing data sets. Figure 4 shows an example of a sub-trajectory with a duration of 10 min, with a simulated MEMS IMU that includes biases and random noise, as well as scale factors and misalignment errors. The performance of the MBKF critically depends on an appropriate selection of the process noise covariance matrix Q and the measurement noise covariance matrix R (Hofacker et al., 2025). For the simulated data set, because the raw GNSS and IMU measurements were generated using the Spirent GSS7000 simulator under certain error models and noise characteristics, both Q and R were configured according to the known simulation parameters.

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FIGURE 4

Example of a simulated 10-min sub-trajectory with a MEMS IMU The labels X, Y, and Z denote the three orthogonal axes of the IMU sensor frame.

4.2 Real Data Set

In addition to simulated data, the proposed algorithm was also tested on real data recorded in various scenarios, including urban canyons, highways, short tunnels, and underground parking lots in the cities of Hildesheim, Hannover, and Frankfurt, Germany. During these tests, the vehicle was equipped with a sensor box including two GNSS receivers, an IMU, and a reference system. To reduce the communication load, all IMU measurements were downsampled to 50 Hz. The reference system and one u-blox F9P receiver operating in the RTK mode were connected to a NavXperience 3G+C antenna, and another u-blox F9P receiver operating in the standard mode was connected to a Tallysman TW7972 antenna. A comprehensive description of each sensor and its functionality is provided in Table 2. The table outlines the specific sensors used for each data type, the frequency at which the sensor data were generated, and the antenna setup employed for data acquisition.

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TABLE 2 Sensor Configuration Details for the Test Vehicle

Figure 5 presents top and side views of the test vehicle setup, where the antennas were mounted on the top of the vehicle in a straight line. This configuration positioned the NavXperience antenna directly above the center rear axle, while the Tallysman antenna was installed at a distance of 0.69 m in the x-direction from the center rear axle. Both antennas were installed at the top of the vehicle; the height difference between the antenna setup and the center rear axle was 1.34 m. The sensor box was installed in the trunk of the vehicle.

A total of 75 test drives were conducted in 2021, 2022, and 2023, generating data with a total duration of approximately 42 h and covering 1300 km. The raw sensor data were collected by a control area network and recorded in robot operating system (ROS) bag format. We converted the ROS bag data to MATLAB files, and all sensor data were time-synchronized based on their timestamps. Duplicate data were removed to ensure consistency and maintain data quality. The processed sensor data were merged to a single data set based on the synchronized timestamps, and two additional flags were introduced: GNSS precise and non-precise availability, indicating complete GNSS blockage situations, such as those that arise in underground garages and tunnels. The GNSS solutions were transformed from the antenna positions to the center rear axle, corresponding to the location for the reference system solutions. For the real data set, we collected IMU data under static conditions for 24 h and determined the process noise covariance matrix Q based on Allan variance analysis. Noise power spectral densities (PSDs) are used to configure the diagonal elements of the MBKF process noise covariance matrix Q. Specifically, the accelerometer noise PSD and accelerometer bias random walk PSD are as follows:

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FIGURE 5

Test vehicle setup

qa=[2.56×10−6,3.24×10−6,4.84×10−6]m2s−3,qba=[2.42×10−9,2.53×10−8,7.31×10−10]m2s−510

The corresponding gyroscope noise PSD and gyroscope bias random walk PSD are as follows:

qg=[3.86×10−8,6.56×10−8,1.79×10−8]rad2/s,qbg=[8.91×10−13,1.31×10−11,1.42×10−13]rad2s−311

The PSD terms are further scaled to overbound unmodeled effects. The measurement noise covariance matrix R was configured empirically, with centimeter-level standard deviation assigned for RTK-based GNSS solutions and meter-level standard deviation assigned for non-RTK GNSS solutions. For example, in the RTK-based case, the position measurement standard deviations in R are set to 0.5 m for all axes, whereas the velocity measurement standard deviations are set to 0.05 m/s. The off-diagonal elements are set to zero, assuming that the position and velocity solutions are uncorrelated.

