⚡ electrical engineering

The Setting of IMU Parameters in Kalman Filtering-based Information Fusion

This paper proposes a method for setting IMU parameters within a Kalman filtering framework by leveraging the relationship between power spectral density and Allan variance to address the challenges of tuning under complex working conditions, demonstrating its effectiveness through two typical sensor fusion systems.

Original authors: Qiang Hu, Yanhua Zou, Shuaiyi Huo, Haibo Ge, Wei Ouyang

Published 2026-08-25

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CC BY 4.0

Original authors: Qiang Hu, Yanhua Zou, Shuaiyi Huo, Haibo Ge, Wei Ouyang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to navigate a car through a dense fog where the only map you have is a compass and a speedometer that update a hundred times a second. This is the reality for modern machines, from self-driving cars to drones, that rely on an inertial measurement unit, or IMU, to know where they are. These small devices contain spinning gyroscopes and sensitive accelerometers that track every twist and turn. However, these sensors are not perfect; they drift over time, accumulating tiny errors that can send a vehicle miles off course if left unchecked. To fix this, engineers combine the IMU's rapid data with slower, more reliable signals from satellites or laser scanners. The challenge lies in telling the computer exactly how much to trust the IMU versus the other sensors. If the computer thinks the IMU is too perfect, it will ignore the helpful corrections from the outside world. If it thinks the IMU is too unreliable, it will discard the valuable high-speed motion data. Getting this balance right is the difference between a smooth journey and a crash.

For years, engineers have struggled with a specific part of this balancing act: how to set the rules for the sensor's internal "noise" and "bias." The noise is the random static that makes a reading jitter, while the bias is a slow, creeping drift that causes the sensor to think it is moving when it is actually still. Standard practice involves testing the sensor while it sits perfectly still on a table. This static test produces a chart called an Allan variance plot, which reveals the sensor's characteristics. However, the real world is rarely still. When a sensor is vibrating in a moving vehicle, the conditions change, and the numbers derived from the quiet table test often fail to predict how the sensor will behave in motion. This mismatch has left many navigation systems relying on guesswork or deep, intuitive experience rather than a solid method to tune their settings.

In a recent study, researchers from Tongji University in Shanghai tackled this problem by creating a clear, mathematical bridge between the quiet table tests and the chaotic reality of a moving vehicle. They focused on the Kalman filter, a sophisticated algorithm that acts as the brain of the navigation system, constantly deciding how to blend different streams of information. The team realized that the standard way of translating the static test results into the filter's settings was often imprecise. They developed a new method to convert the data from the Allan variance plot directly into the specific numbers the filter needs to understand the sensor's uncertainty. Instead of guessing, their approach calculates the exact "power spectral density," a measure of how the sensor's errors behave over time, based on the proven characteristics found during the static calibration.

The researchers tested this new tuning method on two very different navigation systems. First, they used a setup that combines inertial sensors with satellite signals, a common configuration for vehicles. They compared their tuned settings against the factory defaults and found that their method allowed the system to maintain a more accurate position, especially when the satellite signal was briefly lost. The system was better at resisting the slow drift that usually plagues these sensors. Next, they tested the method on a more difficult scenario: a system that uses laser scanners and inertial sensors to navigate through large, open areas where there are few landmarks to lock onto. In these "degenerate" environments, the laser scanner often struggles, forcing the system to rely heavily on the inertial sensor. Here, the difference was stark. While the standard settings used in popular open-source software performed adequately in some cases, they performed significantly worse in the most challenging test sequences compared to the tuned parameters. In the most difficult test sequence, the standard settings resulted in an average position error of over 11 meters, whereas the system using the researchers' tuned parameters reduced this error to approximately 3.2 meters.

The study highlights that the way we currently set these sensors is often too rigid. By using the specific relationship between the sensor's static test results and its behavior in motion, the researchers showed that it is possible to extract much better performance from existing hardware. They did not invent a new sensor or a new type of computer; they simply provided a better way to tell the computer how to listen to the sensor it already has. Their work suggests that for many navigation tasks, the key to better accuracy lies not in buying more expensive equipment, but in understanding the subtle language of the noise and drift that every sensor produces. This approach offers a practical path forward for engineers who need their machines to navigate the real world with greater confidence and precision.

Technical Summary: Setting of IMU Parameters in Kalman Filtering-based Information Fusion

Problem Statement
The setting and tuning of Inertial Measurement Unit (IMU) parameters within sensor fusion frameworks, particularly those based on Kalman filtering, present a significant challenge. While the noise and bias instabilities of IMUs are typically characterized under static calibration conditions using Allan variance, these specifications often fail to accommodate the complexity of dynamic working conditions. Consequently, the effective tuning of process uncertainty parameters (specifically noise and bias instability) in the Kalman filter's covariance matrix frequently relies on empirical experience rather than a rigorous theoretical link between calibration data and filter parameters. This gap is exacerbated by the distinction between continuous-time noise modeling (Power Spectral Density, PSD) and discrete-time filter implementation, leading to sub-optimal information fusion if parameters are set incorrectly.

