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Tracking in-silico Lagrangian sensors in a lab-scale stirred tank reactor

This paper demonstrates that trajectories of in-silico Lagrangian sensors in a lab-scale stirred tank reactor can be reconstructed with errors below 10% by processing inertial measurement unit data through filtering algorithms like the extended Kalman filter, particle filter, and unscented Kalman filter.

Original authors: Vamika Rathi, Fatima Sehar, Finn Sommer, Sebastian Goetschel, Eike Steuwe, Alexandra von Kameke, Daniel Ruprecht

Published 2026-06-12

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CC BY 4.0

Original authors: Vamika Rathi, Fatima Sehar, Finn Sommer, Sebastian Goetschel, Eike Steuwe, Alexandra von Kameke, Daniel Ruprecht

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 you are trying to follow a tiny, invisible swimmer inside a giant, churning bathtub. You can't see the swimmer, but you want to know exactly where it is at every second. This is the challenge scientists face when trying to track "Lagrangian sensors" inside chemical reactors. These sensors are like little floating robots that drift with the liquid to measure temperature, pressure, or chemical levels.

The problem? Once they are inside the swirling liquid, they get lost.

This paper presents a new "GPS system" for these tiny robots, but instead of satellites, it uses math and a special magnetic field. Here is how they did it, explained simply:

The Problem: The "Drunk" Swimmer

Usually, if you want to know where something is, you might just guess based on how fast it was moving last second (like trying to walk in the dark by counting your steps). But in a chemical reactor, the liquid swirls violently. If you just guess, your error grows huge very quickly, like a drunk person stumbling further and further off course.

Also, these sensors aren't perfect swimmers. They are heavy and "lazy" (scientists call this having "inertia"). When the water spins one way, the sensor might keep going the other way for a split second before catching up. This makes predicting their path even harder.

The Solution: A Magnetic Compass and a Smart Brain

The researchers proposed a setup with two main parts:

The Magnetic Beacon: Imagine placing a giant magnet above the bathtub. This creates a unique magnetic "fingerprint" everywhere in the water. The closer the sensor gets to the magnet, the stronger the signal. The sensor has a built-in compass (magnetometer) to read this signal.
The Smart Brain (Filters): The sensor also has an accelerometer (like the one in your phone that knows when you tilt it). The researchers fed this data into a "Smart Brain"—a computer algorithm that acts like a detective.

The detective doesn't just look at the data; it uses a physics-based rulebook. It knows how heavy the sensor is, how thick the liquid is, and how the liquid swirls. It combines the "I feel a magnetic pull this way" data with the "I am spinning this fast" data to guess where the sensor should be.

The Experiment: The "Virtual" Test

Since they couldn't easily test this on real, expensive sensors inside a real chemical plant yet, they built a digital twin (a video game version) of the sensor and the reactor.

The Setup: They created two scenarios in their computer:
1. A perfect, swirling mathematical vortex (like a tornado).
2. A realistic, messy flow from a real lab-scale stirred tank reactor.
The Trick: They made the "game" sensor move exactly according to complex physics laws. Then, they added "static" (noise) to the sensor's readings to make it look like a real, imperfect device.
The Test: They asked three different types of "Smart Brains" (algorithms) to figure out the sensor's path from the noisy data:
- The Extended Kalman Filter (EKF): A very popular, math-heavy detective.
- The Particle Filter (PF): A detective that runs thousands of "what-if" simulations at once.
- The Unscented Kalman Filter (UKF): A generic, off-the-shelf detective from a software library.

The Results: Who Won the Race?

The results were quite clear:

The Winners: The EKF and the Particle Filter were excellent detectives. Even with the noisy data and the messy physics, they reconstructed the sensor's path with very high accuracy (less than 5% error). They were like a seasoned navigator who knows the terrain and can ignore the static on the radio.
The Loser: The generic UKF worked okay in the simple, perfect vortex, but it completely failed in the messy, real-world reactor. It got confused by the complex swirls and its error exploded (over 1800% error). It was like trying to use a simple map to navigate a chaotic city during a storm.

The Catch (Limitations)

The paper is very honest about what they didn't do yet:

It's still a simulation: They tracked a "virtual" sensor, not a real physical one. The real sensor might behave slightly differently than the math predicts.
Magnetic Walls: This system only works if the reactor is made of non-magnetic materials (like glass or plastic). If the tank is steel, the magnet won't work.
Not Real-Time Yet: The computer took time to do the math. To use this in a real factory where decisions happen in milliseconds, they would need to make the code much faster (like switching from a bicycle to a race car).

The Bottom Line

The paper proves that by combining a magnetic field with a smart, physics-based computer algorithm, we can theoretically track these floating sensors in 3D space with high precision. It's a major step toward giving chemical engineers a "live camera" view of what's happening inside their reactors, but the real-world hardware and speed challenges still need to be solved.

