Modeling of Linear Object Deformation Including Knotted Shape

Abstract

A modeling of linear object deformation including knotted shape is presented. Deformable linear objects such as tubes, cords, and wires are used widely; data transmission, object transportation, fixing or packing of objects, and so on. Such manipulative tasks include knotting. They can be also knotted in order to make them compact for their storage or transportation. Therefore, it is important for linear object manipulation to analyze knotting. In this paper, we develop an analytical method to model the shape of a deformable linear object, especially its knotted shape. Firstly , a geometric representation to describe the shape of a linear object with flexural and torsional deformation is introduced. The internal energy of the object and the geometric constraints imposed on it are then formulated. The shape of the object in the stable state can be derived by minimizing the internal energy under the geometric constraints. Secondly, procedure to compute the deformed shape is developed by applying a nonlinear programming technique. Thirdly, the above method is applied to the knotted shape of the object. The object contacts with itself at some regions and frictional force yielded there can influence its shape when it is knotted. Therefore, friction by self-contact of the object is considered. Finally, the validity of our method is demonstrated by measuring the knotted shape of rubber cord.