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Pass the Bucket: Efficient, Robust, Local Load Balancing for Teams of Heterogeneous Robots
Tobias Wallner, Dominik Krupke, Arne Schmidt, Sándor P. Fekete
TL;DR
The paper addresses decentralized load balancing for heterogeneous robots that must coordinate in a confined one-dimensional space with minimal sensing and no centralized control. It analyzes collision-driven bucket brigades and adds local token damping to induce convergence toward velocity-proportional partitions. Simulations report reliable reconvergence across tested perturbations, while the authors delimit the evidence to nonformal criteria and mostly small robot teams.
Problem
Heterogeneous robot teams need to form balanced spatial partitions and recover from disturbances despite having only local collision sensing, no communication, and no centralized control.
Method
The paper combines event-based analysis of collision dynamics with token damping that temporarily slows robots after they cross stored collision positions.
Results
Across all tested perturbation types, the system reliably reconverges to the possibly new equilibrium.
Takeaways & Limitations
The approach provides collision-based, velocity-proportional territory partitioning without communication or centralized planning and shows robustness to varied perturbations.
Takeaways & Limitations
The convergence criteria are indicators rather than formal guarantees, and the reported simulations focus mainly on small robot teams.
Abstract
from arXiv · showhide
We study the problem of decentralized, self-organized task sharing for a swarm of heterogeneous robots that collaborate in transportation or other objectives that require coordinated motion planning. To this end, we present theoretical and practical results for the simple but effective mechanism of \emph{bucket brigades} for load balancing, in which a team of heterogenous robots share a spatial task in a confined, one-dimensional space, while only being able to sense collisions with neighbors or walls. The goal is to optimize throughput of the overall system, without central control or information, aiming at an interval partition proportional to robot velocities. We address possible chaotic system behavior by developing a stabilization mechanism based on simple local aid, a ``token'', that temporarily decelerates robots after an encounter. This purely local change eliminates persistent oscillations, resulting in convergence towards a stable system state. We accelerate system convergence by comparing a single boundary token to ubiquitous two-directional tokens and optimizing the deceleration factor. Event-driven simulations report convergence times and robustness: For a large variety of perturbations (such as robot deletion, position or velocity jittering), the system reliably re-converges. The results suggest a local, practical mechanism for robust load balancing for heterogeneous teams of robots that promises an effective tool as basis for more complex scenarios.
I. INTRODUCTION
The paper studies decentralized bucket brigades in which heterogeneous robots self-organize proportional territory partitions using only local collision sensing. It introduces token-based damping to address oscillation and induce convergence under constrained capabilities.
- Multi-robot coordination can provide efficiency and robustness, but heterogeneous teams with minimal sensing face communication, scalability, and perturbation-resilience challenges.
- Bucket brigades model robots moving back and forth on a one-dimensional interval, reversing at neighbor collisions or boundaries to distribute spatial load.The setting applies to confined environments such as warehouse aisles, agricultural rows, and infrastructure corridors.
- Robots operate with only local collision sensing, no shared clock or communication, and minimal memory, yet must recover from velocity changes and robot additions or removals.The model is designed for resource-constrained platforms and infrastructure-limited environments.
- Undamped dynamics can oscillate indefinitely, motivating a token that stores a collision position and temporarily decelerates a robot after it crosses that position.The slowed robot moves at αv_i with 0 < α < 1 until its next reversal, after which the token is dropped again at collision.
- The target steady state partitions the interval into subintervals whose lengths are proportional to robot velocities.This equilibrium supports balanced spatial task sharing among heterogeneous robots.
- For two robots, token damping makes each successive collision position contract toward x*=v1/(v1+v2).The update satisfies p_{k+1}−x* = κ(p_k−x*) for some κ∈(−1,1).
IV. MATHEMATICAL ANALYSIS
The mathematical analysis represents collision dynamics with an affine event-based map and shows that the undamped system is conservative. Consequently, it cannot asymptotically converge to equilibrium unless initialized there.
- An event-based affine map describes how one set of collision abscissas transfers to the next.The analysis uses an affine map to characterize collision dynamics and then examines its key properties.
- The undamped system is stable but cannot converge to the equilibrium unless it starts there.
