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A Control Barrier Function Candidate for Quadrotors with Limited Field of View

Biagio Trimarchi, Fabrizio Schiano, Roberto Tron

TL;DR

This work addresses vision-based control under limited field of view and unreliable depth by formulating bearing-FOV constraints as Control Barrier Functions (CBFs). A key novelty is splitting the standard CBF constraint into two separate constraints with tunable coefficients, enabling robustness to bounded distance estimation errors captured by the ratio $\\tilde{d} = d/\\hat{d}$ and bounded within $[d_m,d_M]$. The authors develop first- and second-order CBF formulations, including a velocity-control and a acceleration-control scheme, and extend to multiple features; they prove forward invariance of the safe set and validate the approach with numerical simulations on both a double integrator and a quadrotor, demonstrating that the features remain in the camera FOV while tracking the reference trajectory. The method offers a practical pathway to robust, vision-based navigation and tracking in UAVs, with potential extensions to moving features and integration with CBF-CLF control frameworks for real-world deployment.

Abstract

The problem of control based on vision measurements (bearings) has been amply studied in the literature; however, the problem of addressing the limits of the field of view of physical sensors has received relatively less attention (especially for agents with non-trivial dynamics). The technical challenge is that, as in most vision-based control approaches, a standard approach to the problem requires knowing the distance between cameras and observed features in the scene, which is not directly available. Instead, we present a solution based on a Control Barrier Function (CBF) approach that uses a splitting of the original differential constraint to effectively remove the dependence on the unknown measurement error. Compared to the current literature, our approach gives strong robustness guarantees against bounded distance estimation errors. We showcase the proposed solution with the numerical simulations of a double integrator and a quadrotor tracking a trajectory while keeping the corners of a rectangular gate in the camera field of view.

A Control Barrier Function Candidate for Quadrotors with Limited Field of View

TL;DR

This work addresses vision-based control under limited field of view and unreliable depth by formulating bearing-FOV constraints as Control Barrier Functions (CBFs). A key novelty is splitting the standard CBF constraint into two separate constraints with tunable coefficients, enabling robustness to bounded distance estimation errors captured by the ratio and bounded within . The authors develop first- and second-order CBF formulations, including a velocity-control and a acceleration-control scheme, and extend to multiple features; they prove forward invariance of the safe set and validate the approach with numerical simulations on both a double integrator and a quadrotor, demonstrating that the features remain in the camera FOV while tracking the reference trajectory. The method offers a practical pathway to robust, vision-based navigation and tracking in UAVs, with potential extensions to moving features and integration with CBF-CLF control frameworks for real-world deployment.

Abstract

The problem of control based on vision measurements (bearings) has been amply studied in the literature; however, the problem of addressing the limits of the field of view of physical sensors has received relatively less attention (especially for agents with non-trivial dynamics). The technical challenge is that, as in most vision-based control approaches, a standard approach to the problem requires knowing the distance between cameras and observed features in the scene, which is not directly available. Instead, we present a solution based on a Control Barrier Function (CBF) approach that uses a splitting of the original differential constraint to effectively remove the dependence on the unknown measurement error. Compared to the current literature, our approach gives strong robustness guarantees against bounded distance estimation errors. We showcase the proposed solution with the numerical simulations of a double integrator and a quadrotor tracking a trajectory while keeping the corners of a rectangular gate in the camera field of view.
Paper Structure (16 sections, 7 theorems, 55 equations, 5 figures)

This paper contains 16 sections, 7 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Preliminaries
  5. Control Barrier Functions
  6. Control Barrier Function Constraint for Bearing Measurements
  7. Agent kinematics
  8. Sensing model
  9. Control task
  10. Candidate CBF function
  11. Velocity Control
  12. Acceleration Control
  13. Multiple Features
  14. Numerical Simulation
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 1

Consider $x \in \mathbb{R}^{}{}$ and the first order differential equation where $\alpha: \mathbb{R}^{}{} \times \mathbb{R}^{}{} \to \mathbb{R}^{}{}$ is continuously differentiable, locally Lipschitz in both arguments and, for each $t \in \mathbb{R}^{}{}$, $\alpha(t, x)$ is a class $\mathcal{K}$ function of $x$. Then $x = 0$ is a globally uniformly asymptotically stable eq

Figures (5)

  • Figure 1: A 2-D visual task problem: a quadrotor traversing a race circuit needs to keep the gates (depicted in red) as long as possible in its field of view (the blue cone) to orient itself and race through the circuit.
  • Figure 2: The figure shows the relation between the allowable error ration $\tilde{d}$ and the resulting range for $c_2$ in Theorem \ref{['th:split_second_order']}. As we can see, a trade-off exists between the bounds on $\tilde{d}$ and the resulting range for $c_2$.
  • Figure 3: Snapshot of the simulation scenario of the numerical experiments. We can see that the agent (represented by a blue cross) needs to turn to keep all the features (red points) in its field of view (blue cone) to follow the reference trajectory (black line). (a) Front view. (b) Side view (c) Top-down view. (d) Tilted view.
  • Figure 4: Acceleration Control: Plot (a) shows the position tracking error of a rigid body actuated both in linear and angular acceleration along the prescribed path. Plot (b) shows the minimum value among the barrier functions associated with the features.
  • Figure 5: Quadrotor: Plot (a) shows the position tracking error of a quadrotor actuated both in trust and torques along the prescribed path. Plot (b) shows the minimum value among the barrier functions associated with the features.

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5: HOCBF xiao2022HighOrder
  • Theorem 1: xiao2022HighOrder
  • ...and 12 more