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arxiv: 2606.23901 · v1 · pith:ABKBS6PPnew · submitted 2026-06-22 · 💻 cs.RO · cs.SY· eess.SY

Topological Online Learning for Displacement-based Formation Control

Saksham Sharma , Shubhankar Gupta , Sumant A Gunagi , Suresh Sundaram This is my paper

Pith reviewed 2026-06-26 07:45 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords formation controlonline learningdisplacement-based controlmulti-robot systemsdirected graphsgradient flowtopology adaptation
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The pith

TOLD updates interaction weights online to minimize displacement-based formation distortion for single-integrator agents.

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

The paper presents TOLD as a method that adapts the weights on the edges of the communication graph in real time instead of only adjusting individual robot controls. Two online gradient-based update rules are defined: unconstrained Online Gradient Flow and constrained Online Exponential Gradient Flow that keeps weights non-negative and convex. For single-integrator dynamics on directed graphs the exponential version is proved to drive the agents to asymptotic consensus while the unconstrained version keeps formation distortion bounded. Simulations with twelve agents under disturbances and hardware flights with Crazyflie quadrotors both record lower median distortion when the weight adaptation is added to existing node-level controllers.

Core claim

TOLD performs real-time edge-level weight adaptation via Online Gradient Flow or Online Exponential Gradient Flow; for single-integrator agents over directed graphs the exponential variant guarantees asymptotic consensus while the unconstrained variant guarantees bounded formation distortion.

What carries the argument

TOLD framework that updates interaction topology weights online with gradient flows to directly minimize formation distortion.

If this is right

  • OExpGF produces asymptotic consensus on directed graphs for single-integrator agents.
  • OGF keeps formation distortion bounded under the same dynamics.
  • Adding TOLD to node-level controllers yields 1.2 to 33.14 percent median cumulative Root Mean Distortion Error reduction in twelve-agent simulations.
  • Hardware trials with Crazyflie 2.0 quadrotors show greater than 62 percent median distortion reduction for OGF and 31.4 percent for OExpGF versus fixed-weight consensus.

Where Pith is reading between the lines

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

  • The same edge-weight adaptation could be tested on tasks such as flocking or area coverage by replacing the distortion cost with the corresponding objective.
  • Performance under packet loss or time-varying graphs remains unexamined and would require separate stability analysis.

Load-bearing premise

Agents obey single-integrator dynamics and weight updates can be performed without communication delays or unmodeled higher-order effects.

What would settle it

Deploy the controllers on agents with double-integrator dynamics or with added communication delays and check whether the reported consensus and distortion-reduction guarantees still hold.

Figures

Figures reproduced from arXiv: 2606.23901 by Saksham Sharma, Shubhankar Gupta, Sumant A Gunagi, Suresh Sundaram.

Figure 1
Figure 1. Figure 1: Overview of the TOLD formation control framework. OGF and OExpGF adapt the robot-robot interaction topology [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cumulative Root Mean Distortion Error (RMDE) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Top view of actual drone trajectories (solid) and velocity-based references (dashed); references are computed from [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Component and magnitude distributions of formation [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Temporal evolution of weight matrix Frobenius norms [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

This paper addresses the problem of robust formation control by introducing Topological Online Learning for Displacement-based (TOLD) formation control, a real-time edge-level adaptation framework. Unlike conventional node-level robust controllers that regulate individual robot inputs without modifying the interaction topology, TOLD updates the interaction topology weights online to directly minimize formation distortion. Two strategies are proposed under the TOLD formation control framework: Online Gradient Flow (OGF) with unconstrained weights and Online Exponential Gradient Flow (OExpGF) with non-negative convex weights. Theoretical analysis establishes that, for single-integrator agents over directed graphs, OExpGF guarantees asymptotic consensus, while OGF ensures bounded formation distortion. Simulations with twelve robots under intermittent disturbances show 1.2%-33.14% median cumulative Root Mean Distortion Error reduction when augmenting TOLD with node-level controllers. Hardware experiments with Crazyflie 2.0 quadrotors demonstrate over 62% (OGF) and 31.4% (OExpGF) reduction in median formation distortion compared to fixed-weight consensus.

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 / 0 minor

Summary. The manuscript introduces the Topological Online Learning for Displacement-based (TOLD) formation control framework with two online weight-adaptation strategies—Online Gradient Flow (OGF) and Online Exponential Gradient Flow (OExpGF)—that update edge weights in real time to minimize formation distortion for displacement-based control. For single-integrator agents on directed graphs, the paper claims OExpGF guarantees asymptotic consensus while OGF ensures bounded distortion; simulations with twelve robots under disturbances report 1.2%–33.14% median cumulative Root Mean Distortion Error reduction when combined with node-level controllers, and hardware trials on Crazyflie 2.0 quadrotors claim >62% (OGF) and 31.4% (OExpGF) reductions in median formation distortion versus fixed-weight consensus.

