pith. sign in

arxiv: 2307.08571 · v2 · pith:6QM64EDVnew · submitted 2023-07-17 · 📡 eess.SY · cs.SY

Parametric and State Estimation of Stationary MEMS-IMUs: A Tutorial

Daniel Engelsman , Yair Stolero , Itzik Klein This is my paper

Pith reviewed 2026-05-24 07:15 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords inertial navigationMEMS IMUmultiple sensorsstationary arrayerror modelingparametric estimationstate estimation
0
0 comments X

The pith

A stationary array of multiple MEMS-IMUs reduces instrumental errors in a manner that scales with sensor count and elapsed time, as shown by analysis and experiments.

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

The paper sets out to show that multiple inertial sensors placed in a fixed, levelled configuration improve measurement quality and slow the growth of navigation errors. It builds an analytical description of how error depends on the number of sensors and on time, then checks that description against data collected from real hardware. A reader would care because the result points to a practical way to make inertial navigation more accurate by adding inexpensive sensors rather than by adding external references or complex filters.

Core claim

For a stationary and levelled array of MEMS inertial measurement units the authors derive and test an analytical relationship in which measurement and state errors decrease with both increasing sensor count and with time; the model is shown to match experimental observations across signal-level metrics and the resulting navigation estimates of position, velocity, and orientation.

What carries the argument

The analytical model that expresses the robustness of a stationary sensor array against instrumental errors as a function of sensor number and elapsed time.

If this is right

  • Signal accuracy, frequency resolution, and noise rejection all improve as more sensors are added.
  • Redundancy increases, allowing the system to tolerate individual sensor faults.
  • Navigation-state drift is reduced in proportion to the same factors that improve the raw signals.
  • The improvement can be obtained without external aiding or frequency-domain filtering.

Where Pith is reading between the lines

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

  • The same scaling idea could be tested in mildly dynamic conditions if motion compensation is first applied to the raw data.
  • The approach may complement existing Kalman-filter or frequency-domain methods rather than replace them.
  • Similar array-level error reduction might appear in other low-cost sensor types once their dominant error sources are modeled.

Load-bearing premise

The sensors stay perfectly stationary and level for the entire duration of the measurements.

What would settle it

Repeating the experiment with a moving or tilted array and finding that the observed error does not follow the predicted dependence on sensor count and time would falsify the central relationship.

Figures

Figures reproduced from arXiv: 2307.08571 by Daniel Engelsman, Itzik Klein, Yair Stolero.

Figure 1
Figure 1. Figure 1: The Xsens-DOT, a dedicated apparatus for alignment and synchronization of five inertial sensors [36]. B. Experimental setup Under laboratory conditions, data from the stationary sen￾sors was acquired in a time span of about 100 seconds, sampled at 100 Hz. Overall, recordings were obtained from a total of ten sensors, using two Xsens-Dot cases 1 carrying five identical sensors each, as shown in [PITH_FULL_… view at source ↗
Figure 3
Figure 3. Figure 3: Raw biased distributions of the inertial measurements; [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Biased measurements of the inertial sensors; [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bias-free distributions of the inertial measurements; [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimated noise density (RMS) in a log-log plane. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Estimated noise density (RMS) in a semi-log plane. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Propagation of the error states, comparing single sensor [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time-varying distribution; state estimates in black [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Error ellipsoid of position error after 100 seconds. [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Effects of sensor count on precision and accuracy. [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
read the original abstract

