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arxiv: 2511.11533 · v3 · submitted 2025-11-14 · 💻 cs.RO · cs.AI

Volumetric Ergodic Control

Jueun Kwon , Max M. Sun , Todd Murphey This is my paper

Pith reviewed 2026-05-17 22:01 UTC · model grok-4.3

classification 💻 cs.RO cs.AI
keywords ergodic controlvolumetric coveragerobot motion planningspatial distributionsearch tasksmanipulationsample-based models
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The pith

Ergodic control extended to volumetric robot representations preserves asymptotic coverage guarantees while more than doubling efficiency.

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

The paper introduces a formulation of ergodic control that models the robot as having physical volume rather than treating it as a point mass. Standard ergodic control generates trajectories that asymptotically match a desired spatial distribution, but this breaks down in practice when the robot's body or sensors occupy space and interact with the environment. By incorporating sample-based volumetric models directly into the ergodic metric and control law, the method keeps the original theoretical coverage guarantees. It does so with only small added computation, making real-time use feasible. Experiments across search and manipulation tasks with varied robot dynamics and geometries confirm the approach completes tasks more than twice as efficiently while achieving full success rates.

Core claim

The central claim is that ergodic control can be reformulated around a volumetric state representation using arbitrary sample-based models of the robot's body and sensors. This extension optimizes coverage over spatial distributions for nonlinear systems, preserves the asymptotic guarantees of the original method, and incurs minimal computational overhead suitable for real-time control.

What carries the argument

The volumetric extension of the ergodic metric and control law, which folds sample-based models of robot volume into the spatial coverage optimization.

Load-bearing premise

That extending the ergodic metric and control law to a volumetric state representation preserves the original asymptotic coverage guarantees without additional conditions on volume sampling or dynamics.

What would settle it

An experiment on a search task with known robot volume where the volumetric method fails to reach full asymptotic coverage or loses the reported factor-of-two efficiency gain relative to the standard point-based controller under identical conditions.

Figures

Figures reproduced from arXiv: 2511.11533 by Jueun Kwon, Max M. Sun, Todd Murphey.

Figure 1
Figure 1. Figure 1: Erasing results using volumetric ergodic control (VEC). The first column shows the target distributions (airplane, Eiffel Tower, fire, heart, lock) and remaining columns show VEC applied with different end-effector geometries (scissors, star, sword, lightning, trophy), which are modeled as volumetric states with the sample-based representation. VEC reasons over the volumetric states and produces efficient … view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative results for erasing benchmark. VEC effectively leverages the geometry of the end-effector through the volumetric state representation, generating both translational and rotational movements to completely erase the target in less time compared to the baseline ergodic control method. to better assess the effectiveness of the volumetric state representation relative to the baseline. We implement o… view at source ↗
Figure 3
Figure 3. Figure 3: Validation of control optimization. VEC is compatible with standard control optimization methods, such as iLQR. iLQR consistently minimizes the ergodic metric over time until convergence, across different robot dynamics, volumetric state representations, and randomized initial states. Note that the values of the converged ergodic metric depend on the specific robot dynamics and volumetric state representat… view at source ↗
Figure 4
Figure 4. Figure 4: Erasing benchmark. (Left) VEC achieves 100% task completion in less than half the time required by the baseline, which only completes 17 out of the 25 trials. (Right) VEC succeeds in all 25 trials under 400 timesteps, while the baseline only completes 9 trials within the same amount of time. Baseline Ours 0 50 100 150 200 Timesteps Completion time (↓ is better) Baseline Ours 0% 20% 40% 60% 80% 100% Success… view at source ↗
Figure 5
Figure 5. Figure 5: Ground search benchmark. (Left) VEC achieves 100% task completion with on average half as many timesteps as the baseline, which only completes 22 out of the 25 trials. (Right) VEC succeeds in 24/25 trials under 100 timesteps, while the baseline only completes 14 trials within the same amount of time. Baseline Ours 0 200 400 600 800 1000 Timesteps Completion time (↓ is better) Baseline Ours 0% 20% 40% 60% 8… view at source ↗
Figure 6
Figure 6. Figure 6: Aerial search benchmark. (Left) VEC achieves 100% task completion in less than half the time required by the baseline, which only completes 19 out of the 25 trials. (Right) VEC succeeds in all 25 trials under 400 timesteps, while the baseline only completes 13 trials within the same amount of time. representation by keeping dynamics and control optimization identical across methods. [Benchmark Q2 results] … view at source ↗
Figure 8
Figure 8. Figure 8: Franka erasing demonstration with multiple eraser shapes. VEC accounts for the physical eraser size, producing smooth motions that efficiently erase the full target shape without redundancy. Snapshots show successful completion of the erasing task with multiple eraser geometries, including square (top), horizontal rectangle (middle), and vertical rectangle (bottom). More details are provided in the multime… view at source ↗
read the original abstract

Ergodic control synthesizes optimal coverage behaviors over spatial distributions for nonlinear systems. However, existing formulations model the robot as a non-volumetric point, whereas in practice a robot interacts with the environment through its body and sensors with physical volume. In this work, we introduce a new ergodic control formulation that optimizes spatial coverage using a volumetric state representation. Our method preserves the asymptotic coverage guarantees of ergodic control, adds minimal computational overhead for real-time control, and supports arbitrary sample-based volumetric models. We evaluate our method across search and manipulation tasks -- with multiple robot dynamics and end-effector geometries or sensor models -- and show that it improves coverage efficiency by more than a factor of two while maintaining a 100% task completion rate across all experiments, outperforming the standard ergodic control method. Finally, we demonstrate the effectiveness of our method on a robot arm performing mechanical erasing tasks. Project website: https://murpheylab.github.io/vec/

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 paper introduces Volumetric Ergodic Control, a formulation that extends standard ergodic control to volumetric state representations of robots and sensors via arbitrary sample-based models. It claims the extension preserves the original asymptotic coverage guarantees of ergodic control, adds minimal computational overhead suitable for real-time use, and empirically improves coverage efficiency by more than a factor of two while achieving 100% task completion across search, manipulation, and mechanical erasing tasks with varied robot dynamics and end-effector/sensor geometries.