5 EXPERIMENTS AND RESULTS

In this section, we present a comprehensive analysis of the experiments conducted to evaluate the proposed algorithm. The results are categorized into IMU error estimation, KG analysis, and the impact of GNSS outages.

5.1 IMU Error Estimation in Simulated Data

To demonstrate the ability of the DLKF to learn IMU errors accurately, we first verified this approach using simulated data. We simulated IMU measurements with a constant bias and random noise, given in terms of the square root PSD, as shown in Table 3. The standard deviation of random noise samples is computed as the product of the square root of the corresponding PSD and the square root of the sampling rate (Groves, 2013). The scale factor, misalignment error, and g-dependent gyroscope error were not simulated. In this simulation, the MBKF can theoretically estimate the IMU errors accurately, as only the constant bias is included in the state vector, and the Q matrix can be tuned according to the simulated random noise.

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TABLE 3 Simulated IMU Constant Bias and Random Noise Values

After training the DNN from scratch under this error setting, we tested the DNN on 10 sub-trajectories, each with a duration of 1 min. The IMU error estimation and PVA accuracy from both the DLKF and the MBKF were analyzed; GNSS outages were not simulated in this experiment. The average accuracy of the 10 sub-trajectories shows that both the DLKF and MBKF achieve comparable performance in position, with root mean square (RMS) errors of 0.45 m for the DLKF and 0.49 m for the MBKF, and in velocity, with RMS errors of 0.02 m/s for the DLKF and 0.04 m/s for the MBKF. However, the DLKF demonstrates superior performance in attitude estimation, achieving an RMS error of 0.07 rad compared with 0.27 rad for the MBKF. This significant improvement is attributed to the use of ground-truth attitude data by the DLKF during training, whereas the MBKF indirectly estimates the attitude owing to a lack of direct attitude measurements. When reference attitude is not available during training, a reduction in attitude accuracy is expected. However, during our development, we observed that the DLKF can still indirectly learn attitude information from the motion patterns encoded in the raw IMU measurements and INS solutions. Additionally, based on its design and its observed behavior on short sub-trajectories, the DLKF is expected to remain stable over longer data sets once it has converged, such as 30 min or 1 h without GNSS outages. While our preliminary tests align with this expectation, this long-term capability has not been comprehensively evaluated and requires further assessment in future studies. Furthermore, the steady-state conditions of the traditional MBKF have not yet been theoretically analyzed for the DLKF.

Figure 6 presents the estimated IMU errors of the DLKF and MBKF. Compared with the MBKF, the DLKF provides better and more stable estimates for both accelerometer and gyroscope errors. An offset is observed in the MBKF estimates, even after careful tuning. This offset is primarily attributed to the lack of a diverse motion pattern. In this case, the sub-trajectory spans only 1 min, leading to relatively simple and uniform motion. This uniform motion pattern limits the diversity of dynamic information available to the filter, restricting its ability to observe certain states and preventing it from leveraging dynamic corrections through state coupling, which is crucial for accurate IMU error estimation. The accelerometer and gyroscope bias variance derived from the state covariance matrix P of MBKF is consistent with the observed offsets in the bias estimates. In contrast, the DLKF benefits from training the DNN on diverse motion scenarios. In Figure 6, the term “IMU error” is used instead of the term “bias” for consistency. Because only the bias is modeled in the state vector of the MBKF, the estimated bias is interpreted as the IMU error for comparison purposes. It is also important to note that the MBKF may not be fully optimized. However, it is considered sufficiently mature to serve as a reference for evaluating the performance of the DLKF.

