Timer-Based Coverage Control for Mobile Sensors

TL;DR

This work investigates the coverage control problem over a static, compact, and convex workspace and develops a hybrid extension of the continuous-time Lloyd algorithm, which is posed as a set attractivity problem for hybrid systems.

Abstract

This work investigates the coverage control problem over a static, compact, and convex workspace and develops a hybrid extension of the continuous-time Lloyd algorithm. Each agent in a multi-agent system (MAS) is equipped with a timer mechanism that generates intermittent measurement and control update events, which may occur asynchronously between agents. Between consecutive event times, as determined by the corresponding timer mechanism, the controller of each agent is held constant. These controllers are shown to drive the configuration of the MAS into a neighborhood of the set of centroidal Voronoi configurations, i.e., the minimizers of the standard locational cost. The combination of continuous-time dynamics with intermittently updated control inputs is modeled as a hybrid system. The coverage objective is posed as a set attractivity problem for hybrid systems, where an invariance-based convergence analysis yields sufficient conditions that ensure maximal solutions of the hybrid system asymptotically converge to a desired set. A brief simulation example is included to showcase the result.

Figures (10)

• Figure 1: (Top Left) Illustration of the heptagonal workspace $\mathcal{D}$ overlaid with the initial MAS configuration (magenta dots) and the corresponding Voronoi tessellation of $\mathcal{D}$ using the initial MAS configuration as a generator. (Bottom Left) Illustration of the heptagonal workspace $\mathcal{D}$ overlaid with the final MAS configuration (magenta dots) and the corresponding Voronoi tessellation of $\mathcal{D}$ using the final MAS configuration as a generator. (Top Right) Depiction of the workspace $\mathcal{D}$, the heat map resulting from the density function $\varphi(s)$, and the initial MAS configuration. The most important points in the workspace are those closest to the peak of the Gaussian density, colored in yellow. (Bottom Right) Depiction of the workspace $\mathcal{D}$, the heat map resulting from the density function $\varphi(s)$, and the final MAS configuration.
• Figure 2: Image of the workspace $\mathcal{D}$, the initial MAS configuration in magenta dots, the final MAS configuration in green dots, the Voronoi tessellation of $\mathcal{D}$ using the final MAS configuration as a generator, and the trajectories of all agent positions in black curves.
• Figure 3: Depiction of the point-wise Euclidean norm of the centroid tracking error $e_p$, defined in \ref{['eqn: Tracking error']}, for each agent $p\in\mathcal{V}$ versus time. The horizontal and vertical axes are in linear and logarithmic scales, respectively. Since the magnitudes of the centroid tracking errors are bounded by $0.7$ for all time $t\geq 60$, the MAS achieves $\nu$-approximate coverage with the desired $\nu$ of $0.7$. Moreover, the timer-based coverage controller provided better error regulation than afforded by the theory, speaking to the conservatism of the result.
• Figure 4: Portrayal of the point-wise Euclidean norm of the error $\tilde{\eta}_p$, defined in \ref{['eqn: eta_p tilde']}, for each agent $p\in\mathcal{V}$ versus time. The horizontal blue line represents the equation $y=\tilde{\eta}_{\max}$. Under the selected simulation parameters, one can see that $\Vert \textcolor{black}{\tilde{\eta}_p} \Vert \leq \tilde{\eta}_{\max}$ for all time $t\in[0,150]$, as guaranteed by the maximum dwell-time condition.
• Figure 5: Illustration of the locational cost in \ref{['eqn: Locational Cost']} versus time, where the horizontal and vertical axes are in linear and logarithmic scales, respectively. The steady state value of the locational cost is about $201$.

Theorems & Definitions (8)

• Definition 1
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof