Optimal flock formation induced by agent heterogeneity

TL;DR

The paper investigates how inter-individual heterogeneity in agent parameters can enhance flocking stability and convergence. By formulating real-time, per-agent gain optimization under time-varying communication networks and proving Lyapunov-based bounds on the tracking error, it demonstrates that heterogeneous parameter assignments can yield 20–40% faster convergence across target-tracking, time-delay, and free-flocking tasks, while also broadening delay robustness and improving obstacle maneuvering. The authors connect the approach to optimal-control concepts, provide analytical insights into homogeneous versus heterogeneous optima via Laplacian spectra, and show that distributed optimization can achieve comparable performance to centralized heterogeneous schemes. These results suggest that controlled heterogeneity acts as an adaptive, scalable mechanism to promote robust collective behavior in drone swarms and other multi-agent systems.

Abstract

The study of flocking in biological systems has identified conditions for self-organized collective behavior, inspiring the development of decentralized strategies to coordinate the dynamics of swarms of drones and other autonomous vehicles. Previous research has focused primarily on the role of the time-varying interaction network among agents while assuming that the agents themselves are identical or nearly identical. Here, we depart from this conventional assumption to investigate how inter-individual differences between agents affect the stability and convergence in flocking dynamics. We show that flocks of agents with optimally assigned heterogeneous parameters significantly outperform their homogeneous counterparts, achieving 20-40% faster convergence to desired formations across various control tasks. These tasks include target tracking, flock formation, and obstacle maneuvering. In systems with communication delays, heterogeneity can enable convergence even when flocking is unstable for identical agents. Our results challenge existing paradigms in multi-agent control and establish system disorder as an adaptive, distributed mechanism to promote collective behavior in flocking dynamics.

Figures (16)

• Figure 1: Optimal formation in flocks of heterogeneous and homogeneous agents. (a) Tracking error as a function of time for an optimal flock of $N=30$ agents in the 2D space. The blue and orange lines represent flocks of heterogeneous and homogeneous agents, respectively, in which feedback gains are optimized in real time at every time step $w = T = 0.1$. For reference, the black line represents the non-optimal case of randomly assigned time-independent feedback gains. The solid lines represent the median over 100 realizations with different initial conditions (and parameters in the non-optimal case), while the shaded areas indicate the first and third quartile. The insets show a snapshot of the agents position at $t=4$ for the homogeneous flock (top inset) and heterogeneous flock (bottom inset). Agents are color coded by their velocity feedback gain $\bm c_i$. In this simulation, the agents start at a random stationary positions around the origin ($\bm q_i(0)\sim \mathcal{U}[-2,2]^2$) and are tasked to track a virtual target (red dot) that starts far away from the agents ($\bm q_{\rm t}(0) = [100,\, 100]$) and moves with constant velocity ($\bm p_{\rm t}(t) = [100,\, 0]$, $\forall t\geq 0$). (b) Settling time $t_{\rm s}$ as a function of the tolerance $\epsilon$, where the relationship $t_{\rm s}$ vs. $\epsilon$ is illustrated by the dashed lines in panel a. (c) Histogram of the steady-state error $\norm{\bm e(t)}$ of the heterogeneous (blue) and homogeneous (orange) flock across all realizations in panel a for $20\leq t\leq 30$. (d,e) Settling time $t_{\rm s}$ as a function of the interaction range $\beta$ (d) and the number of agents $N$ (e) for heterogeneous (blue) and homogeneous (orange) flocks. The dots and error bars represent the average and one standard deviation over 100 realizations, respectively. All simulations implement Eq. \ref{['eq.flockmodel']} with additive Gaussian noise to probe the robustness to small perturbations. See Methods for details on the simulation parameters and Supplementary Movie 1 for an animation of the dynamics.
• Figure 2: Real-time optimization performance. (a) Settling times $t_{\rm s}$ as functions of the time interval $T$ between communication events (left for $w=T, \epsilon = 0.01$) and the optimization window size $w$ (right for $T = 0.1, \epsilon = 0.01$). The data points represent an average over 100 independent realizations of the initial conditions, and the error bars indicate one standard deviation. (b) Lyapunov exponent $\Lambda_{\rm max}$ of the optimal flock as a function of time for different parameter choices: $T=0.1$ (left), $T=1$ (middle), and $T=10$ (right); $w=T$ in all cases. The color scheme is the same as in panel a, with the blue and orange lines representing optimal flocks of heterogeneous and homogeneous agents, respectively. The solid lines represent the median over 100 realizations and the shaded areas indicate the first and third quartiles. (c) Optimal position (top) and velocity (bottom) feedback gains as functions of time in a representative realization in a heterogeneous flock (a colored line for each agent) and a homogeneous flock (a common black line for all agents). The different columns correspond to the choices of $T$ and $w$ in panel b. In all cases, the number of agents is $N=30$ and the simulation time is 30 (panels b and c are zoomed in to facilitate visualization). See Methods for details on the system parameters.
• Figure 3: Stability landscape for the pre-assigned flocking model. (a) Lyapunov exponent $\Lambda_{\rm max}(J')$ as a function of the feedback gains ($b_1, b_2, b_3$) for $N=3$ agents. The color-coded section shows the stability landscape on a plane containing both the optimal homogeneous gain $\bm b^*_{\rm hom}$ (orange dot) and the optimal heterogeneous gain $\bm b^*_{\rm het}$ (green dot). The flock formation is stable (unstable) for $\Lambda_{\rm max}<0$ ($\Lambda_{\rm max}>0$). (b) Lyapunov exponent $\Lambda_{\rm max}(J')$ for $N=10$ on a plane ($\bm \xi_1,\bm \xi_2$) containing $\bm b^*_{\rm hom}$ and $\bm b^*_{\rm het}$. In both panels, the white curves indicate the cross-sections of a hypersurface (codimension 1) corresponding to single degeneracy of the real parts of the eigenvalues of the Jacobian $J'$.
• Figure 4: Distributed versus centralized optimization. (a) Schematic diagram of the distributed optimization approach, where each agent has access only to local information within a specified range $R$ and the associated subnetwork $\mathcal{G}_i$ (green edges). (b) Tracking error over time for a flock of $N= 30$ agents using heterogeneous distributed optimization with $R=2$ (green), heterogeneous centralized optimization (blue), and homogeneous centralized optimization (orange). (c, d) Average neighborhood size $|\mathcal{N}_i|$ (c) and optimal Lyapunov exponent $\Lambda_{\rm max}$ (d) as functions of $R$ for the distributed method (green curve). In panel d, the blue and orange lines show the optimal values obtained by the centralized heterogeneous and homogeneous approaches, respectively.
• Figure 5: Heterogeneity-induced stability in consensus dynamics with time delay. (a) Lyapunov exponent $\Lambda_{\rm max}$ as a function of the time delay $\tau$ for an optimal flock of heterogeneous (blue) and homogeneous (orange) agents for $N=4$. Consensus is stable (unstable) for all initial conditions $\bm x(0)$ if $\Lambda_{\rm max} > 0$ ($\Lambda_{\rm max} > 0$). A flock is said to be optimal if the choice of parameters $k_i$ minimizes $\Lambda_{\rm max}$. (b) Dynamical evolution of the agents' positions $\bm q_i(t)$ for an optimal flock of heterogeneous (top) and homogeneous (bottom) agents. The time delay is set as $\tau = 0.6$. See SI, Section \ref{['sec.dde']}, for details on the optimization problem.