Swarming Without an Anchor (SWA)

• Swarming Without an Anchor (SWA) is a robust paradigm that enables UAV swarms to maintain cohesive, synchronized behavior without external positioning anchors.
• It employs distributed state estimation with mutual perception, leveraging local observations and linear Kalman filters to achieve velocity consensus and relative state stabilization.
• While SWA ensures formation and synchronization in denied environments, it inherently tolerates a global drift, necessitating re-acquisition of absolute localization for bounded positioning.

Swarming Without an Anchor (SWA) is a systems-theoretic and algorithmic paradigm for enabling multi-robot swarms—primarily, but not exclusively, aerial UAV swarms—to maintain cohesive, synchronized collective behavior in scenarios where all absolute positioning anchors (e.g., GNSS/GPS, motion capture, visual odometry) are lost or unreliable. Unlike traditional approaches that rely on at least one agent having access to a fixed or global reference frame, SWA exclusively leverages mutual local observations and distributed estimation to achieve velocity consensus and relative state stabilization, while inherently tolerating a global unobservable drift. This methodology underpins a new class of robust, infrastructure-free swarm systems applicable in adversarial, denied, or highly dynamic environments (Horyna et al., 22 Aug 2025).

1. Mathematical Framework and System Assumptions

SWA models each agent, specifically UAVs, as point-masses governed by double-integrator dynamics in the lateral plane, extended with state estimators and sensor fusion strategies:

• Observed UAVs (oUAVs):

$\vx^i_{o,k} = \vF_o \vx^i_{o,k-1} + \vw^i_{o,k},\quad \vw^i_{o,k} \sim \mathcal{N}(0, \vQ_o)$

with state $\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$.

• Focal UAV (fUAV):

$\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$

The matrices $\vF_o, \vF_f, \vB_f$ include time discretization $\Delta t$ and a damping constant $e_d = e^{-\Delta t/\tau}$, modeling inertial and control lag.

• Sensing and Communication:
  • Relative measurements $\vz^i_k$ stem from a Mutual Perception System (MPS), e.g., visual fiducials or UWB, providing agent-identified range/bearing.
  • All agents may lose GNSS/VIO simultaneously during dropout events; only unique IDs and relative measurement links are utilized—no sharing of absolute poses (Horyna et al., 22 Aug 2025).
• Reference Frames:
  • Local body $\cB^i$, stable initial $\cS^i$, and floating centroid $\cF^i$ frames are defined per agent, with mutual orientation.

This formalism creates a fully relative, internally consistent estimation space, batched across the swarm.

2. Distributed State Estimation and Mutual Perception

SWA employs a bank of Linear Kalman Filters (LKFs):

• Neighbor Tracking:

For each neighbor, an LKF predicts and corrects the neighboring agent's position and velocity using locally acquired MPS measurements. The prediction step extrapolates based on double-integrator kinematics, while the correction step incorporates new relative observations:

$\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$0

The focal UAV’s own state is recursively estimated by fusing these filtered neighbor positions (translated into a centroid of local formation via circle-fitting) and IMU measurements, using asynchronous data fusion and Mahalanobis-gated outlier rejection.

• Circle-Fitting and the Floating Frame:

Filtering neighbor positions yields the local formation "center" $\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$1, which serves as the reference for the fUAV’s position update [(Horyna et al., 22 Aug 2025), §2.3].

• Mutual Perception System (MPS):

Data association and track maintenance rely on sensor-specific IDs, with per-detection gating and lifetime-based pruning for robust operation under occlusion and intermittent dropout.

3. Velocity Consensus and Synchronization

SWA targets velocity consensus—synchronization of lateral velocities across the swarm—using a second-order error feedback law:

$\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$2

where the errors $\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$3 are computed with respect to the floating frame. This is a distributed double-integrator consensus protocol, with closed-loop dynamics:

$\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$4

Here, $\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$5 is the Laplacian of the neighbor-graph. The nullspace of $\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$6 corresponds to uniform translation; thus, synchronization is achieved up to a global drift. LaSalle invariance ensures all agents reach velocity consensus ($\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$7), while the overall centroid may translate at constant (unobservable) drift [(Horyna et al., 22 Aug 2025), §4.3].