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FIGURE 6

Comparison of estimated IMU errors from the DLKF and MBKF in the simulated data set, shown over 10 independent sub-trajectories, each with a duration of 1 min

5.2 KG Analysis in Simulated Data

In the MBKF, the uncertainty of the measurements and the current predicted states must be carefully considered. This uncertainty is reflected in the KG, as described in Section 2, which determines the weight given to the measurements and predictions. Consequently, the performance of the MBKF is highly dependent on the accuracy of the KG. As shown in Equation (5), if the KG value is close to 0, the filter relies primarily on the prediction, whereas a KG value close to 1 indicates that the filter relies heavily on the measurements. The diagonal elements of the KG represent the direct relationship between predictions and measurements, while the off-diagonal elements capture the indirect relationships. If the states are under-determined, the KG matrix will have fewer columns than rows. In the DLKF, the state vector includes the 3D PVA, with the GNSS PV solutions serving as measurements. Consequently, the KG matrix has dimensions of 9×6.

To analyze the DLKF performance under different levels of GNSS solution quality, we started with GNSS PV solutions computed from raw GNSS measurements generated by the Spirent simulator. These GNSS PV solutions achieve centimeter-level accuracy and do not include any additional simulated measurement noise. To simulate degraded GNSS conditions, we added zero-mean Gaussian noise with varying scale factors to the computed GNSS PV solutions:

y∼p=yp+sp⋅N(0,1)12

y∼v=yv+sv⋅N(0,1)13

where y∼p and y∼v are the noisy GNSS PV solutions, yp and yv are the original GNSS PV solutions, and sp and sv are the corresponding PV scale factors. The simulated IMU measurements used in this experiment are identical to those described in the previous section. We simulated six different sets of sp and sv, as shown in Table 4.

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TABLE 4 Scale Factors for GNSS Position and Velocity Measurements

For each set, the MBKF was tuned according to the simulated errors, and in this case, the KG from the MBKF can be used as a reference. The DLKF was trained from scratch on each set separately, and we then compared the results from the MBKF and DLKF for each set across 10 sub-trajectories. The average PVA RMS errors were computed as the mean values over all sub-trajectories. For the KG comparison, we randomly selected one sub-trajectory from the 10 and visualized the KG for the last time epoch of the selected sub-trajectory. Each visualized KG matrix consists of nine rows and six columns, where the rows correspond to the 3D PVA error states and the columns correspond to the 2D GNSS PV solutions. Each colored block in the visualized KG matrix represents the value of the corresponding KG element, with the magnitude encoded by both color and numerical annotation. For the MBKF, although the full KG matrix includes additional rows associated with accelerometer and gyroscope bias states, we only display the upper 9×6 sub-matrix to enable a direct and fair comparison with the 9-state DLKF configuration.

From set 1 to sets 2 and 3, we increased s_p to analyze the performance of the DLKF. As the noise in position measurements increases, the corresponding dependency should theoretically decrease, which is reflected in the KG by a reduction in the first three diagonal values. Figure 7 shows the KG from the DLKF and MBKF. It is evident that the KG estimated by the DLKF is consistent with the KG manually tuned from the MBKF. As s_p increases, the position-related KG elements decrease as expected, indicating that the filter relies more on the predicted position. The sub-diagonal elements are also clearly visible. Because s_v is 0 in these three sets, the filter attempts to use the precise velocity measurements to indirectly estimate the position.

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FIGURE 7

Comparison of KG from the DLKF (top) and MBKF (bottom) across sets 1, 2, and 3: GNSS position noise level analysis

Subsequently, we increased s_v and conducted similar experiments to evaluate the performance of the DLKF with velocity measurements subjected to different levels of noise in sets 1, 4, and 5. The KG from the DLKF and MBKF is presented in Figure 8, showing that the KG assigns less weight to the GNSS velocity measurements as the GNSS velocity error increases. The performance of the DLKF remains consistent with the finely-tuned MBKF. Consequently, the sub-diagonal elements show that the velocity is indirectly estimated through precise position measurements.

