Coverage and field estimation on bounded domains by diffusive swarms

Abstract

In this paper, we consider stochastic coverage of bounded domains by a diffusing swarm of robots that take local measurements of an underlying scalar field. We introduce three control methodologies with diffusion, advection, and reaction as independent control inputs. We analyze the diffusionbased control strategy using standard operator semigrouptheoretic arguments. We show that the diffusion coefficient can be chosen to be dependent only on the robots' local measurements to ensure that the swarm density converges to a function proportional to the scalar field. The boundedness of the domain precludes the need to impose assumptions on decaying properties of the scalar field at infinity. Moreover, exponential convergence of the swarm density to the equilibrium follows from properties of the spectrum of the semigroup generator. In addition, we use the proposed coverage method to construct a time-inhomogenous diffusion process and apply the observability of the heat equation to reconstruct the scalar field over the entire domain from observations of the robots' random motion over a small subset of the domain. We verify our results through simulations of the coverage scenario on a 2D domain and the field estimation scenario on a 1D domain.

Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition

“…Proof. The first estimate is just a restatement of [13][Theorem IV.4], where it was shown that f is the eigenvector of A a corresponding to simple principal eigenvalue 0, and the rest of spectrum is in the left-half complex plane, left to some negative number −λ. For this estimate, the assumption of the regularity of the initial condition is not required.…”

Quantifying Robotic Swarm Coverage

“…In many applications of swarm robotics, the swarm must spread across a domain according to a target distribution in order to achieve its goal. Some examples are in surveillance and area coverage [8,20,23,31], achieving a heterogeneous target distribution [4,10,14,16,17,34,41], and aggregation and pattern formation [28][29][30][36][37][38]. Despite the importance of assessing performance, some studies such as [28,29,34,38,41] rely only on qualitative methods such as visual comparison.…”

Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms

“…In addition, we demonstrate the effect of the maximum sensing radius on the swarm performance and show that an optimal radius length exists for a given swarm size. Notably, the analysis performed here can also be applied to other stochastic control strategies for robotic swarms, such as [12], [17], and [27].…”

“…This is because the L 1 norms of ρ δ 1 and ρ δ 2 directly measure the numbers of Flying robots and Hovering robots, respectively [see (12) and (13) measures the cumulative number of crop visits [see (17) and (18)], which is the metric of interest in the application. Note that in the optimal control method in Section VI, we use the L 2 norm in the objective function since it is convenient for optimal control.…”

“…In recent years, there have been various applications of control-theoretic techniques to PDE macroscopic models of multiagent systems for the purpose of synthesizing agent controllers that produce desired collective behaviors. ADR PDE models, in particular, have been used by the authors to design robot control policies that achieve target spatial distributions of robot activity over a bounded domain [13] and that drive the swarm to a distribution that is proportional to a locally measured scalar field [12]. ADR PDEs have also been used to control the probability density functions (pdfs) of multidimensional stochastic processes [2], develop multiagent coverage and search strategies that are inspired by bacterial chemotaxis [22], and maximize the probability of swarm robotic presence in a desired region [23].…”