The Problem

Suppose a bounded region R is given, containing an Evader whose position $e(t) \in R$ at any time t is unknown, and the movement of the evader (i.e. e) is continuous. The goal is finding a continuous path $\gamma : [t_i, t_f] \rightarrow R$ (a solution) that guarantees that for some t_d e(t_d) is visible from $\gamma(t_d)$. It has been shown in a paper by Steven LaValle and John Hinrichsen that such a path can be found by searching an information state graph induced by the cell decomposition of R, where the cells are bounded by inflections and bitangents of $\partial R$ (the boundary of R).

To understand this method, note that from any position in R, the pursuer is able to see some parts of R, but not others. Also note that this correspods to seeing some intervals or $\partial R$,nut not seeing others. These gaps in visibility of $\partial R$correspond to subregions of R that may contain an unseen evader. A gap is called contaminated if there is a possibility of an evader hiding in the gap's corresponding subregion. If we know that an evader can't be in an unseen region, we label its corresponding gap cleared. One suggested method of detecting the presence and position of the gaps for an actual robot is performing a 360 degree sweep of distances to the wall, and finding discontinuities in the measurements.

The cells are partitioned by inflections and bitangents because the relevant pursuit events, such as cleared/contaminated gaps appearing, dispearring, merging, or splitting, occur only when the pursuer crosses an inflection or a bitangent. We therefore use an information state determined by the cell location of the pursuer and the status of the gaps (whether they are cleared or contaminated), and can build an information state graph whose nodes are a set all possible information states, with edges between them determined by information state changes possible by crossing cell boundaries. This graph can then be searched to find what sequence of cells must be visited to end up in an information state where all gaps have been cleared.

In this project, we will build on these results to show that a solution appropriate for a very limited robot can be exctracted from the cell decomposition. We will also speculate about the possibility of the topology of the cell decomposition graph being inferred by the same limited robot without a priori knowledge of the map, and without the need to build the map using robot sensor data. These two results can lead to a complete algorithm for such a robot in an unknown enviroment.