A classical approach to the multibody systems (MBS) modeling is to use absolute coordinates, i.e., a set of (possibly redundant) coordinates that describe the absolute position and orientation of the individual bodies with respect to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as an alternative approach to the singularity-free time integration. At the same time, Lie group formulations of EOM naturally respect the geometry of spatial motions during integration. Lie group integration methods, operating directly on the configuration space Lie group, are incompatible with standard formulations of the EOM, and cannot be implemented in existing MBS simulation codes without a major restructuring. The contribution of this paper is twofold: (1) A framework for interfacing Lie group integrators to standard EOM formulations is presented. It allows describing MBS in terms of various absolute coordinates and at the same using Lie group integration schemes. (2) A method for consistently incorporating the geometry of rigid body motions into the evaluation of EOM in absolute coordinates integrated with standard vector space integration schemes. The direct product group and the semidirect product group SO(3)xR3 and the semidirect product group SE(3) are used for representing rigid body motions. The key element is the local-global transitions (LGT) transition map, which facilitates the update of (global) absolute coordinates in terms of the (local) coordinates on the Lie group. This LGT map is specific to the absolute coordinates, the local coordinates on the Lie group, and the Lie group used to represent rigid body configurations.

翻译：多体系统建模的一种经典方法是采用绝对坐标，即一组（可能冗余的）坐标，用于描述各个物体相对于惯性参考系的绝对位置和姿态。在运动方程的时间积分中，一个众所周知的问题是缺乏空间运动的无奇点参数化，通常通过使用单位四元数来解决。李群积分方法被提出作为无奇点时间积分的替代方案。同时，运动方程的李群表述在积分过程中自然地尊重空间运动的几何结构。直接在构型空间李群上操作的李群积分方法与运动方程的标准表述不兼容，并且无法在不进行重大重构的情况下在现有的多体系统仿真代码中实现。本文的贡献有两点：（1）提出了一个将李群积分器与标准运动方程表述对接的框架。该框架允许使用各种绝对坐标描述多体系统，同时采用李群积分方案。（2）提出了一种方法，用于将刚体运动的几何结构一致地纳入到采用标准向量空间积分方案进行积分的、基于绝对坐标的运动方程评估中。文中使用直积群SO(3)xR3和半直积群SE(3)来表示刚体运动。其中的关键要素是局部-全局转换映射，它促进了基于李群上的（局部）坐标来更新（全局）绝对坐标。该LGT映射是针对绝对坐标、李群上的局部坐标以及用于表示刚体构型的李群而特定的。