Energy-efficient flocking with nonlinear navigational feedback

TL;DR

A generalization of the multi-agent model from Olfati-Saber, R with nonlinear navigational feedback forces with nonlinear navigational feedback forces is presented, showing existence of a broad class of nonlinear control forces for which the attractor does not contain periodic trajectories.

Abstract

Modeling collective motion in multi-agent systems has gained significant attention. Of particular interest are sufficient conditions for flocking dynamics. We present a generalization of the multi-agent model of Olfati--Saber with nonlinear navigational feedback forces. Unlike the original model, ours is not generally dissipative and lacks an obvious Lyapunov function. We address this by proposing a method to prove the existence of an attractor without relying on LaSalle's principle. Other contributions are as follows. We prove that, under mild conditions, agents' velocities approach the center of mass velocity exponentially, with the distance between the center of mass and the virtual leader being bounded. In the dissipative case, we show existence of a broad class of nonlinear control forces for which the attractor does not contain periodic trajectories, which cannot be ruled out by LaSalle's principle. Finally, we conduct a computational investigation of the problem of reducing propulsion energy consumption by selecting appropriate navigational feedback forces.

Figures (7)

• Figure 1: A stabilized group of $N = 25$ agents moving in a two-dimensional space. (a) illustrates a "tight packing" configuration with a high group density $\rho$, where the followers (small green arrows) are highly concentrated around the virtual leader (large red arrow). Both ambient conservative forces (blue lines attached to the arrow tips) and self-propulsion conservative forces (orange lines attached to the arrow tips) are non-zero for the agents that are farther than $r_0$ from the virtual leader. (b) illustrates a "loose packing" configuration with a low group density $\rho$. In this case, all conservative forces are zero. The radius of the circles enclosing the followers is equal to the cut-off distance $r_{c}$ of the conservative force. (Color figure online)
• Figure 2: Plots of $\bar{q}_{dev}(t^{\prime})$, $\bar{v}_{dev}(t^{\prime})$ and $\bar{U}(t^{\prime})$ for the dissipative scenario for a group of $N=100$ agents.
• Figure 3: Plots of $\bar{q}_{dev}(t^{\prime})$, $\bar{v}_{dev}(t^{\prime})$ and $\bar{U}(t^{\prime})$ for the non-dissipative scenario for a group of $N=100$ agents.
• Figure 4: Differences of agents' positions and agents' velocities for a group of $N=3$ agents moving in regimes (1) and (3) with $r_{0}=0$. The colored lines represent corresponding measurements for different agents.
• Figure 5: Values of $\bar{U}$, $\bar{Q}_{dev}$, and $\bar{V}_{dev}$ evaluated on $\Xi$, averaged across all but the $k$-th component of $\bm{\theta}$ for $k = 1, 2, 3, 4$.

Theorems & Definitions (44)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Remark 4
• Remark 5
• Remark 6
• Theorem 1
• proof
• Definition 2