Source-linked AI summary
Probabilistic Multi-Robot Gas Source Localization with Uncalibrated Sensors: A Distributed Estimation Approach
Wanting Jin, Marc Zoel Arias Mitjà, Alcherio Martinoli
TL;DR
Heterogeneous, uncalibrated gas sensors make reliable multi-robot source localization and information fusion difficult. The paper introduces distributed rank-based belief estimation, Product of Experts fusion, and region-aware path planning; high-fidelity simulations show reliable localization despite strong sensor heterogeneity and outperform raw-measurement aggregation. The framework is currently most suitable for small- to medium-sized teams and still requires real-world validation.
Problem
Heterogeneous sensors have inconsistent nonlinear responses, limiting direct measurement fusion and coordinated information gathering in multi-robot gas source localization.
Method
Each robot estimates a local belief from rank-based gas features, fuses beliefs through Product of Experts, and plans within an allocated informative region.
Results
The proposed method significantly outperforms raw-measurement aggregation and achieves reliable localization despite strong sensor heterogeneity in high-fidelity simulations.
Takeaways & Limitations
Calibration-free features can support distributed robotic estimation with heterogeneous sensors while coordinated regions reduce redundant observations and balance exploration with exploitation.
Takeaways & Limitations
The current all-to-all communication design is suitable for small- to medium-sized teams, and real-world experiments remain needed for further validation.
Abstract
from arXiv · showhide
Estimating environmental states with multi-robot systems becomes particularly challenging when robots are equipped with uncalibrated and therefore heterogeneous sensors, whose nonlinear and inconsistent responses prevent reliable information fusion. In this paper, we propose a distributed probabilistic framework for source localization tasks that enables calibration-free estimation in the presence of sensor heterogeneity. The key idea is that each robot independently estimates a local belief using a rank-based feature that captures the relative evolution of observations and is invariant to sensor scaling and nonlinearities. These local beliefs are then fused through a product of experts formulation to obtain a consistent global estimate across the team. To further improve the efficiency of team coordination, we introduce an informative region allocation and path planning strategy that reduces redundant exploration while balancing exploration and exploitation. We validate the proposed framework using high-fidelity simulations with realistic gas sensor models. Results demonstrate that our method significantly outperforms a benchmark method based on standard measurement aggregation, achieving reliable source localization accuracy despite strong sensor heterogeneity. More broadly, this work demonstrates how calibration-free sensing representations can be effectively extended to distributed robotic systems, paving the way for their application to other estimation tasks involving heterogeneous sensors.
1 Introduction
Multi-robot gas source localization is difficult because heterogeneous, nonlinear sensors hinder information fusion and coordination must avoid redundant exploration. The paper proposes distributed calibration-free estimation with coordinated motion and evaluates it in high-fidelity simulation.
- Heterogeneous gas sensors vary with manufacturing, nonlinear responses, temperature, humidity, and sensor age, making long-term calibration difficult.
- Existing multi-robot approaches often aggregate measurements directly while assuming homogeneous sensors or simple noise models.
- The proposed framework uses rank-based gas features, local belief estimation, Product of Experts fusion, and informative region allocation for heterogeneous sensors.
- Robots gather samples during motion rather than following a stop-sense-go path, while region allocation reduces redundant exploration.
- The method is evaluated against a benchmark in a high-fidelity environment with realistic MOX sensor models under calibrated and uncalibrated conditions.
2 Problem Statement
The problem is to estimate a stationary gas source’s posterior location from measurements collected by multiple robots in a discretized, obstacle-containing workspace. Estimation uses a belief map, uncertainty entropy, and a MAP source estimate, with termination based on entropy or iteration limits.
- The workspace contains obstacles and a single stationary gas source at an unknown position in free space, discretized into Ne grid cells.
- A team of M robots collects measurements, and each robot’s measurement history is represented as a set Zm_k.
- The objective of gas source localization is estimating the posterior distribution of the source location from collected measurements.
