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arxiv: 2606.13099 · v1 · pith:XLJRCPNYnew · submitted 2026-06-11 · 🧮 math.NA · cs.NA

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

Vamika Rathi , Fatima Sehar , Finn Sommer , Sebastian Goetschel , Eike Steuwe , Alexandra von Kameke , Daniel Ruprecht This is my paper

Pith reviewed 2026-06-27 06:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Lagrangian sensorsstirred tank reactortrajectory reconstructionKalman filterparticle filterinertial measurement unitMaxey-Riley-Gatignol equationsnumerical tracking
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The pith

In-silico Lagrangian sensors in a stirred tank reactor can be tracked in three dimensions from noisy accelerometer and magnetometer data, with position errors below 10%.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether filtering algorithms can turn readings from an inertial measurement unit into reliable three-dimensional position estimates for small sensors moving inside a chemical reactor. It applies an extended Kalman filter, a particle filter, and an unscented Kalman filter, each built around a custom dynamical model, to synthetic data generated from both an analytical vortex and measured flow inside a lab-scale stirred tank. The Maxey-Riley-Gatignol equations supply the ground-truth motion of inertial particles for validation. The work shows that trajectories can be recovered even when the input signals contain noise. This matters because three-dimensional tracking of such sensors has remained difficult, yet improved internal awareness could help operators monitor conditions in reactors where direct observation is limited.

Core claim

Filtering algorithms that use a bespoke dynamical model to convert accelerometer and magnetometer readings into position estimates can reconstruct the trajectories of in-silico Lagrangian sensors moving in a three-dimensional vortex or in the experimentally measured flow field of a lab-scale stirred tank reactor, with errors below 10% when the Maxey-Riley-Gatignol equations serve as ground truth.

What carries the argument

Filtering algorithms (extended Kalman filter, particle filter, unscented Kalman filter) driven by a bespoke dynamical model that converts noisy accelerometer and magnetometer readings into position estimates.

If this is right

  • The same filtering pipeline works for both analytically prescribed flow and experimentally recorded flow inside the tank.
  • All three filter types (extended Kalman, particle, unscented Kalman) produce usable position estimates under the tested noise levels.
  • The approach is tied to the specific sensor design that supplies only accelerometer and magnetometer data.
  • Reconstruction remains possible when the input data are generated from the Maxey-Riley-Gatignol description of inertial particle motion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the dynamical model remains accurate when the sensors are made physical, the method could move from synthetic tests to live reactor monitoring without additional hardware.
  • The same IMU-plus-filter combination might apply to particle tracking in other confined flows where optical access is poor.
  • Performance could be checked by varying the noise level or the tank impeller speed to map the region where errors stay below 10%.

Load-bearing premise

The bespoke dynamical model inside the filters accurately describes how the sensors actually move through the reactor flow field.

What would settle it

Release physical versions of the sensors in the real stirred tank, record their true positions with an independent method such as high-speed imaging, and check whether the filter outputs still stay within 10% error of those measured positions.

Figures

Figures reproduced from arXiv: 2606.13099 by Alexandra von Kameke, Daniel Ruprecht, Eike Steuwe, Fatima Sehar, Finn Sommer, Sebastian Goetschel, Vamika Rathi.

Figure 1
Figure 1. Figure 1: Illustration of a sensor inside a stirred tank reactor and position of the magnetic [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The effect of history term on the trajectory of inertial particle. The figure on [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructed trajectories over the time interval [0 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative error (35) over time for the EKF, PF and UKF (Pykalman) for an [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative Euclidean error (35) over time for the extended Kalman filter and [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
read the original abstract

Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates three-dimensional tracking of in-silico Lagrangian sensors inside a lab-scale stirred tank reactor. Synthetic IMU data (accelerometer and magnetometer) are generated from trajectories obeying the Maxey-Riley-Gatignol equations in both an analytic vortex and an experimentally measured flow field; these data are then processed by an extended Kalman filter, an unscented Kalman filter, and a particle filter that employ a bespoke dynamical model to produce position estimates. The central claim is that trajectories can be reconstructed with errors below 10 percent.

Significance. If the bespoke model accurately captures the sensor dynamics, the numerical experiments would constitute a controlled demonstration that standard nonlinear filters can recover particle paths from noisy IMU measurements in a realistic reactor flow. The use of an independent ground-truth generator (Maxey-Riley-Gatignol) rather than self-consistent synthetic data is a methodological strength that allows direct assessment of tracking error.

major comments (1)
  1. [Abstract] Abstract (paragraph on filtering algorithms): the manuscript does not state whether the bespoke dynamical model inside the EKF/UKF/particle filter is identical to, or a close approximation of, the Maxey-Riley-Gatignol equations used to generate the ground-truth trajectories. This distinction is load-bearing for the headline claim; identical models would render the reported sub-10% errors unsurprising and would not demonstrate robustness to the model mismatch that must occur when the filter is applied to real sensors.
minor comments (1)
  1. The abstract mentions the pykalman implementation but supplies no information on filter initialization, process-noise covariance tuning, or the precise form of the measurement model; these details are needed for reproducibility even if the model-equivalence issue is resolved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the opportunity to clarify the relationship between our filter model and the ground-truth generator. We address the point below and will revise the manuscript to make the modeling assumptions explicit.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on filtering algorithms): the manuscript does not state whether the bespoke dynamical model inside the EKF/UKF/particle filter is identical to, or a close approximation of, the Maxey-Riley-Gatignol equations used to generate the ground-truth trajectories. This distinction is load-bearing for the headline claim; identical models would render the reported sub-10% errors unsurprising and would not demonstrate robustness to the model mismatch that must occur when the filter is applied to real sensors.

    Authors: We agree that the distinction is important and that the current wording leaves it ambiguous. The bespoke dynamical model inside the filters is a reduced-order approximation of the Maxey-Riley-Gatignol equations: it retains the leading inertial, drag, and added-mass terms but drops higher-order Faxén corrections and uses a simplified interpolation of the fluid velocity field. This deliberate mismatch relative to the full MRG ground-truth generator is what allows us to claim robustness. We will revise both the abstract and the methods section to state this approximation explicitly, include a brief quantification of the omitted terms, and discuss the implications for real-sensor deployment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external ground truth

full rationale

The paper generates synthetic data from the Maxey-Riley-Gatignol equations as independent ground truth and feeds noisy IMU readings into filters whose internal dynamics are described as a separate 'bespoke dynamical model.' No equation or section reduces the reported <10% tracking error to a fit performed on the same data, nor does any load-bearing step rely on a self-citation chain or imported uniqueness result. The validation therefore tests the filters against an external generator rather than reproducing an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated beyond the use of standard filtering methods and the Maxey-Riley-Gatignol equations as ground truth.

axioms (1)
  • domain assumption The Maxey-Riley-Gatignol equations accurately describe the motion of inertial particles in the flow field.
    Used as ground-truth for validating reconstructed trajectories.

pith-pipeline@v0.9.1-grok · 5708 in / 1050 out tokens · 15930 ms · 2026-06-27T06:13:57.073266+00:00 · methodology

discussion (0)

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Reference graph

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