the parameterization of rotation and rigid motion€¦ · proper rotation): results translate...

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The Parameterization of The Parameterization of Rotation and Rigid MotionRotation and Rigid Motion

An Attempt at a Systematic An Attempt at a Systematic FrameworkFramework

Lorenzo TrainelliDepartment of Aerospace Engineering

Politecnico di Milano, Italy

6th World 6th World CongressCongress on on ComputationalComputational MechanicsMechanics

BeijingBeijing, , SeptemberSeptember 66--1010, 200, 20044

Politecnico di Milano – Aerospace Engineering Dept.

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Presentation OutlinePresentation Outline

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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Step 1Step 1

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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IntroductionIntroduction• The representation and parametrization of (finite) rotation and

general rigid motion impacts on the analysis of systems characterized by an independent rotation fieldindependent rotation field along with the customary (linear) displacement field (e.g. nonlinear structural dynamics and multibody dynamics)

•• ImplicationsImplications range from the theoretical/didactical to the computational

• A comprehensive view on the subject is hardly commonhardly common within the scientific community– Limited perception of the relationshipsrelationships between a specific

technique and others possible– Specific features of a methodology can be undulyunduly related to the

way rotations are formulated, eventually ending up with either under- or overestimating the impactimpact of a certain choice

– Extremely limited awareness of the extensionextension of rotation parameterization to arbitrary rigid motion


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MotivationMotivation

• The parameterization of rotation and its extension to complete rigid motion play a crucial role in the design of practical geometric integratorsgeometric integrators for nonlinear dynamics, i.e. algorithms that are capable of preserving qualitativequalitative features of the solution–– Frame indifferenceFrame indifference (wrt global frame, material reference entity, …)–– Configuration manifoldConfiguration manifold (position and hidden constraints, …)– Linear and angular momentamomenta (constraints: Newton’s 3rd law, …)– Mechanical energyenergy (constraints: ideal scleronomic behavior)

• We try to present an up-to-date comprehensivecomprehensive picture– We deal with rotation and rigid motion within a common frameworkcommon framework,

summing up the available parameterization techniques, matching together ‘classical’ results with recent developments

– We present the novel vectorialvectorial parameterization of rotationparameterization of rotation– We systematically extendextend rotation results to arbitrary rigid motion


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Step 2Step 2

• Introduction and Motivation•• Basic concepts for RotationBasic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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Rotation Rotation BasicsBasics –– 11

• Consider a rotation of a body about a fixed point o:point x is brougth towhere R is the rotation tensorrotation tensor (i.e. a special orthogonal transformation)

• The velocity of y iswhere ω is the angular angular velocityvelocity

• The angular velocity enters the relation

– is the skew-symmetric tensor corresponding to vector ω


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Rotation Rotation BasicsBasics –– 22

•• Euler’s fundamental theoremEuler’s fundamental theorem on rotation states that:“any rigid motion leaving a point fixed may be represented by a planar rotation about a suitable axis passing through that point”Therefore, a minimalminimal representation is given by the pair (ϕ,e)where ϕ is the rotation angle and e the unit vector of the rotation axis (3 scalar parameters)

•• EulerEuler--RodriguesRodrigues’ formula’ formula provides the link between R and (ϕ,e):

– Note that and– The angular velocity in terms of the pair (ϕ,e) reads


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Step 3Step 3

• Introduction and Motivation• Basic concepts for Rotation•• VectorialVectorial parameterizations of Rotationparameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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ExponentialExponential ParameterizationParameterization(Rotation)(Rotation)

• Euler-Rodrigues’ formula inspiresinspires a naturalnatural parameterization based on the exponential map– Define the rotation vectorrotation vector as– The exponential map of yields Euler-Rodrigues’ formula

– Angular velocity recovered by the associatedassociated differential map

– The exponential map and its associated differential map enjoy special propertiesspecial properties:


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VectorialVectorial ParameterizationParameterization(Rotation)(Rotation)

• The exponential map can be considerably generalized:– Define the generating functiongenerating function

