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 and 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 ( n ) 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, that is, 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 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.
Highlights
Computational multibody system (MBS) dynamics aims at mathematical formulations and efficient computational algorithms for the kinetic analysis of complex mechanical systems
The displacement field of a Cosserat beam, for instance, is a proper motion in E3 and modeled as motion in SE(3). This is an extension of the original work on geometrically exact beams and shells by Simo [70, 71], where the displacement field is modeled on SO(3) × R3. Another topic where Lie group theory is recently applied in MBS dynamics is the time integration
Lie group integration schemes were modified and applied to MBS models in absolute coordinate formulation [17], where the motions of individual bodies are described as a general screw motion that are constrained according to the interconnecting joints
Summary
Computational multibody system (MBS) dynamics aims at mathematical formulations and efficient computational algorithms for the kinetic analysis of complex mechanical systems. This is an extension of the original work on geometrically exact beams and shells by Simo [70, 71], where the displacement field is modeled on SO(3) × R3 Another topic where Lie group theory is recently applied in MBS dynamics is the time integration. To this end, Lie group integration schemes were modified and applied to MBS models in absolute coordinate formulation [17], where the motions of individual bodies are described as a general screw motion that are constrained according to the interconnecting joints. The aim of this paper is to provide a comprehensive summary of the basic concepts for modeling MBS in terms of relative coordinates using joint screws and to relate them to existing formulations that are scattered throughout the literature.
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