Inertial navigation systems (INS) are widely used in almost any operational environment, including aviation, marine, and land vehicles. Inertial measurements from accelerometers and gyroscopes allow the INS to estimate position, velocity, and orientation of its host vehicle. However, as inherent sensor measurement errors propagate into the state estimates, accuracy degrades over time. To mitigate the resulting drift in state estimates, different approaches of parametric and state estimation are proposed to compensate for undesirable errors, using frequency-domain filtering or external information fusion. Another approach uses multiple inertial sensors, a field with rapid growth potential and applications. The increased sampling of the observed phenomenon results in the improvement of several key factors such as signal accuracy, frequency resolution, noise rejection, and higher redundancy. This study offers an analysis tutorial of basic multiple inertial operation, with a new perspective on the error relationship to time, and number of sensors. To that end, a stationary and levelled sensors array is taken, and its robustness against the instrumental errors is analyzed. Subsequently, the hypothesized analytical model is compared with the experimental results, and the level of agreement between them is thoroughly discussed. Ultimately, our results showcase the vast potential of employing multiple sensors, as we observe improvements spanning from the signal level to the navigation states. This tutorial is suitable for both newcomers and people experienced with multiple inertial sensors.

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

2 major / 2 minor

Summary. The manuscript is a tutorial analyzing the use of multiple stationary and levelled MEMS-IMUs to reduce instrumental errors. It derives an analytical model relating error reduction to integration time and sensor count N, validates the model against experiments on a stationary array, and claims that the resulting improvements extend from the raw signal level through to the navigation states (position, velocity, orientation).

Significance. If the error-vs-time-vs-N relationship holds under the stated conditions, the tutorial usefully quantifies the benefit of sensor redundancy for stationary inertial applications and provides a clear pedagogical treatment of basic multi-IMU averaging. The experimental comparison and explicit discussion of model-experiment agreement are positive features.

major comments (2)
  1. [Abstract] Abstract and concluding section: the central claim that improvements 'span from the signal level to the navigation states' is load-bearing yet unsupported. All analysis and data are restricted to a stationary, levelled array; no derivation, simulation, or experiment addresses how the hypothesized error scaling continues when dynamic errors (scale-factor, misalignment, vibration rectification, g-sensitivity) are present.
  2. [Analytical model derivation] The analytical model section (presumably the derivation relating instrumental-error variance to time and N): the model is stated to apply only to stationary/levelled conditions, but the navigation-state claim implicitly assumes the same scaling governs INS error propagation under motion. This assumption is not tested or bounded.
minor comments (2)
  1. Notation for the multi-sensor averaging operator and the precise definition of 'instrumental errors' should be introduced earlier and used consistently.
  2. Figure captions should explicitly state the number of sensors N, integration time, and whether data are raw or averaged.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our tutorial. We address the major comments point by point below, agreeing that clarifications are needed to better reflect the manuscript's stationary scope.

read point-by-point responses

  1. Referee: [Abstract] Abstract and concluding section: the central claim that improvements 'span from the signal level to the navigation states' is load-bearing yet unsupported. All analysis and data are restricted to a stationary, levelled array; no derivation, simulation, or experiment addresses how the hypothesized error scaling continues when dynamic errors (scale-factor, misalignment, vibration rectification, g-sensitivity) are present.

    Authors: We agree that the central claim requires qualification. The manuscript title, experimental setup, and model derivation are restricted to stationary and levelled conditions, with navigation-state improvements demonstrated only via integration of the averaged signals to position, velocity, and orientation (which remain constant in the stationary case). No analysis of dynamic errors is included, as this is outside the tutorial's scope. We will revise the abstract and concluding section to explicitly state that all claims and results apply to stationary/levelled operation and that extension to dynamic conditions is not addressed. revision: yes

  2. Referee: [Analytical model derivation] The analytical model section (presumably the derivation relating instrumental-error variance to time and N): the model is stated to apply only to stationary/levelled conditions, but the navigation-state claim implicitly assumes the same scaling governs INS error propagation under motion. This assumption is not tested or bounded.

