Dr. Chris Kirtley

05-03-2001, 07:48 AM

Dear all,

I guess it's time I summarized the great quaternion debate. To remind

you, I

asked why quaternions (otherwise known as Euler parameters) seem to be

used as the standard method for representaing motion in computer

animation and video games, but are not so often used in biomechanics. It

seems that, whilst quaternions have advantages in terms of lack of

gymbal lock, and insensitivity to round-off errors, they suffer from

problems of interpretation in terms of meaningfully clinical or

anatomical angles.

As Joe Sommer and Bruce MacWilliams suggest, the best compromise is

perhaps to use

quaternions for intermediate calculations, then convert to Euler angles

at the end. I have made a summary of the various methods and provided

equations for conversion here:

http://guardian.curtin.edu.au/cga/faq/angles.html

Please let me know if you have any corrections or additions to this page

in the future.

Many thanks indeed to all contributors!

Chris Kirtley

--

Do you mean by quaternions so called Euler parameters? If so, you are

quite right. The four Euler parameters are extensively used in 3D

multibody simulations (instead of Euler angles) for the reasons you've

mentioned.

An example the software that uses the Euler parameters would be MADYMO.

It is a 3D package extensively used in so-called impact biomechanics.

Here, at NHTSA, we use it as one of the tools for injury assessment.

Erik

Erik G. Takhounts, Ph.D.

Senior Research Engineer

Conrad Technologies Inc.

501 School Street, S.W., Ste. 401

Washington, D.C. 20024

Ph. (202) 863-1931

Fax (202) 863-0909

Email: ETakhounts@NHTSA.DOT.GOV

--

There are Euler Angles (EA) vs. Euler Parameters (EP)

(Euler Parameters are sometimes known as normalized quarternions).

EA have only three independent rotational coordinates (the angles)

EP have nine independent direction cosines.

One would think EA simpler.

But the EA involve trigonometric functions and that is quite a

disadvantage.

Futher, still, there is a more severe problem:

for certain values of one of the angles (current notation calls it

theta)

the second and third angles cannot be distingquished and numerical

difficulties exists for these angles, making the programming more

difficult.

Other wise there really is no problem except for the implementation.

Tom Impelluso

--

In animation, the problems with Euler angles mainly arise when doing

motion editing. As long as you leave them alone they are fine.

Motion editing could be: resampling (interpolation), amplification,

motion blending etc. If you perform these operations on Euler angles

directly, you can get strange results, especially near gimbal lock.

The quaternion representation seems to behave better.

It seems that quaternions are the same as the "euler parameters" which

are often used in computational kinematics:

http://www.cs.berkeley.edu/~laura/cs184/quat/quaternion.html

I have also heard the term "angle-axis representation", i.e. the

rotation

is represented as a rotation of magnitude A about an axis (Ux,Uy,Uz).

Euler parameters are defined as follows

e0 = cos(A/2)

e1 = Ux*sin(A/2)

e2 = Uy*sin(A/2)

e3 = Uz*sin(A/2)

The sum of squares of these parameters is exactly one.

See also "Computer Aided Kinematics and Dynamics of Mechanical Systems",

by E.J. Haug.

Note that this representation is closely related to the three "helical

angles" proposed by Herman Woltring. The helical angle representation

is:

h1 = A*Ux

h2 = A*Uy

h3 = A*Uz

Why quaternions are not used more in biomechanics? This probably has

something to do with interpretation. Euler angles can be associated

with the rotations in a mechanical linkage or 3-D goniometer (Grood

and Suntay, J Biomech Eng, 1983). The other representations work well

for computation but are not so easily interpreted.

On the other hand, Woltring makes some good points on error propagation

in his 1994 paper (J Biomech 27:1399-1414). Near gimbal lock, Euler

angles

become increasingly sensitive to measuring errors.

For the newcomers on Biomch-L, I also recommend reading the debate

between

Grood and Woltring, about 10 years ago on Biomch-L:

http://isb.ri.ccf.org/biomch-l/files/angles3d.topic

Ton van den Bogert

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

--

I think Ton has explained it very well. Quaternions are not used because

they are difficult to interpret. Even the engineers will have difficulty

in

visualising what is happening, let alone doctors, physios etc. A

favourite

book of mine which explains everything is JM McCarthy, Introduction to

Theoretical Kinematics, MIT Press, 1990.

Raymond Lee

--

> So, I gather the problem is that the clinician (or sportsman) needs to

> interpret the results in terms of flexion-extension, ab-adduction and

> transverse rotation.

Yes, I think that's why the helical angles were never accepted.

