Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS) while at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows for use of arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper recursive $O\left( n\right) $ Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. The MBS motion equations are derived in closed form using the Lie group formulation. One are the so-called 'Euler-Jourdain' or 'projection' equations, of which Kane's equations are a special case, and the other are the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

翻译：螺旋理论与李群方法能够实现多体系统的友好建模，同时衍生出计算高效的递归算法。此类公式固有的参考系不变性允许在运动学建模中使用任意参考系（而非遵循诸如Denavit-Hartenberg约定等建模规范），并避免引入关节坐标系。其计算效率源于旋量、加速度与力旋量的表示方法最大限度地降低了计算开销，这一特性可直接推广至动力学公式。本文针对旋量的四种最常用表示方法推导了递归$O\left( n\right) $牛顿-欧拉算法，并讨论了各自特性，这些公式与文献中已有算法相关联。通过李群公式推导了多体系统运动方程的闭式形式，其一为"欧拉-茹尔丹"（即"投影"）方程（凯恩方程为其特例），另一为拉格朗日方程。递归运动学公式可便捷地推广至高阶以计算运动方程导数，为此推导了加速度与加加速度的递归公式，并简要讨论了如何将其应用于线性化运动方程及其时间导数的推导。几何建模方法可直接应用李群积分技术，本文对此亦作简要探讨。