This is anyany odd function of ϕ with limit behavior

– Define the parameter rotation vectorparameter rotation vector as

• The vectorialvectorial parameterization mapparameterization map reads

(Bauchau and Trainelli 2003)

– coefficients are defined as


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VectorialVectorial ParameterizationParameterization(Rotation)(Rotation)

• The vectorial parameterization map possesses qualitative properties identicalidentical to the exponential map, therefore gives rise to a parallel formalism– Angular velocity recovered by the associatedassociated differential map

where yielding

– The vectorial parameterization map and its associated differential map enjoy special propertiesspecial properties:


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SpecificSpecific TechniquesTechniques• The vectorial parameterization of rotation defines a general classgeneral class

encompassing a number of known techniquesknown techniques– The generating function is the only choice to be made– Known parameterizations that fall into the class:

rotation vector/Euler vector (exponential map)Gibbs-Rodrigues parameters (Cayley transform)Wiener-Milenkovic parameters (CRV)Linear parametersreduced Euler parameters (→ quaternions)

............................... others (e.g. Tsiotras et al. 1997)

• The vectorial parameterization framework allows the design design of new parameterizationsof new parameterizations at will– For example, to satisfy given algorithmic requirements

(Bauchau and Trainelli 2003)


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Step 4Step 4

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation•• Representation of Rigid MotionRepresentation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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RigidRigid MotionMotion BasicsBasics –– 11

•• Various choices Various choices to describe complete rigid motion (i.e. coupled translation and rotation). Among them:–– screw displacementscrew displacement (same axis for rotation and translation)–– base pole descriptionbase pole description (fixed point as the center of rotation)

• Base pole description of rigid motion:any rigid displacement can be described by a rotation R about a fixed pointfixed point o (the ‘base pole’) followed by a uniform translation t

• Point x is brought to

• The velocity of y is

where v is the base pole velocitybase pole velocity

– This is an ‘Eulerian’ quantity: it coincides with the velocity of the material point that passes through position o

– The instantaneous motioninstantaneous motion is characterized by


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A A usefuluseful formalismformalism: : dualdual numbersnumbers

• Some kind of formalism is in order for a compact representation of motion that preserves the intrinsic couplingintrinsic coupling between linear and angular components.Different (virtually equivalent) choices are at hand:

– 6-D vectors and tensors (Borri, Trainelli and Bottasso 1994 → 2003)– 3-D dual vectors and tensors (Study 1901, Bottema and Roth 1979)

Here we choose the latter, based on dual numbersdual numbers (Clifford 1873).

•• Dual number formalismDual number formalism: a dual number is defined aswhere the dual unity satisfies and

• Dual numbernumber productDual vectorvector productsEtc.


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•• RepresentationRepresentation of rigid motion using the dual number formalism• Define the (rigid) displacement tensor as

and the generalized velocity as

• Rigid kinematics, i.e. the pair of equations(1)(2)

is synthesized in the following equation:– Latter equation simply the dualizationdualization of (1)– This analogy and the following results can be formally justified in

terms of Lie groupLie group theory


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•• MozziMozzi--ChaslesChasles’ theorem’ theorem on rigid motion states that:“any rigid motion may be represented by a planar rotation about a suitable axis passing through that point, followed by a uniform translation along that same axis”Therefore, a minimalminimal representation is given by the set (ϕ,e,τ,a)where τ is the scalar translation and a any point along the Mozziaxis (6 scalar parameters in all)

• A rearrangement of (ϕ,e,τ,a) in dual format yields two quantities:–– Screw magnitudeScrew magnitude (a dual angle)–– Screw axis vectorScrew axis vector (a dual vector)

where is the moment of the line (e,a)Note: h coincides with the Plücker coordinates of the line (e,a)

• The minimalminimal representation is then given by the pair (Φ,h)


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RigidRigid MotionMotion BasicsBasics –– 44• The link between D and (Φ,h) is elegantly provided by the

generalized Eulergeneralized Euler--RodriguesRodrigues formulaformula:

– dualization of

• Starting from this formula, the going gets easy:all formulae developed for rotation are valid for complete rigidmotion by simply substitutingsubstituting corresponding quantities.

