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Here, we propose a planning method for knotting/unknotting of deformable linear objects. First, we propose a topological description of the state of a linear object. Second, transitions between these states are defined by introducing four basic operations. Then, possible sequences of crossing state transitions, i.e. possible manipulation processes, can be generated once the initial and the objective states are given. Third, a method for determining grasping points and their directions of movement is proposed to realize derived manipulation processes. Our proposed method indicated that it is theoretically possible for any knotting manipulation of a linear object placed on a table to be realized by a one-handed robot with three translational DOF and one rotational DOF. Furthermore, criteria for evaluation of generated plans are introduced to reduce the candidates of manipulation plans. Fourth, a planning method for tying knots tightly is established because they fulfill their fixing function by tightening them. Finally, we report knotting/unknotting manipulation performed by a vision-guided system to demonstrate the usefulness of our approach.

We describe the modeling of linear object deformation based on differential geometry and its applications to manipulative operations. A particle-based approach, the finite element method, and the Cosserat theory have been applied to the modeling of linear object deformation. In this paper, we establish an alternative modeling approach based on an extension of differential geometry. First, we extend differential geometry to describe linear object deformation including flexure, torsion, and extension. Secondly, we show computational results to demonstrate the feasibility of the proposed modeling technique, and we compare computational and experimental results to demonstrate the accuracy of the model. Next, we apply the proposed approach to the grasping of a deformable linear object. We propose a disturbance force margin to indicate the stability of the grasping and we describe the computation of the margin using the proposed approach. Finally, we apply the proposed approach to the deformation path planning of a linear object. We formulate the minimization of potential energy during a deformation path. We compute the optimal deformation path and a feasible deformation path, which are compared with an experimental result.

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