Closed-Form Solutions and Solvability of Line-Geometric Paden–Kahan Problems

TitleClosed-Form Solutions and Solvability of Line-Geometric Paden–Kahan Problems
Publication TypePresentation
Year of Publication2021
AuthorsBongardt B
Conference NameConference on Geometry: Theory and Applications
Date Published09/2021(skipped)

Paden-Kahan (PK) problems represent "geometric subproblems" [1] "in terms of which the inverse kinematics solution of a large number of manipulators can be decomposed” [3]. Based on the original problems by Bradley Paden and William Kahan, generalized variants have been formulated and solved [4, 5, 6]. Within the future talk, the novel line-geometric extension of the second PK problem is introduced and its analytic solution is derived: Indicating a dual number by [x_tilde] with [..], and its matrix form [7] by [..] the problem reads to find dual angles [phi_tilde_12] and [phi_tilde_23] such that the constraint [..] holds, where [..] denotes the adjoint representation [7] of a Plücker vector [Lambda_hat], the terms [..] and [..] indicate skew, intersecting, or parallel rotation axes, and the vectors [..] and [..] represent the unit spears to be matched. The talk explains how the second line-geometric PK problem is solved in closed form by employing the transference principle [2] appropriately. Further it reflects the conditions that determine the problem's solvability in the real domain. The presentation shall conclude with a comparative overview of Paden-Kahan problems and their generalizations.