Direct kinematics, inverse kinematics, and motion planning of 1-DoF rational linkages

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. • Set of methods tailored for rational linkages. • Direct Kinematics based on dual quaternions and rational parameterization. • Numerical Inverse Kinematics algorithm using the Gauss–Newton search. • Motion planning for smooth travel of a tool through arc-length reparameterization.