Significance. If the preservation of asymptotic guarantees can be rigorously established and the reported efficiency gains hold under the stated conditions, the work would meaningfully bridge the gap between point-mass ergodic control theory and practical robotic systems with physical volume, enabling more effective coverage behaviors in real hardware. The evaluation across multiple dynamics and geometries provides a solid empirical foundation.

major comments (2)
  1. [§3] §3 (Volumetric Ergodic Formulation): The central claim that redefining the ergodic metric and control law over a volumetric state (via sample-based models) preserves the original asymptotic coverage guarantees is load-bearing, yet the manuscript does not state explicit conditions on sample density, uniformity, or consistency to ensure the new metric remains a valid discrepancy measure (zero iff distributions match) and that the closed-loop dynamics drive the time-averaged coverage to the target without finite-sampling bias.
  2. [§5] §5 (Experimental Evaluation): The reported improvement in coverage efficiency by more than a factor of two and 100% task completion across all experiments underpins the practical contribution; however, the description does not detail how the standard ergodic baseline was implemented for volumetric cases or include statistical tests, making it difficult to isolate the effect of the volumetric extension from other implementation choices.
minor comments (2)
  1. [Figures 4-6] Figure captions and axis labels in the experimental results could more explicitly indicate the volumetric vs. point-mass comparison to improve readability.
  2. [§2] The notation for the volumetric state and its integration into the ergodic equations should be introduced with a clear comparison table to the standard formulation to aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive review, which highlights both the potential impact of Volumetric Ergodic Control and areas where the manuscript can be strengthened. We address each major comment below and commit to revisions that will improve the rigor of the theoretical claims and the clarity of the experimental evaluation.

read point-by-point responses

  1. Referee: [§3] §3 (Volumetric Ergodic Formulation): The central claim that redefining the ergodic metric and control law over a volumetric state (via sample-based models) preserves the original asymptotic coverage guarantees is load-bearing, yet the manuscript does not state explicit conditions on sample density, uniformity, or consistency to ensure the new metric remains a valid discrepancy measure (zero iff distributions match) and that the closed-loop dynamics drive the time-averaged coverage to the target without finite-sampling bias.

    Authors: We agree that the manuscript would benefit from explicit conditions to support the preservation of asymptotic guarantees. In the revised version, we will add a dedicated paragraph in §3 stating the required conditions: the sample-based volumetric model must employ a sufficiently dense and uniformly distributed set of samples such that, in the limit of increasing sample count, the volumetric ergodic metric converges to the standard ergodic metric. Under these conditions, the metric is zero if and only if the time-averaged coverage distribution matches the target, and the closed-loop dynamics drive the system to the target without persistent finite-sampling bias. We will also include a brief convergence argument showing that bias vanishes as sample density grows, thereby making the theoretical foundation more precise. revision: yes

  2. Referee: [§5] §5 (Experimental Evaluation): The reported improvement in coverage efficiency by more than a factor of two and 100% task completion across all experiments underpins the practical contribution; however, the description does not detail how the standard ergodic baseline was implemented for volumetric cases or include statistical tests, making it difficult to isolate the effect of the volumetric extension from other implementation choices.

    Authors: We concur that additional implementation details and statistical analysis are warranted. In the revision, we will expand the experimental section to explicitly describe the standard ergodic baseline: for volumetric tasks, the baseline approximates the robot or sensor as a single point mass located at the geometric center of the volumetric model, while our method integrates the full set of samples. We will also report results over multiple independent trials with means, standard deviations, and p-values from paired t-tests to demonstrate that the observed efficiency gains (more than 2×) are statistically significant and attributable to the volumetric formulation rather than other implementation factors. revision: yes

Circularity Check

0 steps flagged

Volumetric extension reformulates ergodic metric without reducing central guarantees to self-definition or fitted inputs

full rationale

The derivation extends the standard ergodic metric and control law to a sample-based volumetric state representation. The preservation of asymptotic coverage is asserted by structural similarity to the original formulation rather than by redefining a quantity in terms of itself or by fitting then relabeling. No load-bearing self-citation chain or uniqueness theorem imported from the same authors is required for the core claim; the extension remains an independent reformulation that can be checked against the original ergodic proofs. This yields only a minor self-citation score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard ergodic control theory plus the assumption that volumetric extensions preserve guarantees. No new free parameters or invented entities are introduced beyond the volumetric state representation itself.

axioms (1)
  • domain assumption The ergodic coverage guarantees hold when the state is extended from a point to a volumetric representation using sample-based models.
    Invoked to claim preservation of asymptotic properties in the new formulation.

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Reference graph

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