| 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. |
