纠正刚体运动建模中配置空间错误，确保约束精确满足
A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body
- 从几何角度分析刚体运动的配置空间选择问题
- 证明仅当约束构成子群时数值积分才能精确保持约束
- 建议使用SE(3)而非SO(3)×R³作为刚体建模的配置空间
全息约束刚体的动力学可用带几何约束的牛顿-欧拉方程建模，常被表述为1阶微分代数方程（DAE）系统。在多体系统（MBS）动力学中，通常采用常微分方程积分方法求解，并将刚体运动置于直接积李群SO(3)×R³上，尽管刚体运动实际构成半直积李群SE(3)。已有观察表明，约束满足性依赖于所选的配置空间（c-space）。本文从几何视角分析该问题，证明：若约束在配置空间中构成子群，则数值积分方案可精确满足约束。SE(3)的子群对机械系统建模具有重要意义，包括低副（Reuleaux对）结构，且在标准MBS建模中已被隐式使用。结论是：对于约束刚体的数值DAE建模，应采用SE(3)作为合适的配置空间。但此结论不直接适用于整体多体系统。
原文摘要 · Abstract (English)
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. Inmultibody 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 SO (3)R3, although rigid body motions form the semidirect product Lie group 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 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 SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.
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