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arxiv: 2304.03057 · v4 · submitted 2023-04-06 · 💻 cs.RO

Distributed UAV Formation Control Robust to Relative Pose Measurement Noise

Viktor Walter , Matou\v{s} Vrba , Daniel Bonilla Licea , Matej Hilmer , Martin Saska This is my paper

Pith reviewed 2026-05-24 09:35 UTC · model grok-4.3

classification 💻 cs.RO
keywords UAV formation controlrelative localization noiseFormation-Enforcing Controlgraph rigidityrobust gradient descentsensor noise handlingoscillation reduction
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The pith

Decomposing gradient descent commands and modifying them for estimated noise distribution lets UAVs hold tight formations despite relative pose errors.

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

The paper introduces a method to adapt Formation-Enforcing Control derived from graph rigidity for use with noisy onboard relative localization on lightweight UAVs. It breaks the control command into interpretable parts and adjusts each part according to the measured noise distribution to reduce the chance of overshooting the target shape. This change targets the oscillations and drifts that appear when standard gradient-descent FEC meets real sensor noise. Real-flight tests show the adjusted commands produce measurably smaller oscillations and state deviations than the unmodified version.

Core claim

The proposed solution decomposes the gradient descent-based FEC command into interpretable elements and modifies these individually based on the estimated distribution of sensory noise, such that the resulting action limits the probability of overshooting the desired formation.

What carries the argument

Decomposition of the gradient descent-based FEC command into interpretable elements, each modified according to the estimated distribution of relative pose measurement noise.

If this is right

  • Oscillations and drifts in sensor-driven UAV formations are reduced when command elements are adjusted to noise statistics.
  • Tight formations become feasible with existing relative localization hardware that carries non-negligible measurement noise.
  • The approach maintains the convergence guarantees of the original graph-rigidity FEC while adding noise robustness.
  • Practical deployment no longer requires noise-free sensors or external positioning infrastructure.

Where Pith is reading between the lines

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

  • Similar decomposition-plus-modification steps could be applied to other gradient-based formation or consensus controllers that suffer from sensor noise.
  • Onboard real-time noise distribution estimation becomes a necessary supporting capability for any noise-robust formation system.
  • The method may generalize to ground robots or manipulators that rely on relative pose measurements for coordinated motion.

Load-bearing premise

The distribution of sensory noise can be estimated accurately enough onboard in real time, and the separate modifications preserve convergence and stability of the original formation control.

What would settle it

A set of real-world flight trials in which the modified controller produces equal or greater oscillations and position deviations than pure gradient-descent FEC would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2304.03057 by Daniel Bonilla Licea, Martin Saska, Matej Hilmer, Matou\v{s} Vrba, Viktor Walter.

Figure 1
Figure 1. Figure 1: An example of our fully autonomous Unmanned Aerial Vehicles (UAVs) [ [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Measures involved in the construction of the formation for the presented case. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The velocity set with our proposed controller in 1D case. The region within [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: If the measured relative value of a target state is subject to observation noise, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: On the left, we show the comparison of the trade-off between stable-state standard [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simple case of a 2D relative position measurement with Gaussian noise. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Special case of 2D relative pose measurement for the orientation command based [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simulated formation flight of various configurations. [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Three UAV platforms used in the real-world experiments. [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The desired formations used in our experimental verification. The red propellers denote the tail side of the UAVs. The larger formations Al and Cl contain mutual distances [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Flight with four desired formations. Photographs of the top and side view of the formation are from the time of the closest convergence of each case. The top line of [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Plots of selected parameters measured during the flight of three UAVs where the desired formations were dynamically switched between formation A, formation B, and [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Plots of selected parameters measured during the flight of three UAVs where the desired formations were switched dynamically between formation A and the larger formation [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Plots of selected parameters measured during the flight of three UAVs where the desired formations were switched dynamically between formation A, formation B, formation [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Conditional stopping probability conditioned on the offset [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
read the original abstract