- Belief-map entropy quantifies estimation uncertainty, while the MAP estimate identifies the estimated source location.
- The process stops when belief entropy falls below a predefined threshold or the maximum iteration count is reached.
3 Methodology
The methodology combines distributed Bayesian source estimation with belief fusion and coordinated informative path planning. Robots update local beliefs from observations, share them into a global belief, allocate regions, and replan iteratively.
- The workflow has robots independently estimate local source-location beliefs, share and fuse them, partition the map, and assign regions for path planning.
- At each iteration, robots collect measurements, update beliefs, fuse information, and replan trajectories until entropy or iteration limits terminate the process.
- Source Term Estimation Framework: Source Term Estimation formulates inference over the discrete source location using a plume model and gas concentration measurements.
- Source Term Estimation Framework: For each source hypothesis, predicted and measured gas features are compared; their alignment defines likelihoods used to update posterior beliefs.
- Gas plume model: A data-driven plume model uses environment configuration and candidate source location to predict gas concentration fields in built environments.
- Gas plume model: Predicted concentrations are sampled at robot positions along trajectories for comparison with observations.
Rank-based gas feature -
The paper uses rank-based gas features to compare measured and predicted concentrations despite heterogeneous, nonlinear sensor responses. The representation captures relative observation magnitudes and assumes sensor responses are monotonic transformations of true concentration.
- The rank-based feature captures the relative magnitude evolution of gas observations rather than directly comparing raw concentrations.
- Each measurement receives a normalized rank through an Empirical Distribution Function computed over the robot’s measurement set.
- Predicted concentrations are converted into the same rank-based representation before comparison with measurements.
- The representation avoids requiring consistent scaling between sensor measurements and model predictions.
- The approach assumes each sensor response is a monotonic transformation of true gas concentration, with ranks computed independently for each robot.
Probabilistic distribution estimation -
Each robot evaluates source hypotheses by modeling discrepancies between measured and predicted rank-based features. Historical measurements update a normalized local belief from a uniform free-space prior.
- Each robot evaluates the likelihood of its observations under every candidate source hypothesis.
- A Gaussian discrepancy model compares rank-based features while accounting for measurement and model uncertainties.
- At each iteration, all historical measurements available to a robot are used to update its local belief map.
- The updated local belief is normalized to sum to one across candidate source locations.
- The prior is uniform over free space, while obstacle-occupied cells receive zero probability.
3.2 Belief Sharing and Merging
Robots share their local belief maps and fuse them with a Product of Experts formulation. Equal treatment reflects shared models and environmental information, while consensus can suppress individual noise or bias.
- Robots exchange local belief maps and fuse them to obtain a global belief.
- The Product of Experts formulation multiplies the robots’ local beliefs before normalization.
- Because robots share the same forward model and environmental information, their beliefs are treated equally during fusion.
- The fusion assumes conditional independence between robot observations.
- A location receives high probability only when multiple robots consistently support it, improving robustness to individual noise or bias.
3.3 Informative Region Allocation and Path Planning
The path-planning strategy uses the fused belief to coordinate exploration while balancing spatial coverage against the need for locally diverse gas samples.
- Robots plan trajectories from the fused belief to maximize collective information gain.
- Because measurements are collected continuously during motion, robots are encouraged to explore spatially distinct regions rather than overlap trajectories.
- The rank-based feature requires each robot to sample a sufficiently diverse range of gas concentrations.
- The strategy balances global exploration against local exploitation.
Region allocation method -
The method partitions informative areas into connected, non-overlapping regions assigned to individual robots, balancing coverage and workload while reducing redundant exploration.
- The environment is partitioned into M spatially connected, non-overlapping regions, with one region assigned to each robot.
- Top-M highest-probability active cells seed the regions, with each seed assigned to its nearest robot.Active cells satisfy b_i > ϵ, where ϵ marks sufficiently informative locations.
- A multiseed breadth-first search expands regions on an 8-connected grid while prioritizing smaller regions.Expansion excludes obstacles and previously assigned cells and applies a soft size constraint.