    Authors: The analytical model is derived and validated exclusively under stationary/levelled assumptions for bias and noise averaging. Navigation-state results are obtained by direct double integration of the averaged accelerometer and gyroscope signals from the stationary array, not via a full dynamic INS error-propagation analysis. We acknowledge that no bounding or testing under motion is provided. We will add explicit statements in the model section and conclusion clarifying that the scaling applies only to the stationary case. revision: yes

Circularity Check

0 steps flagged

No significant circularity: analytical model for stationary array validated against independent experiments

full rationale

The paper derives an analytical error model for a stationary, levelled multi-sensor array relating instrumental error reduction to time and sensor count N, then compares the hypothesized model directly to separate experimental measurements. No equations or claims reduce by construction to fitted inputs, no self-citation chains are invoked as load-bearing uniqueness theorems, and the central claim of signal-to-navigation improvements rests on this external comparison rather than renaming or self-definition. The derivation chain is therefore self-contained against the provided experimental benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions about inertial sensor error propagation and the benefits of averaging across multiple units; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption The sensors are stationary and levelled
    The study takes a stationary and levelled sensors array for the analysis of error robustness as stated in the abstract.

pith-pipeline@v0.9.0 · 5772 in / 1155 out tokens · 44326 ms · 2026-05-24T07:15:27.591440+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Underwater MEMS Gyrocompassing: A Virtual Testing Ground

    eess.SY 2024-02 unverdicted novelty 4.0

    Machine learning framework refines disturbed inertial measurements to enable accurate gyrocompassing for UUVs by focusing on Earth's rotation vector.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    An introduction to inertial navigation,

    O. J. Woodman, “An introduction to inertial navigation,” University of Cambridge, Computer Laboratory, Tech. Rep., 2007

    work page 2007

  2. [2]

    P. D. Groves, Principles of GNSS, Inertial and Multi- sensor Integrated Navigation Systems . Artech House, 2013

    work page 2013

  3. [3]

    Titterton and J

    D. Titterton and J. L. Weston, Strapdown Inertial Naviga- tion Technology. American Institute of Aeronautics and Astronautics and the Institution of Electrical Engineers, 2004

    work page 2004

  4. [4]

    Data fusion algo- rithms for multiple inertial measurement units,

    J. B. Bancroft and G. Lachapelle, “Data fusion algo- rithms for multiple inertial measurement units,” Sensors, vol. 11, no. 7, pp. 6771–6798, 2011

    work page 2011

  5. [5]

    An open-source multi inertial measurement unit (MIMU) platform,

    I. Skog, J.-O. Nilsson, and P. H ¨andel, “An open-source multi inertial measurement unit (MIMU) platform,” in 2014 International Symposium on Inertial Sensors and Systems (ISISS). IEEE, 2014, pp. 1–4

    work page 2014

  6. [6]

    A lightweight and accurate localization algorithm using multiple iner- tial measurement units,

    M. Zhang, X. Xu, Y . Chen, and M. Li, “A lightweight and accurate localization algorithm using multiple iner- tial measurement units,” IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 1508–1515, 2020

    work page 2020

  7. [7]

    Sensor fusion to improve state estimate accuracy using multiple inertial measure- ment units,

    U. N. Patel and I. A. Faruque, “Sensor fusion to improve state estimate accuracy using multiple inertial measure- ment units,” in 2021 IEEE International Symposium on Inertial Sensors and Systems (INERTIAL) . IEEE, 2021, pp. 1–4

    work page 2021

  8. [8]

    A unified filter for fusion of multiple inertial measurement units,

    Y . Libero and I. Klein, “A unified filter for fusion of multiple inertial measurement units,” arXiv preprint arXiv:2208.14524, 2022

    work page arXiv 2022

  9. [9]

    Multiple inertial measurement unit fusion for pedestrian navigation,

    J. B. Bancroft, “Multiple inertial measurement unit fusion for pedestrian navigation,” University of Calgary , 2010

    work page 2010

  10. [10]

    Pedestrian track- ing using an IMU array,

    I. Skog, J.-O. Nilsson, and P. H ¨andel, “Pedestrian track- ing using an IMU array,” in 2014 IEEE International Conference on Electronics, Computing and Communica- tion Technologies (CONECCT). IEEE, 2014, pp. 1–4

    work page 2014

  11. [11]

