# Algorithmic approach to geometric solution of generalized Paden–Kahan subproblem and its extension

Dimovski, Igor and Trompeska, Mirjana and Samak, Samoil and Dukovski, Vladimir and Cvetkoska, Dijana (2018) Algorithmic approach to geometric solution of generalized Paden–Kahan subproblem and its extension. International Journal of Advanced Robotic Systems (IJARS), 15 (1).

## Abstract

Kinematics as a science of geometry of motion describes motion by means of position, orientation, and their time
derivatives. The focus of this article aims screw theory approach for the solution of inverse kinematics problem. The
kinematic elements are mathematically assembled through screw theory by using only the base, tool, and workpiece
coordinate systems—opposite to conventional Denavit–Hartenberg approach, where at least n þ 1 coordinate frames
are needed for a robot manipulator with n joints. The inverse kinematics solution in Denavit–Hartenberg convention is
implicit. Instead, explicit solutions to inverse kinematics using the Paden–Kahan subproblems could be expressed. This
article gives step-by-step application of geometric algorithm for the solution of all the cases of Paden–Kahan subproblem 2
and some extension of that subproblem based on subproblem 2. The algorithm described here covers all of the cases that
can appear in the generalized subproblem 2 definition, which makes it applicable for multiple movement configurations.
The extended subproblem is used to solve inverse kinematics of a manipulator that cannot be solved using only three basic
Paden–Kahan subproblems, as they are originally formulated. Instead, here is provided solution for the case of three
subsequent rotations, where last two axes are parallel and the first one does not lie in the same plane with neither of the
other axes. Since the inverse kinematics problem may have no solution, unique solution, or many solutions, this article
gives a thorough discussion about the necessary conditions for the existence and number of solutions.
Keywords
Screw motion, Paden–Kahan subproblem, geometric algorithm, inverse kinematics, mathematical foundations

Item Type: Article 1.482 Natural sciences > Computer and information sciencesEngineering and Technology > Materials engineeringEngineering and Technology > Mechanical engineeringNatural sciences > Other natural sciences Faculty of Technology Svetlana Risteska 15 Sep 2020 07:23 15 Sep 2020 07:23 https://eprints.ugd.edu.mk/id/eprint/24439