- The allocation produces spatial connectivity, balanced workload, and no overlap between regions.
- One planning iteration combines local beliefs, Product of Experts fusion, and non-overlapping region allocation.The true source location is marked by a star in the figure.
Path planning within each region -
Each robot plans within its assigned region by moving toward an information-weighted centroid rather than a single target cell. This supports broad exploration when beliefs are diffuse and focused exploitation as they concentrate.
- Each robot plans its trajectory within its assigned region and moves toward an information-weighted centroid instead of a single cell.
- The centroid strategy approximates information-driven exploration while balancing coverage and exploitation.
- Diffuse beliefs place the centroid toward broad exploration, whereas concentrated beliefs shift it toward high-probability regions for focused exploitation.
- The planning iteration shows robot trajectories, sensor outputs, local beliefs, global PoE fusion, and resulting navigation goals.Darker trajectory markers indicate higher sensor outputs, while darker belief cells indicate higher source probabilities.
4 Experiments and Results
The evaluation uses high-fidelity Webots simulations with heterogeneous MOX sensors across three obstructed indoor environments and compares Belief Sharing with Measurement Aggregation. Belief Sharing remains effective with uncalibrated sensors, whereas the baseline fails consistently in that setting; the framework is suitable for small- to medium-sized teams but has communication-scaling limits.
- 4.1 Simulation Setup: The evaluation uses Webots, a gas-dispersion plugin, three Khepera IV robots, and MOX sensors in three indoor environments with differing obstacles and airflow.Thirty randomized trials per setting and environment produce 360 simulation runs.
- 4.1 Simulation Setup: The MOX simulation models nonlinear response, additive noise, and slow response-recovery dynamics to generate heterogeneous sensor outputs.Robots receive baseline resistances of 100 Ω, 500 Ω, and 1500 Ω.
- 4.2 Benchmark Method: Belief Sharing is compared with Measurement Aggregation, which combines all robots’ measurements and assumes they are directly comparable across sensors.Both methods use the same region allocation and path planning strategies for benchmarking fairness.
- 4.2 Evaluation Metrics: Performance is evaluated by localization accuracy, measured as Euclidean error to the true source, and exploration efficiency, measured by total team trajectory length.
- 4.2 Results: With calibrated sensors, Belief Sharing and Measurement Aggregation achieve comparable localization accuracy and trajectory lengths.The comparison distinguishes calibrated results from uncalibrated results in Fig. 6.
- 4.2 Results: In uncalibrated settings, Measurement Aggregation consistently fails across map configurations, while Belief Sharing localizes the source in most cases across all environments.The baseline’s failure is attributed to mismatches between predicted concentrations and heterogeneous, nonlinear raw sensor measurements.
- 4.2 Results: Belief Sharing achieves source localization with moderate trajectory lengths, balancing exploration and exploitation, and is robust to heterogeneous sensor responses in complex environments.
- 4.3 Scalability: The all-to-all architecture communicates an N_e-dimensional belief map from every robot each iteration, costing O(M^2N_e), which limits suitability to small- and medium-sized teams.Belief fusion at each robot requires O(MN_e) operations.
5 Conclusion and Outlook
The framework combines rank-based local beliefs and global fusion for distributed gas source localization with uncalibrated sensors. Simulations show reliable localization under strong sensor heterogeneity, while future work targets communication efficiency and real-world validation.
- Each robot estimates a local belief using rank-based gas features invariant to sensor scaling and nonlinearities, then fuses beliefs into a global source-location estimate.
- Informative region allocation and path planning balance exploration and exploitation while reducing redundant observations.
- The framework supports continuous measurement collection during motion and collaborative refinement of the source-location belief.
- Extensive high-fidelity simulations with realistic MOX sensor models show significant improvement over a baseline that aggregates raw sensor measurements.
- Extending validation to real-world experiments and improving communication efficiency through reduced data exchange are identified as future directions.