DANCeRS: Consensus in Robot Swarms

• DANCeRS is a unified distributed consensus framework for robot swarms, leveraging Gaussian Belief Propagation on factor graphs for continuous and discrete decision making.
• It models swarms as dynamic factor graphs where peer-to-peer Gaussian message passing enables scalable formation, path planning, and collision avoidance.
• Experimental results show faster convergence and robust performance compared to traditional methods by efficiently managing local communication and dynamic networks.

Searching arXiv for the named system and closely related work on distributed swarm consensus, factor graphs, and Gaussian belief propagation. DANCeRS is a distributed consensus and coordination framework for robot swarms that uses Gaussian Belief Propagation on factor graphs to unify consensus in continuous and discrete decision spaces (Patwardhan et al., 25 Aug 2025). In the formulation introduced in "DANCeRS: A Distributed Algorithm for Negotiating Consensus in Robot Swarms with Gaussian Belief Propagation" (Patwardhan et al., 25 Aug 2025), a swarm is represented as a factor graph and robots rely on purely peer-to-peer message passing to achieve agreement on global behavior while operating under local communication constraints. The framework is presented through two applications: shape formation with path planning and collision avoidance, and consensus over a set of discrete decisions (Patwardhan et al., 25 Aug 2025).

1. Definition and scope

DANCeRS stands for Distributed Algorithm for Negotiating Consensus in Robot Swarms with Gaussian Belief Propagation (Patwardhan et al., 25 Aug 2025). It is described as a general, fully distributed consensus and coordination framework for robot swarms, built on Gaussian Belief Propagation (GBP) over factor graphs, and explicitly designed to unify consensus over continuous global quantities and discrete choices within a single mathematical and algorithmic structure (Patwardhan et al., 25 Aug 2025).

The central problem is to enable a swarm of $N$ robots to agree on a global parameter $\chi$, jointly plan trajectories that respect non-holonomic dynamics and avoid collisions, and reach agreement on a discrete choice when needed (Patwardhan et al., 25 Aug 2025). The method does so by representing swarm state and coupling constraints as a factor graph, running GBP in a peer-to-peer and asynchronous way, using Lie-group GBP for continuous consensus on manifolds such as $SE(2)$, and embedding discrete decisions into a continuous latent space so they can be handled by the same inference machinery (Patwardhan et al., 25 Aug 2025).

A defining feature of the framework is that it does not treat consensus in discrete and continuous spaces as separate problems. This suggests a more unified view of swarm coordination in which formation pose agreement, trajectory coordination, and best-of-$N$ decision making can be expressed through the same factor-graph formalism (Patwardhan et al., 25 Aug 2025).

2. Factor-graph and GBP formulation

At any time $t$, the swarm is modeled as a sparse undirected communication graph

$G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$

where an edge $E_{ij} \in E$ exists if

$\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$

with $r_C$ the communication radius (Patwardhan et al., 25 Aug 2025).

Each robot maintains a local factor graph stack with two layers (Patwardhan et al., 25 Aug 2025):

Layer	Variables	Purpose
Global Consensus Layer $\mathcal{G}$	$\chi$0	Consensus on a global parameter
Path Planning Layer $\chi$1	$\chi$2	Local trajectory optimisation

The Global Consensus Layer contains a sliding window of robot $\chi$3's belief about the global parameter $\chi$4, together with prior, inter-robot consensus, and temporal factors (Patwardhan et al., 25 Aug 2025). The Path Planning Layer contains planned states over a horizon, along with dynamics factors, a unicycle model factor, and inter-robot collision avoidance factors (Patwardhan et al., 25 Aug 2025).

The joint distribution over all variables is factorized as

$\chi$5

with each factor taking Gaussian form

$\chi$6

where

$\chi$7

and $\chi$8 is the precision matrix (Patwardhan et al., 25 Aug 2025).

Beliefs and messages are represented in information form: $\chi$9 where $SE(2)$0 and $SE(2)$1 (Patwardhan et al., 25 Aug 2025). Messages passed between factor and variable nodes are likewise Gaussians in information form (Patwardhan et al., 25 Aug 2025).

For variables in Lie groups such as $SE(2)$2, the Gaussian belief lives on the tangent space, and residuals use the right-minus operation

$SE(2)$3

so that consensus over pose variables can be handled by Lie-group GBP rather than Euclidean averaging (Patwardhan et al., 25 Aug 2025).

3. Consensus mechanisms

Continuous consensus

For continuous consensus, each robot holds a local estimate $SE(2)$4 of a global quantity $SE(2)$5 (Patwardhan et al., 25 Aug 2025). Consensus is enforced through three classes of factors (Patwardhan et al., 25 Aug 2025).

The prior factor anchors robot $SE(2)$6's belief to its initial estimate: $SE(2)$7

The inter-robot consensus factor couples neighboring robots: $SE(2)$8

The temporal factor links consecutive consensus-window variables and preserves past consensus information during disconnections (Patwardhan et al., 25 Aug 2025).

