Formation Control for CRLB-Optimal Cooperative Sensing in Low-Altitude Wireless Networks

1. Problem Statement

The paper addresses the challenge of cooperative target sensing using Uncrewed Aerial Vehicles (UAVs) in Low-Altitude Wireless Networks (LAWNs). While UAV swarms offer superior sensing accuracy compared to single platforms due to spatial diversity, their performance is heavily dependent on their spatial configuration (formation geometry).

Existing research often focuses on static sensor placement or individual trajectory optimization but fails to address:

• How multiple UAVs should cooperatively form and maintain a specific geometric configuration dynamically.
• How to achieve this optimal geometry from arbitrary initial deployments while adhering to physical constraints like obstacle avoidance and coordinated motion.

The core objective is to design a formation that minimizes the Cramér–Rao Lower Bound (CRLB) of the target location estimation error and to develop a distributed control strategy to steer UAVs into this formation safely.

2. Methodology

A. System Modeling

• Scenario: A formation of $M \geq 3$ UAVs cooperatively senses a ground target at location $s$. UAVs fly at a fixed altitude $H$.
• Measurement Model: UAVs measure round-trip delay converted to range ($d_m$). Measurements are subject to Additive White Gaussian Noise (AWGN).
• Noise Characteristics: The noise variance $\sigma_m^2$ is inversely proportional to the Signal-to-Noise Ratio (SNR), which follows a quartic distance loss model ($d_m^{-4}$) typical for two-way sensing channels.
• Dynamics: UAV motion is modeled as a first-order Gaussian Markov process, combining global translational movement (formation center) and local adjustments (formation shape).

B. CRLB-Optimal Formation Geometry Derivation

The authors derive the optimal geometry by minimizing the trace of the inverse Fisher Information Matrix (FIM), utilizing the A-optimality criterion (minimizing average estimation variance).

1. FIM Construction: The FIM for the target coordinates is derived from range measurements.
2. Optimization Condition: The trace of the inverse FIM is minimized if and only if the FIM is a scalar matrix (isotropic). This requires:
   • $J_{xx} = J_{yy}$ (Equal information in x and y axes).
   • $J_{xy} = 0$ (No correlation between axes).
3. Optimal Geometry Results:
   • Azimuth: To satisfy isotropy in the horizontal plane, UAVs must be arranged in a regular $M$-polygon centered on the target's vertical projection.
   • Elevation: The optimal elevation angle $\phi^*$ is determined by a trade-off between path loss (favoring lower angles) and geometric diversity (favoring higher angles).
     • At low altitudes, $\phi^* \approx 54.74^\circ$ (maximizing projected SNR).
     • At high altitudes, $\phi^* \to 45^\circ$ (prioritizing geometric robustness).
   • Final Shape: The optimal formation $F^*$ is an inverted cone with the target at the vertex, where UAVs are uniformly distributed on a horizontal circle at altitude $H$.

C. Distributed Formation Control Strategy

To steer UAVs from arbitrary positions to $F^*$ while avoiding obstacles, a distributed control law is proposed:

1. Consensus Mechanism: A leader-follower architecture ensures all UAVs align to a global velocity $v^*$ using only local neighbor information.
2. Shape Control: A decentralized cost function penalizes deviations from desired relative displacements ($\delta_{pm}$) and velocity misalignment. The control input is derived via a negative gradient descent on this cost.
3. Obstacle Avoidance:
   • A repulsive potential field is integrated to generate a repulsion vector when UAVs approach obstacles within a safety radius.
   • Adaptive Scaling: A global scaling factor $\eta_k$ dynamically shrinks the formation when passing through narrow gaps (bottlenecks) to prevent collisions, then expands back to the optimal shape once the obstacle is cleared.

3. Key Contributions

1. Analytical Characterization of Optimal Geometry: The paper proves that the CRLB-optimal formation for cooperative sensing is a regular polygon with a specific elevation angle determined by the path-loss vs. geometric diversity trade-off.
2. Distributed Control Framework: Development of a control strategy that achieves the optimal formation from any initial deployment using only local neighbor information, without requiring a central controller.
3. Safety and Adaptability: Integration of obstacle avoidance and formation scaling mechanisms, allowing the swarm to deform temporarily to navigate constraints and recover the optimal sensing geometry afterward.
4. Performance Validation: Demonstration that the proposed scheme consistently outperforms benchmark formations (e.g., fixed angles, random azimuths) in terms of CRLB and convergence speed.

4. Numerical Results

Simulations were conducted with $M=6$ UAVs targeting a ground point at $s=[80, 90]^T$ with a flight altitude of $H=20$m.

• CRLB Performance: The proposed "CRLB-optimal formation" consistently achieved the lowest CRLB across all altitudes compared to benchmarks (fixed elevation/random azimuth, fixed elevation/regular polygon). The performance gap widened at higher altitudes where path loss and geometric sensitivity increased.
• Formation Evolution: In obstacle-rich environments, the UAVs successfully:
  1. Coordinated to form the initial structure.
  2. Adaptively contracted (scaled down) to pass through narrow gaps.
  3. Expanded and reconfigured back to the regular polygon shape after bypassing obstacles.
• Convergence: The CRLB decreased rapidly as the formation approached the optimal geometry, stabilizing near the theoretical lower bound. The control cost increased temporarily during obstacle avoidance (due to deformation) but converged as the formation stabilized.

5. Significance

This work bridges the gap between theoretical sensing limits and practical swarm control.

• Theoretical Impact: It provides a rigorous geometric characterization of how UAVs should be arranged to maximize sensing accuracy under realistic path-loss models, moving beyond idealized static placement.
• Practical Impact: The proposed distributed control algorithm is highly relevant for real-world applications (surveillance, emergency response) where UAVs must operate in dynamic, cluttered environments. It ensures that sensing performance is not compromised by the need to navigate obstacles, offering a robust solution for autonomous cooperative sensing in Low-Altitude Wireless Networks.