Grood and Suntay is still the only rotation representation that is

used in human movement analysis.

By the way, this not only applies to joints but also to motion

relative to a global reference frame. Yeadon has defined the

terms somersault, tilt, and twist, which are Euler angles that

are consistent with existing coaching terminology.

> But couldn't quaternions be used until the final stage and then

> converted? Or would this still result in problems from the Euler

> representation?

No, then there would be no problem.

> Also, I confess I've never understood the difference between Grood and

> Suntay and Euler - is there a difference?

First, purists make a distinction between Euler angles and Cardanic

angles. Euler rotations are XYX, XZX, YXY, YZY, ZXZ, ZYZ, i.e. all

six sequences where the first and last rotation are about the same

coordinate axis. These were originally developed for celestial

mechanics,

i.e. the first rotation would be the orbit, the last the spin, and the

second the tilt of the axis. These have a singularity when the second

rotation is zero. Cardanic representations are XYZ, XZY,

YZX, YXZ, ZXY, ZYX. These six sequences have a singularity (Gimbal

lock)

when the second rotation is 90 degrees. All of these (Euler and Cardan

angles) are now loosely referred to as "Euler angles". In mechanics

textbooks you usually still find the distinction.

Yes, they are the same as Grood/Suntay. The matrix representation is

exactly the same. Grood really confused the issue by insisting that

this was not a sequence of rotations, but three simultaneous rotations.

But by arranging the mechanical linkage in a certain way, you

effectively

create a sequence. And the mathematics is exactly the same.

A.J. (Ton) van den Bogert, PhD

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

--

I standardly use quarternions for my measurements for all the reasons

you

indicated. But I would never consider publishing data or even casually

presenting results expressed in that form. I think the reason they don't

catch on, is you (or at least I) can't look at them and get an intuitive

feel for what's going on. You could probably make the same argument for

Euler anlges as you have to think about the sequence, but a degree unit

falling between -180 to +180 is somehow more intuitive. Probably just

conditioning, but I think we're adverse to "new tricks" such as this.

Bruce MacWilliams, Ph.D.

--

There is another reason why the animation world prefers quaternions over

Euler angles. Supposedly when you key frame (i.e., interpolate positions

and

orientations between several specified positions and orientations),

Euler

angles produce an unrealistic jerky motion whereas quaternions produce a

smooth motion. I have not actually tried this before to compare, but I

have

heard this comment from several sources in the past.

Also, as you point out, quaternions are not usually taught in most

biomechanics (or even engineering) curriculums. They require a redundant

parameter, which can be slightly inconvenient, but the benefit is the

elimination of gimbal lock (at least from a dynamic simulation

perspective).

Also, the four parameters are related to the screw axis between two

positions, so that can be helpful conceptually.

Computationally, you have to be careful with some singularity problems

when

converting between quaternions and direction cosine matrices (I can't

remember in which direction the problems occur). But other than that,

the

extra constraint equations relating the four parameters is a small price

to

pay for the benefits.

B.J.

B.J. Fregly, Ph.D.

Assistant Professor

Department of Aerospace Engineering,

Mechanics, and Engineering Science,

Biomedical Engineering Program, and

Dept. of Orthopaedics and Rehabilitation

University of Florida

Tel: (352) 392-8157

Fax: (352) 392-7303

E-mail: fregly@aero.ufl.edu

Home page: www.aero.ufl.edu/~fregly

--

1) Minor correction on nomenclature for rotation angle

sequences

The six possible Euler angle sets are XYX, XZX, YXY, YZY,

ZXZ and ZYZ where the first and third rotation axes repeat

(cyclic). Hence Euler angle representations experience

problems when the second rotation angle is zero or 180

degrees (i.e. the first and third axes are parallel). It

has been many years since I saw Euler's work, but I believe

that he used the ZXZ set.

The six possible Cardan-Bryant angle sets are XYZ, XZY, YXZ,

YZX, ZXY and ZYX where no axis repeats (anticyclic). Hence

Cardan-Bryant angle representations experience problems when

the second rotation is 90 or 270 degrees (e.g. the first and

third axes are parallel).

Craig (1989) provides 3x3 rotation matrices for all of the

rotation sets. He, like many others, refers to all twelve sets

simply as Euler angles.

2) Comments on rotation angles versus quaternions

In biomechanics, we face two major problems in regard to

describing both absolute attitude of an anatomical segment

or a camera with respect to an inertial frame or relative

attitude between two segments across an anatomical joint.

Our descriptions must be both mathematically tractable and

clinically relevant.

