Position-Based Flocking for Persistent Alignment without Velocity Sensing

1. Problem Statement

Collective motion in robotic swarms (e.g., drones, ground robots) traditionally relies on velocity alignment rules (inspired by Reynolds' Boids), where agents adjust their velocities to match neighbors. However, this approach faces significant challenges in real-world applications:

• Sensing Limitations: Direct velocity measurement is often noisy, unreliable, or unavailable in low-cost robotic systems.
• Estimation Errors: Deriving velocity from position data (differentiation) amplifies sensor noise and introduces latency.
• Communication Overhead: Explicitly sharing velocity vectors increases bandwidth requirements.
• Decay of Alignment: Standard position-based approximations of velocity often rely on a $1/t$ gain factor, which decays to zero over time, causing the flock to lose directional coherence and fail to maintain a persistent heading.

The core problem addressed is how to achieve persistent, coherent collective motion and stable alignment using only relative position data, without direct velocity sensing or communication, while overcoming the decay issue inherent in simple position-difference approximations.

2. Methodology

The authors propose a Position-Based Flocking Model that approximates relative velocity using historical and current position data, augmented by a novel time- and density-dependent alignment gain.

A. Baseline Model

The paper starts with a standard second-order agent model (position $p_i$, velocity $v_i$, control $u_i$) using a velocity-alignment baseline:
$$u_i = \sum_{j \in N_i} \psi(\|p_j - p_i\|)(p_j - p_i) + \sum_{j \in N_i} (v_j - v_i)$$
Where the first term handles cohesion/separation, and the second enforces velocity alignment.

B. Position-Based Approximation

To remove velocity sensing, the authors approximate the relative velocity term $\sum (v_j - v_i)$ using the change in relative position over time:
$$v_j - v_i \approx \frac{(p_j - p_i) - (p_j(0) - p_i(0))}{t}$$
Here, $(p_j(0) - p_i(0))$ represents the initial relative position, acting as a "weak memory imprint." This allows the system to infer motion direction from the displacement vector relative to the start configuration.

C. The Threshold Mechanism (Key Innovation)

The simple approximation above includes a $1/t$ gain, which vanishes as $t \to \infty$, leading to a loss of alignment. To solve this, the authors introduce a piecewise gain function $\phi$:
$$\phi = \begin{cases} \frac{|N_i|}{t} & \text{if } 0 < t \leq \frac{1}{k} \\ k|N_i| & \text{if } t > \frac{1}{k} \end{cases}$$

• Transient Phase ($t \leq 1/k$): The gain decays as $1/t$, allowing rapid initial alignment based on the "imprinting" of the initial configuration.
• Persistent Phase ($t > 1/k$): The gain saturates at a constant non-zero value $k|N_i|$. This ensures that the alignment force does not vanish, maintaining persistent velocity consensus indefinitely.

The final control law becomes:
$$u_i = \sum_{j \in N_i} (\psi(\|p_j - p_i\|) + \phi)(p_j - p_i) - \phi \sum_{j \in N_i} (p_j(0) - p_i(0))$$

3. Key Contributions

1. Velocity-Free Persistent Alignment: A novel control law that achieves long-term directional consensus using only relative position measurements, eliminating the need for velocity sensors or estimators.
2. Non-Vanishing Gain Strategy: The introduction of a threshold mechanism that switches from a transient decaying gain to a constant steady-state gain, solving the problem of alignment decay in position-based models.
3. Bio-Inspired Memory Imprint: The model utilizes the initial relative configuration as a transient "memory" to drive early alignment, mimicking biological mechanisms where initial spatial relationships influence group dynamics before stabilizing into a rigid flock structure.
4. Experimental Validation: Successful implementation and validation on nine real wheeled mobile robots (GRITSBots) in a Robotarium testbed, proving the algorithm's robustness to real-world noise and actuation inaccuracies.

4. Results

The performance was evaluated through simulations (50 agents) and physical experiments (9 robots), comparing the proposed Position-Based with Threshold (P-based (thr.)) against a standard Velocity-Based (V-based) model and a Position-Based without Threshold variant.

• Alignment Persistence:
  • V-based: Achieved strong alignment but exhibited spatial flexibility with varying collective direction over time.
  • P-based (no thr.): Showed strong initial alignment that degraded over time as the $1/t$ gain decayed, leading to a loss of directional coherence.
  • P-based (thr.): Achieved rapid convergence to near-perfect alignment ($\gamma \approx 1$) and sustained this alignment throughout the simulation and experiment.
• Formation Structure:
  • The P-based (thr.) model produced more compact formations with narrower inter-agent separations (1.5–2m vs. 2–3m in V-based).
  • The resulting formations resembled the stable, directional structures seen in migratory bird flocks (e.g., V-shapes), whereas the V-based model resembled flexible foraging groups.
• Robustness: The real-robot experiments confirmed that the model functions effectively despite sensor noise, communication constraints, and actuation errors, maintaining coherent flocking behavior for 120 seconds.

5. Significance

This work bridges a critical gap between theoretical swarm robotics and practical deployment:

• Hardware Agnosticism: It enables the deployment of swarm algorithms on low-cost robots lacking expensive velocity sensors (e.g., LiDAR-only or UWB-based systems).
• Communication Efficiency: By relying solely on position data (which is often already required for collision avoidance), it reduces the bandwidth needed for coordination compared to exchanging velocity vectors.
• Biological Plausibility: The mechanism offers a plausible explanation for how biological swarms (birds, fish) maintain coherent, persistent motion without explicit velocity matching, relying instead on spatial relationships and anticipatory cues.
• Scalability: The approach scales well to larger teams and is robust to network topology changes, provided the network remains connected.

In conclusion, the paper demonstrates that persistent collective alignment is achievable without velocity sensing by leveraging a time-dependent gain with a non-zero threshold, offering a robust solution for real-world robotic swarms operating under sensing and communication constraints.