Morse Decompositions for Coverage Tasks

Exact cellular decompositions represent a robot's free space by dividing it into regions with simple structure such that the sum of the regions fills the free space. These decompositions have been widely used for path planning between two points, but can be used for mapping and coverage of free spaces. In this paper, we define exact cellular decompositions where critical points of Morse functions indicate the location of cell boundaries. Morse functions are those whose critical points are non-degenerate. Between critical points, the structure of a space is effectively the same, so simple control strategies to achieve tasks, such as coverage, are feasible within each cell. This allows us to introduce a general framework for coverage tasks because varying the Morse function has the effect of changing the pattern by which a robot covers its free space. In this paper, we give examples of different Morse functions and comment on their corresponding tasks. In a companion paper, we describe the sensor-based algorithm that constructs the decomposition.