Title: Screw and Lie Group Theory in Multibody Dynamics -- Recursive Algorithms and Equations of Motion of Tree-Topology Systems Show Chinese title

Title: 螺钉和李群理论在多体动力学中的应用——树结构系统的递归算法和运动方程

Abstract: 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. Show Chinese abstract

Abstract: 螺旋和李群理论允许用户友好的多体系统（MBS）建模，同时它们也产生了计算高效的递归算法。这种公式的固有框架不变性允许在运动学建模中使用任意参考框架（而不是遵循如Denavit-Hartenberg约定等建模惯例），并避免引入关节框架。计算效率来自于对螺旋、加速度和力偶的表示，这种表示最小化了计算工作量。这可以直接应用于动力学公式。在本文中，针对螺旋的四种最常用表示法推导了递归$O\left( n\right) $牛顿-欧拉算法，并讨论了它们的特定特征。这些公式与文献中提出的相应算法有关。使用李群公式推导了MBS运动方程的显式形式。一种是所谓的“欧拉-若尔当”或“投影”方程，其中Kane方程是一个特例，另一种是拉格朗日方程。递归运动学公式可以轻松扩展到更高阶以计算运动方程的导数。为此，推导了加速度和急动度的递归公式。简要讨论了如何利用这一点来推导线性化运动方程及其时间导数。几何建模允许直接应用李群积分方法，这被简要讨论了。