Significance. If the central claims hold after addressing the dynamics gap, the work would provide a concrete edge-level online adaptation mechanism that augments existing node-level robust controllers, with explicit theoretical statements under single-integrator dynamics and quantitative hardware validation. The absence of free parameters in the core flows and the direct comparison to fixed-weight baselines are positive features that would strengthen the contribution to real-time topological learning in multi-robot systems.

major comments (2)
  1. [Theoretical Analysis] Theoretical Analysis section: the asymptotic consensus guarantee for OExpGF and bounded-distortion guarantee for OGF are derived exclusively under single-integrator dynamics on directed graphs. The hardware experiments apply the identical update laws to Crazyflie 2.0 quadrotors whose closed-loop dynamics are fourth-order cascaded attitude/position systems; no singular-perturbation argument, reduction lemma, or robustness margin is supplied showing that the online flows remain contractive or bounded under the increased plant order.
  2. [Hardware Experiments] Hardware Experiments section: the reported 62% (OGF) and 31.4% (OExpGF) median formation-distortion reductions are presented as evidence of TOLD efficacy, yet the manuscript supplies no analysis separating the contribution of the online weight updates from the inner-loop attitude controllers, communication delays, or the particular fixed-weight baseline chosen; without such separation the attribution to the topological adaptation cannot be rigorously established.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments regarding the scope of the theoretical guarantees and the interpretation of the hardware results. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Theoretical Analysis] Theoretical Analysis section: the asymptotic consensus guarantee for OExpGF and bounded-distortion guarantee for OGF are derived exclusively under single-integrator dynamics on directed graphs. The hardware experiments apply the identical update laws to Crazyflie 2.0 quadrotors whose closed-loop dynamics are fourth-order cascaded attitude/position systems; no singular-perturbation argument, reduction lemma, or robustness margin is supplied showing that the online flows remain contractive or bounded under the increased plant order.

    Authors: We agree that the asymptotic consensus guarantee for OExpGF and the bounded-distortion guarantee for OGF are derived exclusively under single-integrator dynamics, as stated in the Theoretical Analysis section. The hardware experiments apply the same update laws to the velocity references of the quadrotors' inner-loop controllers and serve as empirical validation rather than a formal extension of the theory. No singular-perturbation or robustness analysis bridging the dynamics gap is provided in the manuscript. In the revised version we will add a clarifying remark in the Theoretical Analysis section stating the kinematic-level scope of the guarantees and noting that the experimental results are presented as practical evidence without claiming direct transfer of the asymptotic properties. This is a partial revision limited to improved exposition. revision: partial

  2. Referee: [Hardware Experiments] Hardware Experiments section: the reported 62% (OGF) and 31.4% (OExpGF) median formation-distortion reductions are presented as evidence of TOLD efficacy, yet the manuscript supplies no analysis separating the contribution of the online weight updates from the inner-loop attitude controllers, communication delays, or the particular fixed-weight baseline chosen; without such separation the attribution to the topological adaptation cannot be rigorously established.

    Authors: The hardware trials compare the TOLD framework (online weight adaptation) directly against a fixed-weight consensus baseline while keeping all other elements identical: the same Crazyflie 2.0 firmware, inner-loop attitude controllers, communication delays, and experimental conditions. The only difference between the two cases is the edge-weight policy. Consequently the observed reductions in median formation distortion are attributable to the online adaptation. We will revise the Hardware Experiments section to explicitly state this controlled comparison setup, thereby strengthening the attribution without requiring new experiments. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained

full rationale

The provided abstract and context describe a new TOLD framework whose OGF and OExpGF weight-update laws are defined directly, with theoretical guarantees (asymptotic consensus for OExpGF, bounded distortion for OGF) proved under single-integrator dynamics on directed graphs. Simulations and hardware results are presented as separate empirical checks rather than as inputs to the theory. No self-definitional equations, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the given material. The derivation chain therefore does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on standard domain assumptions from multi-agent systems without introducing free parameters or new entities; the central claims depend on single-integrator dynamics and directed graphs.

axioms (2)
  • domain assumption Agents are modeled as single-integrator dynamics
    Invoked explicitly for the theoretical analysis establishing guarantees over directed graphs.
  • domain assumption Interaction topology is a directed graph
    Used as the setting for both OExpGF consensus guarantee and OGF bounded distortion result.

pith-pipeline@v0.9.1-grok · 5725 in / 1271 out tokens · 38349 ms · 2026-06-26T07:45:43.747784+00:00 · methodology

discussion (0)

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Reference graph

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