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.
The goal of coverage path planning is to determine a path that passes a detector over all points in an environment. This work prescribes a provably complete coverage path planner for robots in unknown spaces. We achieve coverage using Morse decompositions which are exact cellular decompositions whose cells are defined in terms of critical points of Morse functions. Generically, two critical points define a cell. We encode the topology of the Morse decomposition using a graph that has nodes corresponding to the critical points and edges representing the cells defined by pairs of critical points. The robot simultaneously covers the space while incrementally constructing this graph. To achieve this, the robot must sense all the critical points. Therefore, we first introduce a critical point sensing method that uses range sensors. Then we present a provably complete algorithm which guarantees that the robot will encounter all the critical points, thereby constructing the full graph, i.e., achieving complete coverage. We also validate our approach by performing experiments on a mobile robot equipped with a sonar ring.
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