Technical Summary: Tracking in-silico Lagrangian Sensors in a Lab-Scale Stirred Tank Reactor

Problem Statement
Lagrangian sensors (LS), which travel freely within a fluid to provide spatially resolved measurements of reactor conditions, offer significant advantages over fixed Eulerian sensors, particularly in bio-reactors where vigorous mixing is detrimental. However, the three-dimensional tracking of these sensors remains a largely unsolved challenge. While sensors can measure acceleration and magnetic fields via built-in inertial measurement units (IMUs), standalone inertial data suffers from rapid drift, and traditional tracking methods (e.g., video, radio signals) are often infeasible for standard industrial operations. Furthermore, LS do not perfectly follow fluid streamlines due to inertial effects (governed by the Stokes number), necessitating physics-based dynamic models rather than simple kinematic assumptions. This paper addresses the challenge of reconstructing the 3D trajectories of LS using only accelerometer and magnetometer readings fused with a physics-based dynamic model.

Methodology
The authors propose a tracking framework that fuses sensor data (acceleration and magnetic field strength/direction) with a bespoke dynamic model using sequential filtering algorithms.

Physical Setup: The study utilizes in-silico replicas of a recently proposed LS design (Geläschus et al., 2026). The sensor is equipped with an accelerometer and a vector magnetometer. An external artificial magnetic dipole field is generated above the reactor to serve as a reference beacon.
Ground Truth & Dynamics: The "ground truth" trajectories are generated by numerically solving the Maxey-Riley-Gatignol equations (MaRGE), which account for inertial effects, added mass, and the Basset history force. The study considers two flow fields:
1. An analytically defined 3D vortex.
2. An experimentally measured flow field from a lab-scale stirred tank reactor (STR), obtained via Shake-the-Box particle image velocimetry.
Filtering Algorithms: Three filtering approaches are implemented and compared:
1. Extended Kalman Filter (EKF): Uses a state vector including position, velocity, and acceleration. The process model is derived from the MaRGE equations but intentionally excludes the Basset history term to simulate a controlled model error. The system is linearized via Jacobians (computed using automatic differentiation) for state prediction and update.
2. Particle Filter (PF): A sequential Monte Carlo method using a bootstrap approach. The state vector includes position and velocity. It employs adaptive tempering to maintain particle diversity and systematic resampling with roughening to prevent sample impoverishment.
3. Unscented Kalman Filter (UKF): Implemented using the pykalman library as a baseline comparison, utilizing the same noise covariances and initial conditions as the EKF.
Noise & Validation: Synthetic data is generated with 5% Gaussian noise added to measurements. The filters are tested against this noisy data to reconstruct the ground truth. Performance is evaluated using the relative Euclidean error normalized by the trajectory arc length.

Key Contributions

Physics-Based Tracking Framework: The paper introduces a framework that integrates an artificial magnetic field with IMU data, specifically utilizing a dynamic model based on the MaRGE equations to account for the inertial nature of the sensors.
Model Error Robustness: The study demonstrates that accurate tracking is achievable even when the filter's process model (which ignores the Basset force) differs from the ground truth dynamics (which includes it), simulating a realistic scenario of model mismatch.
Algorithmic Comparison: A systematic comparison of EKF, PF, and UKF in both idealized (vortex) and complex, experimentally derived flow fields.
Open Source Reproducibility: All tracking results and code are made available for reproduction.

Results

Vortex Flow (Idealized): Both the EKF and PF successfully reconstructed trajectories with relative errors below 1%. The UKF initially performed well but diverged progressively after $t \approx 2$ , resulting in errors more than three times larger than the other filters by $t=5$ . The UKF was found to be significantly more sensitive to the initial state covariance.
Stirred Tank Reactor (Experimental): In the complex, experimentally measured flow field, the EKF outperformed the PF, maintaining relative errors below 6% (average ~1.1%) compared to the PF's average of ~2.2%. The PF exhibited higher volatility due to the difficulty of representing the posterior distribution in complex flows with a finite particle set. The UKF failed completely in this scenario, producing a mean relative error exceeding 1800%, attributed to the strong nonlinearities in the experimental flow field which the generic UKF implementation could not handle.
Model Error Handling: Despite the intentional omission of the Basset force in the filter models (introducing a significant model error), both EKF and PF were able to track the in-silico particles effectively, suggesting the framework is robust to certain types of dynamic model inaccuracies.

Significance and Claims
The paper claims that the proposed framework demonstrates the potential for online 3D tracking of Lagrangian sensors in chemical reactors. The authors emphasize that while the results are promising, they are based on in-silico particles and simulated trajectories, not real-world experimental tracking.

The study highlights several limitations and future steps:

Real-World Validation: The gap between the dynamic models used in filters and the actual motion of real-world LS (which may have geometric center/mass misalignment) is not yet fully understood. Future work must validate the approach with real particles in the lab-scale reactor.
Material Constraints: The artificial magnetic field approach is limited to reactors made of non-magnetic materials; steel vessels would require alternative positioning strategies.
Real-Time Capability: Current implementations are offline. Achieving real-time tracking requires a computational speedup of two orders of magnitude, likely necessitating a migration from Python to compiled languages (C/C++) and parallel processing on GPUs.
Noise Characteristics: The assumption of additive Gaussian noise may not reflect the complexities of low-power communication in industrial settings, though machine learning denoising techniques are suggested as a potential remedy.

In conclusion, the authors present a robust proof-of-concept that physics-based filtering of IMU and magnetic data can reconstruct trajectories with relative errors below 10% (and often below 5%) in complex flow environments, paving the way for future operational deployment of Lagrangian sensors.