A technique that allows a Formation-Enforcing Control (FEC) derived from graph rigidity theory to interface with a realistic relative localization system onboard lightweight Unmanned Aerial Vehicles (UAVs) is proposed in this paper. The proposed methodology enables reliable real-world deployment of UAVs in tight formations using relative localization systems burdened by non-negligible sensory noise. Such noise otherwise causes undesirable oscillations and drifts in sensor-based formations, and this effect is not sufficiently addressed in existing FEC algorithms. The proposed solution is based on decomposition of the gradient descent-based FEC command into interpretable elements, and then modifying these individually based on the estimated distribution of sensory noise, such that the resulting action limits the probability of overshooting the desired formation. The behavior of the system was analyzed and the practicality of the proposed solution was compared to pure gradient-descent in real-world experiments where it presented significantly better performance in terms of oscillations, deviation from the desired state

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 / 3 minor

Summary. The manuscript proposes a method to make graph-rigidity-derived Formation-Enforcing Control (FEC) robust to relative-pose measurement noise on UAVs. The gradient-descent command is decomposed into interpretable elements that are then individually scaled or clipped using an onboard estimate of the noise distribution, with the goal of limiting overshoot probability. The authors state that the modified system behavior was analyzed and report that real-world experiments show significantly reduced oscillations and deviation from the desired formation relative to unmodified gradient-descent FEC.

Significance. If the per-element modifications preserve convergence and stability of the original rigidity-based FEC, the work would address a practical barrier to tight sensor-based UAV formations. The real-world experiments constitute a concrete strength, supplying empirical evidence of improved behavior under realistic noise levels.

major comments (2)
  1. [§4] §4 (System Analysis): The claim that the modified commands still drive the system to the desired formation rests on an analysis whose details are not supplied; no Lyapunov function, invariance set, or contraction bound is given that accounts for the effect of noise-estimate error on the modified vector field.
  2. [§5] §5 (Experiments): The headline performance improvement is demonstrated only for the tested noise levels and formation graphs; the manuscript does not report how the onboard noise-distribution estimate is obtained in real time or provide a sensitivity analysis when that estimate deviates from ground truth, leaving the central robustness claim without a clear domain of validity.
minor comments (3)
  1. [§3] The decomposition of the FEC command into elements is described at a high level; explicit equations showing each element before and after the noise-based modification would improve reproducibility.
  2. Figure captions and axis labels in the experimental plots could more clearly indicate the noise levels and estimation method used in each trial.
  3. [§6] A short discussion of how the method scales with formation size or graph connectivity would help readers assess generality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address the two major points below and indicate the revisions that will be made.

read point-by-point responses

  1. Referee: [§4] §4 (System Analysis): The claim that the modified commands still drive the system to the desired formation rests on an analysis whose details are not supplied; no Lyapunov function, invariance set, or contraction bound is given that accounts for the effect of noise-estimate error on the modified vector field.

    Authors: We acknowledge that §4 provides only a qualitative description of how the decomposed and scaled commands affect the closed-loop vector field and does not contain a formal stability argument that incorporates errors in the onboard noise-distribution estimate. In the revised manuscript we will augment §4 with an explicit Lyapunov analysis that treats the estimation error as a bounded disturbance and shows convergence to a neighborhood of the desired formation whose size depends on the bound. revision: yes

  2. Referee: [§5] §5 (Experiments): The headline performance improvement is demonstrated only for the tested noise levels and formation graphs; the manuscript does not report how the onboard noise-distribution estimate is obtained in real time or provide a sensitivity analysis when that estimate deviates from ground truth, leaving the central robustness claim without a clear domain of validity.

    Authors: The current manuscript does not describe the real-time computation of the noise-distribution estimate nor does it include a sensitivity study. We will add both: a concise subsection explaining the onboard estimator and a set of additional simulation and flight results that quantify performance degradation as the estimate deviates from ground truth, thereby clarifying the operating domain of the robustness guarantee. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is a direct constructive modification using external noise estimates

full rationale

The paper's core contribution decomposes a standard gradient-descent FEC command (from graph rigidity theory) into elements and scales them using an onboard estimate of relative-pose noise distribution. This is an explicit engineering construction, not a reduction of any claimed prediction or result to its own fitted inputs or self-citations. No self-definitional loops, fitted parameters renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors, or smuggled ansatzes appear in the derivation chain. The method remains self-contained against external benchmarks (rigidity theory and real-time noise estimation) with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identifiable; the method implicitly relies on the ability to estimate noise distributions but does not detail any fitted values or new postulated entities.

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Reference graph

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