4. Performance Metrics, Error Bounds, and Theoretical Guarantees

SWA’s performance is quantified by several metrics:

• Cohesion: Mean nearest-neighbor distance

$\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$8

• Drift Velocity: The velocity of the swarm’s centroid

$\vx^i_{o,k} = [\vr^i_k,\, \dot\vr^i_k,\, \ddot\vr^i_k]^T$9

• Estimator Consistency: Normalized Estimation Error Squared (NEES, ANEES)

$\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$0

The LKF’s steady-state covariance $\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$1 is analytically characterized, and the velocity synchronization rate is controlled by the algebraic connectivity $\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$2 of the neighbor graph:

$\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$3

Only a uniform translation mode (unobservable drift) persists. All other disturbances and estimation errors are attenuated [(Horyna et al., 22 Aug 2025), §5].

5. Experimental Validation and Implementation

Extensive experiments, both simulated (Gazebo + ROS) and real-world (DJI F450, Pixhawk 4, Intel NUC), validate SWA:

• Simulation: Four UAVs lose GNSS at $\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$4 s; the swarm with baseline control rapidly disintegrates, while SWA preserves cohesion and synchronizes velocity. Larger swarms experience reduced drift due to IMU bias averaging.
• Real-world: Three UAVs operating in a field undergo complete GNSS jamming; over >100 m drift, no collisions occur and the local formation is maintained. Peak drift speed ≈7 m/s is observed, but swarm structure persists [(Horyna et al., 22 Aug 2025), §6].

Implementation guidelines:

• Use a reliable MPS (e.g., UWB, UVDAR, LiDAR) sampling at 5–20 Hz.
• Tune each Kalman filter’s process and measurement covariances to match sensor characteristics, validated with NEES/ANEES.
• Deploy a sufficiently dense neighbor graph (large $\vx_{f,k} = \vF_f \vx_{f,k-1} + \vB_f \vu_{f,k} + \vw_{f,k},\quad \vu_{f,k} = \vv_{d,k},\quad \vw_{f,k} \sim \mathcal{N}(0, \vQ_f)$5) for rapid consensus.
• Employ the circle-center estimation for compute-efficient centroid localization.
• Apply a simple PD-type velocity controller.

SWA is compatible with existing multi-UAV platforms, provided they support inter-agent ranging, minimal communication for data association, and robust onboard estimation.

6. Core Insights and Fundamental Limitations

• Decentralization and Robustness: SWA exploits mutual relative sensing to build a local, drift-synchronizing frame, making each agent independent of absolute localization. Cohesion and velocity synchronization are preserved even under complete anchor dropout.
• Unobservable Drift: The swarm frame drifts uniformly in translation; this drift is harmless when absolute localization is not needed or when GNSS is restored downstream.
• Scalability: Synchronization performance and drift mitigation improve with swarm size due to bias averaging and increased neighbor redundancy.
• Limitation: SWA cannot eliminate drift; global position grows unbounded unless absolute localization is reacquired. Only internal shape and velocities are regulated (Horyna et al., 22 Aug 2025).

7. Context Within the Literature and Outlook

SWA formalizes and systematizes a rapidly evolving direction in swarm robotics emphasizing infrastructure denial, minimal inter-agent communication, and resilience to sensor faults. Related work includes:

• Ergodic and spectral coverage-based SWA (scale-invariant, decentralized, graceful degradation) (Meyer et al., 2022).
• Communication-less Lloyd-type coverage and flocking SWA with provable safety and convergence under sensor uncertainty (Boldrer et al., 26 Apr 2025).
• Bio-inspired and anticipation-driven swarming schemes yielding condensed, cohesive aggregations without global references (Petracek et al., 2023, Morin et al., 2015).
• SWA in distributed satellite systems for autonomous planetary characterization, using only inter-spacecraft and landmark observations (Dennison et al., 2022).
• Consensus-driven SWA for GNSS-degraded environments, leveraging blockchain-like replicated logs for resilient state management (Dev et al., 1 Aug 2025).

SWA’s principles are being applied to ground, aerial, and space domains, with ongoing research focused on reducing drift through opportunistic external updates, improving robustness to high observation noise, and extending theoretical guarantees to more complex dynamics and environment classes (Horyna et al., 22 Aug 2025).