A. Original System
The original collision-driven system is modeled as a linear map on collision abscissas centered at the velocity-proportional equilibrium. In the rotational regime, its dynamics preserve a quadratic invariant, so trajectories oscillate rather than converge unless initialized at equilibrium.
- Dynamics: In the rotational regime, only alternating head-on encounters occur, while same-direction catch-ups are excluded after the system enters that regime.Every adjacent pair then collides infinitely often, defining the collision-position sequences.
- Dynamics: Collision abscissas are updated by a sparse, neighbor-coupled system Ax′ = Bx + c, centered as u′ = Mu around the unique velocity-proportional equilibrium.The centered map has the equilibrium as its unique fixed point.
- Invariant: A symmetric positive-definite tridiagonal matrix G makes the sweep map a G-isometry, preserving the quadratic form associated with collision-abscissa deviations.The proof sketch constructs G so each single-event operator preserves the form.
- Invariant: The resulting trajectories lie on invariant ellipsoids and can be interpreted as compositions of planar rotations in the G-inner product.Figure 5 illustrates this rotational evolution for an exemplary configuration.
- Convergence: u_k → 0 if and only if u_0 = 0, so the undamped system cannot asymptotically converge from a non-equilibrium start.The preserved positive quadratic value prevents deviations from tending to zero when the initial deviation is nonzero.
B. Damped System: Token Mechanics
The damped system introduces a local token that temporarily slows a robot when it crosses a stored collision position. This changes only local pair timing and induces convergence toward the equilibrium in the illustrated one-directional case.
- Motivation: The undamped transfer is conservative and provides no contraction toward equilibrium.This motivates adding damping to the collision dynamics.
- Token mechanics: A token stores a collision abscissa between adjacent robots and slows a crossing robot by factor α ∈ (0,1) until its next reversal.At the next collision, the token is dropped; it remains between the original robot pair.
- Token mechanics: In the one-directional version, a single token between robots 1 and 2 affects only robot 1’s velocity while robot 2 carries and drops it without deceleration.This modifies only the first pair’s relation.
1) Single, one-directional token ("Token-1"):
A single one-directional token modifies collision dynamics through local deceleration, producing determinant contraction and empirically inducing convergence, while a full spectral proof remains open.
- Single, one-directional token: The token changes the collision update map when the first robot carries it, replacing M with the damped map Mα.Otherwise, the original undamped map M applies.
- Single, one-directional token: For every 0 < α < 1, the one-token damped map has determinant magnitude below one.This establishes determinant contraction, not full spectral contraction.
- Single, one-directional token: The main convergence result remains conjectural because a full spectral contraction proof has not been provided.The conjecture asserts ρ(Mα) < 1 and u →0 for 0 < α < 1, supported by simulations.
- Single, one-directional token: A token placed between the first pair induces collision abscissas to converge toward equilibrium in an exemplary configuration.The corresponding figure shows rapid approach for the illustrated start state.
- Single, one-directional token: The proposed extension places tokens between every adjacent pair, requiring each robot to store collision abscissas with both neighbors and adjust velocity.This is the Token-all arrangement used in later experiments.
V. SIMULATION EXPERIMENTS
The simulations examine whether collision-driven systems enter a catch-up-free regime, how token placement and velocity heterogeneity affect convergence, how α shapes dynamics, and how perturbations affect recovery.
- Simulation Experiments: The experiments are designed to address five questions covering rotational-state convergence, token placement, velocity heterogeneity, slow-down factor α, and perturbation response.These questions organize the simulation study.
A. Experiment Design
The study uses event-driven Monte Carlo simulations with randomized robot configurations, explicit convergence criteria, and comparisons across token settings, velocity ranges, damping factors, and perturbations.
- Experiment Design: The simulator uses event-driven updates because velocities change only at predictable collision events.Metric tracking and termination checks are restricted to these events.
- Experiment Design: Convergence requires the previous 10 collision abscissas for every adjacent pair to remain within ε = 5×10−4 of equilibrium.This criterion is applied to each adjacent robot pair.
- Experiment Design: RQ1 measures catch-up-event ratios over time across no-token, Token-1, and Token-all settings with α = 0.5.RQ2 compares convergence-time CCDFs for both token variants and n ∈{3,5,7}.
- Experiment Design: RQ3 varies the upper velocity bound, while RQ4 sweeps α from (0,1) in 0.01 increments for n ∈{3,5,7,9}.RQ5 evaluates robot addition/removal and velocity or position jitter, including amplitudes from 0% to 25%.
