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

Paden-Kahan (PK) problems represent "geometric subproblems" "in terms of which the inverse kinematics solution of a large number of manipulators can be decomposed”. Based on the original problems by Bradley Paden and William Kahan, generalized variants have been formulated and solved. Within the 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 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 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 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.