Screw and Lie Group Theory in Multibody Kinematics -- Motion Representation and Recursive Kinematics of Tree-Topology Systems

TL;DR

This paper advocates for using screw and Lie group theory to model multibody system kinematics, offering more efficient, frame-invariant recursive algorithms and multiple formulations that avoid restrictive conventions like Denavit-Hartenberg parameters.

Contribution

It introduces three variants for describing tree-topology multibody systems using Lie group theory, enhancing modeling flexibility and computational efficiency.

Findings

Presented three formulations for MBS kinematics without joint frames.

Derived recursive expressions for twists and Jacobians.

Reviewed multiple definitions of twists within Lie group framework.

Abstract

After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies as well as the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O\left( n\right) $ algorithms, for which the so-called 'spatial operator algebra' is one example, and allows for use of readily available geometric data. In this paper three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, i.e. joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit-Hartenberg parameters, redundant. Four different definitions of twists are recalled and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of the different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.