QP-Based Angle-Aware Coverage Controller

• The paper introduces a QP-based controller that jointly optimizes drone positions and camera orientations to maximize multi-view coverage for improved 3D reconstruction.
• It employs control barrier functions and feasible-set reshaping to ensure safety and robustness against actuation uncertainties in dynamic multi-agent environments.
• Real-time implementation is achieved with JIT compilation and GPU acceleration, demonstrating substantial improvements in coverage efficiency and reconstruction quality.

Quadratic Programming (QP)-Based Angle-Aware Coverage Controller

A QP-based angle-aware coverage controller is a numerical optimization framework designed for multi-agent systems, such as drone networks, with the joint objective of achieving region coverage and capturing images from diverse camera orientations. Such controllers extend classical coverage strategies by integrating camera attitude (e.g., gimbal angles, yaw/pitch) as explicit control variables, and employ quadratic programming to solve, at each control step, for agent velocities and camera rotational rates that maximize coverage measures while adhering to safety and operational constraints. The frameworks are distinguished by their mathematical formulation, use of control barrier functions (CBFs) for safety, and computational tools enabling real-time implementation under high-dimensional and nonlinear constraints.

1. Mathematical Formulation and State Space Extension

The distinguishing innovation in angle-aware coverage control is the extension of conventional state space to include camera orientation, resulting in a joint agent configuration vector

$p_i = [x_i, y_i, z_i, \theta_i^h, \theta_i^v]^T$

where $(x_i, y_i, z_i)$ denotes the translational position, and $(\theta_i^h, \theta_i^v)$ specifies the horizontal (yaw) and vertical (pitch) attitude of the camera or gimbal (Lu et al., 22 Apr 2024, Hanif et al., 21 Oct 2025). This formalism enables the explicit control not only of the agent’s motion but also of the direction from which observations are collected, establishing the capacity for multi-view coverage essential for applications such as 3D reconstruction.

A key performance function $h(p_i, q_j)$ is defined on discretized observation points $q_j$, quantifying the quality of coverage contributed by agent $i$. For coverage tasks over a spatial-angular grid, $h$ typically combines terms encoding (i) positional inclusion within the field of view, given as a function of the cosine of the angle between the camera axis and the target, and (ii) mismatch between desired and actual viewing orientation, both embedded in a Gaussian form: $$h(p_i, q_j) = \exp\left(-\frac{h_1(p_i, q_j)}{2\sigma_1^2}\right) \exp\left(-\frac{h_2(p_i, q_j)}{2\sigma_2^2}\right)$$ where $h_1$ quantifies angular proximity to the field of view and $h_2$ evaluates the orientation match.

2. Quadratic Programming-Based Controller with Barrier Function Constraints

At each step, the controller solves a quadratic optimization problem to determine control inputs (velocity and angular rate) that:

• Minimize an objective function reflecting control effort and slack variables,
• Ensure sufficient progress toward coverage goals,
• Satisfy safety and hardware constraints as affine inequalities.

The canonical QP is structured as: $\min_{u_i, w_i} \; \epsilon \|u_i\|^2 + |w_i|^2$ subject to barrier-function-derived constraints, for agent $i$:

• Sampling performance constraint:

$\xi_1^T u_i + \xi_2 \geq w_i$

enforcing a required minimum decay rate of the global coverage objective $J$ (where $J$ is a sum over importance indices $\varphi_j$ or $\psi_j$ for each grid cell),

• Gimbal pitch constraint:

$\chi_1^T u_i + \chi_2 \geq 0$

maintaining camera angles within mechanical limits,

• Collision avoidance constraint:

$\rho_1^T u_i + \rho_2 \geq 0$

ensuring minimum separation between agents (Hanif et al., 21 Oct 2025).

Barrier function candidates are expressed, for example, as: $b_{i,I} = I_i - |\mathcal{V}_i(p)| \cdot \gamma$ where $$I_i = \sum_{j \in \mathcal{V}_i(p)} \delta_1 h_1(p_i, q_j) \varphi_j$$ quantifies the contribution of agent $i$, and $\gamma$ is the prescribed decay rate.

