Tunable Virtual IMU Frame by Weighted Averaging of Multiple Non-Collocated IMUs

Tim Barfoot; Yizhou Gao

REVIEW 3 major objections 5 minor 28 references

Tunable Virtual IMU Frame by Weighted Averaging of Multiple Non-Collocated IMUs

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By choosing weights so that the weighted sum of IMU offsets from a chosen virtual frame is zero, the paper shows that lever-arm terms vanish exactly, producing a single tunable virtual IMU.

desk verdict A clean, correct lever-arm-cancelling IMU averaging recipe with a useful closed-form weight solver; the main gap is unexamined sensitivity to extrinsic calibration error and overstrong real-data claims. read the letter →

arxiv 2506.00371 v1 pith:QQYE4TPO submitted 2025-05-31 cs.RO

Yizhou Gao , Tim Barfoot This is my paper

classification cs.RO

keywords virtualIMUaveraginglever-armeffectmulti-IMUfusionsensorvisual-inertialodometryquadraticprogramminginertialmeasurementunits

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that several rigidly connected IMUs placed far apart can be combined into one virtual IMU simply by weighted averaging, provided the weights make the weighted sum of the IMU offsets from the chosen virtual frame vanish. With that one condition, every lever-arm term—the extra accelerations a rotating body imposes on a sensor offset from the rotation centre—cancels exactly, so the fused measurement looks like it came from a single IMU at the chosen frame. This matters because state estimators can then treat the array as one IMU located at, say, a camera or GNSS antenna, with no lever-arm corrections in the propagation model. The paper also supplies a closed-form quadratic-programming weight choice that places the virtual frame where the user wants while minimizing output noise, and it demonstrates reduced pose error in simulation and on a real multi-IMU dataset.

What carries the argument

The central object is a virtual IMU defined by weighted averages of the aligned sensor outputs, with the accelerometer weights constrained by $\sum_j w_j r_j = 0$. The mechanism is linearity of the skew-symmetric operator, which lets the offset sum $\sum_j w_j r_j$ factor out of the lever-arm terms so that the constraint cancels them. For frame placement, the paper solves a quadratic program that minimizes combined variance $\frac{1}{2}\sum_j (w_j\sigma_j)^2$ subject to $\sum_j w_j r_j = 0$ and $\sum_j w_j = 1$, giving a closed-form KKT solution $\hat w = \Sigma^{-1}(1 - R^T(\bar R \bar R^T)^+ \bar r)$ that is then normalized to obtain the final weights.

What would settle it

Mount two IMUs roughly a metre apart on a rigid bar, calibrate the offsets, choose weights satisfying $\sum_j w_j r_j = 0$, and spin the bar about its centre with no translation; if the weighted accelerometer average shows a signal growing with the square of angular rate, the lever-arm cancellation failed, meaning the calibration or the rigid-body assumption is wrong.

Watch

Extended reading notes

Core claim

The paper's central claim is that averaging multiple non-collocated accelerometers is simple and exact once the weights satisfy a single constraint. For IMU $j$ with known fixed pose $(C_j, r_j)$ relative to the desired virtual frame, the aligned accelerometer contains lever-arm terms $\omega^\wedge \omega^\wedge r_j$ and $\alpha^\wedge r_j$. Weighted averaging pulls these terms out as $\omega^\wedge \omega^\wedge \sum_j w_j r_j + \alpha^\wedge \sum_j w_j r_j$, so imposing $\sum_j w_j r_j = 0$ removes them identically, reducing the averaged measurement to the specific force at the virtual frame plus a combined bias and noise. The same alignment-and-average treatment works for gyroscopes, whose weights need only sum to one and can differ from the accelerometer weights.

Load-bearing premise

The derivation assumes the fixed position and orientation of each IMU relative to the chosen virtual frame is known exactly, because the lever-arm cancellation depends on weights satisfying $\sum_j w_j r_j = 0$ for those true offsets.

