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Gaussian Processes for Modelling Spatial Fields with Robot Swarms
Guillermo Legarda Herranz, Gianpiero Francesca, Mauro Birattari
TL;DR
Spatial-field modelling with robot swarms is difficult when robots lack external positioning and standard GPR scales poorly with accumulated data. LU-GPR combines belief-propagation-based frame alignment with online, adaptive GP regression and peer-model fusion. The reported results show scalability with swarm size and robustness to limited communication ranges, while demonstrating crowd-flow estimation in an evacuation scenario.
Problem
Standard GPR has O(n^3) inference, while decentralised multi-robot GPR methods generally assume external positioning that swarms may lack.
Method
LU-GPR combines local GBP-based frame alignment with online sparse GP regression, adaptive weighting, and product-of-experts fusion of robots’ models.
Results
LU-GPR is reported to scale with robot number, remain robust to limited communication ranges and varying input noise, and estimate crowd behaviour in an evacuation scenario.
Takeaways & Limitations
Robot swarms can estimate spatial fields without external positioning while maintaining decentralised operation and producing a global model from peer estimates.
Takeaways & Limitations
Systematic validation of adaptivity and robustness in time-varying scenarios and with outliers remains unresolved.
Abstract
from arXiv · showhide
Robot swarms, by virtue of their decentralised architecture, are a natural tool for scalable, robust modelling of spatial fields, such as water temperature, wind velocity, or terrain elevation. However, existing methods rely on external positioning systems that allow each robot to determine its own position in space. Here, we introduce location-unaware Gaussian process regression (LU-GPR) as a solution to the modelling of spatial fields in the absence of such positioning systems. LU-GPR allows each robot to infer the posterior mean and variance of the field in space, while simultaneously agreeing on a common frame of reference with its peers, using only local sensing and communication. We propose an online algorithm that allows each robot to consistently infer local estimates as its local frame of reference converges to the common one. By means of a product of experts model, each robot also combines the estimates of its peers with its own to obtain a global model. Our results show that LU-GPR scales well with the number of robots and is robust to limited communication ranges. We also demonstrate how it can be used in real-world monitoring scenarios to estimate the flow of an evacuating crowd.
I. INTRODUCTION
Spatial-field modelling matters across natural and collective phenomena, but standard GPR scales poorly and decentralised multi-robot methods generally assume external positioning. LU-GPR addresses this gap with online, fully decentralised regression for position-unaware robot swarms.
- I. INTRODUCTION: The approach targets spatial fields ranging from ocean currents and terrain elevations to crowd motion, including evacuation monitoring by robot swarms.
- I. INTRODUCTION: O(n^3) inference in standard GPR makes modelling increasingly impractical as collected samples grow.
- I. INTRODUCTION: Existing decentralised multi-robot GPR methods assume access to positioning systems, unlike swarms operating with only local perception and communication.
- I. INTRODUCTION: LU-GPR combines local frame alignment, online GP regression, and peer-model fusion without external positioning.It uses GBP for common-frame estimation, random Fourier features for sample-independent inference complexity, and adaptive weighting for unreliable or older samples.
B. Gaussian belief propagation
Gaussian belief propagation performs marginal inference through iterative message passing on factor graphs. In robot swarms, it enables common-frame estimation without global positioning and with limited-range communication.
- B. Gaussian belief propagation: GBP updates Gaussian variable beliefs using neighbouring factors and converges exactly on tree-structured graphs, while remaining empirically effective on graphs with cycles.
- B. Gaussian belief propagation: In a prior robot-swarm implementation, GBP enabled robots without global positioning and with limited-range communication to converge to a common frame of reference.Robots maintained bounded factor graphs whose variables represented positions relative to that shared frame and were updated from odometry.
III. SPATIAL FIELD MODELLING
LU-GPR lets robots model a spatial field without knowing their global positions, while using local communication to maintain a shared reference and construct a global model.
- III. SPATIAL FIELD MODELLING: Each robot simultaneously runs GBP for frame estimation and updates its GP model from temporally collected field data.
- III. SPATIAL FIELD MODELLING: Robots fuse peers’ local models with their own to obtain a global model of the latent spatial function.
A. Location-unaware GPR
Location-unaware GPR corrects stored GP inputs as local frames move, while weighting samples by age and reliability. Its updates preserve bounded computation and support adaptive, robust estimation.
- A. Location-unaware GPR: Stored sample locations become unreliable as robots move because local-frame alignment changes while each robot retains only a bounded position graph.
- A. Location-unaware GPR: The online model must correct functions of stored inputs and reduce the influence of older or unreliable measurements.
- A. Location-unaware GPR: A forgetting factor λ∈(0,1] increases the effective noise of earlier samples, making them less reliable over time.
- A. Location-unaware GPR: Feature-space corrections represent spatial displacement as rotations, allowing the design matrix to be updated in O(m) time.The rotation acts in fixed 2×2 blocks corresponding to random Fourier features.
- A. Location-unaware GPR: Forgetting alone drives the model toward its prior as λ→0, whereas displacement alone applies a similarity transformation that rotates paired mean components.
B. Ensemble learning
LU-GPR exchanges local posterior models among communicating robots and fuses them into a system-wide prediction despite partial observations and changing local frames.
- B. Ensemble learning: Each robot exchanges timestamped local posterior mean and variance models with neighbours, prioritising models the responder has not recently received.The protocol uses unique robot identifiers and locally incremented timestamps; a responder returns the model with the largest timestamp difference when no new model is available.
- B. Ensemble learning: Robots update received models in their changing local frames with the same regression rules and apply forgetting to account for odometry drift.The communication protocol runs at the same rate as GBP messaging, while the regression and GBP implementations remain modularly independent.
