Research on robot manipulation has focused, in recent years, on grasping everyday objects, with target objects almost exclusively rigid items. Non–rigid objects, as textile ones, pose many additional challenges with respect to rigid object manipulation. In this seminar we will present how we can employ topology to study the ``state'' of a rectangular textile using the configuration space of $n$ points on the plane. Using a CW-decomposition of such space, we can define for any mesh associated with a rectangular textile a vector in an euclidean space with as many dimensions as the number of regions we have defined. This allows us to study the distribution of such points on the cloth and define meaningful states for detection and manipulation planning of textiles. We will explain how such regions can be defined and computationally how we can assign to any mesh the corresponding region. If time permits, we will also explain how the CW-structure allows us to define more than just euclidean distance between such mesh-distributions.

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