Editorial extensions

If this is right

The VIMU can be placed coincident with a camera, GNSS antenna, or any target frame inside the convex hull of the IMUs, so propagation requires no lever-arm terms.
With $n$ identical IMUs and equal weights, the combined noise and bias-drift standard deviation scale as $\sigma/\sqrt{n}$, so pose error shrinks as IMUs are added, matching the simulation trends.
Because gyroscope averaging is location-independent, gyroscope and accelerometer weights can be chosen separately, allowing simultaneous noise minimization and frame placement.
The closed-form weight solution lets a user pick both the virtual frame location and the noise level without iterative optimization, making the method a drop-in replacement for a single IMU in existing estimators.
Placing the VIMU outside the convex hull forces some weights to become negative, and the real-data experiments show this amplifies noise and increases pose error.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the fixed poses of the IMUs were estimated online instead of assumed known, the same zero-sum condition could be re-enforced continuously, yielding a virtual frame whose lever-arm cancellation survives slow mounting flex or thermal drift.
The convex-hull result doubles as a sensor-placement criterion: to keep all weights nonnegative and noise low, the desired virtual frame should lie inside the IMU array, so mounting IMUs around a camera rather than off to one side is preferable.
The derivation generalizes to any array of vector-valued sensors whose outputs contain a term linear in sensor offset, such as magnetometers on a rotating rigid body, where the same weighted-average condition would cancel the offset term.
Fault detection could be implemented by watching residuals from the weighted average and re-solving the weights with outlier IMUs down-weighted while preserving the zero-offset constraint, an extension the paper flags as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper has a genuinely clean algebraic core. If you choose accelerometer weights so that the weighted sum of the IMU lever arms vanishes, the averaged accelerometer measurement collapses to the specific force at your chosen virtual frame, with no lever-arm terms and no need to estimate angular acceleration. The closed-form QP solution for those weights is valid, and the authors deserve credit for turning a simple observation into a practical design rule. The simulation is sensible, and the PennCOSYVIO results mostly support the noise-reduction benefit, though not uniformly.

The math in Section IV is correct under the stated assumptions: rigid body, known and fixed extrinsics, and the usual IMU noise/bias models. The KKT derivation is fine, aside from a small presentation wobble in Eq. (24) where they write ŵ = w/(wᵀΣw) as if it were an equality; it is really an unnormalized vector that gets normalized in (25). Minor.

The real soft spot is calibration sensitivity. The stress-test note is on target: the 'no lever-arm' claim is exact only when {Cⱼ, rⱼ} are perfect. With cm-level position error and typical rotation rates, the residual lever-arm error is on the order of 0.01–0.02 m/s², comparable to MEMS noise. The paper neither bounds this residual nor runs a perturbed-extrinsics simulation. That does not destroy the contribution—all IMU fusion methods need calibration—but it means the advertised benefit is an idealization, and the reader should know how fragile it is in practice.

The second issue is the real-data interpretation. Table III shows that the 'Averaged' configuration is not uniformly better than the single IMU: AS rotational error and BS positional error are worse. The text says 'we can clearly see that the averaged and centered configurations work better than using a single IMU,' which is too strong. The centered configuration wins on those runs, and the averaged is mixed. That is a claim-vs-evidence mismatch the authors can fix by qualifying the conclusion or analyzing why averaging sometimes hurts.

There is no circularity problem: the weights are computed from known geometry and noise parameters, and the experiments are on an external benchmark. The citation pattern looks reasonable, and the comparison to prior multi-IMU fusion methods is fair.

Who is this for? Roboticists doing visual-inertial or GNSS-inertial fusion who want a simple, computationally cheap way to leverage multiple IMUs without rewriting their estimator. It deserves a serious referee; the core result is correct and the paper is short enough that a careful review is low cost. I would ask the authors to add a calibration-sensitivity analysis and to temper the real-data claims before acceptance.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method for combining several rigidly connected but physically separated IMUs into a single virtual IMU (VIMU) by a weighted average of their measurements. The central derivation (Section IV) shows that if the weights satisfy sum_j w_j r_j = 0, where r_j is the position of IMU j in the desired VIMU frame, then the lever-arm terms in the averaged accelerometer cancel exactly and the VIMU measurement model reduces to that of a single IMU located at the VIMU frame (Eqs. (12)-(14)). The authors then pose a quadratic program that minimizes the combined measurement noise subject to this placement constraint and derive a closed-form KKT solution (Eqs. (18)-(25)). The method is evaluated in a monocular IEKF simulation with symmetric and asymmetric IMU configurations, and on the PennCOSYVIO dataset using OpenVINS, where a single IMU, an equal-weight average, an optimized VIMU placed near the reference camera, and a negative-weight configuration are compared. The main claims are noise reduction through averaging and the freedom to place the VIMU at a convenient location (e.g., a camera frame) without lever-arm terms in the propagation model.

