A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of parallel Kinematic Manipulators

TL;DR

This paper introduces a novel recursive Lie-group method to efficiently compute second derivatives of inverse dynamics in parallel kinematic manipulators, enabling advanced control strategies for such systems.

Contribution

It presents the first Lie-group based recursive algorithm for second derivatives of inverse dynamics in PKMs, exploiting their topology for computational efficiency.

Findings

Efficient computation of second derivatives demonstrated on a 6-DOF Gough-Stewart platform.

Application of the method to a planar PKM with flatness-based control.

Numerical results validate the proposed recursive approach.

Abstract

Series elastic actuators (SEA) were introduced for serial robotic arms. Their model-based trajectory tracking control requires the second time derivatives of the inverse dynamics solution, for which algorithms were proposed. Trajectory control of parallel kinematics manipulators (PKM) equipped with SEAs has not yet been pursued. Key element for this is the computationally efficient evaluation of the second time derivative of the inverse dynamics solution. This has not been presented in the literature, and is addressed in the present paper for the first time. The special topology of PKM is exploited reusing the recursive algorithms for evaluating the inverse dynamics of serial robots. A Lie group formulation is used and all relations are derived within this framework. Numerical results are presented for a 6-DOF Gough-Stewart platform (as part of an exoskeleton), and for a planar PKM when a flatness-based control scheme is applied.