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Distributed Stochastic Optimal Control for Pattern-Oriented Swarms

Qingrui Zhang, Chenghao Yu, Feng Xue, Xintong Wang

arXiv:2609.12959v2cs.RO

TL;DR

Pattern-oriented swarms must coordinate geometry, self-organization, and safe navigation amid dynamic uncertainty. The paper combines temporal GRF-based stochastic control with uncertainty-aware collision avoidance and density-guided pattern control. In fully dynamic obstacle simulations, its CVaR formulation achieved a 99.6% survival rate, 3.2% above the baseline without uncertainty modeling.

  • Problem

    Pattern-oriented swarms must preserve task geometry while navigating dynamic, uncertain environments and adapting to changing swarm populations.

    Source: §I · p. 1

  • Method

    The paper uses a temporal GRF-based stochastic-control framework with risk-aware collision avoidance and density-field pattern control that avoids explicit robot-to-position assignments.

    Source: §I · p. 2

  • Results

    99.6% survival rate in fully dynamic obstacle environments for the CVaR-based formulation, improving by 3.2% over the baseline without uncertainty modeling.

    Source: §VII · p. 23

  • Takeaways & Limitations

    Density-guided pattern control supports self-healing and reconfiguration without explicit robot-to-position assignments, while the framework's pattern coverage stays above 91.9% across the reported shapes.

    Source: 3) Summary · p. 17, §VII · p. 23

Abstract

from arXiv · show

While offering significant promise for diverse applications, pattern-oriented swarms encounter multifaceted challenges in geometric control, self-organization, and safe navigation through dynamic environments. In this paper, we present a GRF-based stochastic optimal control framework to address these challenges within a unified probabilistic architecture. By extending the GRF into the temporal domain, the proposed framework casts collective coordination as a Bayesian inference task, enabling swarms to accommodate environmental uncertainty, satisfy non-convex constraints, and reconcile heterogeneous dynamics across diverse platforms. We develop an uncertainty- and safety-aware collision avoidance module for navigation in the presence of stochastic obstacle motion. The unscented transform is employed to propagate state uncertainty for both dynamic obstacles and neighboring agents, yielding principled confidence bounds for collision avoidance. In addition, density-guided pattern control is introduced, which encodes geometric patterns as implicit density fields. This representation decouples pattern specification from explicit agent-to-target assignments, thereby facilitating intrinsic self-healing and elastic reconfiguration in a distributed manner. The proposed framework is extensively evaluated through Monte Carlo simulations across diverse scenarios. Its model-agnostic nature is demonstrated on both quadrotor and fixed-wing UAV swarms, highlighting its generalizability across platforms with heterogeneous dynamics. Finally, the efficacy and robustness of the proposed method are validated through indoor experiments with a 15-quadrotor swarm and outdoor deployments involving 4 custom-built autonomous quadrotors. These experiments substantiate the proposed framework's capacity to maintain reliable geometric pattern transitions and safety-aware navigation within real-world environments.

I. INTRODUCTION

Pattern-oriented swarms must maintain task-specific geometry while navigating uncertain, dynamic environments and adapting to changing populations. The paper unifies swarm coordination, uncertainty-aware collision avoidance, and density-guided pattern control in a GRF-based stochastic-control framework.

A. Swarm Control Methods

  • Rule-based, optimization-based, and learning-based swarm controllers have limits in handling coupled objectives, formal safety, or adaptation outside their design assumptions [14]–, –,,.The paper identifies modular cost weighting and black-box policies as obstacles to modeling objective interdependencies and providing safety guarantees.Source: §I · pp. 1–2

I. INTRODUCTION

  • The framework represents swarm coordination as finite-horizon GRF inference and uses sampling-based optimization to accommodate uncertainty, non-convex constraints, and heterogeneous dynamics.It models interactions through clique potentials and connects probability maximization to path integral control.Source: §I · p. 2

B. Geometric Pattern Control

  • Density-guided control encodes patterns as implicit fields, avoiding explicit robot-to-target assignments and supporting self-healing and population-dependent reconfiguration.Remaining robots redistribute into vacated regions, and pattern scale adapts to swarm population.Source: §I · p. 3, B. Geometric Pattern Control · p. 4

