Technical Brief

Control of Stochastic Boundary Coverage by Multirobot Systems

[+] Author and Article Information
Theodore P. Pavlic

School of Life Sciences,
Arizona State University,
Tempe, AZ 85281
e-mail: tpavlic@asu.edu

Sean Wilson

School for Engineering of Matter, Transport and Energy,
Arizona State University,
Tempe, AZ 85281
e-mail: Sean.T.Wilson@asu.edu

Ganesh P. Kumar

School for Computing, Informatics,
and Decision Systems Engineering,
Arizona State University,
Tempe, AZ 85281
e-mail: Ganesh.P.Kumar@asu.edu

Spring Berman

Assistant Professor
School for Engineering of Matter, Transport and Energy,
Arizona State University,
Tempe, AZ 85281
e-mail: Spring.Berman@asu.edu

To obtain the code, contact Dr. Theodore Pavlic (tpavlic@asu.edu).

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 1, 2014; final manuscript received August 14, 2014; published online October 21, 2014. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(3), 034504 (Oct 21, 2014) (9 pages) Paper No: DS-14-1058; doi: 10.1115/1.4028353 History: Received February 01, 2014; Revised August 14, 2014

This technical brief summarizes and extends our recently introduced control framework for stochastically allocating a swarm of robots among boundaries of circular regions. As in the previous work, a macroscopic model of the swarm population dynamics is used to synthesize robot control policies that establish and maintain stable predictable team sizes around region boundaries. However, this extension shows that the control strategy can be implemented with no robot-to-robot communication. Moreover, target team sizes can vary across different types of regions, where a region's type is a subjective characteristic that only needs to be detectable by each individual robot. Thus, regions of one type may have a higher equilibrium team size than regions of another type. In other work that predicts and controls stochastic swarm behaviors using macroscopic models, the equilibrium allocations of the swarm are sensitive to changes in the mean robot encounter rates with objects in the environment. Thus, in those works, as the swarm density or number of objects changes, the control policies on each robot must be retuned to achieve the desired allocations. However, our approach is insensitive to changes in encounter rate and therefore requires no retuning as the environment changes. In this extension, we validate these claims and show how the convergence rate to the target equilibrium allocations can be controlled in swarms with a sufficiently large free-robot population. Furthermore, we demonstrate how our framework can be used to experimentally measure the rates of robot encounters with occupied and unoccupied sections of region boundaries. Thus, our method can be viewed both as an encounter-rate-independent allocation strategy as well as a tool for accurately measuring encounter rates when using other swarm control strategies that depend on them.

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Grahic Jump Location
Fig. 1

Example scenario with two types of disk-shaped regions, labeled 1 and 2. The unlabeled circles are robots that are allocating themselves to the region boundaries.

Grahic Jump Location
Fig. 2

Control flow chart. Unbound robots move randomly until encountering other robots, which are avoided, or unbound zones of disk boundary. On encountering an unbound zone, a robot compares a generated pseudo–random number Rb~unif(0,1) to pb to determine whether to bind to that zone. Once bound, the robot waits to detect the close proximity of another free robot. On that event, the robot generates a second pseudo–random number Ru~unif(0,1) to compare to pu to determine whether to unbind from the disk and begin the random walk again.

Grahic Jump Location
Fig. 3

Effect of encounter ratio and avoidance. For each eb/eu ratio and avoidance distance a, the plot shows the predicted relationship between the control factor pb/(pb+pu) and the predicted equilibrium mean allocation. The predictions of the mean come from Eq. (3b). The star in each plot represents the levels corresponding to the (pb,pu) = (1,0) case if robots could re-assort on boundaries to optimally pack and eliminate wasted space. Consistent with expectations from the classical linear sequential parking problem, the curves in (a) all predict the parking constant of approximately 75% in the saturated case.

Grahic Jump Location
Fig. 4

Example simulated trajectories. In (a), a single execution of a simulation of 300 robots allocating to three disks is shown. In (b), the average trajectory is shown across ten simulations. The (pb, pu) policy chosen was picked to achieve a mean [B]/([B]+[U]) allocation of 37.5%, shown as a dashed horizontal line.

Grahic Jump Location
Fig. 5

Effect of varying disk mixture. Ten trials were generated for each experimental treatment (i.e., number of big disks, number of small disks, big allocation ratio, and small allocation ratio); the big-disk type is twice as large as the small-disk type. The averages shown for each big-disk allocation in (a) are taken across the pool of 110 trials that include all eleven small-disk allocation ratios for the corresponding big-disk allocation ratio and big–small mixture (and similar for the statistics in (b)). Error bars show ±1 standard error of the mean (SEM). The broken line in the plot shows the expected allocation curve from theory.

Grahic Jump Location
Fig. 6

Effect of encounter ratio on time constant. The time constant τ described in Eq. (7) cannot be predicted without knowledge of encounter rates eu and eb. However, the shape of τ is set by the ratio only eb/eu. In (a), eu is arbitrarily assumed to be 1/r0 and the ratio eb/eu is varied. In (b), time-constant data are shown that were generated from simulations of 300 robots across a range of control factors. For each factor, ten trials were combined into an average step response, as in Fig. 4(b), which was then used to estimate a corresponding time constant. Using an eb/eu ratio estimated from allocation data, a corresponding eu was fit to the time-constant data.



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