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FIGURE 8

Comparison of KG from the DLKF (top) and MBKF (bottom) across sets 1, 4, and 5: GNSS velocity noise level analysis

In set 6, we introduced errors to both position and velocity in the GNSS solutions. As shown in Figure 9, both the DLKF and MBKF exhibit similar weights for the PV in the diagonal and sub-diagonal elements of the KG. Table 5 presents the PVA RMS errors across all six sets. The results indicate that the performances of the DLKF and MBKF in set 6 are comparable.

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FIGURE 9

Comparison of KG from the DLKF (left) and MBKF (right) in set 6, with noise in both GNSS position and velocity measurements

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TABLE 5 Comparison of the DLKF and MBKF with Different Scale Factors All errors presented in this table are RMS errors.

Overall, the trends of the KG for both the DLKF and MBKF are consistent, although the specific values of the elements are slightly different. This discrepancy is also reflected in the PVA RMS errors. As indicated in Table 5, the PVA RMS errors increase for higher scale factors. The DLKF and MBKF demonstrate comparable performance; however, the DLKF generally performs slightly better in terms of PV errors. Notably, the DLKF consistently outperforms the MBKF in attitude estimation, for the same reasons explained in Section 5.1.

5.3 GNSS Outage Analysis

We simulated artificial GNSS outages based on the training strategy described in Section 3. We conducted experiments on both simulated and real data sets to analyze the performance of the DLKF under GNSS outage conditions. In the simulated data set, we used centimeter-level GNSS PV solutions and integrated a MEMS IMU. GNSS outages were simulated for durations of 30–50 s across all 10 sub-trajectories in the test data set. We then compared the average PVA RMS errors between the DLKF and MBKF. The DLKF achieves PVA accuracies of 0.63 m, 0.13 m/s, and 0.17 rad, respectively, compared with the MBKF accuracies of 1.43 m, 0.23 m/s, and 0.23 rad. These results demonstrate that the performance of the DLKF is superior during GNSS outages. We visualized the estimated sub-trajectories from both the MBKF and DLKF with the ground truth in Figure 10. It can be observed that the DLKF maintains better accuracy during GNSS outage conditions than the MBKF. This result aligns with our earlier findings that the DLKF can estimate IMU errors more accurately than the MBKF, particularly in the case of MEMS IMU simulations, as the MBKF only considers bias in the state vector and addresses other errors by increasing the Q matrix.

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FIGURE 10

Comparison of estimated sub-trajectories from the DLKF and MBKF under GNSS outage conditions in the simulated data set

Next, we proceeded to analyze the real data set. We conducted initial experiments on a subset of the entire real data set, with a duration of 2 h. Following the same training strategy utilized for the simulated data, the selected data set was divided into several sub-trajectories and shuffled, resulting in 110 training and 12 testing sub-trajectories. The GNSS solutions were obtained from the u-blox F9P in RTK mode. GNSS outages were simulated for durations of 30–50 s for all sub-trajectories in the data set. The estimated IMU errors from the DLKF and MBKF are displayed in Figure 11. We plotted the average IMU errors of three sub-trajectories, focusing on initial convergence and reconvergence after the outage. It is evident that the DLKF achieves an initial convergence more rapidly than the MBKF. Furthermore, after the GNSS outage, the DLKF quickly reconverges and maintains stability, indicating its robustness in the absence of GNSS signals. In contrast, the MBKF requires additional time to reconverge, showing a delay in re-establishing accurate IMU error estimation after a GNSS outage. This behavior is consistent across different sub-trajectories.

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FIGURE 11

Comparison of estimated IMU errors from the DLKF and MBKF in the real data set

The same conclusion can also be observed in the 2D position map. Figure 12 illustrates four sub-trajectories from the test data set. It is evident that the DNN is capable of learning the appropriate IMU errors, thereby maintaining the navigation system at an acceptable level of accuracy during GNSS outages. This ability to retain accuracy ensures reliable navigation performance even when GNSS signals are temporarily unavailable. After the GNSS outages, the DLKF demonstrates rapid reconvergence, restoring accurate position estimates. In contrast, the MBKF shows slower reconvergence and struggles to maintain the same level of accuracy during the outage. The rapid reconvergence of the DLKF allows it to swiftly recover and continue providing precise navigation solutions, which is critical for applications requiring continuous and reliable positioning information.

