Derivatives of Screw Systems in Body-Fixed Representation

Terze

2016

Acta Mech

Self Cite

Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

Section: Acceleration Of Tree Topology Mbsmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Terze

2016

Acta Mech

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“…(4) and (5) is straightforward. The configuration of B α is the combina- (Brockett, 1984) that gave rise to very compact formulations for the mechanism kinematics and algorithms for the dynamics of MBS (Ploen and Park, 1997;Park, 1994) (Müller, 2014b, a). Furthermore, this provides the basis for higher-order kinematic analysis of mechanisms.…”

Section: Mechanism Configurationmentioning

confidence: 99%

Representation of the kinematic topology of mechanisms for kinematic analysis

2015

Mech. Sci.

Abstract. The kinematic modeling of multi-loop mechanisms requires a systematic representation of the kinematic topology, i.e. the arrangement of links and joints. A linear graph, called the topological graph, is used to this end. Various forms of this graph have been introduced for application in mechanism kinematics and multibody dynamics aiming at matrix formulations of the governing equations. For the (higher-order) kinematic analysis of mechanisms a simple yet stringent representation of the topological information is often sufficient. This paper proposes a simple concept and notation for use in kinematic analysis. Upon a topological graph, an order relation of links and joints is introduced allowing for recursive computation of the mechanism configuration. An ordering is also introduced on the topologically independent fundamental cycles. The latter is indispensable for formulating generically independent loop closure constraints. These are presented for linkages with only lower pairs, as well as for mechanisms with one higher kinematic pair per fundamental cycle. The corresponding formulation is known as cut-body and cut-joint approach, respectively.

“…Clearly, the closed-form expressions become very involved. Their explicit determination can be avoided by recursive evaluation [55].…”

Section: Body Fixed Representationmentioning

confidence: 99%

“…The same applies to higher derivatives (which, for instance, are required for motion planning). The explicit form of the νth-order partial derivative was presented in [55]:…”

Section: Body Fixed Representationmentioning

confidence: 99%

Screw and Lie group theory in multibody dynamics

2017

Multibody Syst Dyn

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Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS), and at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows to use 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(n) 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. Two forms of MBS motion equations are derived in closed form using the Lie group formulation: the so-called Euler-Jourdain or "projection" equations, of which Kane's equations are a special case, and 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.