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Asynchronous Cooperative Online Learning for Multi-Robot Control under Computational Delays

Xiaobing Dai, Zewen Yang, Wei Ren, Sandra Hirche

arXiv:2608.29562v1cs.LGcs.MAcs.ROeess.SY

TL;DR

The paper addresses cooperative multi-agent control when unknown dynamics, heterogeneous computational delays, and differing query points complicate distributed Gaussian-process learning. It proposes asynchronous aggregation based on prediction accuracy, query-point differences, and delays, together with an adjoint-MAS distributed controller. Simulations on unmanned surface vehicles report improved learning and control performance relative to existing approaches.

  • Problem

    Computational delays and differing prediction query points make cooperative Gaussian-process learning for multi-agent control difficult, while existing work largely focuses on communication delays.

  • Method

    The paper combines asynchronous cooperative Gaussian-process learning with an adjoint-MAS-based distributed control law for practical second-order consensus.

  • Results

    Simulations on unmanned surface vehicles demonstrate improved learning and control performance compared with existing cooperative approaches.

  • Takeaways & Limitations

    The framework provides theoretical approximate-consensus guarantees while explicitly incorporating prediction accuracy, query-point differences, and computational delays.

Abstract

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Ensuring the safe operation of multi-agent systems (MASs) under uncertain environments is crucial for cooperative robotic, where external disturbances and inaccurate dynamic models can significantly compromise performance and reliability. To address this challenge, calibrated machine learning models, particularly Gaussian process (GP) regression, are extensively employed due to their interpretable performance quantification. As the interconnected communication of MASs facilitates cooperative learning, agents are able to enhance learning performance by exchanging local GP inferences with their neighbors and aggregating the received information via distributed GP strategies. However, variations in computational power and prediction tasks among agents inevitably lead to heterogeneous computational delays and differences in query points, which are often overlooked in existing aggregation methods. To overcome these limitations, this work proposes an asynchronous cooperative learning strategy that explicitly accounts for prediction accuracy, query point variations and delay effects. Additionally, a distributed control law based on an adjoint MAS is developed to ensure the desired control performance. Simulations on unmanned surface vehicles validate the effectiveness of the proposed approach, demonstrating substantial improvements in both learning and control performance compared to the state-of-the-art approaches.

I. INTRODUCTION

The paper addresses cooperative online learning and control for uncertain multi-agent systems when agents experience heterogeneous computational delays and differing prediction query points. It proposes asynchronous cooperative learning and distributed control to support practical second-order consensus.

  • Unknown agent dynamics and environmental uncertainties make model-based multi-agent control difficult for safety-critical robotic applications.
  • Gaussian-process cooperative learning offers flexible modeling and interpretable error quantification, but neighbor prediction and aggregation can impose substantial computational burden.
  • Existing delayed-learning research mainly addresses communication delays, leaving computational delays in cooperative online learning-based multi-agent control largely unexplored.
  • The paper asks whether delayed neighbor predictions improve learning, how to aggregate asynchronous predictions at different query points, and how to design and formally analyze the resulting controller.
  • Each agent locally predicts unknown dynamics, while the proposed asynchronous strategy accounts for prediction accuracy, query-point differences, and delay effects during aggregation.

II. PROBLEM SETTING

The problem setting models homogeneous Euler–Lagrange agents with known nominal dynamics and unknown shared disturbances connected by an undirected communication graph. The objective is stationary approximate second-order consensus despite uncertain dynamics and estimation error.

  • The agent state combines generalized coordinates and velocities in a 2n-dimensional state vector.
  • Each agent follows Euler–Lagrange dynamics with generalized coordinates, velocities, known nominal terms, control input, and an unknown continuous disturbance shared across agents.
  • The communication network is an undirected connected graph, enabling information transmission between agents through neighboring links.
  • Approximate consensus requires bounded position and velocity disagreements, while additionally requiring all agent velocities to converge to zero.
  • Approximate rather than exact consensus is used because uncertain dynamics and practically non-vanishing disturbance-estimation errors remain.

B. Data-driven Distributed Control

The framework combines local online machine-learning estimates, asynchronous neighbor information, and an adjoint-MAS correction within a distributed controller. Its stated objective is practical second-order consensus under heterogeneous computational delays.

  • The distributed control law combines consensus, model-based feedforward, learned disturbance compensation, and an adjoint-MAS correction term.The consensus term uses position and velocity errors, while the model-based term compensates known dynamics.
  • Each agent estimates its unknown disturbance using an online machine-learning model with a time-varying local data set.
  • The learning process accommodates heterogeneous computation times through agent-specific availability times and delay functions.
  • The proposed framework transmits learned information to neighbors and receives neighbor information, with initialization performed once at t = 0.
  • The objective is to design asynchronous cooperative learning and distributed control that achieve practical second-order consensus over the uncertain multi-agent system.