B. Results
Across simulations, tokens eliminate persistent catch-up behavior and improve convergence, while velocity range, robot count, damping factor, and perturbation type shape convergence speed and robustness.
- Results: Catch-up ratios rapidly reach zero for all tested n, and tokens reduce the initial ratio and accelerate its decline.No tested run exhibited a same-direction catch-up beyond a certain time, but this is not a formal proof of permanence.
- Results: Token-all converges faster than Token-1, while Token-1 convergence time increases noticeably as n grows.The results support broader damping distribution for practical convergence.
- Results: Increasing the maximum uniformly sampled velocity accelerates convergence across all tested n and both token configurations.The paper suggests increased collision frequency may explain this pattern, while asymptotic behavior with increasing n remains unexplored.
- Results: The empirically optimal α values are approximately 0.38, 0.35, 0.33, and 0.29 for n = 3, 5, 7, and 9, respectively.Mean convergence time grows super-linearly when α deviates from the optimum in either direction.
- Results: The system reliably reconverges after all tested perturbation types, including robot changes, velocity jitter, and position jitter.After removal reconvergence is faster than after addition, and Token-all is faster than Token-1.
- Results: Position-jitter convergence time grows at most linearly, with no catastrophic degradation even at 25% displacement.Velocity-perturbation reconvergence is significantly faster than random-start convergence, with all corrected p < 0.0125.
D. Threats to Validity
The study reports robust reconvergence across several operational perturbations, but its convergence criteria are indicators rather than formal guarantees and its simulations emphasize small teams.
- Threats to Validity: The convergence criteria are strong indicators, not formal guarantees, and random initialization may under-sample highly symmetric edge cases.Manually crafted scenarios and stricter conditions revealed no counterexamples, but larger-robot simulations were not reported.
- Threats to Validity: Velocity perturbations of factor 0.5 or 2 yielded faster reconvergence than the random-start baseline for n = 5.The comparison was statistically significant under a Mann–Whitney U test with p < 0.0125.
- Application Scope: The approach supports decentralized fleet adjustment after robot failures or demand surges through reconvergence after perturbations.The warehouse discussion identifies dynamic fleet adjustments as a use case for the reported robustness.
- Application Scope: The proposed setting targets confined one-dimensional environments such as warehouse aisles, agricultural rows, and infrastructure corridors.These application domains motivate the spatial coordination model but do not establish performance beyond the studied setting.
- Robustness Evidence: Reported robustness includes robot addition or removal, velocity changes, and position jitter in heterogeneous teams.These perturbations model fleet scaling, battery or payload variation, and localization uncertainty.
B. From Model to Physical Systems
The model can be translated to physical systems with minimal sensing, memory, and control requirements, while simulations indicate robustness without communication. Physical deployment remains constrained by idealized point-agent assumptions and acceleration limits.
- Physical Robot Constraints: The idealized model assumes point agents, instantaneous velocity changes, and precise collision detection, unlike physical robots with body radius and finite sensing range.In practice, model collisions correspond to detections at separation d, using sensors or compliant bumpers.
- Physical Robot Constraints: Acceleration limits require token-triggered velocity ramps rather than instantaneous deceleration.This is a direct adjustment needed when implementing the token mechanism on physical robots.
- Robustness: Simulation experiments report resilience to robot addition or removal, velocity changes, and position jitter.These disturbances represent fleet scaling, battery or payload variation, and localization uncertainty.
- Alternative Approaches: Communication-based alternatives can compute boundaries directly or eliminate collisions, but require velocity knowledge, computation, navigation, or infrastructure.The comparison frames communication-free coordination as a trade-off against added capabilities and infrastructure.
- Coordination Mechanism: The approach provides decentralized, communication-free coordination with velocity-proportional territory partitioning for heterogeneous robots.Its mechanism relies on collision-based coordination rather than centralized planning.
- Outlook: Graph-topology extensions remain preliminary because some traversal rules can create unreachable subgraphs and alter convergence behavior.Boundary markers are suggested for partitioning graphs into linear segments.
- Deployment Requirements: Physical deployment requires velocity measurement, acceleration control, collision detection, and single-value memory for token storage.These requirements support infrastructure-limited operation without communication overhead or centralized control.