Screw and Lie group theory in multibody dynamics

A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input-output passive. We propose passivity-preserving splitting methods for the numerical solution of a class of problems which includes this system as a special case. We prove the passivitypreservation property for the splitting methods, and we investigate stability and energy behaviour in numerical experiments. Implementation is discussed in detail for a special case where the splitting gives rise to the subsequent integration of two completely integrable flows. The equations for the attitude are reformulated on SO(3) using rotation matrices rather than local parametrizations with Euler angles.

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“…and from (24) and (25) we deduce that the rigid body Coriolis-centripetal matrix in this case can be rewritten in the form…”

Section: A Vessel Rigid Body Modelmentioning

confidence: 97%

“…is the hat-map, and so (3) is the Lie algebra of SO (3), consisting of 3 × 3 skew-symmetric matrices, for further details see [25]. Following [19] (p. 151) and [20], equations (1) can be obtained defining the kinetic energy of the system to be…”

Section: A Vessel Rigid Body Modelmentioning

confidence: 99%

Passivity-preserving splitting methods for rigid body systems

Celledoni¹,

Høiseth²,

Ramzina³

2018

Multibody Syst Dyn

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“…His original applications were kinematics and one of the most important theorems in the area, Chasles' theorem, states that the most general rigid body displacement can be described by a screw transformation. More recently screw theory, and the highly related study of dual quaternions, has been applied to robotics, computational geometry and multibody dynamics [3,10,13]. Screw transformations consist of a translation along an axis and a rotation around that axis.…”

Section: Screw Theorymentioning

confidence: 99%

Direct Linear Interpolation of Geometric Objects in Conformal Geometric Algebra

Hadfield

Lasenby

2019

Adv. Appl. Clifford Algebras

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Typically we do not add objects in conformal geometric algebra (CGA), rather we apply operations that preserve grade, usually via rotors, such as rotation, translation, dilation, or via reflection and inversion. However, here we show that direct linear interpolation of conformal geometric objects can be both intuitive and of practical use. We present a method that generates useful interpolations of point pairs, lines, circles, planes and spheres and describe algorithms and proofs of interest for computer vision applications that use this direct averaging of geometric objects. * Corresponding author. 1 There are a range of different notations in use in the literature to define the basis of CGA: e is sometimes referred to in other works as e + ,ē as e − , n∞ as e∞ and an eo is sometimes defined which is equal to −n 0 .

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“…. It is used in MBS dynamics modeling [68], basically because when using the mixed twist, the Newton-Euler equations with respect to the COM are decoupled and because the body-fixed inertia tensor is constant (see also the companion paper [50]). The mixed twist is readily found as…”

Section: Mixed Twistsmentioning

confidence: 99%

“…with X m := X h and the matrix A m as in (44) but with the Ad r i,j replaced by the matrix in (50). This allows for a closed-form inversion of A m analogous to that of A h .…”

Section: Mixed System Jacobian and Its Decompositionmentioning

confidence: 99%

Screw and Lie group theory in multibody kinematics

Müller

2017

Multibody Syst Dyn

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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 and 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(n) 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, that is, 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 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.

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Screw and Lie group theory in multibody dynamics

Cited by 50 publications

References 89 publications

Passivity-preserving splitting methods for rigid body systems

Passivity-preserving splitting methods for rigid body systems

Direct Linear Interpolation of Geometric Objects in Conformal Geometric Algebra

Screw and Lie group theory in multibody kinematics

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