3. Feasible-Set Reshaping and Robustness Guarantees

Classical QP-based controllers for safety-critical systems may fail under actuation uncertainty, causing infeasibility or non-smooth control solutions due to imperfect velocity tracking (Wu et al., 2022). A refined QP framework introduces:

• Relaxation parameter $\delta_i$: guarantees QP feasibility even with disturbances (since $[v; \delta]=0$ is always admissible).
• Feasible-set reshaping via a positive basis $A_i^r$: ensures that the QP solution mapping is locally Lipschitz in the data, preventing chattering and maintaining robustness.

The reshaped feasible set for agent $i$ is defined as: $$\mathcal{P}_i^r(p) = \{ [v; \delta] \in \mathbb{R}^{n+1} : A_i^r v + a_i^r(p) \delta \leq 0, \; \delta \in [0, \delta_{\max}] \}$$ with $A_i^r$ constructed so that any $n$ rows are linearly independent, and the constraint vector $a_i^r(p)$ is derived from original safety constraints through a map $\varphi$, involving design parameters.

This methodology absorbs uncertainties arising from actuation dynamics and maintains barrier-function invariance, critical for collision avoidance in high-dimensional angle-aware systems.

4. Online Map Feedback and Adaptive Objective Index

In adaptive multi-agent imaging frameworks such as Coverage-Recon (Hanif et al., 21 Oct 2025), the importance index $\varphi_j$ for each observation point is updated in real time based on both the agent’s sensing performance and changes in the reconstructed 3D map:

• As agents capture images, $\varphi_j$ decays due to high coverage.
• Feedback from mesh changes (computed via NeuralRecon’s triangle mesh outputs and either grid-based vertex counts or M3C2 distances) detects regions of evolving uncertainty.
• When global coverage reaches a preset threshold ($J < J_{th}$), significant local mesh changes trigger a jump in $\varphi_j$ for those regions:

$$\varphi_j(t^+) = \varphi_j(t^-) + \delta_2 h_2(q_j, t)$$

where $h_2$ quantifies local mesh change. This mechanism dynamically re-prioritizes regions requiring further multi-view sampling, ensuring iterative refinement of reconstruction.

5. Computational Implementation: Real-Time Feasibility

The computational demand of solving high-dimensional QPs, incorporating both position and orientation variables for multiple drones over finely discretized coverage fields (e.g., $15$ million cells), necessitates advanced computational strategies (Lu et al., 22 Apr 2024). Key technologies include:

• JAX for automatic differentiation and array programming, permitting the fast computation of gradients for performance functions and barrier terms,
• Just-in-Time (JIT) compilation to reduce per-step execution time (from $>1$ s to $\sim100$ ms on CPU, $\sim22$ ms with GPU),
• GPU Acceleration for matrix operations, crucial for scaling to larger systems or higher coverage resolution.

These tools enable real-time deployment in environments such as ROS (Robot Operating System) and facilitate practical experimental evaluations with multiple agents.

6. Empirical Verification and Performance Analysis

Simulation and field experiments employing QP-based angle-aware coverage controllers demonstrate:

• Drones initially positioned with downward-pointing cameras gradually adjust both position and orientation to systematically cover previously unobserved regions and perspectives (Lu et al., 22 Apr 2024).
• The prescriptive QP controller yields near-linear decay in the global coverage objective, achieving higher rates of coverage than traditional methods excluding camera attitude optimization.
• Coverage-Recon demonstrates superior 3D reconstruction quality—quantified by higher F-Score against ground-truth mesh—when integrating online map feedback compared to fixed or sensor-only objective selection (Hanif et al., 21 Oct 2025). The computational overhead of mesh-change evaluation is negligible in practice, especially when employing the efficient grid-based method.

The approach reliably enforces safety constraints (such as inter-agent separation and camera pitch limits) via embedded barrier functions, with no documented violations throughout extensive simulation durations and experimental flights.

7. Significance, Applicability, and Future Considerations

The QP-based angle-aware coverage controller constitutes a robust optimization framework for multi-agent systems tasked with observation-intensive missions. Key advances include the explicit extension of coverage state space to orientation variables, provably safe control via barrier functions and feasible-set reshaping, and the online adaptation of objectives using map-feedback from neural reconstruction algorithms. These developments directly impact real-time 3D mapping, collaborative inspection, and persistent monitoring across domains requiring spatial and angular coverage.

A plausible implication is the scalability of these controllers to persistent monitoring with dynamically evolving objectives and hardware-constrained sensing platforms. Directions for future research include hardware experimentation to further validate simulation results and the refinement of feedback mechanisms to accommodate streaming multi-agent data in complex environmental settings.