Optimal control of stochastic coverage strategies for robotic swarms

Abstract

This paper addresses a trajectory planning and task allocation problem for a swarm of resource-constrained robots that cannot localize or communicate with each other and that exhibit stochasticity in their motion and task-switching policies. We model the population dynamics of the robotic swarm as a set of advection-diffusion-reaction partial differential equations (PDEs), a linear parabolic PDE model that is bilinear in the robots' velocity and task-switching rates. These parameters constitute a set of time-dependent control variables that can be optimized and broadcast to the robots prior to their deployment. The planning and allocation problem can then be formulated as a PDE-constrained optimization problem, which we solve using techniques from optimal control. Simulations of a commercial pollination scenario validate the ability of our control approach to drive a robotic swarm to achieve predefined spatial distributions of activity over a closed domain, which may contain obstacles.

Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms

“…We show that as the number of robots approaches infinity, the discrete density functions converge to the continuous solution of the macroscopic model. We illustrate our approach for a simulated scenario in which a swarm of microaerial vehicles must pollinate a crop field, similar to the problem in [13]. We apply the optimal control approach in [13] to compute vehicle control policies that achieve a target spatial distribution of pollination.…”

“…We illustrate our approach for a simulated scenario in which a swarm of microaerial vehicles must pollinate a crop field, similar to the problem in [13]. We apply the optimal control approach in [13] to compute vehicle control policies that achieve a target spatial distribution of pollination. We also use our derived error bound to estimate the required swarm size that will achieve the target pollination distribution within a specified percentage of accuracy.…”

“…1) We provide a rigorous analysis of the error bound between the aforementioned microscopic and macroscopic models, which is still absent in the literature on stochastic control of multiagent systems with state transitions. This analysis, together with our optimal control approach in [13] which approximates the target distribution using the macroscopic model, provides a formal mathematical validation of our swarm control strategy. 2) Based on the scaling laws that are observed in the error estimates, we propose a principled approach to determine the required number of robots and optimal robot sensing radius that will achieve a target distribution within a specified error.…”

“…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].…”

“…Toward this end, we employ a methodology that is based on models of the robots' decision making and motion at multiple levels of abstraction. The multilevel modeling framework is adopted from the disciplines of stochastic chemical kinetics and fluid dynamics, and it has been used by the authors and others, e.g., in [13], [17], [27], to describe the population dynamics of large numbers of robots. This framework has also been used to model collective behaviors in biological swarms, such as flocking, schooling, chemotaxis, pattern formation, and predator-prey interactions [24].…”