To analyze machinery, engineers often prefer unit

quaternions for absolute attitude of components (Haug, 1989,

and Kuipers, 1999) or rotation matrices across mechanical

joints (Denavit and Hartenberg, 1955) in that they often do

not need to describe the motion clinically.

As many biomechanics researchers know, rotation angles are

interesting for clinical description but can become

intractable for absolute attitude and for relative attitude

across joints that permit large angular excursions in all

three rotations.

Grood and Suntay's (1983) mechanical analog for the knee was

an important contribution that helped describe relative

attitude in clinically relevant terms. Their approach can

be adopted for many other joints (ankle, wrist) but has

difficulty for others (shoulder).

Concomitantly, I strongly support the efforts of ISB to

develop working groups to recommend "best practice" for

standardized description of attitude across specific joints.

If a working group of scholars can provide clinically

relevant descriptions of attitude using either angle sets or

quaternions, I agree with you that we should take their

"advice as to when a particular method is best".

Have any members of the ISB International Shoulder Group

(http://isb.ri.ccf.org/groups.html) provided input?

The following link also provides a simple discussion

oriented more toward camera imaging but it is still quite

applicable and also provides code snippets.

http://www.gamasutra.com/features/19980703/quaternions_01.htm

3) Comments on screw "helical" axes

For some reason, the biomechanics community has steadfastly

adopted the nomenclature "helical axes" to describe screw

kinematics as postulated by Poinsot and Chasles and later

formalized by Ball (1900). Unfortunately the rich

literature bases in both mathematics and engineering do not

use this "helical" nomenclature at all.

The axis direction and rotation of the displacement screw

axis (DSA) - also know as the finite helical axis (FHA) -

is the same as the direction and rotation of unit quaternion

components. Hence screw kinematics are not superior to

quaternions for representing attitude (absolute or relative)

or attitude displacements. They do however provide an

elegant method to combine the description of location and

attitude (absolute or relative) or location and attitude

displacements.

In particular, geometry of the screw axode surface swept by

the DSA as an anatomical joint moves through its range of

motion should be invariant to size and direction of

displacements as well as joint speed and acceleration.

Hence inspecting anatomical axodes (or deviation from

expected norms) should be able to help identify kinematic

irregularities or joint laxity. Unfortunately, while screw

axodes are quite useful to analyze machine joints, axode

geometric invariants are far too sensitive to experimental

measurement noise for routine clinical biomechanics today.

Further, axode invariants are much harder to visualize and

describe clinically than even quaternions.

4) Personal recommendation

As an engineer, I prefer a combination of Euler parameters

which are unit quaternions and orthonormal rotation matrices

for analysis. Unfortunately, I still prefer rotation angle

sets for clinical discussion.

After 25 years of studying both theoretical and experimental

kinematics (including Chasles' original paper for historical

inspiration), I still cannot say that any one method for

representation of attitude is absolutely the best.

Thanks for posting the summary of representations. If you

feel brave, you may wish to check Rooney's (1977, 1978)

summaries also.

Best wishes,

Joe Sommer

REFERENCES

Ball, R.S. (1900) A Treatise on the Theory of Screws,

Cambridge University Press

Craig, J.J. (1989) Introduction to Robotics, Addison-Wesley

Denavit, J. and Hartenberg, R.S. (1955) A kinematic notation

for lower pair mechanisms based on matrices. ASME J.

Applied Mechanics, 22:215-221

Grood, E.S. and Suntay, W.J. (1983) A joint coordinate

system for the clinical description of three-dimensional

motions: applications to the knee. ASME J. Biomech. Eng.,

105:136-144

Haug, E.J. (1989) Computer-Aided Kinematics and Dynamics of

Mechanical Systems, Allyn and Bacon

Kuipers, J.B. (1999) Quaternions and Rotation Sequences,

Princeton University Press

Rooney, J. (1977) A survey of representations of spatial

rotation about a fixed point. Environment and Planning B,

4:185-210

Rooney, J. (1978) A comparison of representations of general

spatial displacement. Environment and Planning B, 5:45-88

H.J. Sommer III, Ph.D., Professor of Mechanical Engineering

The Pennsylvania State University

337 Leonhard Building, University Park, PA 16802

(814)863-8997 FAX (814)865-9693

hjs1@psu.edu http://www.me.psu.edu/sommer

--

Hi Chris:

The advantages of Quaternions were described in my presentation in the

last 3D conference in South Africa.

The full presentation is at:

http://www.macrosport.com/sportsci/apasweb/presentations/capetown/demo_files

/frame.htm

Of course, the Quaternions is the way to go.