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FIGURE 12

Comparison of estimated sub-trajectories from the DLKF and MBKF under GNSS outage conditions in the real data set

To conduct a more comprehensive analysis of GNSS outage conditions, we performed several experiments on larger data sets, consisting of 9 h of training data and 2 h of test data. The length of each sub-trajectory was increased to 120 s, with outages simulated from 60 to 120 s. The training and testing data span from 2021 to 2023 and utilize the same sensor setups. We computed the average 2D position RMS errors (north and east) of all sub-trajectories and compared the results from the DLKF and MBKF. The left panel in Figure 13 shows the average 2D position RMS errors for the first 60 s, during which GNSS solutions are available. The DLKF demonstrates better accuracy compared with the MBKF. After the MBKF converges, the trends of the DLKF and MBKF become similar. After 60 s, the integration system encounters a GNSS outage scenario. Given that the INS is sensitive to noise, the position error grows rapidly over time. As shown in the right panel, the average 2D position RMS errors from both the DLKF and MBKF increase over time. However, the position error from the DLKF grows more slowly than that from the MBKF, resulting in a lower error. The DLKF outperforms the MBKF, achieving an improvement of approximately 50% in position accuracy.

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FIGURE 13

Average position errors (2D RMS error) of all sub-trajectories for the DLKF and MBKF: without GNSS outages (left) and with GNSS outages (right)

To evaluate the individual performance of the sub-trajectories rather than just the average, we recorded the accumulated position RMS errors for each sub-trajectory every 10 s during the GNSS outage and presented the data in a histogram, as shown in Figure 14. The results consistently indicate that a greater proportion of sub-trajectories from the DLKF achieve a smaller position error compared with the MBKF.

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FIGURE 14

Histogram of accumulated position errors (2D RMS error) for individual subtrajectories during GNSS outages from 10 s to 60 s

6 CONCLUSION

In this study, we proposed a novel DLKF for GNSS/INS integration, specifically designed to address challenges in GNSS outage conditions. Compared with other data-driven approaches, the DLKF offers greater interpretability owing to its integration with the MBKF framework. The DNN does not replace the MBKF; instead, the DNN complements the MBKF by learning complex nonlinear system dynamics and noise characteristics. The proposed DNN architecture, which combines CNN and LSTM networks, effectively captures both spatial and temporal dependencies in IMU measurements, providing robust and reliable navigation solutions. Experimental results obtained from both simulated and real-world data sets show that the DLKF can accurately estimate IMU errors and enables the system to maintain stable navigation solutions and rapidly reconverge once GNSS measurements become available again. The KG learned by the DLKF closely matches that of a carefully tuned MBKF, demonstrating consistency and interpretability in balancing measurement and prediction. By merging data-driven and model-based methods, the DLKF offers a promising approach to address the limitations of traditional GNSS/INS integration. Future research will evaluate the stability and performance of the DLKF over longer trajectories and extended GNSS outages. Additionally, evaluating the proposed DLKF against more complex MBKF formulations, such as a 21-state MBKF or AEKF, would provide further insight into the performance of explicit physical modeling and data-driven approaches. Beyond these evaluations, subsequent work could explore optimizing the DNN structure, investigating varying GNSS conditions, and extending this architecture to integrity monitoring and other sensor integrations.

HOW TO CITE THIS ARTICLE:

Li, S., Mikhaylov, M., Pany, T., Menghani, R., & Mikhaylov, N. (2026). Deep-learning-assisted Kalman filter for GNSS/MEMS IMU integration in GNSS-denied environments. NAVIGATION, 73. https://doi.org/10.33012/navi.787