C. Uncertainty-Aware Collision Avoidance

  • Unscented-transform uncertainty propagation and probability-based safety potentials integrate chance constraints or CVaR into the GRF framework for dynamic-obstacle navigation.The paper presents these formulations as providing probabilistic safety guarantees under non-deterministic obstacle motion.Source: §I · p. 2, C. Uncertainty-Aware Collision Avoidance · p. 5

A. Gibbs Random Field

  • A GRF represents robot states through local, undirected interactions and clique potentials, with lower-energy configurations receiving greater probability.Unary and pairwise potentials encode individual objectives and neighboring-robot interactions.Source: A. Gibbs Random Field · p. 5

B. Mean Field Approximation

This section develops a factorized approximation for coupled distributions and formulates swarm control as risk-constrained stochastic optimization over a finite horizon. It also describes sigma-point uncertainty propagation and temporal GRF inference for control selection.

B. Mean Field Approximation

  • The mean-field approximation replaces a coupled joint distribution with factorized marginals and chooses them to minimize KL divergence from the true distribution.The resulting local distribution depends on the expected log-likelihood under other robots' variational beliefs, supporting distributed control design.Source: B. Mean Field Approximation · p. 5

C. Stochastic Optimal Control

  • For stochastic swarm control, the objective is to reach a prescribed geometry while satisfying state, input, and risk-based safety constraints over a finite horizon.The risk functional can encode a chance constraint on violation probability or CVaR on tail severity.Source: C. Stochastic Optimal Control · p. 6

D. Unscented Transform

  • The unscented transform propagates a distribution through nonlinear dynamics using deterministic sigma points, then reconstructs its mean and covariance by weighted summation.For an n-dimensional state, 2n + 1 points suffice to capture the true mean and covariance; the method provides at least second-order accuracy.Source: D. Unscented Transform · p. 6
  • The spatial GRF is extended across a control-conditioned trajectory horizon, and the optimal control sequence is formulated as a maximum a posteriori estimate.The spatial model uses unary and pairwise energies to balance individual objectives and local coupling constraints.Source: D. Unscented Transform · pp. 6–7
  • The section identifies distributed optimization of the stochastic optimal-control problem as the remaining challenge.The preceding formulation specifies the control problem; a distributed solver is still needed.Source: D. Unscented Transform · p. 7

IV. METHODOLOGY

The GRF-MPPI architecture combines local robot interactions, a GRF-based coordination layer, and a stochastic trajectory optimizer. Its components connect MAP estimation with importance-weighted sampling and uncertainty propagation for collision avoidance.

IV. METHODOLOGY

  • The architecture begins with individual robots interacting with their immediate neighbors.This multi-robot system supplies the local interactions represented in the coordination layer.Source: §IV · p. 7
  • A GRF-based coordination layer formalizes robot interactions as clique potentials and converts energy minimization into a MAP estimation problem.Source: §IV · p. 7
  • The trajectory optimizer uses MPPI importance-weighted sampling with UKF-based uncertainty propagation for each robot's robust collision avoidance.Source: §IV · p. 7

1) Probabilistic Trajectory Modeling:

The framework turns swarm coordination into probabilistic inference over a finite horizon, then uses mean-field factorization and sampling-distribution optimization to make control distributed and tractable.

1) Probabilistic Trajectory Modeling:

  • Mean-field factorization converts the joint trajectory objective into local subproblems that robots solve independently, reducing computational overhead while maintaining collective swarm behavior.Source: 1) Probabilistic Trajectory Modeling · p. 7
  • Each robot can evaluate its local energy from task costs and expected pairwise interaction costs using uncertainty-aware neighbor distributions received through communication.Source: 1) Probabilistic Trajectory Modeling · p. 7
  • The GRF maps swarm interactions and spatial constraints through clique potentials into a probability distribution, casting optimal control as Bayesian inference.Source: Figure 1 · p. 8

2) Optimal Sampling Distribution:

  • Minimizing expected trajectory cost without regularization degenerates to a point mass on the single best trajectory, precluding effective sampling-based optimization.Source: 2) Optimal Sampling Distribution · p. 8
  • KL-divergence regularization keeps the sampling distribution near the control prior, while temperature λ trades broader exploration against concentration on low-cost trajectories.Source: 2) Optimal Sampling Distribution · p. 8
  • The optimal sampling distribution reweights the control prior toward lower-cost trajectories; direct sampling from it is intractable, motivating Monte Carlo importance sampling.Source: 2) Optimal Sampling Distribution · p. 9

3) Importance Sampling and Monte Carlo Estimation:

The method estimates optimal control by sampling trajectories from a tractable distribution and weighting them to approximate expectations under the optimal distribution.

