Abstract
Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.
Highlights
Space kinematics is solely based on screw theory, and so is the kinematics of multibody systems (MBS)
Even though such concepts are not so widely known in the MBS community, which is an obstacle hindering a fruitful exploitation of such concepts for computational MBS dynamics
A recent trend in MBS dynamics is to employ the terminology and certain concepts of Lie groups noting that rigid body motions, i.e., finite frame transformations, form such a group possessing certain desirable properties
Summary
Space kinematics is solely based on screw theory, and so is the kinematics of multibody systems (MBS). The motivation to apply Lie group methods to the absolute and relative coordinate formulations is different The former aims to apply geometric Lie group integration methods, whereas the latter aims to employ the geometry of screw motions for the modeling purposes. This will be discussed in the paper. 2, the geometry of the rigid body motion is discussed and the configuration space is identified with a Lie group. This is the basis for modeling MBS with tree structure in terms of relative coordinates in Sect. The current trend and open issues are discussed
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