Methodology
The authors propose a systematic method for setting IMU parameters by bridging the gap between Allan variance calibration results and the continuous-time process noise covariance matrix used in Kalman filtering.

Theoretical Formulation: The paper revisits the discretization of continuous-time error-state models for inertial navigation. It establishes the mathematical relationship between the continuous-time Power Spectral Density (PSD) matrix ( $S$ ) and the discrete-time process noise covariance matrix ( $Q$ ). The authors derive that for small time intervals, $Q \approx G S \Delta t G^T$ , where $G$ is the input matrix.
Allan Variance to PSD Conversion: The core of the methodology involves converting standard Allan variance specifications (Noise Random Walk $N$ $N$ and Bias Instability $B$ $B$ ) into the PSD coefficients required for the filter.
- For Noise Random Walk, the PSD is directly related to the coefficient $N^2$ .
- For Bias Instability, the authors model the bias as a first-order Gauss-Markov process. They derive a specific relationship between the Allan variance value at the "flat" region of the plot and the PSD coefficient ( $S_B$ ). Through integration and analysis of the function $F(\lambda)$ , they determine that the PSD for bias instability is approximately $S_B \approx 5.25 \frac{\sigma_{zB}^2}{T_B}$ , where $\sigma_{zB}$ is the bias instability value from the Allan variance plot and $T_B$ is the correlation time.
Tuning Strategy: The authors introduce scaling factors ( $k_{bg}$ and $k_{ba}$ ) to adjust the calculated bias instability parameters. This allows for fine-tuning based on specific system performance, acknowledging that the correlation time ( $T_B$ ) extracted from Allan variance plots can be non-isotropic and sensitive to the chosen integration period.

Key Contributions
The paper makes two primary contributions:

Reformulation of Noise Matrices: The authors revisit and clarify the derivation of continuous-time and discrete-time noise matrices, explicitly proposing a method to set the process covariance matrix based on Allan variance calibration data.
Validation in Degenerated Scenarios: The proposed parameter setting method is experimentally verified in two distinct systems:
- INS/GNSS Integrated Navigation: Using the KFGINS open-source framework.
- LiDAR-Inertial Odometry: Using the FASTLIO2 open-source framework, specifically tested under large-scale degenerated scenarios where LiDAR features are scarce, forcing reliance on IMU propagation.

Experimental Results

INS/GNSS: Experiments with the ADIS16465 IMU showed that the system generally preferred the standard setting of calibrated parameters (scaling factor $k=1$ ). While slightly reducing the bias instability parameters (scaling factor $k=0.5$ ) also achieved satisfactory positioning accuracy, the results indicated a preference for the standard calibrated values.
LiDAR-Inertial Odometry: Tests on the Xsens MTi-600 IMU using degenerated datasets (e.g., Seq4) demonstrated the critical importance of parameter tuning.
- The default FASTLIO2 settings resulted in high errors in challenging sequences (e.g., RMSE of 11.7m in Seq4).
- Applying the proposed tuning method showed that the choice of scaling factor was critical. A scaling factor of $k=10$ caused the system to diverge in Seq4. Increasing the factor to $k=25$ resulted in a high average RMSE of 18.4m, while further increasing to $k=50$ improved the result to 5.58m.
- The most effective tuning was achieved by selecting specific companion factors ( $k_{bg}=20, k_{ba}=40$ ), which significantly reduced the average RMSE in Seq4 to 3.26m.
- The study noted that the default discrete covariance values in FASTLIO2 were orders of magnitude different from those derived via the proposed Allan variance-to-PSD conversion, highlighting the necessity of the proposed tuning approach for industrial-grade sensors.

Significance and Claims
The paper claims that the effective tuning of IMU parameters is not merely an empirical task but can be grounded in the rigorous relationship between Allan variance and Power Spectral Density. By providing a clear derivation for converting calibration data into filter parameters, the authors offer a method to improve the reliability of inertial-based navigation, particularly in scenarios where external measurements (like GNSS or LiDAR) are degraded or unavailable. The work emphasizes that while noise random walk parameters are often reliable, bias instability parameters require careful derivation and potential scaling to match the specific dynamics of the fusion system. The authors conclude that their method demonstrates feasibility in both INS/GNSS and LiDAR-Inertial systems, with future work aimed at developing even more efficient tuning methodologies.

Technical Summary: Setting of IMU Parameters in Kalman Filtering-based Information Fusion

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