B. Uncertainty-Aware Collision Avoidance

The framework models collision-relevant entities as uncertain and predicts their trajectories with UKF-based propagation. It then incorporates probabilistic or tail-risk collision criteria into the GRF energy used for sampling-based control.

B. Uncertainty-Aware Collision Avoidance

1) Uncertainty Modeling and Propagation:

2) Risk Constraint Formulation:

  • Risk constraints are enforced through potential energies: violating trajectories receive higher energy and therefore lower importance weights during sampling-based optimization.Source: 2) Risk Constraint Formulation · p. 10

3) Chance Constraints:

Chance constraints bound collision probability using uncertainty-dependent safety margins, while CVaR targets the severity of outcomes in the worst part of the risk distribution.

3) Chance Constraints:

  • The chance-constrained formulation bounds the probability of safety-constraint violation by a specified maximum collision probability α.Source: 3) Chance Constraints · p. 11
  • Under a Gaussian obstacle-position assumption and linearized collision constraint, the deterministic safety margin expands as position uncertainty increases or risk tolerance α decreases.Source: 3) Chance Constraints · p. 11

4) Conditional Value at Risk:

C. Density-Guided Swarm Pattern Control

The section encodes target patterns as density fields within the GRF, combining boundary enforcement with density-guided control. Mean Shift mechanisms disperse agents from crowded regions or guide them toward unoccupied areas.

1) Signed Distance Function for Pattern Representation:

  • The target region is represented implicitly by a signed distance field, with composite patterns formed from multiple subregions.The field distinguishes distance to the pattern boundary from the clique potential.Source: 1) Signed Distance Function for… · p. 12

2) Boundary Potential Energy:

  • A boundary potential penalizes agents outside the target region and attracts them back across its boundary.It is a unary potential because it depends on an individual agent’s position relative to the boundary.Source: 2) Boundary Potential Energy · p. 12

3) Density Potential Energy:

  • Neighbor-based Mean Shift repulsion favors sparse configurations by assigning lower energy to agents with larger displacement from the local density peak.A small displacement indicates placement near a crowded density peak.Source: 3) Density Potential Energy · p. 12
  • Mean Shift over unoccupied grid points guides agents toward under-populated areas, supporting coverage in complex non-convex patterns.Grid points inside the pattern are classified as unoccupied when no robot lies within the occupation radius.Source: 3) Density Potential Energy · pp. 12–13

4) Total Pattern Potential Energy:

The pattern-related Gibbs energy combines boundary enforcement with a selected density strategy, optionally adding pairwise pattern interactions. The algorithm uses trajectory prediction and sampled controls to evaluate costs and compute an optimal control.

4) Total Pattern Potential Energy:

  • When neighbor repulsion is selected, the formulation may include an additional pairwise pattern potential between neighboring agents.This is optional rather than part of every selected density strategy.Source: 4) Total Pattern Potential Energy · p. 13
  • The total Gibbs energy combines basic swarm and pattern potentials, with lower energy corresponding to configurations supporting collision avoidance, velocity alignment, and pattern maintenance.The pattern energy integrates the boundary potential with either neighbor repulsion or unoccupied-region attraction; a weighting coefficient balances boundary enforcement against density uniformity.Source: 4) Total Pattern Potential Energy · p. 13
  • The algorithm predicts neighbor and obstacle trajectories with the UKF, evaluates sampled control trajectories using collision and pattern potentials, then computes control by importance sampling.The pseudocode includes unoccupied-region computation and trajectory-cost accumulation across the prediction horizon.Source: 4) Total Pattern Potential Energy · p. 13

5) Dynamic Pattern Adaptation:

The simulations assess pattern adaptation, formation metrics, and the integrated framework across increasingly complex scenarios. Survival-rate comparisons show CVaR-MPPI leading across obstacle scenarios, with a larger gap as environmental uncertainty grows.

