دانلود رایگان مقاله رویکرد کواترنیونیک برای equiform سینماتیک و خط عناصر اقلیدسی 4-فضا و 3-فضایی

Abstract

We extend the quaternionic kinematic mapping of Euclidean displacements of Euclidean 4-space E4 to the group of equiform transformations S(4). As a consequence the equiform motions of basic elements (points, oriented lines, oriented planes, oriented hyperplanes) of E4 can be written compactly in terms of 2×2 quaternionic matrices. This representation is extended to oriented line-elements of E4 and to instantaneous screws of S(4), for which a classification (incl. corresponding normal forms) is given. Based on this preparatory work we study the relation between instantaneous equiform motions and the geometry of line-elements (path normal-elements, path tangent-elements) in E4. Finally, we show that the line-elements of projective 3-space can be mapped bijectively on the Segre variety Σ3,2.

7. Conclusion and outlook

In Section 1 we provided a detailed literature review on equiform kinematics in Euclidean spaces of dimension 3 and higher. In Section 2 a quaternion based kinematic mapping for equiform transformations of Euclidean spaces of dimension 3 and 4 was introduced. Based on this mapping and a study on instantaneous equiform motions of E4 in Section 3, we discussed the set of path normal-elements and path tangent-elements in the Sections 4 and 5. Moreover in Section 6 we presented an improved point-model for unoriented line-elements of the projective 3-space. Point-models for the set of oriented line-elements of E3 are of practical interest for the motion representation of robots with an axial symmetric end-tool (e.g. a milling cutter). Therefore a forthcoming publication of the author is devoted to the study and comparison of such models; e.g. the quadric of Eq. (23) sliced along the hyperplane l0 = 0 represents one possible point-model.