Through GBP, neighboring robots’ beliefs are iteratively pulled toward each other and toward prior structure. The sliding window and temporal factors are especially important in dynamic communication graphs, because they allow robots to retain consensus information when links disappear and reappear (Patwardhan et al., 25 Aug 2025).

Discrete consensus

For discrete decisions, DANCeRS chooses $SE(2)$9 and maps discrete options into $N$0 with the quantization function

$N$1

with inverse

$N$2

(Patwardhan et al., 25 Aug 2025).

A robot’s initial discrete decision $N$3 becomes a prior via

$N$4

and the prior factor is

$N$5

Inter-robot agreement in the discrete setting is enforced by the scalar factor

$N$6

(Patwardhan et al., 25 Aug 2025). After GBP converges, the continuous consensus value is mapped back to a discrete decision by $N$7 (Patwardhan et al., 25 Aug 2025).

This suggests that discrete consensus in DANCeRS is a continuous relaxation solved by Gaussian inference rather than a separate categorical protocol (Patwardhan et al., 25 Aug 2025).

4. Robot-local algorithm and communication model

DANCeRS is purely peer-to-peer: robots maintain local variables and factors, exchange small Gaussian messages with neighbors, and never construct the global factor graph centrally (Patwardhan et al., 25 Aug 2025). Each robot maintains a “webpage” that stores outgoing messages for neighbors’ variables and factors (Patwardhan et al., 25 Aug 2025).

At runtime, a robot updates its neighborhood using the communication radius $N$8, adds inter-robot consensus and collision-avoidance factors when new robots enter range, deletes them when robots leave range, and runs a fixed number $N$9 of GBP iterations in each layer (Patwardhan et al., 25 Aug 2025). After message passing, it updates the current and horizon path variables and slides the Global Consensus window by deleting the oldest variable, creating a new variable, and attaching a new temporal factor and prior (Patwardhan et al., 25 Aug 2025).

The framework is therefore asynchronous and distributed by construction. Its computational and communication costs scale with local degree and variable dimension rather than with total swarm size, which is presented as one basis for its scalability (Patwardhan et al., 25 Aug 2025).

5. Shape formation, path planning, and collision avoidance

One of the paper’s two principal applications is shape formation in a $t$0 m$t$1 environment using robots of radius 1 m and non-holonomic unicycle dynamics (Patwardhan et al., 25 Aug 2025). The swarm is assumed to know a canonical formation shape defined by $t$2 points $t$3, with minimum spacing $t$4, and must agree on the global pose of this formation while planning collision-free trajectories to occupy those points (Patwardhan et al., 25 Aug 2025).

The formation pose is represented in $t$5 by

$t$6

(Patwardhan et al., 25 Aug 2025). A canonical formation point $t$7 is transformed to the global frame through

$t$8

and inverse transformation is given by

$t$9

(Patwardhan et al., 25 Aug 2025).

The path-planning state is

$G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$0

(Patwardhan et al., 25 Aug 2025). The unicycle model factor enforces non-holonomic motion through

$G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$1

(Patwardhan et al., 25 Aug 2025). Collision avoidance between robots $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$2 and $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$3 at horizon step $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$4 is modeled by the factor

$G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$5

where $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$6 and $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$7 is the minimum separation distance (Patwardhan et al., 25 Aug 2025).

DANCeRS also introduces an occupancy weighting mechanism for selecting formation points. Each formation point $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$8 is augmented with an occupancy weight $G = (V, E), \quad V = \{V_i\}_{i=0}^{N-1},$9,

$E_{ij} \in E$0

and stored in a KD-tree to support fast nearest-neighbor queries in 3D (Patwardhan et al., 25 Aug 2025). The weights are increased for occupied points, decreased over time, and used to bias each robot toward nearby, low-occupancy targets (Patwardhan et al., 25 Aug 2025). The chosen point is then mapped back into the global frame to define the motion goal (Patwardhan et al., 25 Aug 2025).

The paper reports that this occupancy-aware selection, combined with GBP-based planning, enables formation assembly even for shapes with disconnected components such as a “wifi” symbol, an exclamation mark, and a smiley face (Patwardhan et al., 25 Aug 2025).

6. Experimental results and performance

For continuous consensus and shape formation, DANCeRS is evaluated in a $E_{ij} \in E$1 m$E_{ij} \in E$2 environment with factor strengths $E_{ij} \in E$3, $E_{ij} \in E$4, $E_{ij} \in E$5, a consensus prior $E_{ij} \in E$6, a sliding window $E_{ij} \in E$7, update interval $E_{ij} \in E$8, and $E_{ij} \in E$9 GBP iterations per timestep (Patwardhan et al., 25 Aug 2025). Fifty trials are run per condition (Patwardhan et al., 25 Aug 2025).

