Direct Kinematics, Inverse Kinematics, and Motion Planning of 1-DoF Rational Linkages

Abstract

This study presents a set of algorithms that deal with trajectory planning of rational single-loop mechanisms with one degree of freedom (DoF). Benefiting from a dual quaternion representation of a rational motion, a formula for direct (forward) kinematics, a numerical inverse kinematics algorithm, and the generation of a driving-joint trajectory are provided. A novel approach using the Gauss-Newton search for the one-parameter inverse kinematics problem is presented. Additionally, a method for performing smooth equidistant travel of the tool is provided by applying arc-length reparameterization. This general approach can be applied to one-DoF mechanisms with four to seven joints characterized by a rational motion, without any additional geometrical analysis. An experiment was performed to demonstrate the usage in a laboratory setup.

This is the final published version of the article, available as open access in Mechanism and Machine Theory, Volume 213, 2025. DOI: 10.1016/j.mechmachtheory.2025.106074. Licensed under CC BY license