Second-order Taylor Stability Analysis of Isolated Kinematic Singularities of Closed-chain Mechanisms

摘要

When the geometric design of a closed-chain mechanism is non-generic, the singularity locus of the mechanism may exhibit isolated points. It is well known that these isolated points are unstable since they disappear or generate/reveal cusps when the geometric design of the mechanism slightly deviates from a non-generic design, possibly affecting the ability of the mechanism to reconfigure without crossing undesirable singularities. This paper presents a method based on second-order Taylor expansions to determine how these isolated singularities transform when perturbing the different geometric parameters of a non-generic mechanism. The method consists in approximating the singularity locus by a conic section near the isolated singularity, and classifying the resulting conic in terms of the perturbations of the different geometric parameters. Two non-generic closed-chain mechanisms are used to illustrate the presented method: an orthogonal 3R serial arm with specified position for its tip, and the planar Stewart parallel platform.