Dipartimento di Ingegneria Aerospaziale
Politecnico di Milano
The Parameterization of
Rotation and Rigid Motion
An Attempt at a Systematic
Framework
Lorenzo Trainelli
Department of Aerospace Engineering
Politecnico di Milano, Italy
6th World Congress on Computational Mechanics
Beijing, September 6-10, 2004
Parametrization of Rotation and Rigid Motion
Presentation Outline
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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
Politecnico di Milano – Aerospace Engineering Dept.
2/27
Parametrization of Rotation and Rigid Motion
Step 1
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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
Politecnico di Milano – Aerospace Engineering Dept.
3/27
Parametrization of Rotation and Rigid Motion
Introduction
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The representation and parametrization of (finite) rotation and
general rigid motion impacts on the analysis of systems
characterized by an independent rotation field along with the
customary (linear) displacement field (e.g. nonlinear structural
dynamics and multibody dynamics)
• Implications range from the theoretical/didactical to the
computational
• A comprehensive view on the subject is hardly common within
the scientific community
– Limited perception of the relationships between a specific
technique and others possible
– Specific features of a methodology can be unduly related to the
way rotations are formulated, eventually ending up with either
under- or overestimating the impact of a certain choice
– Extremely limited awareness of the extension of rotation
parameterization to arbitrary rigid motion
Politecnico di Milano – Aerospace Engineering Dept.
4/27
Parametrization of Rotation and Rigid Motion
Motivation
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The parameterization of rotation and its extension to complete
rigid motion play a crucial role in the design of practical
geometric integrators for nonlinear dynamics, i.e. algorithms that
are capable of preserving qualitative features of the solution
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Frame indifference (wrt global frame, material reference entity, …)
Configuration manifold (position and hidden constraints, …)
Linear and angular momenta (constraints: Newton’s 3rd law, …)
Mechanical energy (constraints: ideal scleronomic behavior)
We try to present an up-to-date comprehensive picture
– We deal with rotation and rigid motion within a common framework,
framework
summing up the available parameterization techniques, matching
together ‘classical’ results with recent developments
– We present the novel vectorial parameterization of rotation
– We systematically extend rotation results to arbitrary rigid motion
Politecnico di Milano – Aerospace Engineering Dept.
5/27
Parametrization of Rotation and Rigid Motion
Step 2
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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
Politecnico di Milano – Aerospace Engineering Dept.
6/27
Parametrization of Rotation and Rigid Motion
Rotation Basics – 1
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Consider a rotation of a body about a fixed point o:
point x is brougth to
where R is the rotation tensor (i.e. a special orthogonal
transformation)
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The velocity of y is
where ω is the angular velocity
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The angular velocity enters the relation
–
is the skew-symmetric tensor corresponding to vector ω
Politecnico di Milano – Aerospace Engineering Dept.
7/27
Parametrization of Rotation and Rigid Motion
Rotation Basics – 2
• Euler’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 minimal 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)
• Euler-Rodrigues’ formula provides the link between R and (ϕ,e):
– Note that
and
– The angular velocity in terms of the pair (ϕ,e) reads
Politecnico di Milano – Aerospace Engineering Dept.
8/27
Parametrization of Rotation and Rigid Motion
Step 3
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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
Politecnico di Milano – Aerospace Engineering Dept.
9/27
Parametrization of Rotation and Rigid Motion
Exponential Parameterization
(Rotation)
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Euler-Rodrigues’ formula inspires a natural parameterization
based on the exponential map
– Define the rotation vector as
– The exponential map of
yields Euler-Rodrigues’ formula
– Angular velocity recovered by the associated differential map
– The exponential map and its associated differential map enjoy
special properties:
properties
Politecnico di Milano – Aerospace Engineering Dept.
10/27
Parametrization of Rotation and Rigid Motion
Vectorial Parameterization
(Rotation)
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The exponential map can be considerably generalized:
– Define the generating function
This is any odd function of ϕ with limit behavior
– Define the parameter rotation vector as
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The vectorial parameterization map reads
(Bauchau and Trainelli 2003)
– coefficients are defined as
Politecnico di Milano – Aerospace Engineering Dept.
11/27
Parametrization of Rotation and Rigid Motion
Vectorial Parameterization
(Rotation)
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The vectorial parameterization map possesses qualitative
properties identical to the exponential map, therefore gives rise
to a parallel formalism
– Angular velocity recovered by the associated differential map
where
yielding
– The vectorial parameterization map and its associated differential
map enjoy special properties:
properties
Politecnico di Milano – Aerospace Engineering Dept.
12/27
Parametrization of Rotation and Rigid Motion
Specific Techniques
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The vectorial parameterization of rotation defines a general class
encompassing a number of known 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 parameters
reduced Euler parameters (→ quaternions)
............................... others (e.g. Tsiotras et al. 1997)
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The vectorial parameterization framework allows the design
of new parameterizations at will
– For example, to satisfy given algorithmic requirements
(Bauchau and Trainelli 2003)
Politecnico di Milano – Aerospace Engineering Dept.
13/27
Parametrization of Rotation and Rigid Motion
Step 4
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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
Politecnico di Milano – Aerospace Engineering Dept.
14/27
Parametrization of Rotation and Rigid Motion
Rigid Motion Basics – 1
• Various choices to describe complete rigid motion (i.e. coupled
translation and rotation). Among them:
– screw displacement (same axis for rotation and translation)
– base pole description (fixed point as the center of rotation)
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Base pole description of rigid motion:
any rigid displacement can be described by a rotation R about a
fixed point o (the ‘base pole’) followed by a uniform translation t
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Point x is brought to
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The velocity of y is
where v is the base pole velocity
– This is an ‘Eulerian’ quantity: it coincides with the velocity of the
material point that passes through position o
– The instantaneous motion is characterized by
Politecnico di Milano – Aerospace Engineering Dept.
