Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms

Abstract

Abstract-This paper presents a novel procedure for computing parameters of a robotic swarm that guarantee coverage performance by the swarm within a specified error from a target spatial distribution. The main contribution of this paper is the analysis of the dependence of this error on two key parameters: the number of robots in the swarm and the robot sensing radius. The robots cannot localize or communicate with one another, and they exhibit stochasticity in their motion and task-switching policies. We model the population dynamics of the swarm as an advectiondiffusion-reaction partial differential equation (PDE) with time-dependent advection and reaction terms. We derive rigorous bounds on the discrepancies between the target distribution and the coverage achieved by individual-based and PDE models of the swarm. We use these bounds to select the swarm size that will achieve coverage performance within a given error and the corresponding robot sensing radius that will minimize this error. We also apply the optimal control approach from our prior work in [13] to compute the robots' velocity field and task-switching rates. We validate our procedure through simulations of a scenario, in which a robotic swarm must achieve a specified density of pollination activity over a crop field.

Quantitative Assessment of Robotic Swarm Coverage

“…Our method is independent of the controller used to generate the swarm distribution, and thus has the potential to be used in a diverse range of robotics applications. In (Li et al, 2017) and (Zhang et al, 2018), error metrics similar to the one presented here are used, but their properties are not discussed in sufficient detail for them to be widely adopted. In particular, although the error metric that we study always takes values somewhere between 0 and 2, these values are, in general, not achievable for an arbitrary desired distribution and a fixed number of robots.…”

Quantifying Robotic Swarm Coverage

“…This means that by using the L 1 norm we capture the idea that discretizing the domain provides a measure of error, but avoid the pitfalls of discretization methods described in Subsection 2.3. Studies in optimal control of swarms often use the L 2 norm due to the favorable inner product structure [40]. We point out that the L 1 norm is bounded from above by the L 2 norm: indeed, according to the Cauchy-Schwarz inequality, for any function f we have,…”

“…We apply this idea but with the opposite aim: namely, we represent discrete points (the robots' positions) with a continuous function. A similar strategy was alluded to in [18] and used in [25,40] to measure the effectiveness of a certain robotic control law, but to our knowledge, our work here and in [1] is the first to develop any such method in a form sufficiently general for common use. This section is devoted to our definition of the error metric and to its basic properties and computational considerations.…”

“…Remarks and Basic Properties Our error is defined as the L 1 norm of the difference between the swarm blob function ρ δ N and the desired robot distribution ρ. One could use another L p norm; however, p = 1 is a standard choice in applications that involve particle transportation and coverage such as [18,40]. Moreover, the L 1 norm has a key property: for any two integrable functions f and g,…”

“…Therefore, in subsequent sections we restrict our attention to the extrema and PDF of the instantaneous formulation without loss of generality. In addition, [40] considers a one-sided notion of error, in which a scarcity of robots is penalized but an excess is not, that is,…”