Collisionless and Decentralized Formation Control for Strings

TL;DR

The paper addresses collisionless, decentralized formation control for a 1D chain of $N$ agents with singular nearest-neighbor interactions. It develops a Cucker-Smale–inspired model with decentralized feedback, defines a total energy $E(x,v)$ and a dissipation mechanism, and proves global existence, velocity flocking under a $\phi$-integral condition, and exponential pattern formation via a modified energy $E_\gamma$, all validated by numerical experiments. Key contributions include precise, verifiable conditions guaranteeing collision avoidance, consensus in velocity, and convergence to prescribed inter-agent spacings, plus a finite-time blow-up example for $N=2$ when $\alpha<1$, and comprehensive simulations that illuminate parameter sensitivity. The results advance safe, scalable platooning by providing analytical guarantees that rely on local interactions and decentralized control rather than long-range communication.

Abstract

A decentralized feedback controller for multi-agent systems, inspired by vehicle platooning, is proposed. The closed loop resulting from the decentralized control action has three distinctive features: the generation of collision-free trajectories, flocking of the system towards a consensus state in velocity, and asymptotic convergence to a prescribed pattern of distances between agents. For each feature, a rigorous dynamical analysis is provided, yielding a characterization of the set of parameters and initial configurations where collision avoidance, flocking, and pattern formation are guaranteed. Numerical tests assess the theoretical results presented.

Figures (6)

• Figure 1: Diagram at a particular instant of three consecutive agents for the considered MAS. The singular interactions, providing barriers to the agents, are indicated with the radii $\delta_i$ of the semicircles. Note that with this nomenclature, in steady state, the agents will not be considered to have collided whenever $|x_{i-1}-x_i|-\delta_{i-1}>0$. Also note that the first and last agent only have a barrier on one of their sides
• Figure 2: Positions over time of 5 particles on the line. Left: case 1) agents initially at rest and located at non-collided positions. Right: case 2) agents with non-zero initial velocities and located at close proximity but not collided. Thin lines represent the volumes of the agents. No collisions occur due to the singular interactions and the desired formation is acquired in steady state. It can also be noted that the velocities seem to converge exponentially fast. Note that the volumes seem to change in time but this is only due to the chosen visualization style.
• Figure 3: 5 particles on the line with non-zero average velocity. Left-Top: Energy decomposition; Left-Bottom: Dissipation; Right-Top: Minimal distance between agents; Right-Bottom: Positions over time of the particles achieving a consensus speed and the desired spatial formation as the system evolves. Note that the energies satisfy statement (ii) in Lemma \ref{['lem_energy']}.
• Figure 4: 5 particles on the line with zero average velocity. Left-Top: Energy decomposition; Left-Bottom: Dissipation; Right-Top: Minimal distance between agents; Right-Bottom: Positions over time of the particles achieving a consensus speed of zero and the desired spatial formation as the system evolves. Note that the initial configuration violates \ref{['eq:condT2']}.
• Figure 5: 10 particles on the line with zero average velocity when $\beta=4.1$. Left-Top: Positions over time of the particles when the control is used; Left-Bottom: Energy decomposition and flocking condition for $\beta>1$; Right-Top: Errors from the desired formation $x_i-x_{i+1}-z_i$; Right-Bottom: Positions over time of the particles when the control is not used. Flocking does not occur, although collisions are still avoided.

Theorems & Definitions (17)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Theorem 3
• proof
• Theorem 4