15/27
Parametrization of Rotation and Rigid Motion
A useful formalism: dual numbers
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Some kind of formalism is in order for a compact representation
of motion that preserves the intrinsic 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 numbers (Clifford 1873).
• Dual number formalism:
formalism a dual number is defined as
where the dual unity satisfies
and
• Dual number product
Dual vector products
Etc.
Politecnico di Milano – Aerospace Engineering Dept.
16/27
Parametrization of Rotation and Rigid Motion
Rigid Motion Basics – 2
• Representation of rigid motion using the dual number formalism
• Define the (rigid) displacement tensor as
and the generalized velocity as
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Rigid kinematics, i.e. the pair of equations
(1)
(2)
is synthesized in the following equation:
– Latter equation simply the dualization of (1)
– This analogy and the following results can be formally justified in
terms of Lie group theory
Politecnico di Milano – Aerospace Engineering Dept.
17/27
Parametrization of Rotation and Rigid Motion
Rigid Motion Basics – 3
• Mozzi-Chasles’ 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 minimal representation is given by the set (ϕ,e,τ,a)
where τ is the scalar translation and a any point along the Mozzi
axis (6 scalar parameters in all)
• A rearrangement of (ϕ,e,τ,a) in dual format yields two quantities:
– Screw magnitude (a dual angle)
– Screw 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 minimal representation is then given by the pair (Φ,h)
Politecnico di Milano – Aerospace Engineering Dept.
18/27
Parametrization of Rotation and Rigid Motion
Rigid Motion Basics – 4
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The link between D and (Φ,h) is elegantly provided by the
generalized Euler-Rodrigues formula:
formula
– dualization of
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Starting from this formula, the going gets easy:
all formulae developed for rotation are valid for complete rigid
motion by simply substituting corresponding quantities.
For example, the generalized velocity in terms of the pair (Φ,h) reads
– dualization of
Politecnico di Milano – Aerospace Engineering Dept.
19/27
Parametrization of Rotation and Rigid Motion
Step 5
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•
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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
Politecnico di Milano – Aerospace Engineering Dept.
20/27
Parametrization of Rotation and Rigid Motion
Vectorial Parameterization
(Rigid Motion)
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The exponential map of rotation is immediately generalized to rigid
motion, mimicking the case of rotation with dual quantities
– Define the displacement vector as
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–
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yields the generalized Euler-Rodrigues’ formula
yields the generalized velocity
The same holds for the vectorial 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
Politecnico di Milano – Aerospace Engineering Dept.
21/27
Parametrization of Rotation and Rigid Motion
Step 6
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•
•
•
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•
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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
Politecnico di Milano – Aerospace Engineering Dept.
22/27
Parametrization of Rotation and Rigid Motion
Non-Vectorial Parameterizations
• The parameters do not form a frame-invariant vector or tensor
• Rotation:
Rotation 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)
• Eulerian angles represent a minimal parameterization of rotation
(i.e. 3 scalars)
– It can be extended to full rigid motion (minimal: 6 scalars)
• Unit quaternions represent a 1-redundant parameterization of
rotation (i.e. 4 scalars)
– It can also be extended to rigid motion (2-redundant: 8 scalars)
– Closely related to the
vectorial parameterization
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Next slides focus on the ‘dualization’ of Eulerian angles:
Eulerian ‘screws’ (Bottema and Roth 1979)
Politecnico di Milano – Aerospace Engineering Dept.
23/27
Parametrization of Rotation and Rigid Motion
Eulerian Angles
(Rotation)
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We refer to the classical Euler sequence (i.e. precession, nutation,
proper rotation): results translate easily to any sequence
– Initial triad
– Rotated triad
– Nodal line
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, i.e.
The angular velocity reads
where
– The angular velocity is obtained by sum of the Euler angle derivatives
about their corresponding axis
Politecnico di Milano – Aerospace Engineering Dept.
24/27
Parametrization of Rotation and Rigid Motion
Eulerian ‘Screws’
(Rigid Motion)
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An arbitrary rigid motion can be decomposed into subsequent
screw motions along the Euler sequence rotation axes defined
by its rotational part
– The dual angles
screw magnitudes with
– Remarkably, this implies
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are
as scalar translations along
The generalized velocity reads
– This entails
Politecnico di Milano – Aerospace Engineering Dept.
25/27
Parametrization of Rotation and Rigid Motion
Last step
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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
Politecnico di Milano – Aerospace Engineering Dept.
26/27
Parametrization of Rotation and Rigid Motion
Concluding Remarks
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We presented, within a coherent, unified setting,
setting the most
important parameterizations of rotation
– Including the vectorial parameterization of rotation,
rotation a class of
techniques that encompasses and extends many known
parameterizations developed to date
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We systematically generalized the parameterizations of
rotation to the case of general rigid motion, involving coupled
translation and rotation
– We reviewed some known results that have not penetrated, to
date, the scientific community at large
– Besides being elegant and theoretically/didactically relevant,
these extended parameterizations can be usefully employed in
applications in computational mechanics without heavy coding
overheads, given their common formalism with rotation
– In particular, they can be employed in the design of
nonconventional invariant-preserving numerical procedures
(Borri, Bottasso, Trainelli 1994 → today), due to the preservation
of the inherent coupling between translation and rotation (this
gets lost when separate discretizations are used)
Politecnico di Milano – Aerospace Engineering Dept.
27/27