Screw and Lie Group Theory in Multibody Dynamics -- Recursive Algorithms and Equations of Motion of Tree-Topology Systems

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.

References in corpus (3)

• Screw and Lie Group Theory in Multibody Kinematics -- Motion Representation and Recursive Kinematics of Tree-Topology Systems
• The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems
• A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

Cited by in corpus (9)

• Screw and Lie Group Theory in Multibody Kinematics -- Motion Representation and Recursive Kinematics of Tree-Topology Systems
• An Overview of Formulae for the Higher-Order Kinematics of Lower-Pair Chains with Applications in Robotics and Mechanism Theory
• Review of the Exponential and Cayley Map on SE(3) as relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems
• Dynamics of Parallel Manipulators with Hybrid Complex Limbs -- Modular Modeling and Parallel Computing
• An $O(n$)-Algorithm for the Higher-Order Kinematics and Inverse Dynamics of Serial Manipulators using Spatial Representation of Twists
• A Constraint Embedding Approach for Dynamics Modeling of Parallel Kinematic Manipulators with Hybrid Limbs
• Closed Form Time Derivatives of the Equations of Motion of Rigid Body Systems
• A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of parallel Kinematic Manipulators
• Passivity-preserving splitting methods for rigid body systems