For example, the generalized velocity in terms of the pair (Φ,h) reads

– dualization of


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Step 5Step 5

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion•• VectorialVectorial parameterizations of Rigid Motionparameterizations of Rigid Motion• Non-vectorial Parameterizations• Concluding remarks


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VectorialVectorial ParameterizationParameterization((RigidRigid MotionMotion))

• The exponential mapexponential map of rotation is immediately generalized to rigid motion, mimicking the case of rotation with dual quantities– Define the displacement vectordisplacement vector as

– yields the generalized Euler-Rodrigues’ formula

– yields the generalized velocity

• The same holds for the vectorialvectorial parameterization– Define the parameter displacement vector as

where is the screw parameter magnitude– The vectorial parameterization map is

– Every other result simply follows by dualization of rotation formulae


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Step 6Step 6

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion•• NonNon--vectorialvectorial ParameterizationsParameterizations• Concluding remarks


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NonNon--VectorialVectorial ParameterizationsParameterizations

• The parameters do notdo not form a frame-invariant vector or tensor•• RotationRotation: among all possible non-vectorial parameterizations 2

classes of techniques have been widely employed to date:– Eulerian angles (Euler angles, Cardan angles, Bryant angles, etc.)– Unit quaternions (Euler-Rodrigues parameters)

•• EulerianEulerian anglesangles represent a minimalminimal parameterization of rotation (i.e. 3 scalars)– It can be extended to full rigid motion (minimal: 6 scalars)

•• Unit Unit quaternionsquaternions represent a 11--redundantredundant parameterization of rotation (i.e. 4 scalars)– It can also be extended to rigid motion (2-redundant: 8 scalars)–– CloselyClosely related to the vectorial parameterization

• Next slides focus on the ‘dualization’ of Eulerian angles:EulerianEulerian ‘screws’‘screws’ (Bottema and Roth 1979)


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EulerianEulerian AnglesAngles(Rotation)(Rotation)

• We refer to the classical EulerEuler sequence (i.e. precession, nutation, proper rotation): results translate easily to any sequence

– Initial triad– Rotated triad , i.e.– Nodal line

• The angular velocity reads

where

– The angular velocity is obtained by sum of the Euler angle derivatives about their corresponding axis


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EulerianEulerian ‘‘Screws’Screws’((RigidRigid MotionMotion))

• An arbitrary rigid motion can be decomposed into subsequent screw motions along the EulerEuler sequence rotation axes defined by its rotational part

– The dual angles are screw magnitudes with as scalar translations along

– Remarkably, this implies

• The generalized velocity reads

– This entails


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Last stepLast step

• Introduction and Motivation• Basic concepts for Rotation• Vectorial parameterizations of Rotation• Representation of Rigid Motion• Vectorial parameterizations of Rigid Motion• Non-vectorial Parameterizations•• Concluding remarksConcluding remarks


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ConcludingConcluding RemarksRemarks• We presented, within a coherent, unified settingcoherent, unified setting, the most

important parameterizations of rotation– Including the vectorialvectorial parameterization of rotationparameterization of rotation, a class of

techniques that encompasses and extends many known parameterizations developed to date

• We systematically generalizedgeneralized the parameterizations of rotation to the case of general rigid motion, involving coupledcoupledtranslation and rotation– We reviewed some known results that have nothave not penetrated, to

date, the scientific community at large– Besides being elegant and theoretically/didactically relevant,

these extended parameterizations cancan be usefully employed in applications in computational mechanics without heavy coding overheads, given their commoncommon formalism with rotation

– In particular, they can be employed in the design of nonconventional invariantinvariant--preservingpreserving numerical procedures (Borri, Bottasso, Trainelli 1994 → today), due to the preservation of the inherent couplinginherent coupling between translation and rotation (this gets lost when separate discretizations are used)

the parameterization of rotation and rigid motion€¦ · proper rotation): results translate...

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