- B. Ensemble learning: At inference, each robot combines partial local models using a generalised product of experts to form a predictive posterior.The complete dataset is partitioned across robots, so each robot initially has access only to a subset of observations.
- B. Ensemble learning: Using β_i = 1/|M_k| recovers the prior outside the data range, while complete model dissemination gives overall complexity O(km^3).Here, M_k denotes the collected models and k the number of robots.
IV. EXPERIMENTAL VALIDATION
LU-GPR is validated in synthetic-field and evacuating-crowd scenarios to assess swarm-scale spatial-field modelling and crowd-flow estimation.
- IV. EXPERIMENTAL VALIDATION: The validation uses a randomly generated synthetic function to test robot-number and communication-range effects, then models an evacuating crowd.Both scenarios assume zero GP means.
A. Simulation environment
The experiments simulate kinematic robots that explore by ballistic random walks, communicate locally, and measure environmental or observed-agent states with specified noise levels.
- A. Simulation environment: Analyses run in Unity with cylindrical 0.25-m-radius robots modelled as kinematic agents.Robot controls determine position evolution, while obstacle detection and local messaging support movement and communication.
- A. Simulation environment: Each robot moves straight until encountering a robot or obstacle, then selects a random direction for ballistic random-walk exploration.This exploration policy is used for all experiments.
- A. Simulation environment: Robots move at v_max = 0.5 m s^-1 and measure positions and velocities with uncertainties σ_mp = 0.02 m and σ_mv = 0.01 m s^-1.Odometry noise is set to σ_v = 0.1 m/m travelled; communication and update rates match the cited GBP implementation.
- A. Simulation environment: The experiments fix σ_n = 0.1, λ = 0.98, and m = 50 random Fourier features, while sampling every 1 s and updating inference every 5 s.The forgetting parameter is selected from an approximation relating odometry noise to GP forgetting, with the distinction that they affect different variables.
B. Synthetic function analysis
Synthetic experiments vary swarm size and communication radius in an 8 m × 8 m arena, finding that larger swarms improve convergence while communication range mainly affects convergence speed.
- B. Synthetic function analysis: The study uses an 8 m × 8 m walled arena, a GP-generated zero-mean toy field, and posterior inference on a 20×20 grid.The setup isolates LU-GPR behaviour from model identification, with four robots and 1 m communication illustrated in Fig. 2.
- B. Synthetic function analysis: All samples receive equal weight, with ρ_i = 1, and initial robot positions are random because their selection is treated as arbitrary.The experiments use 10 independent runs for each swarm-size and communication-range combination.
- B. Synthetic function analysis: Larger swarms reduce estimation bias and variance and converge faster, while communication range has little apparent effect on final estimation but affects convergence rate.Robots converge even with small swarm sizes; the result compares ranges of 1 m, 2 m, and 4 m with fully connected swarms.
- B. Synthetic function analysis: Reducing λ increases the converged RMSE, while larger swarms lower bias through more persistent environmental monitoring that offsets sample forgetting with new data.Some bias remains inevitable because the kernel function is approximated.
C. Modelling crowd behaviour
The study evaluates LU-GPR for estimating pedestrian flow in an evacuation environment using a six-robot swarm. Both weighted and unweighted approaches converge, while the weighted variant appears less biased but remains inconclusive because of high variance.
- The six-robot crowd-monitoring experiment estimates expected pedestrian direction from local velocity measurements during a random walk.Each robot averages normalized pedestrian velocities within its sensing range to obtain an orientation estimate for GP updates.
- The simulated evacuation environment contains a symmetric 10 m × 10 m room with three entrances and one exit, producing pedestrian motion toward the right.The symmetry enforces a zero-mean distribution, and the crowd is modeled using the cognitive model of Moussaïd et al.
- Samples weighted by the number of observed pedestrians are intended to give greater influence to averages computed from larger groups.This weighting is introduced because samples based on more pedestrians are expected to be more reliable.
- The posterior covers a larger domain than the ground truth because pedestrians do not traverse every part of the environment.Robot presence affects expected pedestrian behaviour near domain edges, although the overall patterns remain similar.
- Both weighted and unweighted approaches converge, but overlapping confidence intervals prevent a conclusive comparison.The weighted approach appears to achieve lower estimation bias, while high experimental variance limits inference.
V. DISCUSSION
The discussion presents LU-GPR as a scalable solution for spatial-field estimation without external positioning, while identifying limitations in validation, exploration, uncertainty handling, and ensemble design.
- LU-GPR addresses spatial-field estimation with robot swarms lacking access to a positioning system.
- Systematic validation of the claimed adaptive and robust update rules remains open for time-varying scenarios and outliers.
- The ballistic random walk may be suboptimal, while naive acquisition strategies could cause disconnected robots to choose the same exploration behaviour.
- Odometry noise is treated unrealistically because output-noise variance grows exponentially while odometry uncertainty grows linearly in the input.
- LU-GPR ignores uncertainties in GBP position beliefs, which may produce overconfident posterior variances; mapping them to samples would destroy correlations.
- The GPoE is suitable for decentralised swarms because robots need not share priors and can, in principle, learn individual hyperparameters.
- Despite these limitations, LU-GPR is reported as scalable and robust to limited communication ranges and varying input-noise levels.
APPENDIX
The appendix derives the update rules by substituting the model expressions into the matrix definitions and using orthogonality and independent, identically distributed weights.
- APPENDIX: The update rule for B in (12) is obtained by substituting (10) and (11) into B = Φ^TG2Φ + σ2.
- APPENDIX: Assuming independent, identically distributed weights and using R^T = R^-1 for orthogonal R yields (12).
- APPENDIX: The update rule (13) follows by the analogous substitution of (10), (11), and the stacked observation vector.