Ergodic Control

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9 papers2 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Oct 15, 2026

Time-Optimal Ergodic Retrieval

Retrieval tasks are a form of search task where a robot must explore, find an object, and return with it as quickly as possible. Recent work in ergodic control has demonstrated time-optimal path planning for search tasks where visitation is proportionally matched, within some tolerance, to a given target distribution. However, these planners minimize the duration of a full deterministic search, while retrieval missions end their search at an unknown point in the trajectory and must return to a given base. In this work, we propose a new formulation of the time-optimal ergodic search problem that models detection as a Poisson process and minimizes total mission time by including the return phase in the objective. Experimental results show that time-optimal ergodic retrieval (TOER) achieves up to 19%19\% shorter mission durations relative to ergodic baselines and that this advantage grows as the return phase becomes more costly. Simulation and hardware implementations validate TOER's compatibility with environments of non-convex geometry as well as both 2D and 3D search domains.

Christian Sulana, Christian Hughes, Ian Abraham

Christian Sulana is with Saddleback College, Mission Viejo, California, USA · Christian Hughes is with the Department of Mechanical Engineering, Yale University, New Haven, CT 06520, USA · Ian Abraham is with the School of Electrical and Computer Engineering, The University of Sydney, Sydney, Australia

Ergodic ControlRobotic Object SearchRobot Motion PlanningRobotic OptimizationAutonomous Robot ExplorationStochastic Optimal Control
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Oct 4, 2026

Safe Ergodic Control for Multi-Robot Systems via Quadratic Programming

Ergodic control drives robots to spend time in each region in proportion to a spatial distribution of interest, making it well suited for dense spatiotemporal environmental monitoring. Existing safe ergodic controllers rely on offline trajectory optimization or hierarchical architectures, which limit real-time applicability and decouple the ergodicity objective from the safety constraint. This paper presents a quadratic programming (QP)-based controller that treats ergodicity and safety jointly. We first introduce a Gaussian-kernel ergodic metric that, unlike the classical indicator-based metric, is time differentiable. This allows the exponential decay of the metric to be imposed as a time-varying control barrier function (CBF) constraint, relaxed by a slack variable, alongside hard CBF constraints for region containment and inter-robot collision avoidance. We establish that the resulting QP remains feasible from any safe initial configuration and that, up to the slack term, the ergodic metric decays exponentially. Simulations show improved performance over baseline methods, and experiments with three aerial vehicles validate the work on real hardware.

Yo Toyomoto, Mohamed Elobaid, Bryce L. Ferguson +2

Electrical and Computer Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia · Thayer School of Engineering at Dartmouth College, Hanover, NH, USA · Department of Computer Science, Dartmouth College, Hanover, NH, USA

Multi-Robot SystemsControl Barrier FunctionsRobot SafetyQuadratic ProgrammingErgodic Control
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Sep 9, 2026

Ergodic Control and Controlled Diffusion for Robot Learning: Review and Tutorial

Diffusion learning leverages the statistical mechanism of diffusion processes for learning, reasoning, and inferring complex distributions from data. Recent advances in diffusion learning have been transformative, with robot learning emerging as a key opportunity area, with applications spanning perception, control, and decision-making. At the same time, the statistical mechanism of diffusion processes can be controlled to shape the temporal evolution of the state distribution underlying robot trajectories, inducing ergodic behavior in robotic systems. The frameworks of controlled diffusion and ergodic control were developed around the same time as diffusion learning, and their theories and algorithms have increasingly converged. Ergodicity induced by controlled diffusion has several significant implications for robot learning, distinct from applying diffusion learning to robotics problems: it formally enforces statistical properties required to ensure optimality of robot learning, enables non-myopic search over uncertain information landscapes for data collection, and enables behavior specification based on spatial rather than temporal characteristics of trajectories. This survey introduces the intuition behind controlled diffusion for robot learning, explores its connection to diffusion learning, presents theoretical foundations and numerical tutorials for solving controlled diffusion and ergodic control problems, and reviews applications across robotics. Finally, we discuss key challenges and future opportunities in leveraging controlled diffusion for robot learning.

