Noise-enabled goal attainment in crowded collectives

TL;DR

Simulation, theory, and experiments are used to study how adding stochasticity to agent motion can reduce traffic jams and help agents travel more quickly to prescribed goals, and identify new directions for emergent traffic research.

Abstract

In crowded environments, individuals must navigate around other occupants to reach their destinations. Understanding and controlling traffic flows in these spaces is relevant for coordinating robot swarms and designing infrastructure for dense populations. Here, we use simulations, theory, and experiments to study how adding stochasticity to agent motion can reduce traffic jams and help agents travel more quickly to prescribed goals. A computational approach reveals the collective behavior. Above a critical noise level, large jams do not persist. From this observation, we analytically approximate the swarm's goal attainment rate, which allows us to solve for the agent density and noise level that maximize the goals reached. Robotic experiments corroborate the behaviors observed in our simulated and theoretical results. Finally, we compare simple, local navigation approaches with a sophisticated but computationally costly central planner. A simple reactive scheme performs well up to moderate densities and is far more computationally efficient than a planner, motivating further research into robust, decentralized navigation methods for crowded environments. By integrating ideas from physics and engineering using simulations, theory, and experiments, our work identifies new directions for emergent traffic research.

Figures (11)

• Figure 1: Noisy swarm setting.a) Robots begin in random positions and individually receive random goal positions (star shapes of matching color), which represent locations for dropping off deliveries or construction parts. See SI Video 1 for animated examples of robot trajectories. b) Robots cease forward motion, but can still rotate in place, when an occupant is detected in their sensing cones (blue). The sensing cone is a $\gamma = 120^{\circ}$ sector of a circle with radius $r$. At time $t$, robots have heading $\theta(t)$ and an optimal travel angle $\theta^*(t)$ to the goal (green star). c) A robot with no rotational noise moves directly from one goal position to the next. d) A robot with rotational noise $\sigma$ moves in steps. For each one, it draws the step's random length from $\textit{Unif}(\frac{1}{2} b, \frac{3}{2} b)$ and its heading from the Gaussian distribution $\mathcal{N}(\theta^*(t), \sigma^2)$. e, f) At high densities, visible traffic jams occur in simulation and experiment. Green dots mark robots' goals. In f), which shows a view of our robotic experiments from the overhead camera, red lights indicate that a robot's vision cone is currently occupied. Robot positions, headings, goals, and sensing cone edges are annotated.
• Figure 2: Simulations and theory.a) Jammed and dilute regimes. Dots show robot positions for various $(n, \sigma)$ after 8000 seconds of simulated time. Color indicates whether robots are blocked (red) or free to move (blue). All subplots in this figure use periodic boundary conditions, $L = 40, r = 2, \gamma = \frac{2\pi}{3}, v = 0.5, b = 0.5, \epsilon = 0.6$. See Table \ref{['table:parameters']} for more parameter details, and SI Video 2 for an animated version of this phase diagram. b) Solid lines plot simulation data for goal attainment rate $G$, the total goals reached per second by the entire swarm. Dashed lines show our theoretical approximations of $G$ for the dilute regime. The dashed lines end where our theory predicts the dilute regime ends. (The absence of dashed lines for $n \in \{128, 256\}$ reflects Eq. \ref{['eq:sigma_star']} diverging at high densities; the theory remains valid in the regime of interest where the optimal parameters lie.) Goal attainment rate is computed over the last 25% of the simulation time to capture steady state behavior. Simulations are run for 50 trials in the high-variance regime ($n \leq 128$ and $\sigma \leq 2.0$), and for 20 trials otherwise. c) Goal attainment across $(n, \sigma)$ space. The 'S' dot marks the optimal $(n, \sigma)$ observed in simulations, the 'T' dot marks the optimal $(n, \sigma)$ predicted by theory, and white dots mark the predicted optimal $\sigma$ predicted by theory for each $n$.
• Figure 3: Robotic experiments. a) Individual robot and schematic of experimental setup. b) Goal attainment rate for experiments vs. simulations with matching parameters. Experimental data have 6 trials per point; simulation data have 100 trials per point. Data includes the last 75% of 300-second trials. Shaded error bands show standard error. See Table \ref{['table:parameters']} for parameter details and SI Video 3 for experimental footage.
• Figure 4: More intelligent controllers. In all subplots, the purple lines for local, constant noise represent different values of $\sigma$ ranging from $0.5$ to $1.5$. Shaded error bands show standard error. Data contains 20 trials per point. See Table \ref{['table:parameters']} for more parameter details. a) Comparing the attainment rate of different kinds of noisy navigation controllers. See SI Video 4 for animated examples of the different controllers. b) Comparing the ratio of runtime to simulated time for local and global controllers. c) Comparing the goals reached per wall-clock time of local and global controllers.
• Figure S1: Statistics of distances on a torus.a) A square of side length $L$ that has periodic boundary conditions, which we also call a torus. Several pairs of points and the shortest paths between them are drawn. For the bottom pair, the point on the left has the same position as the faded point, but wrapped across the periodic boundaries. b) When $x \leq \frac{L}{2}$, the area within $x$ of a point $a$ (shaded) is the area of a circle. c) When $\frac{L}{2} < x \leq \frac{\sqrt{2}}{2}L$, parts of the wrapped circle begin to overlap. To compute the the area within $x$ of a point $a$, we can subtract the double-counted regions of area $J$. d) The area $J$ can be computed by finding the areas of a circle sector and a triangle. e) Comparing analytical results from main text Eqn. \ref{['eq:goal_distance']} to simulations using a torus with $L = 1$ and 50000 randomly generated distances.