Threshold Decision-Making Dynamics Adaptive to Physical Constraints and Changing Environment

TL;DR

This work addresses threshold-based task switching for spatial tasks under varying physical states by coupling nonlinear opinion dynamics (NOD) with the agent's physical dynamics, creating a closed-loop decision mechanism without communication networks. The authors derive and analyze a bifurcation structure, showing a pitchfork-type bifurcation at $u^* = d$ governs the emergence of stable task-choosing equilibria, and that thresholds can be modulated by environmental inputs $b$ and physical gains. They demonstrate adaptive thresholding by tuning parameters such as $u$ and $K_x$, which expand or contract the bistable region via saddle-node bifurcations, enabling environment-driven switching and behavior such as emergent explore-exploit dynamics and declustering. The framework is validated through decentralized two-robot trash-collection simulations, illustrating practical benefits for scalable, robust multi-robot task allocation with no inter-robot communication.

Abstract

We propose a threshold decision-making framework for controlling the physical dynamics of an agent switching between two spatial tasks. Our framework couples a nonlinear opinion dynamics model that represents the evolution of an agent's preference for a particular task with the physical dynamics of the agent. We prove the bifurcation that governs the behavior of the coupled dynamics. We show by means of the bifurcation behavior how the coupled dynamics are adaptive to the physical constraints of the agent. We also show how the bifurcation can be modulated to allow the agent to switch tasks based on thresholds adaptive to environmental conditions. We illustrate the benefits of the approach through a decentralized multi-robot task allocation application for trash collection.

Figures (6)

• Figure 1: Application of the coupled NOD and physical dynamics model in \ref{['eq:decision_making_model']} to multi-robot trash collection across two trash patches. \ref{['subfig:dc_t_0']}-\ref{['subfig:dc_t_150']} show the positions of the robots (depicted as diamonds) at $t=0$, $t=25$ and $t=100$s. The patch boundaries are denoted by the rectangles and the uncollected (collected) trash pieces are shown in black (green). The bottom row shows the evolution of the opinion $z_i$, the position $x_i$ and input $b_i$, for the robots highlighted in red, blue, and green in the top row. Parameters: $q_{\min} = 1.5$, $u = 1.3$, $d = 1$, $K_z = 2$, $l = 1$, $\sigma = 0.1$, $k = 10$, $K_y = 0.15$.
• Figure 2: Bifurcation diagrams for the coupled NOD and physical dynamics of an agent \ref{['eq:decision_making_model']}. Each bifurcation diagram plots the equilibrium value of $z$ as a function of a fixed value of $u$. Top row corresponds to parameters for which the system undergoes a symmetric supercritical pitchfork bifurcation and its unfolding. Bottom row corresponds to parameters for which the system undergoes a symmetric subcritical pitchfork bifurcation with a quintic stabilizing term and its unfolding. Blue (red) lines show stable (unstable) branches of equilibria. The vertical dashed line is $u = u^*$. Regions with only one branch of equilibria for $u > u^*$ are shaded in light blue. Bifurcation diagrams generated with help of MatCont MatCont. Parameters: $d = 1$, $b \in \{0,\pm 0.2\}$, $K_x = 3$, $k = 10$, $\sigma = 0.1$, $\rho = 0.5$.
• Figure 3: Bifurcation diagrams for the coupled NOD and physical dynamics of an agent \ref{['eq:decision_making_model']} illustrating how $u$ and $K_x$ change the task switching threshold by changing the size of the bistable region. Each bifurcation diagram plots the equilibrium value of $z$ as a function of $b$. Left: bifurcation diagram corresponding to two different values of $u$. Right: bifurcation diagram corresponding to two different values of $K_x$. Blue lines show stable branches of equilibria. Red and magenta lines show unstable branches of equilibria. Bifurcation diagrams generated with MatCont MatCont. Parameters: $d = 1$, $k = 10$, $\sigma = 0.1$, $\rho = 0.5$, Left: $K_x = 3$, Right: $u = 1.1$.
• Figure 4: Evolution of $\partial b^*/\partial K_x$ as a function of $z$. Parameters: $d = 1$, $u = 1.2$, $K_x = 0.05$, $k = 10$, $\sigma = 0.1$, $\rho = 0.8$.
• Figure 5: Two robots (red square and a blue triangle) collecting trash (black circles) across two trash patches (black rectangles) adapt to environmental changes. At $t = 20$, the red robot receives a signal from the environment to decrease $u$ thus increasing its switching threshold. Top: spatial trajectory of the two robots. Bottom: time trajectory of the opinion $z$, the position $x$ and input $b$ of each robot. Parameters: $q_{\min} = 1.5$, $d = 1$, $K_z = 2$, $l = 1$, $\sigma = 0.1$, $k = 10$, $K_y = 0.15$, $K_x = 0.11$, blue robot: $u = 1.3$, red robot before $t = 20$: $u = 1.3$ and after $t = 20$: $u = 1.05$.

Theorems & Definitions (10)

• Lemma III.1
• proof
• Theorem III.2
• proof
• Proposition III.3
• proof
• Theorem III.4
• proof
• Proposition III.5
• proof