Max Muchen Sun, Cem Bilaloglu, Ananya Rao +6

Northwestern University · Idiap Research Institute · Carnegie Mellon University +4

Ergodic ControlRobot LearningStochastic Optimal ControlData-Efficient Robot Learning
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Aug 20, 2026

SAGE: Ergodic Control for Autonomous and Adaptive Inspection of Subsea Infrastructure

Subsea Christmas Trees (XTs) are underwater structures that use valves for directing oil flow, needing constant inspection. But not every valve carries the same risk at the same time: a valve with a suspected leak needs to be revisited far more often than one with a clean history, and that risk picture changes during the mission as new leaks are found. To handle this, we present SAGE (Semantic and Adaptive Generative Ergodicity), an ergodic-control architecture that allocates vehicle time in proportion to a live, sensor-derived risk distribution rather than a scripted route. We study a two-XT scenario, with five valves in total, and compare a fixed-loop A* tour against SAGE. Both methods can be tuned to spend similar total time near a high-risk valve, but only ergodic control also checks it more often: in simulation, a dominant-risk valve was revisited every 5.8 s under ergodic control against a fixed 8.1 s for every valve under A*, regardless of risk, so a leak can go unnoticed for barely two-thirds as long. Because the tracked distribution is recomputed rather than planned once, a newly detected leak shifts vehicle behavior on the next control cycle with no explicit re-planning step and no operator in the loop, which a fixed tour cannot do without a discrete re-route. We derive the ergodic control law behind this behavior and report simulation results on the five-valve scenario.

Markus Buchholz, Ignacio Carlucho, Yvan R. Petillot

Norwegian Defence Research Establishment (FFI), Kjeller, Norway. · School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh, UK.

Ergodic ControlRobotic InspectionUnderwater Robotics
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Aug 6, 2026

ErgoSurf: Ergodic Control for the Coverage of Unknown Surfaces

Contact-centric tasks on surfaces, ranging from inspection and cleaning to sanding and polishing, require robots to systematically cover the surface while maintaining stable contact. Ergodic control generates trajectories that spend time at a location proportional to a desired, task-specific spatial distribution, enabling efficient information gathering and coverage. However, traditional ergodic control methods rely on prior knowledge of surface geometry or require a vision sensory input to scan the geometry beforehand, limiting their applicability in real-world scenarios with unknown or dynamic environments. This paper introduces a novel online ergodic control framework that achieves systematic surface coverage while simultaneously reconstructing unknown surface geometry. We employ a Gaussian Process Implicit Surface (GPIS) model that learns global surface geometry from intrinsic tactile sensing during execution. For efficient online planning, we approximate the surface locally using point clouds sampled from tangent planes at observed contact points and iteratively fit them to the Gaussian Process. This approximation simultaneously serves as the sampling domain for both the target and the coverage distributions. We employ a heat-diffusion analogy to compute potential fields that guide ergodic exploration, translating spatial coverage objectives into smooth robot trajectories. We demonstrate our framework through simulation and real-robot experiments, validating simultaneous ergodic coverage and online surface geometry learning with reconstruction errors approaching the ground truth.

Stefan Schneyer, Timo Bachmann, Maged Iskandar +4

German Aerospace Center (DLR), Institute of Robotics and Mechatronics (RM), Münchener Str. 20, 82234 Weßling, Germany · Technical University of Munich (TUM); School of Computation, Information and Technology; Boltzmannstr. 3, 85748 Garching, Germany

Mobile RoboticsErgodic Control
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Aug 3, 2026

