Decentralized Meanshift Control Law for Robotic Swarms

• Decentralized meanshift control law is a distributed feedback mechanism that uses discrete mass-distribution to coordinate robot swarms for precise shape and density formations.
• It employs local communication and consensus protocols to estimate global mass parameters while integrating collision-avoidance, ensuring robust performance despite changes in swarm size.
• Empirical validations demonstrate rapid error decay and adaptability, with simulations and real-world experiments confirming effective convergence and resilience to agent failures.

A decentralized meanshift control law is a fully distributed feedback mechanism for large-scale robot swarms, designed to achieve precise spatial density and shape formation by coordinating agent positions via local information and neighbor communication. In contrast to centralized control or continuous density-based models, this approach employs a discrete mass-distribution formalism over a set of sample points encoding the target geometric configuration. The decentralized meanshift law incorporates a consensus-based distributed estimation protocol for global density parameters, yielding adaptive and scalable shape control robust to variations in swarm size and local agent failures (Cai et al., 1 Feb 2026).

1. Discrete Mass-Distribution Formalism

The discrete mass-distribution model specifies a target formation by $m$ sample points $\{q_k\}_{k=1}^m \subset \mathbb{R}^d$, obtained through uniform discretization of the desired contour or surface in $d$-dimensional space. For a swarm of $n$ robots with configuration $$p = [p_1^\top, \ldots, p_n^\top]^\top \in \mathbb{R}^{dn}$$, one defines the normalized robot mass around each sample point:

$$P_k(p) = \frac{1}{n} \sum_{i=1}^{n} \exp\big(-\beta \|q_k - p_i\|^2\big), \qquad k = 1, \dots, m,$$

where $\beta > 0$ is a kernel bandwidth. The vector $P(p) = [P_1,\ldots, P_m]^\top$ describes the swarm's discrete density profile. Shape-matching performance is quantified by the error metric

$$F(p) = -\frac{1}{m} \sum_{k=1}^m \ln P_k(p) - \frac{1}{2} \ln m,$$

which vanishes only for perfect uniform coverage, i.e., $P_k(p)$ constant across $k$ (Cai et al., 1 Feb 2026).

2. Meanshift Control Law and Decentralization

In the centralized variant, agents follow a gradient-descent step on $F(p)$:

$v_i^{ms} = -\kappa_1 \nabla_{p_i} F(p),$

where $\kappa_1$ is chosen to ensure speed constraints. Substituting the expression for the gradient, the velocity command admits a meanshift structure:

$$v_i^{ms} = \frac{(\sigma_1/m) \sum_{k=1}^m [P_k(p)]^{-1} \exp(-\beta \|q_k - p_i\|^2)(q_k - p_i)} {\sum_{k=1}^m [P_k(p)]^{-1} \exp(-\beta \|q_k - p_i\|^2)}.$$

The update biases movement toward sample points: higher weights accrue to nearby under-covered regions (low $P_k$), enhancing spatial uniformity. For full decentralization, each robot $i$ locally maintains estimates $\hat{P}_{k,i}$ of $P_k$, exchanging these only with neighbors:

$$v_i^{ms} = \frac{(\sigma_1/m) \sum_{k=1}^m [\hat{P}_{k,i}]^{-1} \exp(-\beta \|q_k - p_i\|^2)(q_k - p_i)} {\sum_{k=1}^m [\hat{P}_{k,i}]^{-1} \exp(-\beta \|q_k - p_i\|^2)}.$$

A collision-avoidance velocity $v_i^{cv}$ is added; the input is saturated to enforce speed limits:

$v_i = \text{sat}(v_i^{ms} + v_i^{cv}).$

This fully decentralized law requires only local neighbor communication and knowledge of the sample points $\{q_k\}$ (Cai et al., 1 Feb 2026).