5) Dynamic Pattern Adaptation:

  • Time-varying signed distance fields support moving patterns, while a population-based scale factor adjusts pattern size to maintain consistent robot density.The pattern center and velocity vary over time; the scale factor depends on swarm population relative to a reference population.Source: 5) Dynamic Pattern Adaptation · p. 13

D. Performance Metrics

  • Pattern performance is assessed through conformity, distribution uniformity, coverage, and a composite formation score formed as a weighted sum.The uniformity score uses the coefficient of variation of nearest-neighbor distances.Source: D. Performance Metrics · p. 14

V. NUMERICAL SIMULATIONS

  • The simulation program evaluates components individually before testing integrated performance in multi-formation transitions with quadrotor and fixed-wing UAVs.The evaluation spans progressively more complex scenarios and uses Monte Carlo simulation.Source: §V · p. 14
  • CVaR-MPPI achieves the highest survival rates across methods and obstacle scenarios, with its performance gap increasing as environmental uncertainty grows.Error bars indicate standard deviation over 50 trials.Source: Figure 5 · p. 14

A. Collision Avoidance Performance Evaluation

CVaR-MPPI showed higher survival under dynamic obstacles, while sensitivity tests identify α=0.7 as the safety-efficiency balance across tested obstacle densities and weightings.

1) Simulation Setup:

2) Results and Analysis:

  • CVaR-MPPI reached 99.6% survival in the Dynamic Environment, 1.5% above CC-MPPI and 3.2% above Baseline MPPI.Table I provides the comprehensive performance comparison.Source: 2) Results and Analysis · p. 15, 3) Summary · p. 15, Table I · p. 15
  • In dynamic scenes, CC-MPPI's chance-constraint linearization degraded under nonlinear motion, while CVaR-MPPI's Monte Carlo sampling captured nonlinear collision-risk distributions.Baseline MPPI had the lowest survival across scenarios, with degradation more pronounced as environmental uncertainty increased.Source: 2) Results and Analysis · p. 15, 3) Summary · p. 15

B. CVaR Parameter Sensitivity Analysis

3) Summary:

The preceding summary recommends α=0.7 for safety-efficiency performance; density-guided Unoccupied Attraction improves formation coverage and quality, while self-healing is evaluated under sudden agent loss.

3) Summary:

  • α=0.7 is the recommended default, scoring highest across weighting schemes while using 20% less energy than α=0.9; α=0.9 gives the highest worst-case survival, 95.0%.The comparative collision-avoidance experiments used α=0.9 to establish an upper bound on achievable safety performance.Source: 3) Summary · pp. 16–17

1) Simulation Setup:

2) Results and Analysis:

  • In the 8-shape, Unoccupied Attraction achieved 98.3% coverage versus 92.3% for Neighbor Repulsion and 92.9% for Baseline.Table III presents quantitative comparisons across all five geometric patterns.Source: 2) Results and Analysis · p. 17, Table III · p. 18
  • In the 8-shape, Unoccupied Attraction reached 77.5% uniformity versus 69.0% for Neighbor Repulsion, with convergence taking 32.5 versus 30.0 steps.In the A-shape, the passage reports an 8.3% coverage and 5.4% uniformity improvement alongside a 23% convergence-time difference.Source: 2) Results and Analysis · p. 17

3) Summary:

  • Coverage stayed above 91.9% from Circle to 8-shape, while Neighbor Repulsion ranged from 84.7% to 92.3%.The passage presents this as evidence of performance across the geometric-complexity spectrum.Source: 3) Summary · p. 17

D. Self-Healing Evaluation Under Agent Loss

2) Results and Analysis:

The results show that Unoccupied Attraction sustains formation quality despite agent loss, while the framework also handles complex transitions and safety constraints in simulation and fixed-wing missions. Real-world evaluations use indoor and outdoor quadrotor platforms.