Gideon Ariel, Ph.D.

http://www.arielnet.com

--

as Ton already pointed out, there is an extensive explanation of the use

of quaternions (in their "euler parameter" representation) in "Computer

Aided Kinematics and Dynamics of Mechanical Systems", by E.J. Haug. In

this Book Haug introduces the theory the multi-body-simulation software

DADS is based on.

>From the theoretical point of view euler parameters

e0 = cos(A/2)

e1 = Ux*sin(A/2)

e2 = Uy*sin(A/2)

e3 = Uz*sin(A/2)

are an excellent tool to describe orientations and numerically solve

equations of motion of mechanical multi-body-systems, since there is no

gimbal lock, they are well defined on the unit-sphere (any orientation

of a rigid body can be uniquely/continuesly described by the set of (Ux,

Uy, Uz, A).

Whereas there is no way of integrating angular velocity w to obtain

orientation (since it's not integrable) one may integrate the time-

derivative of euler parameters to calculate e0(t), e1(t), e2(t), e3(t).

On the other hand euler parameters cannot be used to describe multiple

revolutions (uniqueness only on the unit-sphere) which is not too much

of a problem in biomechanics, since no joint range in biological systems

is greater. A problem with euler parameters is though, that it's almost

impossible to set up 3D-torque elements (e.g. for spherical joints).

Except for the trivial case where the joint torque always acts along

(Ux,Uy,Uz) it is very hard to describe a certain experimental behavior

with an euler parameter torque element. I tried this when I wanted to

limit the range of motion for a spherical joint but finally gave up,

since it was simply impossible (at least for me) to find the equations

using euler parameters. So I ended up using angles again.

The interpretaton of euler parameters is simple as long as one looks at

orientation only. But as soon as torque elements or experimental data

are to be described, interpretation seems to be impossible.

Arnim Henze.

Institut f"ur Astronomie und Astrophysik

Universit"at T"ubingen Tel.: ++49 7071 29 78654

Auf der Morgenstelle 10 Fax : ++49 7071 29 5889

D-72076 T"ubingen, Germany email: henze@tat.physik.uni-tuebingen.de

--

Let me remind you that various angular conventions used in biomechanics

are

extensively discussed in the book “Kinematics of Human Motion” by V.

Zatsiorsky. This book was published by Human Kinetics in 1997.

In particular, Section 1.2.6 of the book discusses advantages and

disadvantages of various angular conventions.

Regards,

Alexander Aruin, Ph.D.

Associate Professor of Physical Therapy and Bioengineering (UIC),

Physical Medicine & Rehabilitation (Rush Medical College)

Director of the Knecht Movement Science Laboratory

Department of Physical Therapy (M/C 898)

University of Illinois at Chicago

1919 West Taylor Street,

Chicago, Illinois 60612

Tel: (312) 355-0904 (Office)

(312) 355-0902 (Laboratory)

Fax: (312) 996-4583

E-mail: aaruin@uic.edu

--

A very interesting discussion, and certainly one that is useful to us

here, as we have recently

acquired a "Polaris" system which actually outputs the marker positions

in.....guess

what,....... Quaternions!! So we do have a vested interest to fully

understand this concept.

However, I would like to raise a very important and related issue, i.e.

the practical

(clinical) utilisation of gait/motion data. This I think should be

considered as a basis for

all such discussions.

Does it matter which angular representation is used?

Under many clinical circumstances, gait/ motion information is nearly

unhelpful, and such

mathematical delicacies are ignored completely.

I don't dispute the academic validity of this type of argument, and

perhaps its mathematical

fascination. Detailed discussion of these concepts would be important

for the purposes of

computer simulations and calculations of movements, say in VR

applications or other

computerised processes, where dealing with the errors is not only a

calculation but a

mathematical flirtation. However, when trying to understand the gait

recovery of a 57 year old

stroke patient and decide where to focus the physiotherapy efforts, all

these seem like "star

trek".

I would like to encourage contributions from all colleagues who have the

pleasure of having to

deal with the pragmatic world of health care. At least from a clinical

point of view, it is

important to stimulate discussions about gait data analysis that aims to

enhance patient care,

or at least make it clear that how the discussion could be potentially

relevant to "shop

floor".

