🤖 AI Summary
This work addresses the challenge of distinguishing non-transverse intersecting motion branches in linkage mechanisms, which has hindered fine-grained analysis of their singularities and mobility. By leveraging the intrinsic geometric information of kinematic tangent cones—defined constructively for the first time—the study integrates local singularity analysis with computational algebraic geometry to extend existing algorithmic frameworks. The proposed computable method effectively identifies and separates non-transverse bifurcating motion branches at specific configurations. This approach overcomes the limitations of conventional local analyses and significantly enhances the understanding and modeling of complex kinematic structures.
📝 Abstract
The local analysis is an established approach to the study of singularities and mobility of linkages. Key result of such analyses is a local picture of the finite motion through a configuration. This reveals the finite mobility at that point and the tangents to smooth motion curves. It does, however, not immediately allow to distinguish between motion branches that do not intersect transversally (which is a rather uncommon situation that has only recently been discussed in the literature). The mathematical framework for such a local analysis is the kinematic tangent cone. It is shown in this paper that the constructive definition of the kinematic tangent cone already involves all information necessary to separate different motion branches. A computational method is derived by amending the algorithmic framework reported in previous publications.