Vision-Based Multirotor Control for Spherical Target Tracking: A Bearing-Angle Approach

TL;DR

$The paper addresses tracking a moving spherical target with unknown radius $r$ using a monocular camera by transforming bearing measurements into a bearing-angle pair and introducing the variable $x=\sin(\theta)$ to derive polar-like system dynamics.$ The proposed approach combines an IBVS-inspired nonlinear adaptive controller with backstepping and Lyapunov-based stability, including adaptive estimates for the target radius $\hat{r}$ and a scaled acceleration $\hat{\boldsymbol{\rho}}$, achieving asymptotic convergence of bearing error $\boldsymbol{\delta}_1$, distance error $\delta_2$, and relative velocity error $\boldsymbol{\delta}_3$ under Barbalat's lemma.$ The method explicitly handles camera field-of-view constraints through a Rodrigues-rotation-based attitude strategy to ensure target visibility, and simulations demonstrate robust performance under measurement noise and acceleration of the target.$

Abstract

This work addresses the problem of designing a visual servo controller for a multirotor vehicle, with the end goal of tracking a moving spherical target with unknown radius. To address this problem, we first transform two bearing measurements provided by a camera sensor into a bearing-angle pair. We then use this information to derive the system's dynamics in a new set of coordinates, where the angle measurement is used to quantify a relative distance to the target. Building on this system representation, we design an adaptive nonlinear control algorithm that takes advantage of the properties of the new system geometry and assumes that the target follows a constant acceleration model. Simulation results illustrate the performance of the proposed control algorithm.

Figures (5)

• Figure 1: Top-down view of a 3-dimensional sphere target observed by the tracker vehicle.
• Figure 2: Example level sets of the squared norm of the tracking error, parameterized by Cartesian coordinates (left), versus polar coordinates (right), for observing a static target with radius $r=1\,m$ located at the origin, from a relative reference position $\mathbf{p}^{\ast}=[-4.8, 0,0]^{\top}$.
• Figure 3: Illustration of the visibility model, where the volume in light blue encodes the field of view of the camera sensor.
• Figure 4: Simulation of the target-tracker system with a constant bearing-angle reference. The vehicle and tracker positions are depicted in blue and green, respectively. Each circle represents a time step, with the attitude of the vehicle body axis represented using red, green and blue vectors, corresponding to $\mathbf{x}_{\mathrm{B}}$, $\mathbf{y}_{\mathrm{B}}$ and $\mathbf{z}_{\mathrm{B}}$, respectively.
• Figure 5: Evolution of the bearing tracking error norm $\left\|{\boldsymbol{\delta}}_1\right\|$, the relative distance error $\delta_2$, relative velocity error ${\boldsymbol{\delta}}_3$, desired relative velocity $\mathbf{w}^{d}$ and the adaptive estimators $\hat{{\boldsymbol{\rho}}}$ and $\hat{\mathbf{r}}$.

Theorems & Definitions (3)

• Theorem 1
• proof
• Remark 1