Dr. H Rassoulian BSc, MSc, PhD, MIPEM, SRCS

Head of Clinical Bioengineering Group

Dept. Medical Physics & Bioengineering

Southampton University Hospitals NHS Trust

Southampton SO16 6YD

United Kingdom

Tel: 023 80796945

Fax: 023 80794117

email1: Hamid.Rassoulian@suht.swest.nhs.uk

email2: hamidR@soton.ac.uk

--

Dr. Chris Kirtley MD PhD

Associate Professor

HomeCare Technologies for the 21st Century (Whitaker Foundation)

NIDRR Rehabilitation Engineering Research Center on TeleRehabilitation

Dept. of Biomedical Engineering, Pangborn 105B

Catholic University of America

620 Michigan Ave NE

Washington, DC 20064

Tel. 202-319-6247, fax 202-319-4287

Email: kirtley@cua.edu

http://engineering.cua.edu/biomedical

Clinical Gait Analysis: http://guardian.curtin.edu.au/cga

Send subscribe/unsubscribe to listproc@info.curtin.edu.au

--

Dr. Chris Kirtley MD PhD

Associate Professor

HomeCare Technologies for the 21st Century (Whitaker Foundation)

NIDRR Rehabilitation Engineering Research Center on TeleRehabilitation

Dept. of Biomedical Engineering, Pangborn 105B

Catholic University of America

620 Michigan Ave NE

Washington, DC 20064

Tel. 202-319-6247, fax 202-319-4287

Email: kirtley@cua.edu

http://engineering.cua.edu/biomedical

Clinical Gait Analysis: http://guardian.curtin.edu.au/cga

Send subscribe/unsubscribe to listproc@info.curtin.edu.au

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I guess it's time I summarized the great quaternion debate. To remind

you, I

asked why quaternions (otherwise known as Euler parameters) seem to be

used as the standard method for representaing motion in computer

animation and video games, but are not so often used in biomechanics. It

seems that, whilst quaternions have advantages in terms of lack of

gymbal lock, and insensitivity to round-off errors, they suffer from

problems of interpretation in terms of meaningfully clinical or

anatomical angles.

As Joe Sommer and Bruce MacWilliams suggest, the best compromise is

perhaps to use

quaternions for intermediate calculations, then convert to Euler angles

at the end. I have made a summary of the various methods and provided

equations for conversion here:

http://guardian.curtin.edu.au/cga/faq/angles.html

Please let me know if you have any corrections or additions to this page

in the future.

Many thanks indeed to all contributors!

Chris Kirtley

--

Do you mean by quaternions so called Euler parameters? If so, you are

quite right. The four Euler parameters are extensively used in 3D

multibody simulations (instead of Euler angles) for the reasons you've

mentioned.

An example the software that uses the Euler parameters would be MADYMO.

It is a 3D package extensively used in so-called impact biomechanics.

Here, at NHTSA, we use it as one of the tools for injury assessment.

Erik

Erik G. Takhounts, Ph.D.

Senior Research Engineer

Conrad Technologies Inc.

501 School Street, S.W., Ste. 401

Washington, D.C. 20024

Ph. (202) 863-1931

Fax (202) 863-0909

Email: ETakhounts@NHTSA.DOT.GOV

--

There are Euler Angles (EA) vs. Euler Parameters (EP)

(Euler Parameters are sometimes known as normalized quarternions).

EA have only three independent rotational coordinates (the angles)

EP have nine independent direction cosines.

One would think EA simpler.

But the EA involve trigonometric functions and that is quite a

disadvantage.

Futher, still, there is a more severe problem:

for certain values of one of the angles (current notation calls it

theta)

the second and third angles cannot be distingquished and numerical

difficulties exists for these angles, making the programming more

difficult.

Other wise there really is no problem except for the implementation.

Tom Impelluso

--

In animation, the problems with Euler angles mainly arise when doing

motion editing. As long as you leave them alone they are fine.

Motion editing could be: resampling (interpolation), amplification,

motion blending etc. If you perform these operations on Euler angles

directly, you can get strange results, especially near gimbal lock.

The quaternion representation seems to behave better.

It seems that quaternions are the same as the "euler parameters" which

are often used in computational kinematics:

http://www.cs.berkeley.edu/~laura/cs184/quat/quaternion.html

I have also heard the term "angle-axis representation", i.e. the

rotation

is represented as a rotation of magnitude A about an axis (Ux,Uy,Uz).

Euler parameters are defined as follows

e0 = cos(A/2)

e1 = Ux*sin(A/2)

e2 = Uy*sin(A/2)

e3 = Uz*sin(A/2)

The sum of squares of these parameters is exactly one.

See also "Computer Aided Kinematics and Dynamics of Mechanical Systems",

by E.J. Haug.