Significance. The theoretical part of the paper is clean and largely self-contained. The lever-arm cancellation is an elegant observation, and the closed-form weights are a useful implementation detail, so the result, if accompanied by adequate validation, would offer a simple preprocessing step that makes multi-IMU arrays compatible with existing single-IMU estimators. The simulation results show monotonic error reduction as the number of IMUs grows, and the real-data study includes a sensible negative-weight control that behaves as predicted. The main limitations are that the derivation is exact only under perfect extrinsic calibration, no sensitivity analysis is provided for calibration errors, and the real-data experiment does not cleanly separate the averaging benefit from the frame-placement benefit. The paper is a solid incremental contribution rather than a breakthrough, but it is within the scope of a robotics sensing journal.

major comments (3)

[Section III.C, Eqs. (12)-(14)] The lever-arm cancellation is exact only under the stated assumption that {C_j, r_j} are known without error. The manuscript provides no sensitivity analysis or perturbed-extrinsics simulation, and the PennCOSYVIO experiment does not report calibration uncertainty. For small position errors delta r_j, the residual in the averaged accelerometer contains sum_j w_j (omega^ ^ omega^ delta r_j + alpha^ delta r_j), plus orientation-error terms; with delta r of about 1 cm and omega, alpha of order 1 rad/s and 1 rad/s^2, this residual is comparable to consumer MEMS noise. The abstract's claim that the VIMU 'removes the need to consider any lever-arm terms' should therefore be qualified to the ideal-calibration case, or the paper should add a first-order error analysis and a perturbed-extrinsics simulation to bound the practical regime.
[Section VI.D and Table III] The text states that 'the averaged and centered configurations work better than using a single IMU,' but Table III contains counterexamples: for run BS, the Averaged configuration gives positional MAE/RMSE of 2.1734/2.2367 m, which is worse than the Single-IMU baseline of 1.5926/1.7065 m; for run AS, the Averaged rotational MAE (0.0421 rad) is slightly worse than the Single baseline (0.0400 rad). Since each row appears to be a single-trajectory evaluation, the observed reversals could be due to run-to-run variance. The paper should provide repeated-run statistics or an explicit discussion of single-run variability, and the empirical claim should be moderated accordingly.
[Section VI.C-VI.D] The comparison does not isolate the frame-placement benefit from the averaging benefit. The 'Averaged' configuration (weights [1/3,1/3,1/3]) also satisfies the no-lever-arm condition of Eq. (13) at the centroid of the three IMUs, and in run AF it outperforms the 'Centered' configuration. In addition, OpenVINS is run with online extrinsic and time-offset estimation, so a constant displacement between the VIMU and the camera can be absorbed for any configuration. The experiments therefore demonstrate noise/bias-drift reduction from averaging, but they do not directly validate the claim that placing the VIMU at the camera removes the need for lever-arm modeling. A controlled test (for example, comparing the VIMU placed at the camera against the same averaging with the extrinsics handled differently) would provide the missing evidence.

minor comments (5)

[Table I] The row labeled 'IMU-S6 Asymmetric 6' should be labeled 'IMU-A6' to match Table II and avoid a duplicate label with the symmetric six-IMU row.
[Section II.D] The sentence 'Unlike methods in II-D, [22] proposes...' appears to reference the section it is in; it should refer to Section II-C (or the intended prior class) and should be corrected.
[Section V.D] The phrase 'as shown by (17) and (??)' contains an unresolved cross-reference; replace '(??)' with the correct equation number.
[Section VII] The sentence 'we presented new way to average individual IMUs' should read 'we presented a new way to average individual IMUs.'
[Section IV.E] The statement 'in theory, we can place the VIMU frame anywhere' with four non-coplanar IMUs is true only for signed weights; the subsequent discussion of negative weights explains this, but an explicit qualification in the text would prevent misreading.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lever-arm cancellation is a direct algebraic consequence of the deliberately imposed constraint, and the weights are computed from known calibration and noise parameters, not fitted to outcomes.

