Robotic manipulation of cloth is a complex task because of the infinite-dimensional shape-state space of textiles, which makes their state estimation very difficult. In this paper we introduce the dGLI Cloth Coordinates, a finite low-dimensional representation of cloth states that allows us to efficiently distinguish a large variety of different folded states, opening the door to efficient learning methods for cloth manipulation planning and control. Our representation is based on a directional derivative of the Gauss Linking Integral and allows us to represent spatial as well as planar folded configurations in a consistent and unified way. The proposed dGLI Cloth Coordinates are shown to be more accurate in the representation of cloth states and significantly more sensitive to changes in grasping affordances than other classic shape distance methods. Finally, we apply our representation to real images of a cloth, showing that with it we can identify the different states using a distance-based classifier.This work was developed under the project CLOTHILDE which has received funding from the European Research Council (ERC) under the EU-Horizon 2020 research and innovation programme (grant agreement No. 741930). M. Alberich-Carramiñana is also with the Barcelona Graduate School of Mathematics (BGSMath) and the Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech), and she and J. Amorós are partially supported by the Spanish State Research Agency AEI/10.13039/501100011033 grant PID2019-103849GB-I00 and by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP. J. Borràs is supported by the Spanish State Research Agency MCIN/ AEI /10.13039/501100011033 grant PID2020-118649RB-I00 (CHLOE-GRAPH project).Peer ReviewedPostprint (published version