Note that this representation is closely related to the three "helical

angles" proposed by Herman Woltring. The helical angle representation

is:

h1 = A*Ux

h2 = A*Uy

h3 = A*Uz

Why quaternions are not used more in biomechanics? This probably has

something to do with interpretation. Euler angles can be associated

with the rotations in a mechanical linkage or 3-D goniometer (Grood

and Suntay, J Biomech Eng, 1983). The other representations work well

for computation but are not so easily interpreted.

On the other hand, Woltring makes some good points on error propagation

in his 1994 paper (J Biomech 27:1399-1414). Near gimbal lock, Euler

angles

become increasingly sensitive to measuring errors.

For the newcomers on Biomch-L, I also recommend reading the debate

between

Grood and Woltring, about 10 years ago on Biomch-L:

http://isb.ri.ccf.org/biomch-l/files/angles3d.topic

Ton van den Bogert

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

--

I think Ton has explained it very well. Quaternions are not used because

they are difficult to interpret. Even the engineers will have difficulty

in

visualising what is happening, let alone doctors, physios etc. A

favourite

book of mine which explains everything is JM McCarthy, Introduction to

Theoretical Kinematics, MIT Press, 1990.

Raymond Lee

--

> So, I gather the problem is that the clinician (or sportsman) needs to

> interpret the results in terms of flexion-extension, ab-adduction and

> transverse rotation.

Yes, I think that's why the helical angles were never accepted.

Grood and Suntay is still the only rotation representation that is

used in human movement analysis.

By the way, this not only applies to joints but also to motion

relative to a global reference frame. Yeadon has defined the

terms somersault, tilt, and twist, which are Euler angles that

are consistent with existing coaching terminology.

> But couldn't quaternions be used until the final stage and then

> converted? Or would this still result in problems from the Euler

> representation?

No, then there would be no problem.

> Also, I confess I've never understood the difference between Grood and

> Suntay and Euler - is there a difference?

First, purists make a distinction between Euler angles and Cardanic

angles. Euler rotations are XYX, XZX, YXY, YZY, ZXZ, ZYZ, i.e. all

six sequences where the first and last rotation are about the same

coordinate axis. These were originally developed for celestial

mechanics,

i.e. the first rotation would be the orbit, the last the spin, and the

second the tilt of the axis. These have a singularity when the second

rotation is zero. Cardanic representations are XYZ, XZY,

YZX, YXZ, ZXY, ZYX. These six sequences have a singularity (Gimbal

lock)

when the second rotation is 90 degrees. All of these (Euler and Cardan

angles) are now loosely referred to as "Euler angles". In mechanics

textbooks you usually still find the distinction.

Yes, they are the same as Grood/Suntay. The matrix representation is

exactly the same. Grood really confused the issue by insisting that

this was not a sequence of rotations, but three simultaneous rotations.

But by arranging the mechanical linkage in a certain way, you

effectively

create a sequence. And the mathematics is exactly the same.

A.J. (Ton) van den Bogert, PhD

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

--

I standardly use quarternions for my measurements for all the reasons

you

indicated. But I would never consider publishing data or even casually

presenting results expressed in that form. I think the reason they don't

catch on, is you (or at least I) can't look at them and get an intuitive

feel for what's going on. You could probably make the same argument for

Euler anlges as you have to think about the sequence, but a degree unit

falling between -180 to +180 is somehow more intuitive. Probably just

conditioning, but I think we're adverse to "new tricks" such as this.

Bruce MacWilliams, Ph.D.

--

There is another reason why the animation world prefers quaternions over

Euler angles. Supposedly when you key frame (i.e., interpolate positions

and

orientations between several specified positions and orientations),

Euler

angles produce an unrealistic jerky motion whereas quaternions produce a

smooth motion. I have not actually tried this before to compare, but I

have

heard this comment from several sources in the past.

Also, as you point out, quaternions are not usually taught in most

biomechanics (or even engineering) curriculums. They require a redundant

parameter, which can be slightly inconvenient, but the benefit is the

elimination of gimbal lock (at least from a dynamic simulation

perspective).

Also, the four parameters are related to the screw axis between two

positions, so that can be helpful conceptually.

Computationally, you have to be careful with some singularity problems

when

converting between quaternions and direction cosine matrices (I can't

remember in which direction the problems occur). But other than that,

the

extra constraint equations relating the four parameters is a small price

to

pay for the benefits.

B.J.

B.J. Fregly, Ph.D.

