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

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.

A speedup method for solving the inverse kinematics problem of robotic manipulators

“…7 At present, methods to solve the IKP of redundant DOF robots are mainly numerical, such as pseudo-inverse of Jacobian methods, 8 extended Jacobian methods, 9 task-space augmentation methods, 10 gradient projection methods, 11 damped least square methods, 12 and weighted least-norm methods. 13 Besides, there are analytical methods to solve the IKP of redundant robots with specific geometric properties, such as solution algorithms based on screw theory, 14 geometric methods for robots with unique geometric features, 15,16 and the analytical approach by fixing a certain DOF. 17 We denote the configuration space of an n-joint, seriallink articulated robot as Q, which consists of n-tuples element q ¼ ðq 1 ; q 2 ; .…”

The Use of 3-DOF Laser Interferometer for Rapid Estimation of CNC Drives Settings

“…There are several models useful for describing geometric errors. Among the most popular are the Screw Theory ST [22,23], the Homogeneous Transfer Model HTM [24] and the Differential Motion Matrix DMM [25,26]. Since we found the models too complex to accommodate measurement data from a linear laser interferometer, we decided to build our own model based on DMM.…”

Vision-based measurement for decoupling identification of geometric errors of rotating axes for five-axis platform

“…This is assumed because the position of the axes does not affect the result of the transformation when the directions are rotated. In equation (38), r 1 and r 2 represent unit direction vectors.…”

“…Based on equation (41), η, λ, and µ are obtained accordingly. By employing the Paden-Kahan subproblem adoption [38], α a and γ a are calculated as: Posture calibration of the rotation axis. Note that the coordinate is the 2-norm of the difference between the actual pose matrix and the ideal pose matrix before and after calibration.…”