3. Distributed Mass Estimation and Consensus Protocol

Distributed estimation of all $m$ mass parameters $P_k(p)$ is achieved by augmenting each agent $i$'s state with estimators $\hat{P}_{k,i}$ and auxiliary variables $z_{k,i}$ initialized to zero. The decentralized estimator:

$$\hat{P}_{k,i} = \exp(-\beta \|p_i - q_k\|^2) + z_{k,i}, \qquad \dot{z}_{k,i} = \gamma \sum_{j \in \mathcal{N}_i} \text{sign}(\hat{P}_{k,j} - \hat{P}_{k,i}),$$

implements a distributed average-tracking protocol, where $\gamma > 0$ is the update gain, and $\mathcal{N}_i$ are neighbors of $i$. The estimator tracks the average of the moving local reference $e^{-\beta\|p_i - q_k\|^2}$, converging asymptotically to the global mass value $P_k(p)$ under connectivity and gain conditions:

$\gamma > (n-1) \sqrt{2\beta / e} \, v_{max}.$

This protocol is robust to time-varying references and bounded agent speeds, following established consensus principles (Cai et al., 1 Feb 2026).

4. Convergence and Stability Results

The mass estimator converges to global values provided the communication graph remains connected, and the estimator gain $\gamma$ exceeds a speed-dependent threshold. Swarm formation convergence assumes true global mass knowledge ($\hat{P}_{k,i} = P_k$), convexity of the target shape, and the combined meanshift–collision-avoidance control law. Key properties are:

• $F(p)$ is smooth, radially unbounded, and non-negative.
• The closed-loop system is locally Lipschitz, guaranteeing unique solutions.
• The temporal derivative satisfies $$\dot{F} \leq -\varepsilon \sum_i \|v_i^{ms}\|^2 \leq 0$$.
• By LaSalle’s invariance principle, swarm configurations converge to critical points of $F$, which, for convex shapes, ensure all robots settle within the convex hull of $\{q_k\}$ (Cai et al., 1 Feb 2026).

5. Implementation Considerations and Computational Aspects

Key empirical and algorithmic considerations include:

• Sampling: $m \geq 5n$ sample points mitigate local minima and promote effective coverage.
• Communication: Robustness and convergence require maintenance of a connected communication or sensing graph.
• Kernel bandwidth $\beta$ determines spatial resolution of density estimates; selection involves annealing or cross-validation methods.
• Each robot’s computation and communication per mass-estimation cycle scales as $\mathcal{O}(m + |\mathcal{N}_i|)$, independent of $n$.
• Collision avoidance is built-in to the control law and does not require centralized intervention (Cai et al., 1 Feb 2026).

6. Empirical Validation and Adaptability

Extensive simulations span both convex and highly non-convex target shapes, with swarm sizes up to $n=100$ robots and real-world experiments with 10 TurtleBot3s. The results demonstrate:

• Rapid decay of the error metric $F$ and vanishing estimation error within 10 seconds for large swarms.
• Superior spatial uniformity and convergence rates compared to image-moment-based methods (in the bunny-head shape case with $n=50$).
• Resilience to agent removal and reinsertion: the swarm adapts and re-forms the shape in under 50 seconds after agent loss or reintegration, following $z$-state reset (Cai et al., 1 Feb 2026).

7. Connections to Broader Decentralized Mean-Field Control

While the decentralized meanshift law operates in continuous space with explicit geometric sample points, related mean-field decentralized approaches on discrete state spaces, notably for Markov decision processes on graphs, establish similar global stability via decentralized polynomial feedbacks and Lyapunov certificates (Biswal et al., 2017). Both frameworks harness localized agent information and connectivity-driven consensus/feedback to asymptotically realize global density objectives, though the mathematical models and feedback realizations are distinct.

A plausible implication is that future hybridizations could leverage the geometric flexibility of sample-point-based mass distribution with the mean-field stability guarantees of polynomial feedback for broader classes of agent networks (Cai et al., 1 Feb 2026, Biswal et al., 2017).