A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body
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- 作者：Andreas Müller
- 关键词：Constrained rigid body ; Numerical time integration ; Multibody dynamics ; Absolute coordinate formulation ; Rigid body kinematics ; Screws ; Lie groups ; Isotropy groups
- 刊名：BIT Numerical Mathematics
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：56
- 期：3
- 页码：995-1015
- 全文大小：1,385 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Computational Mathematics and Numerical Analysis
Numeric Computing
Mathematics - 出版者：Springer Netherlands
- ISSN：1572-9125
- 卷排序：56
文摘
The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group \(\textit{SO}\,(3) \times {\mathbb {R}}^{3}\), although rigid body motions form the semidirect product Lie group \(\textit{SE}\,(3) \). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of \(\textit{SE}\,(3) \) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that \(\textit{SE}\,(3) \) is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.