    On the noise and power performance of a shoe-mounted multi-IMU inertial positioning system,

    S. Bose, A. K. Gupta, and P. Handel, “On the noise and power performance of a shoe-mounted multi-IMU inertial positioning system,” in 2017 International Con- ference on Indoor Positioning and Indoor Navigation (IPIN). IEEE, 2017, pp. 1–8

    work page 2017

  12. [12]

    Deep-learning-based pedestrian inertial navigation: Methods, data set, and on-device inference,

    C. Chen, P. Zhao, C. X. Lu, W. Wang, A. Markham, and N. Trigoni, “Deep-learning-based pedestrian inertial navigation: Methods, data set, and on-device inference,” IEEE Internet of Things Journal, vol. 7, no. 5, pp. 4431– 4441, 2020

    work page 2020

  13. [13]

    Localization and velocity tracking of human via 3 IMU sensors,

    Q. Yuan and I.-M. Chen, “Localization and velocity tracking of human via 3 IMU sensors,” Sensors and Actuators A: Physical , vol. 212, pp. 25–33, 2014

    work page 2014

  14. [14]

    Window-warping: a time series data augmentation of imu data for construction equipment activity identification,

    K. M. Rashid and J. Louis, “Window-warping: a time series data augmentation of imu data for construction equipment activity identification,” in ISARC. Proceed- ings of the International Symposium on Automation and Robotics in Construction, vol. 36. IAARC Publications, 2019, pp. 651–657

    work page 2019

  15. [15]

    Online learning of wearable sensing for human activity recognition,

    Y . Zhang, B. Gao, D. Yang, W. L. Woo, and H. Wen, “Online learning of wearable sensing for human activity recognition,” IEEE Internet of Things Journal , vol. 9, no. 23, pp. 24 315–24 327, 2022

    work page 2022

  16. [16]

    Improving accuracy with multiple sensors: Study of redundant MEMS-IMU/GPS configurations,

    S. Guerrier, “Improving accuracy with multiple sensors: Study of redundant MEMS-IMU/GPS configurations,” in Proceedings of the 22nd international technical meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2009) , 2009, pp. 3114–3121

    work page 2009

  17. [17]

    A new approach to better low-cost MEMS IMU perfor- mance using sensor arrays,

    H. Martin, P. Groves, M. Newman, and R. Faragher, “A new approach to better low-cost MEMS IMU perfor- mance using sensor arrays,” in Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) , 2013, pp. 2125–2142

    work page 2013

  18. [18]

    Inertial sensor arrays—a liter- ature review,

    J.-O. Nilsson and I. Skog, “Inertial sensor arrays—a liter- ature review,” in 2016 European Navigation Conference (ENC). IEEE, 2016, pp. 1–10

    work page 2016

  19. [19]

    Iner- tial sensor arrays, maximum likelihood, and Cram ´er–Rao bound,

    I. Skog, J.-O. Nilsson, P. H ¨andel, and A. Nehorai, “Iner- tial sensor arrays, maximum likelihood, and Cram ´er–Rao bound,”IEEE Transactions on Signal Processing, vol. 64, 12 no. 16, pp. 4218–4227, 2016

    work page 2016

  20. [20]

    A precise dead reckoning algorithm based on bluetooth and multiple sensors,

    N. Yu, X. Zhan, S. Zhao, Y . Wu, and R. Feng, “A precise dead reckoning algorithm based on bluetooth and multiple sensors,” IEEE Internet of Things Journal , vol. 5, no. 1, pp. 336–351, 2017

    work page 2017

  21. [21]

    A novel 3-d indoor localization algorithm based on ble and multiple sensors,

    Y . Yu, R. Chen, L. Chen, X. Zheng, D. Wu, W. Li, and Y . Wu, “A novel 3-d indoor localization algorithm based on ble and multiple sensors,” IEEE Internet of Things Journal, vol. 8, no. 11, pp. 9359–9372, 2021

    work page 2021

  22. [22]

