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 as well as 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\left( n\right) $ 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, i.e. 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 the 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.

翻译：经过三十年的计算多体系统（MBS）动力学发展，当前研究的重点在于开发既紧凑用户友好又计算高效的复杂MBS分析公式。其关键在于采用整体几何方法进行运动学建模，认识到刚体的一般运动以及技术关节引起的相对运动均为螺旋运动。此外，螺旋理论提供了几何框架，而李群理论则为直观紧凑的MBS建模提供了分析基础。这种建模方法固有的框架不变性催生了高效的递归$O\left( n\right) $算法（其中"空间算子代数"是典型示例），并允许直接利用现有几何数据。本文提出了三种基于相对坐标（即关节变量）描述树形拓扑MBS构形的变体：使用体固联关节坐标系的标准公式、无关节坐标系的公式、以及既无关节坐标系也无体固联参考系的公式。这些方法无需引入关节参考系即可描述MBS运动学，从而消除了对Denavit-Hartenberg参数等限制性建模惯例的依赖。本文回顾了四种不同的旋量定义并推导了相应的递归表达式，同时给出了对应的雅可比矩阵及其分解形式。本文旨在推广李群建模方法，并从螺旋理论与李群理论的统一视角综述基于相对（关节）坐标的树形拓扑MBS运动学不同公式。