TRACE: Ergodic Trajectory Optimization for Active Scene Reconstruction

Existing active reconstruction systems with Gaussian-splatting maps select observations greedily, optimizing a single next-best-view (NBV) at each step and connecting the chosen views by short-horizon path planning. This greedy decoupling disregards the global structure of scene information, producing inefficient trajectories that waste sensing capacity in transit between selected views. In this work, we study active reconstruction as an ergodic coverage problem: the time-averaged spatial statistics of the sensor trajectory should match a target information distribution induced by the current map. Our approach derives this target distribution online from uncertainty and visibility, and calculates ergodic trajectories via a kernel-ergodic horizon planner with gradient flow and footprint depletion, closing the loop between mapping and trajectory optimization. We thoroughly evaluate TRACE on the Replica dataset against the Next-Best-View (NBV) baselines, improving PSNR by 1.5 dB. Code: https://github.com/spikelab-jhu/trace-active-reconstruction.

Ziyue Zheng, Linli Shi, Bingkun He +2

1Johns Hopkins University, USA · University of Pennsylvania, USA

Trajectory OptimizationTrajectory Generation3D ReconstructionActive View SelectionErgodic Control
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May 13, 2026

Ergodic Imitation for Adaptive Exploration around Demonstrations

In robotics, a common challenge in imitation learning is the mismatch between training and deployment conditions, caused, for example, by environmental changes or imperfect observation and control. When a robot follows a nominal trajectory under such mismatch, it may become stuck and fail to complete the task. This calls for adaptive online exploration strategies that remain grounded in demonstrations. To this end, we propose an adaptive ergodic imitation approach that constructs a target distribution from the geometry of the retrieved demonstrations and uses it to generate trajectories that adaptively interpolate between tracking and exploration. Our method extends ergodic control beyond its traditional role in area-coverage and search by incorporating demonstrations into a retrieval-based receding-horizon framework for adaptive imitation.

Ziyi Xu, Cem Bilaloglu, Yiming Li +1

Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland · Idiap Research Institute, Martigny, Switzerland

Imitation LearningLearning from DemonstrationRobot Imitation LearningErgodic Control
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Apr 1, 2026

Infinite-Horizon Ergodic Control via Kernel Mean Embeddings

This paper derives an infinite-horizon ergodic controller based on kernel mean embeddings for long-duration coverage tasks on general domains. While existing kernel-based ergodic control methods provide strong coverage guarantees on general coverage domains, their practical use has been limited to sub-ergodic, finite-time horizons due to intractable computational scaling, prohibiting its use for long-duration coverage. We resolve this scaling by deriving an infinite-horizon ergodic controller equipped with an extended kernel mean embedding error visitation state that recursively records state visitation. This extended state decouples past visitation from future control synthesis and expands ergodic control to infinite-time settings. In addition, we present a variation of the controller that operates on a receding-horizon control formulation with the extended error state. We demonstrate theoretical proof of asymptotic convergence of the derived controller and show preservation of ergodic coverage guarantees for a class of 2D and 3D coverage problems.

Christian Hughes, Ian Abraham

Ergodic ControlOptimal ControlKernel Mean Embeddings
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Mar 18, 2026

Manufacturing Micro-Patterned Surfaces with Multi-Robot Systems

Applying micro-patterns to surfaces has been shown to impart useful physical properties such as drag reduction and hydrophobicity. However, current manufacturing techniques cannot produce micro-patterned surfaces at scale due to high-cost machinery and inefficient coverage techniques such as raster-scanning. In this work, we use multiple robots, each equipped with a patterning tool, to manufacture these surfaces. To allow these robots to coordinate during the patterning task, we use the ergodic control algorithm, which specifies coverage objectives using distributions. We demonstrate that robots can divide complicated coverage objectives by communicating compressed representations of their trajectory history both in simulations and experimental trials. Further, we show that robot-produced patterning can lower the coefficient of friction of metallic surfaces. This work demonstrates that distributed multi-robot systems can coordinate to manufacture products that were previously unrealizable at scale.

Annalisa T. Taylor, Malachi Landis, Ping Guo +1

Mechanical Engineering Department, Northwestern University

Ergodic ControlRoboticsMulti-Robot SystemsMulti-Robot Coordination
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