Determination of similarity between coupler curves of spatial linkage based on Knot theory

In this paper, the coupler curve is represented as a polygonal curve, thus, the problem of coupler curve similarity is equal to the problem of polygonal curve similarity. Based on Knot theory, a new method is put forward to solve the matching problem between 3-D coupler-curves. Firstly, the fragments of the polygonal curves are classified into different equivalence classes based on the writhing number index which is one of the knot invariants and can describe characters of spatial curves to some extent. Secondly, the paper calculates the transformations (rotation and translation) using unit quaternion method in each equivalence class. Thirdly, the transformations are clustered to find the optimal matching matrix and the similarity between two coupler curve is calculated. Finally, an example of matching coupler-curves of R-U-S-R mechanism was given. This method also can be extended to solve path generation problem of spatial mechanism in the future.