III. ASYNCHRONOUS COOPERATIVE ONLINE LEARNING

This section develops the GP-based cooperative online learning foundation, including calibrated local predictions, deterministic error bounds, and bounded computation under online data management. It prepares the framework for heterogeneous computational delays.

  • Gaussian Process Regression: GP regression models each unknown component in an RKHS using stationary, Lipschitz kernels and bounded RKHS norms.The bounded RKHS assumption imposes smoothness on the unknown functions and supports prediction-error analysis.
  • Gaussian Process Regression: Each agent uses local online data to compute a posterior mean for estimating the unknown function and a posterior variance for quantifying prediction performance.The model is built from local measurements, kernel vectors, Gram matrices, and measurement-noise bounds.
  • Prediction Error Bound: The concatenated GP prediction error has a deterministic RKHS-based bound whose uncertainty term decreases as informative online samples are collected.The coefficient β depends on the maximal data-set size, while σ decreases with additional informative samples.
  • Computational Delay: Online GP computation is kept uniformly bounded by limiting each local data set to a maximum size and deleting the oldest sample before adding a new one.Incremental Cholesky rank-one updates yield computational complexity approximately O(|D_i(·)|^2).
  • Computational Delay: The framework explicitly models computational delays from GP inference and updates, while treating communication delays as negligible relative to computation delays.The stated future-work direction is to account for computational and communication delays simultaneously.

B. Cooperative Learning with Delayed Predictions

This section bounds prediction errors when agents use delayed neighboring predictions at mismatched query points and aggregates them using uncertainty- and mismatch-aware weights. The resulting guarantee is designed for control under asynchronous computation rather than posterior convergence.

  • Delayed Prediction Accuracy: The delayed prediction error of a neighboring agent is bounded by GP uncertainty plus a term for query-state mismatch.The mismatch is between the current state of agent i and the delayed state associated with agent j's prediction.
  • Delayed Prediction Accuracy: The mismatch term has square-root dependence on state error, yielding Hölder continuity of order 1/2 rather than a linear Lipschitz bound.This relaxes the need for bounded derivatives, but produces slower local decay near equilibrium.
  • Cooperative Aggregation: Stored local and neighboring predictions are aggregated with time-varying weights that favor smaller GP uncertainty and smaller query-point mismatch.The weighting construction is inspired by generalized Product-of-Experts and uses informativeness-related terms.
  • Cooperative Aggregation: The weighting information is computable online from each agent’s local state and delayed neighboring state and uncertainty information.This uses only locally available state and transmitted delayed predictions.
  • Cooperative Aggregation: The aggregation theorem provides a time-varying control-oriented error bound based on the individual bounds of an agent and its neighbors.It does not establish posterior contraction or asymptotic GP convergence under heterogeneous asynchronous delays.

IV. DISTRIBUTED SECOND-ORDER CONSENSUS CONTROL

This section designs the distributed correction term through an adjoint multi-agent system. The adjoint system supplies the consensus behavior used to support second-order consensus control.

  • Control-Law Design: The control design answers the question of how to construct the correction term and analyze MAS control performance.The correction term is devised in the following subsection, followed by performance analysis.
  • Adjoint MAS: The correction input is constructed using an adjoint MAS with joint agent coordinates and an adjoint consensus error.The adjoint agents start from the original positions and velocities, while gains determine the adjoint-system performance.
  • Adjoint MAS: Choosing positive gains satisfying c̄1c̄3 ≥ c̄0 guarantees second-order consensus for the adjoint MAS under any topology.The guarantee uses zero position and velocity consensus errors in the stated definition.
  • Adjoint MAS: Because the adjoint dynamics exclude the unknown disturbance function, the adjoint MAS achieves second-order consensus asymptotically.The correction term is then designed from the introduced adjoint system.

B. Analysis of Learning-based Control Performance

The proposed controller combines an adjoint reference dynamics with asynchronous GP predictions to establish bounded tracking and practical second-order consensus under heterogeneous computational delays.

