Behavioral-based circular formation control for robot swarms

TL;DR

This paper aims to deploy a dense robot swarm that mimics the behavior of tornado schooling fish, and employs a combination of a scalable overtaking rule, a guiding vector field, and a control barrier function with an adaptive radius to facilitate smooth overtakes.

Abstract

This paper focuses on coordinating a robot swarm orbiting a convex path without collisions among the individuals. The individual robots lack braking capabilities and can only adjust their courses while maintaining their constant but different speeds. Instead of controlling the spatial relations between the robots, our formation control algorithm aims to deploy a dense robot swarm that mimics the behavior of tornado schooling fish. To achieve this objective safely, we employ a combination of a scalable overtaking rule, a guiding vector field, and a control barrier function with an adaptive radius to facilitate smooth overtakes. The decision-making process of the robots is distributed, relying only on local information. Practical applications include defensive structures or escorting missions with the added resiliency of a swarm without a centralized command. We provide a rigorous analysis of the proposed strategy and validate its effectiveness through numerical simulations involving a high density of unicycles.

Figures (6)

• Figure 1: For two non-collaborative robots, denoted as robot $i$ (blue) and robot $j$ (green), the overtaking rule in Definition \ref{['def: over']} dictates that the robot $i$ must keep $v_{ij}$ outside the collision cone generated by the virtual radius $\rho$ and $p_{ij}$. Here, it is important to clarify that $r\in\mathbb{R}$ represents the actual collision radius and $\rho$ assists with a smoother overtaking.
• Figure 2: O.R. stands for the overtaking rule in Definition \ref{['def: over']}. When the overtaking begins we have $\hat{v}_j^T E \hat{v}_i \geq 0$ as an initial condition.
• Figure 3: Illustration of three robots satisfying Assumption \ref{['asmp: pij_vi']}. The blue semicircle indicates all the admissible directions for $v_i$ regarding $p_{ij}$.
• Figure 4: A pair of robots $(i,j)$ follows an elliptical path, where $s_i > s_j$ with the Definition \ref{['def: over']} as the overtaking rule. Since $\rho,\kappa$ are designed according to Lemma \ref{['lemm: rho_kappa']}, and this scenario fulfills the pre-overtaking conditions of Lemma \ref{['lemm: over_man']} and the Assumption \ref{['asmp: pij_vi']}, the simulation validates the prediction of having $L_gh^i(q_{ij})>0$ during the overtaking (blue colored area). Note that the stage changes at $t=17.7$ and $L_gh^i(q_{ij})$ reach their minimum value at $t=21.2$.
• Figure 5: A mother ship travels vertically up at $X = -80$ and launches during $10$ T units a swarm of $50$ unicycles to orbit around an ellipse with different speeds of around $5$ L units per T unit. A robot is launched once the previous one is at $2.5$ times the safety distance $r$. The arrows show the direction of the GVF (\ref{['eq: gvf']}). Each robot $i$ is provided with the $\omega_j$ of all $j\in\mathcal{N}_i$, and $\mathcal{N}_i$ is generated by design before the simulation starts. Clusters of robots form for some periods when the speed of the robots are similar. On the right side: a) the distances between all the robots and the safety distance (black dashed line), during $t\in[0,10]$ many robots are together in the mother ship explaining the $0$ distance in the plot; b)$L_gh^i$ when $\Psi <0$, which remains positive; c) the proper design of $\kappa$ and $\rho$ as in Remark \ref{['rem: rhokappa']} promote smooth overtaking with non-aggressive $\omega$'s; d) once the last robot is launched, the cases when $\hat{p}_{ij}^\top E \hat{v}_{ij}\leq0$ are isolated and do not compromise the overtaking, i.e., the Assumption \ref{['asmp: pij_vi']} is conservative for this simulation.

Theorems & Definitions (15)

• Definition 1: Safety cbf_theory
• Definition 2: Valid CBF cbf_theory
• Definition 3: Collision between two robots
• Definition 4: Overtaking
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Remark 2: On the design of $\rho$ and $\kappa$