2) Results and Analysis:

  • 93.9% final coverage and 75.7% final uniformity at 50% agent loss; coverage falls only 3.2% from 10% to 50% loss with Unoccupied Attraction.The figure summarizes final coverage across loss conditions; the reported uniformity and degradation figures further characterize recovery and robustness.Source: Figure 9 · p. 18, 2) Results and Analysis · pp. 18–19

3) Summary:

  • Density-gradient attraction guides surviving agents toward unoccupied target regions, supporting rapid compensation for losses and dynamic reconfiguration.The paper presents this mechanism as a basis for maintaining formation quality despite partial system failures.Source: 3) Summary · p. 19

E. Comprehensive Evaluation of the Swarm Pattern Control

  • Across four non-convex formations, the 30-agent swarm achieved 96.9%–99.8% coverage and 100% survival over a 1050-step obstacle-navigation simulation.The swarm transitioned through S-, Y-, S-, and U-shapes while maintaining collision-free navigation.Source: E. Comprehensive Evaluation of the… · p. 19

F. Generalization to Fixed-Wing UAVs

  • The eight-agent fixed-wing mission recorded zero collisions over 420 steps, with 6.90 m average inter-agent spacing against a 3.0 m safety threshold.The mission included formation transitions, a 12 m channel crossing, a constrained U-turn, and dynamic-obstacle avoidance.Source: F. Generalization to Fixed-Wing UAVs · p. 20

VI. REAL-WORLD EXPERIMENTS

  • Indoor tests used 15 Crazyflie quadrotors with 120 Hz motion-capture state estimation, while outdoor tests deployed four custom quadrotors without external positioning.Outdoor vehicles used onboard computation and LiDAR-based localization in unstructured environments.Source: §VI · p. 20

A. Experimental Platform

  • The experimental platforms were Crazyflie 2.1 nano quadrotors indoors and custom-built Q250 quadrotors outdoors.Indoor coordination used Crazyswarm and external motion-capture feedback; outdoor vehicles used Pixhawk 6C controllers, Intel NUC11 computers, and LiDAR-inertial odometry.Source: A. Experimental Platform · pp. 20–21, Figure 12 · p. 21

B. Formation Transition Experiment

The experiments report pattern transitions, self-healing after agent loss, and collision avoidance in indoor and outdoor settings. The conclusion describes a distributed GRF-based control approach and its simulation outcomes.

B. Formation Transition Experiment

C. Self-Healing Experiment

  • After four of 15 agents landed, the remaining 11 redistributed toward newly unoccupied regions and recovered comparable coverage and uniformity.Before the loss event, the A formation had a 77.0% uniformity score.Source: C. Self-Healing Experiment · p. 21, Figure 14 · p. 22

D. Swarm Crossing Experiment

  • During the 12-agent crossing without inter-swarm communication, minimum separation stayed above the 0.4 m threshold and the experiment ended with zero collisions.The closest encounter was 0.498 m; both groups expanded to about 2.1 m during passage and contracted to around 1.5 m afterward.Source: D. Swarm Crossing Experiment · pp. 21–22, Figure 15 · p. 23

E. Outdoor Swarm Navigation Experiment

  • Onboard, MPPI configuration scales down from K=10,000 rollouts with Δt=0.1 s and H=15 in simulation to K=2,000, Δt=0.2 s, and H=7.The reduction narrows sampled control candidates and weakens solver stochastic optimality, resulting in more conservative maneuvers and less aggressive avoidance.Source: E. Outdoor Swarm Navigation Experiment · p. 22
  • The outdoor swarm preserved its triangular pattern through a gate with static and dynamic obstacles, recording zero collisions and 100% survival despite reduced onboard sampling.Trajectories were smoother but less responsive, and pattern metrics fluctuated more around the gate crossing than in simulation.Source: E. Outdoor Swarm Navigation Experiment · pp. 22–23

VII. CONCLUSIONS

  • Mean-field approximation decomposes joint GRF optimization into decoupled local subproblems, enabling fully distributed computation solved with model predictive path integral methods.The framework formulates finite-horizon swarm control as maximum a posteriori estimation over a joint probability distribution.Source: §VII · p. 23
  • In simulation, CVaR achieved 99.6% survival in fully dynamic obstacle environments, 3.2% above baseline; non-convex patterns exceeded 91% coverage, with recovery up to 50% agent loss.The framework uses unscented-transform trajectory prediction with risk-constrained optimization and density-guided pattern control.Source: §VII · p. 23
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