    Multiple inertial measurement units–an empirical study,

    A. Larey, E. Aknin, and I. Klein, “Multiple inertial measurement units–an empirical study,” IEEE Access , vol. 8, pp. 75 656–75 665, 2020

    work page 2020

  23. [23]

    Self-calibration of inertial sensor arrays,

    H. Carlsson, I. Skog, and J. Jald ´en, “Self-calibration of inertial sensor arrays,” IEEE Sensors Journal , vol. 21, no. 6, pp. 8451–8463, 2021

    work page 2021

  24. [24]

    A learning-based approach for bias elimination in low-cost gyroscopes,

    D. Engelsman and I. Klein, “A learning-based approach for bias elimination in low-cost gyroscopes,” in 2022 IEEE International Symposium on Robotic and Sensors Environments (ROSE). IEEE, 2022, pp. 01–05

    work page 2022

  25. [25]

    Principles of GNSS, inertial, and multisen- sor integrated navigation systems, [book review],

    P. D. Groves, “Principles of GNSS, inertial, and multisen- sor integrated navigation systems, [book review],” IEEE Aerospace and Electronic Systems Magazine , vol. 30, no. 2, pp. 26–27, 2015

    work page 2015

  26. [26]

    Calibration of a novel MEMS inertial reference unit,

    J. K. Bekkeng, “Calibration of a novel MEMS inertial reference unit,” IEEE Transactions on instrumentation and measurement, vol. 58, no. 6, pp. 1967–1974, 2008

    work page 1967

  27. [27]

    D. R. Cox and D. V . Hinkley,Theoretical statistics. CRC Press, 1979

    work page 1979

  28. [28]

    Cir- cumventing dynamic modeling: Evaluation of the error- state kalman filter applied to mobile robot localization,

    S. I. Roumeliotis, G. S. Sukhatme, and G. A. Bekey, “Cir- cumventing dynamic modeling: Evaluation of the error- state kalman filter applied to mobile robot localization,” in Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No. 99CH36288C) , vol. 2. IEEE, 1999, pp. 1656–1663

    work page 1999

  29. [29]

    Titterton, J

    D. Titterton, J. L. Weston, and J. Weston, Strapdown inertial navigation technology . IET, 2004, vol. 17

    work page 2004

  30. [30]

    A. H. Jazwinski, Stochastic processes and filtering the- ory. Courier Corporation, 2007

    work page 2007

  31. [31]

    Information and entropy flow in the kalman–bucy filter,

    S. K. Mitter and N. J. Newton, “Information and entropy flow in the kalman–bucy filter,” Journal of Statistical Physics, vol. 118, pp. 145–176, 2005

    work page 2005

  32. [32]

    Pseudo-measurements as aiding to INS during GPS outages,

    I. Klein, S. Filin, and T. Toledo, “Pseudo-measurements as aiding to INS during GPS outages,” Navigation, vol. 57, no. 1, pp. 25–34, 2010

    work page 2010

  33. [33]

    Information Aided Navigation: A Review

    D. Engelsman and I. Klein, “Information aided naviga- tion: A review,” arXiv preprint arXiv:2301.01114, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  34. [34]

    Numerical aspects of different kalman filter implementations,

    M. Verhaegen and P. Van Dooren, “Numerical aspects of different kalman filter implementations,” IEEE Transac- tions on Automatic Control, vol. 31, no. 10, pp. 907–917, 1986

    work page 1986

  35. [35]

    J. S. Bendat and A. G. Piersol, Random data: analysis and measurement procedures. John Wiley & Sons, 2011

    work page 2011

  36. [36]

    [Online]

    Xsens Technologies, Xsens DOT User Manual , Movella, Pantheon, 7521 Enschede, The Netherlands, 2021. [Online]. Available: https://www.movella.com/products/ wearables/movella-dot

    work page 2021