full rationale

The derivation from Eq. (7) to Eq. (13)-(14) is a straightforward algebraic manipulation: aligning accelerometer measurements to the vehicle frame and factoring the skew-symmetric lever-arm terms gives terms proportional to sum_j w_j r_j. Setting that sum to zero is an explicit design constraint in the optimization (18), not a hidden fit. The weights are computed closed-form from the known extrinsic parameters {C_j, r_j} and the assumed noise variances sigma_j^2, and the same weights are then tested on simulated data and on the external PennCOSYVIO benchmark, so no result is fitted to the experimental outcome. There are no load-bearing self-citations: the cited prior work (e.g., IEKF [26], OpenVINS [28]) is used as an external estimator, and the paper's own authors appear only as the authors of this manuscript. The concern that imperfect extrinsic calibration leaves residual lever-arm terms is an assumption/sensitivity issue, not a circularity: the paper states the fixed-and-known calibration assumption explicitly in Section III.C and does not claim robustness to calibration error. The 'no lever-arm terms' statement is conditional on exactly that assumption and on the constraint being satisfied, so it is a designed property rather than a prediction that reduces to its inputs. Overall, the derivation chain is self-contained and no step is circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation relies on standard rigid-body kinematics and a standard IMU measurement model. The only substantive domain assumptions are the rigid-body condition, known extrinsic calibrations, and independent isotropic IMU noise processes. No free parameters are introduced by the paper; the weights are the output of the optimization, and the noise variances are assumed known from the IMU specifications.

assumptions (5)

domain assumption The IMUs are rigidly connected to a single rigid body, so the kinematics of each IMU obeys the transport theorem as in Eq. (7).
Invoked in Section III.C when writing the accelerometer measurement model with lever-arm terms.
domain assumption The extrinsic calibrations {C_j, r_j} between each IMU and the VIMU frame are fixed and known.
Stated in Section III.C; required for the zero-lever-arm condition to hold exactly.
domain assumption IMU biases drift as random walks with independent, isotropic Gaussian increments, so the combined drift covariance is sum_j w_j^2 sigma_j^2.
Used in Section IV.D to set up the quadratic programming objective.
domain assumption The IMU measurement noises are white, zero-mean, and independent across sensors.
Standard model used throughout; needed for the noise reduction formula and the covariance expression.
standard math The quadratic program in Eq. (18) is convex, so the KKT conditions are necessary and sufficient.
Relied upon for the closed-form solution in Section IV.E.

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Cite this review

Pith. "Pith review of Tunable Virtual IMU Frame by Weighted Averaging of Multiple Non-Collocated IMUs." pith.science (2026). https://pith.science/paper/QQYE4TPO

@misc{pith2026250600371,
  author       = {Pith},
  title        = {Pith review of: Tunable Virtual IMU Frame by Weighted Averaging of Multiple Non-Collocated IMUs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQYE4TPO}},
  note         = {Machine review of arXiv:2506.00371}
}

read the original abstract

We present a new method to combine several rigidly connected but physically separated IMUs through a weighted average into a single virtual IMU (VIMU). This has the benefits of (i) reducing process noise through averaging, and (ii) allowing for tuning the location of the VIMU. The VIMU can be placed to be coincident with, for example, a camera frame or GNSS frame, thereby offering a quality-of-life improvement for users. Specifically, our VIMU removes the need to consider any lever-arm terms in the propagation model. We also present a quadratic programming method for selecting the weights to minimize the noise of the VIMU while still selecting the placement of its reference frame. We tested our method in simulation and validated it on a real dataset. The results show that our averaging technique works for IMUs with large separation and performance gain is observed in both the simulation and the real experiment compared to using only a single IMU.

Figures

Figures reproduced from arXiv: 2506.00371 by the authors.

**Figure 2.** Poses of IMUs in simulated configurations. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Example Results for Run AF. Single (blue) Averaged (red), Centered (green), Negative (yellow). We see that averaging [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) shows the type and location of sensors on the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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