Assistant Professor

Department of Aerospace Engineering,

Mechanics, and Engineering Science,

Biomedical Engineering Program, and

Dept. of Orthopaedics and Rehabilitation

University of Florida

Tel: (352) 392-8157

Fax: (352) 392-7303

E-mail: fregly@aero.ufl.edu

Home page: www.aero.ufl.edu/~fregly

--

1) Minor correction on nomenclature for rotation angle

sequences

The six possible Euler angle sets are XYX, XZX, YXY, YZY,

ZXZ and ZYZ where the first and third rotation axes repeat

(cyclic). Hence Euler angle representations experience

problems when the second rotation angle is zero or 180

degrees (i.e. the first and third axes are parallel). It

has been many years since I saw Euler's work, but I believe

that he used the ZXZ set.

The six possible Cardan-Bryant angle sets are XYZ, XZY, YXZ,

YZX, ZXY and ZYX where no axis repeats (anticyclic). Hence

Cardan-Bryant angle representations experience problems when

the second rotation is 90 or 270 degrees (e.g. the first and

third axes are parallel).

Craig (1989) provides 3x3 rotation matrices for all of the

rotation sets. He, like many others, refers to all twelve sets

simply as Euler angles.

2) Comments on rotation angles versus quaternions

In biomechanics, we face two major problems in regard to

describing both absolute attitude of an anatomical segment

or a camera with respect to an inertial frame or relative

attitude between two segments across an anatomical joint.

Our descriptions must be both mathematically tractable and

clinically relevant.

To analyze machinery, engineers often prefer unit

quaternions for absolute attitude of components (Haug, 1989,

and Kuipers, 1999) or rotation matrices across mechanical

joints (Denavit and Hartenberg, 1955) in that they often do

not need to describe the motion clinically.

As many biomechanics researchers know, rotation angles are

interesting for clinical description but can become

intractable for absolute attitude and for relative attitude

across joints that permit large angular excursions in all

three rotations.

Grood and Suntay's (1983) mechanical analog for the knee was

an important contribution that helped describe relative

attitude in clinically relevant terms. Their approach can

be adopted for many other joints (ankle, wrist) but has

difficulty for others (shoulder).

Concomitantly, I strongly support the efforts of ISB to

develop working groups to recommend "best practice" for

standardized description of attitude across specific joints.

If a working group of scholars can provide clinically

relevant descriptions of attitude using either angle sets or

quaternions, I agree with you that we should take their

"advice as to when a particular method is best".

Have any members of the ISB International Shoulder Group

(http://isb.ri.ccf.org/groups.html) provided input?

The following link also provides a simple discussion

oriented more toward camera imaging but it is still quite

applicable and also provides code snippets.

http://www.gamasutra.com/features/19980703/quaternions_01.htm

3) Comments on screw "helical" axes

For some reason, the biomechanics community has steadfastly

adopted the nomenclature "helical axes" to describe screw

kinematics as postulated by Poinsot and Chasles and later

formalized by Ball (1900). Unfortunately the rich

literature bases in both mathematics and engineering do not

use this "helical" nomenclature at all.

The axis direction and rotation of the displacement screw

axis (DSA) - also know as the finite helical axis (FHA) -

is the same as the direction and rotation of unit quaternion

components. Hence screw kinematics are not superior to

quaternions for representing attitude (absolute or relative)

or attitude displacements. They do however provide an

elegant method to combine the description of location and

attitude (absolute or relative) or location and attitude

displacements.

In particular, geometry of the screw axode surface swept by

the DSA as an anatomical joint moves through its range of

motion should be invariant to size and direction of

displacements as well as joint speed and acceleration.

Hence inspecting anatomical axodes (or deviation from

expected norms) should be able to help identify kinematic

irregularities or joint laxity. Unfortunately, while screw

axodes are quite useful to analyze machine joints, axode

geometric invariants are far too sensitive to experimental

measurement noise for routine clinical biomechanics today.

Further, axode invariants are much harder to visualize and

describe clinically than even quaternions.

4) Personal recommendation

As an engineer, I prefer a combination of Euler parameters

which are unit quaternions and orthonormal rotation matrices

for analysis. Unfortunately, I still prefer rotation angle

sets for clinical discussion.

After 25 years of studying both theoretical and experimental

kinematics (including Chasles' original paper for historical

inspiration), I still cannot say that any one method for

representation of attitude is absolutely the best.

Thanks for posting the summary of representations. If you

feel brave, you may wish to check Rooney's (1977, 1978)

summaries also.

Best wishes,

Joe Sommer

REFERENCES

Ball, R.S. (1900) A Treatise on the Theory of Screws,

Cambridge University Press

Craig, J.J. (1989) Introduction to Robotics, Addison-Wesley

Denavit, J. and Hartenberg, R.S. (1955) A kinematic notation

for lower pair mechanisms based on matrices. ASME J.

