Distributed-infrastructure multi-robot routing using a Helmholtz-Hodge decomposition

Abstract

Using graphs and simplicial complexes as models for an environment containing a large number of agents, we provide distributed algorithms based on the HelmholtzHodge decomposition that, given desired flow rates on edges or across faces, produce incompressible approximations to the specified flows. These flows are then "lifted" to produce hybrid controllers for the agents, and a related algorithm is described that computes continuous streamfunctions over the environment, also in a distributed way.

Coverage and field estimation on bounded domains by diffusive swarms

“…CMMI-1436960. Karthik can be beneficial for scalability in controller design [28], [5], [27]. This advantage of permutation invariance has led to multiple works on partial differential equation (PDE)-based multi-agent control, in which the Eulerian perspective of particles/agents is fundamental [9], [19], [40].…”

“…Karthik Elamvazhuthi, Chase Adams, and Spring Berman are with the School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, 85281 USA {karthikevaz, Chase.Adams, Spring.Berman}@asu.edu can be beneficial for scalability in controller design [28], [5], [27]. This advantage of permutation invariance has led to multiple works on partial differential equation (PDE)-based multi-agent control, in which the Eulerian perspective of particles/agents is fundamental [9], [19], [40].…”

Hodge Laplacians on graphs

“…Applications of cohomology and Hodge theory are plentiful, we find them in numerical analysis [3], peridynamics [30], topological data analysis [26], computational topology [33], graphics [66], image processing [70], robotics [45], sensor networks [65], neuroscience [53], and many other areas in physical science and engineering. But these applications are not 'surprising' in the sense that they all concern physics, geometry, or topology -areas that gave birth to cohomology and Hodge theory in the first place.…”

“…Unlike physical problems arising from areas such as continuum mechanics or electromagnetics, where the differentiable Hodge-de Rham theory has been applied with great efficacy for both modeling and computations [3,30,45,53,65,66,70], those arising from data analytic applications are likely to be far less structured [4,15,16,26,40,43,56,69]. Often one could at best assume some weak notion of proximity of data points.…”

Deformable-medium affordances for interacting with multi-robot systems

“…But, it becomes cumbersome as the number of agents increases, as was shown in [3]. Alternative approaches that have been proposed include induced flows across agents [4], boundary value control [5], or behavioral interactions [6], [7], [8]. A related question concerns the appropriate structure of user interfaces that ensures that sufficient situational awareness is provided and that the user is not overloaded with swarm-related inputs [9], [10].…”