Resilient source seeking with robot swarms

TL;DR

The paper tackles source-seeking for an unknown scalar field using a robot swarm by introducing an ascending-direction controller that guides the swarm centroid to the source $p_\sigma$ based on measurements and deployment geometry rather than individual gradient estimates. It develops the theoretical foundations for an ascending direction $L_\sigma(p_c,x)$, provides Taylor-based approximations $L_\sigma^1(p_c,x)$ that are guaranteed to point uphill under non-degenerate deployments, and derives conditions and symmetry-based designs (discrete, regular polygons, and continuous densities) that keep the direction aligned with the gradient. A Lyapunov-based convergence proof for single-integrator dynamics shows centroid convergence to the source within a prescribed region, and simulations with hundreds of robots demonstrate resilience to actuator noise, obstacle morphing, and agent failures. The results suggest a robust, formation-flexible framework for persistent environmental sensing and search tasks, with future work focusing on distributed computation and more constrained robot dynamics such as unicycles.

Abstract

We present a solution for locating the source, or maximum, of an unknown scalar field using a swarm of mobile robots. Unlike relying on the traditional gradient information, the swarm determines an ascending direction to approach the source with arbitrary precision. The ascending direction is calculated from measurements of the field strength at the robot locations and their relative positions concerning the centroid. Rather than focusing on individual robots, we focus the analysis on the density of robots per unit area to guarantee a more resilient swarm, i.e., the functionality remains even if individuals go missing or are misplaced during the mission. We reinforce the robustness of the algorithm by providing sufficient conditions for the swarm shape so that the ascending direction is almost parallel to the gradient. The swarm can respond to an unexpected environment by morphing its shape and exploiting the existence of multiple ascending directions. Finally, we validate our approach numerically with hundreds of robots. The fact that a large number of robots always calculate an ascending direction compensates for the loss of individuals and mitigates issues arising from the actuator and sensor noises.

Figures (6)

• Figure 1: Deployment $x$ of a robot swarm with centroid at $p_c$.
• Figure 2: The 4-robot rectangular deployment $x^\text{4rct}$ are at the corners of the rectangle in red color. The $250$ robots in blue color belong to the deployment $x^\text{rct}$ and they are spread uniformly within the rectangle. The gradient $\nabla\sigma(p_c)$ is arbitrary and forms an angle $\theta$ with the horizontal axis of the deployment.
• Figure 3: Illustration of the symmetries S{1,3} for the surface $\mathcal{A}$ on the left and right respectively, to design $L^1_\sigma(p_c, x)$ parallel to the gradient $\nabla\sigma(p_c)$.
• Figure 4: On the left, a swarm of $1000$ robots satisfying the symmetries $S\{0,1,2,3\}$ from Proposition \ref{['pro: U']}. For the three figures, the black arrows are the gradient $\nabla\sigma$, the green arrows are $L^1_\sigma$, and the red arrows are the direction $L_\sigma$ computed by the robots. Arrows are normalized for representation purposes as we focus only on their direction. As we get far from the continuum, with $500$ and $100$ robots on the middle and right figures, there is no guarantee that $L^1_\sigma$ is parallel to the gradient $\nabla\sigma$. However, in this example, $L_\sigma$ has almost the same direction as $L_\sigma^1$ despite a big$D$ concerning the level curves.
• Figure 5: On the left, a swarm of $1000$ robots satisfying only the symmetries $S\{0,1\}$ from Proposition \ref{['pro: U']}. The arrows have the same meaning as in Figure \ref{['fig: prop5']}. The modification of the variances of $\rho(X,Y)$, e.g., by stretching the swarm shape, stretches $L^1_\sigma$ and $L_\sigma$ accordingly. In this example, the direction of $L_\sigma$ almost does not diverge from $L_\sigma^1$ despite a big$D$ concerning the level curves.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Proposition 1