PDE-based optimization for stochastic mapping and coverage strategies using robotic ensembles

Abstract

This paper presents a novel partial differential equation (PDE)-based framework for controlling an ensemble of robots, which have limited sensing and actuation capabilities and exhibit stochastic behaviors, to perform mapping and coverage tasks. We model the ensemble population dynamics as an advection-diffusion-reaction PDE model and formulate the mapping and coverage tasks as identification and control problems for this model. In the mapping task, robots are deployed over a closed domain to gather data, which is unlocalized and independent of robot identities, for reconstructing the unknown spatial distribution of a region of interest. We frame this task as a convex optimization problem whose solution represents the region as a spatially-dependent coefficient in the PDE model. We then consider a coverage problem in which the robots must perform a desired activity at a programmable probability rate to achieve a target spatial distribution of activity over the reconstructed region of interest. We formulate this task as an optimal control problem in which the PDE model is expressed as a bilinear control system, with the robots' coverage activity rate and velocity field defined as the control inputs. We validate our approach with simulations of a combined mapping and coverage scenario in two environments with three target coverage distributions.

Bilinear Controllability of a Class of Advection–Diffusion–Reaction Systems

“…See [41] for such a convergence analysis. Problem III.1 can be reframed in terms of equation ( In models of robotic swarms, it is useful to consider hybrid variants of the SDE (4) to account for the fact that each robot, in addition to a continuous spatial state Z(t), can be associated with a discrete state Y (t) ∈ V = {1, ..., N } at each time t [32], [16]. The elements of V can correspond to different behavioral states or tasks that can be performed by a robot, such as "searching," "lifting," or "digging."…”

“…Proposition IV. 16. If the graph G = (V, E) is strongly connected, then the system (7) is STLC from every point in int(P(V)).…”

“…Additionally, the results in previous controllability studies were proven with control parameters that are square-integrable. However, in bilinear optimal control of PDEs associated with stochastic processes, the boundedness of the vector fields is a common requirement [16], [19], [20]. Toward this end, we establish controllability with control inputs that are (essentially) bounded in space and time.…”

“…Boundedness of the domain is a common constraint in many problems in swarm robotics, e.g. in [16], [32], where optimal control techniques were used to optimize swarm behavior. Additionally, the results in previous controllability studies were proven with control parameters that are square-integrable.…”

Partial Differential Equation-Based Trajectory Planning for Multiple Unmanned Air Vehicles in Dynamic and Uncertain Environments

“…Furthermore, based on the analysis of particle flow, which is a result of PDE Cucker-Smale technique, the stability of the technique is proven. In other study [18], stochastic mapping and coverage strategy using robotic ensembles by PDE-based optimization is studied. In their study, by using an advection-diffusion-reaction PDE model, they task the agents to gather the data and identify the region of interest.…”