Abstract
This paper presents a screw theory approach for the computation of the instantaneous rotation centers of indeterminate planar linkages. Since the end of the 19th century, the determination of the instantaneous rotation, or velocity centers of planar mechanisms has been an important topic in kinematics that has led to the well-known Aronhold–Kennedy theorem. At the beginning of the 20th century, it was found that there were planar mechanisms for which the application of the Aronhold–Kennedy theorem was unable to find all the instantaneous rotation centers (IRCs). These mechanisms were denominated complex or indeterminate. The beginning of this century saw a renewed interest in complex or indeterminate planar mechanisms. In this contribution, a new and simpler screw theory approach for the determination of indeterminate rotation centers of planar linkages is presented. The new approach provides a simpler method for setting up the equations. Furthermore, the algebraic equations to be solved are simpler than the ones published to date. The method is based on the systematic application of screw theory, isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).
Highlights
The instantaneous rotation center (IRC) of two links in a one-DOF planar linkage is defined as a pair of coincident points that belong to each body and have the same velocity, with respect to another reference frame
The location of the secondary IRCs, obtained from the position vectors, is shown in Therein, one can observe that the three centers associated with three relative movements, namely, i $k, i $ j, and j $k, between three arbitrary links lie on a straight line, according to the Aronhold–Kennedy theorem
The upper diagonal arrangement of IRCs, where the primary centers are marked with the number −1, and the secondary centers are marked with the number −2, are shown in
Summary
The instantaneous rotation center (IRC) of two links in a one-DOF planar linkage is defined as a pair of coincident points that belong to each body and have the same velocity, with respect to another reference frame. There is another line of research of the determination of the IRC of planar linkages The origin of this line of research is due to Chang and Her [15], who in 2008 presented a virtual cam method for locating the IRCs of indeterminate linkages. It shows that the application of the algebraic structure of the Lie algebra, se(3), provides a unifying approach for the determination of the instantaneous screw axes of any spatial linkage; on the other hand, it shows that for planar linkages, the approach is reduced to the application of the Klein form to screws perpendicular to the plane motion—i.e., the classical Aronhold–Kennedy theorem. This paper contains a small Appendix A, which shows how to represent a secondary IRC as a linear combination of two primary IRCs
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