The paper defines convergence for formation parameters as mean inter-robot deviation in formation position belief below 0.1 m and heading deviation below 0.01 rad (Patwardhan et al., 25 Aug 2025). On this basis, DANCeRS is reported to achieve order-of-magnitude faster convergence than the mean-shift based distributed consensus method of Sun et al. 2023, with convergence improving as communication radius $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$0 and number of robots $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$1 increase (Patwardhan et al., 25 Aug 2025). The sliding window also reduces the number of iterations to consensus, especially under sparse connectivity (Patwardhan et al., 25 Aug 2025).

For discrete decision making, robots are placed in a triangular grid with minimum spacing 5 m, initial random decisions in $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$2, and communication radius $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$3 varied in steps of 6 m (Patwardhan et al., 25 Aug 2025). DANCeRS is compared to ECA and PCA (Patwardhan et al., 25 Aug 2025). The reported results show that as $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$4 increases, DANCeRS requires fewer iterations; for larger $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$5, convergence iterations remain nearly constant as swarm size grows (Patwardhan et al., 25 Aug 2025). ECA is reported to fail to converge at $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$6 m, while PCA converges but relies on assumptions the paper characterizes as unrealistic for dynamic networks (Patwardhan et al., 25 Aug 2025).

The paper also studies the effect of the consensus factor strength $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$7 and reports that a useful upper bound is approximately

$\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$8

for discrete consensus (Patwardhan et al., 25 Aug 2025). In a sparse-network experiment with $\|\mathbf{x}_i - \mathbf{x}_j\| < r_C,$9, $r_C$0 m, and a fraction $r_C$1 of informed robots given strong priors with $r_C$2, DANCeRS converges in 80–100% of trials for all tested $r_C$3 values, while ECA converges in 0% of trials and PCA requires higher seed density to reach comparable success (Patwardhan et al., 25 Aug 2025).

7. Scalability, robustness, and limitations

The framework emphasizes four system-level properties: scalability, robustness in dynamic graphs, purely peer-to-peer communication, and a unified treatment of consensus and planning (Patwardhan et al., 25 Aug 2025). Message dimensionality is equal to the variable dimension, such as 3 for $r_C$4 and 1 for discrete consensus, and each robot optimizes only over its local factor graph (Patwardhan et al., 25 Aug 2025). This suggests that DANCeRS is intended for large swarms and low-power platforms where centralized optimization would be impractical.

The paper nevertheless identifies several limitations and assumptions (Patwardhan et al., 25 Aug 2025). The factors are Gaussian or linearized to Gaussian form, so strongly non-linear or non-Gaussian behavior may reduce fidelity (Patwardhan et al., 25 Aug 2025). The continuous formulation assumes access to Lie-group operations such as exponential and logarithm maps (Patwardhan et al., 25 Aug 2025). Static formation shapes require robots to store all formation points, which can be memory-intensive for large shapes (Patwardhan et al., 25 Aug 2025). Discrete consensus assumes a fixed number of options $r_C$5, and the current formulation is not directly adaptive to dynamically changing discrete sets (Patwardhan et al., 25 Aug 2025). The framework also inherits the approximate character of loopy GBP on cyclic graphs, so convergence quality depends on factor strengths and graph structure (Patwardhan et al., 25 Aug 2025).

The paper suggests extensions including non-Gaussian beliefs, more expressive encodings of discrete decisions, heterogeneous robot teams, dynamic decision spaces, and real-world deployment under lossy communications and limited compute (Patwardhan et al., 25 Aug 2025). A plausible implication is that DANCeRS is best understood not as a single-purpose consensus routine, but as a general inference substrate for swarm coordination problems that can be enlarged as the factor graph is enriched (Patwardhan et al., 25 Aug 2025).

8. Position within the literature

DANCeRS is presented as a unified alternative to approaches that treat consensus in discrete and continuous decision spaces as distinct problems (Patwardhan et al., 25 Aug 2025). It is grounded in Gaussian Belief Propagation and factor-graph inference, and its two showcased applications—shape formation with path planning and collision avoidance, and consensus over discrete decisions—are intended to demonstrate the breadth of that formulation (Patwardhan et al., 25 Aug 2025).

In that sense, DANCeRS differs sharply from the dance-generation and dance-analysis systems that share a superficially similar acronym or title fragment elsewhere in recent arXiv literature, such as DANCER for pose-guided single-person dance video synthesis (Xing et al., 31 Oct 2025) or group choreography generators based on diffusion models (Le et al., 2023, Dai et al., 2024). Those systems address visual synthesis or choreography generation, whereas DANCeRS addresses swarm consensus and distributed multi-robot coordination (Patwardhan et al., 25 Aug 2025).

The paper’s central contribution is therefore algorithmic rather than performative: it reinterprets swarm agreement, motion planning, and discrete choice as local inference in a dynamic factor graph, solved by Gaussian message passing (Patwardhan et al., 25 Aug 2025).