  • Control design: The adjoint MAS provides ideal reference dynamics, while the learning-based controller drives the actual system toward that behavior despite delayed GP predictions.The construction is conceptually related to certainty-equivalence control and backstepping.
  • Tracking analysis: Lemma 5 shows that each agent’s tracking-error norm is ultimately bounded, yielding ∥ε_i(t)∥≤¯ε for all t when initial reference coordinates coincide.The bound explicitly characterizes effects of GP uncertainty, asynchronous delays, and cooperative learning.
  • Implementation considerations: The discontinuous sign function introduces chattering, while smoothing it can reduce chattering at the cost of a small residual approximation error.The mismatch between agents is bounded by 2¯ε, and smoother inputs follow from improved learning accuracy or reduced delay.
  • Consensus guarantee: Theorem 2 establishes practical second-order consensus with ϵ0=2¯ε and ϵ1=¯ε under the proposed asynchronous GP-based controller.The guarantee assumes the stated system, GP, and controller conditions, including coefficients satisfying the supporting lemmas.
  • Scope and dependencies: The ultimate bound depends on agent count, posterior-variance bounds, and computational delays; tighter GP accuracy or faster computation can reduce it.The dependence on agent count may potentially be alleviated using local input-to-state stability or normalized consensus gains.

1) Learning Performance:

Cooperative aggregation improves theoretical prediction and control bounds over local learning when neighboring predictions reduce uncertainty or delay-induced query mismatch.

  • Learning Performance: Cooperative aggregation has a smaller theoretical prediction-error bound than local learning because posterior-variance reduction is strict under the stated comparison.The improvement is especially indicated when neighboring predictions have lower uncertainty or smaller query-point mismatch.
  • Learning Performance: Neighboring predictions can reduce the tracking-error bound when they have lower uncertainty or smaller delay-induced mismatch.The relevant factors are smaller ¯σ_i and smaller ∆_i.
  • Control Performance: The cooperative method achieves control-error bounds no larger than local learning, with ϵ0≤ϵloc,0 and ϵ1≤ϵloc,1.The comparison concerns ultimate consensus and velocity errors.
  • Control Performance: The analysis concludes that sharing learning information improves prediction accuracy and enhances control performance relative to local learning.This conclusion follows from the stated smaller error bounds for cooperative learning.

V. SIMULATION

Simulations evaluate six unmanned surface vehicles under unknown environmental effects and heterogeneous computation times, comparing the proposed method with established distributed GP strategies.

  • A. Simulation Setting: The simulation studies second-order consensus for N=6 unmanned surface vehicles with n=3 degrees of freedom and unknown wind and current effects.The vehicles communicate over the graph shown in Figure 2.
  • A. Simulation Setting: The comparison includes MoE, PoE, gPoE, BCM, rBCM, and Local methods using the same random initial values.The proposed aggregation is evaluated against these existing learning approaches.
  • B. Control and Prediction Performances: The proposed aggregation achieves significantly reduced aggregated prediction errors compared with the alternative strategies.The alternatives based on PoE and BCM neglect query-point bias, while cooperative learning outperforms local learning when delay is explicitly included.
  • B. Control and Prediction Performances: The actual aggregated prediction error remains below the Theorem 1 bound for all time, supporting the correctness of the theoretical analysis.Figure 3 compares the actual error with the derived prediction-error bound.
  • B. Control and Prediction Performances: The proposed methodology achieves superior maximal consensus error and maximal velocity compared with existing cooperative learning-based methods.Figure 4 presents the comparative control-performance evaluation.

C. Monte Carlo Test

A 100-trial Monte Carlo evaluation compares cooperative learning-based control strategies across prediction, consensus, and velocity errors. The proposed methodology reports lower errors and theoretical support for approximate consensus.

  • Monte Carlo setup: 100 independent trials use randomly sampled initial positions and velocities to assess algorithm performance across diverse initialization scenarios.Initial position coordinates and velocities are sampled uniformly from [-1, 1]^3.
  • Prediction performance: The proposed methodology consistently achieves smaller aggregated prediction errors than the compared cooperative learning-based control strategies.Figure 5 reports mean prediction error over time, with shaded variances across the Monte Carlo tests.
  • Control performance: The proposed methodology achieves lower maximal consensus and velocity errors than existing methods in the control comparison.Figure 6 evaluates these errors over time using mean values and variances from 100 Monte Carlo tests.
  • Steady-state evaluation: Steady-state evaluation over t ∈ [10, 30] shows substantially reduced disturbance-prediction and consensus-control errors for the proposed approach.The indicators include maximal prediction error, consensus error, and velocity error.
  • Framework: The framework combines cooperative online learning for unknown dynamics with a distributed control law based on an adjoint multi-agent system.Numerical simulations on unmanned surface vehicles compare the framework with existing cooperative approaches.
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