Applied Mechanics, 22:215-221

Grood, E.S. and Suntay, W.J. (1983) A joint coordinate

system for the clinical description of three-dimensional

motions: applications to the knee. ASME J. Biomech. Eng.,

105:136-144

Haug, E.J. (1989) Computer-Aided Kinematics and Dynamics of

Mechanical Systems, Allyn and Bacon

Kuipers, J.B. (1999) Quaternions and Rotation Sequences,

Princeton University Press

Rooney, J. (1977) A survey of representations of spatial

rotation about a fixed point. Environment and Planning B,

4:185-210

Rooney, J. (1978) A comparison of representations of general

spatial displacement. Environment and Planning B, 5:45-88

H.J. Sommer III, Ph.D., Professor of Mechanical Engineering

The Pennsylvania State University

337 Leonhard Building, University Park, PA 16802

(814)863-8997 FAX (814)865-9693

hjs1@psu.edu http://www.me.psu.edu/sommer

--

Hi Chris:

The advantages of Quaternions were described in my presentation in the

last 3D conference in South Africa.

The full presentation is at:

http://www.macrosport.com/sportsci/apasweb/presentations/capetown/demo_files

/frame.htm

Of course, the Quaternions is the way to go.

Gideon Ariel, Ph.D.

http://www.arielnet.com

--

as Ton already pointed out, there is an extensive explanation of the use

of quaternions (in their "euler parameter" representation) in "Computer

Aided Kinematics and Dynamics of Mechanical Systems", by E.J. Haug. In

this Book Haug introduces the theory the multi-body-simulation software

DADS is based on.

>From the theoretical point of view euler parameters

e0 = cos(A/2)

e1 = Ux*sin(A/2)

e2 = Uy*sin(A/2)

e3 = Uz*sin(A/2)

are an excellent tool to describe orientations and numerically solve

equations of motion of mechanical multi-body-systems, since there is no

gimbal lock, they are well defined on the unit-sphere (any orientation

of a rigid body can be uniquely/continuesly described by the set of (Ux,

Uy, Uz, A).

Whereas there is no way of integrating angular velocity w to obtain

orientation (since it's not integrable) one may integrate the time-

derivative of euler parameters to calculate e0(t), e1(t), e2(t), e3(t).

On the other hand euler parameters cannot be used to describe multiple

revolutions (uniqueness only on the unit-sphere) which is not too much

of a problem in biomechanics, since no joint range in biological systems

is greater. A problem with euler parameters is though, that it's almost

impossible to set up 3D-torque elements (e.g. for spherical joints).

Except for the trivial case where the joint torque always acts along

(Ux,Uy,Uz) it is very hard to describe a certain experimental behavior

with an euler parameter torque element. I tried this when I wanted to

limit the range of motion for a spherical joint but finally gave up,

since it was simply impossible (at least for me) to find the equations

using euler parameters. So I ended up using angles again.

The interpretaton of euler parameters is simple as long as one looks at

orientation only. But as soon as torque elements or experimental data

are to be described, interpretation seems to be impossible.

Arnim Henze.

Institut f"ur Astronomie und Astrophysik

Universit"at T"ubingen Tel.: ++49 7071 29 78654

Auf der Morgenstelle 10 Fax : ++49 7071 29 5889

D-72076 T"ubingen, Germany email: henze@tat.physik.uni-tuebingen.de

--

Let me remind you that various angular conventions used in biomechanics

are

extensively discussed in the book “Kinematics of Human Motion” by V.

Zatsiorsky. This book was published by Human Kinetics in 1997.

In particular, Section 1.2.6 of the book discusses advantages and

disadvantages of various angular conventions.

Regards,

Alexander Aruin, Ph.D.

Associate Professor of Physical Therapy and Bioengineering (UIC),

Physical Medicine & Rehabilitation (Rush Medical College)

Director of the Knecht Movement Science Laboratory

Department of Physical Therapy (M/C 898)

University of Illinois at Chicago

1919 West Taylor Street,

Chicago, Illinois 60612

Tel: (312) 355-0904 (Office)

(312) 355-0902 (Laboratory)

Fax: (312) 996-4583

E-mail: aaruin@uic.edu

--

A very interesting discussion, and certainly one that is useful to us

here, as we have recently

acquired a "Polaris" system which actually outputs the marker positions

in.....guess

what,....... Quaternions!! So we do have a vested interest to fully

understand this concept.

However, I would like to raise a very important and related issue, i.e.

the practical

(clinical) utilisation of gait/motion data. This I think should be

considered as a basis for

all such discussions.

Does it matter which angular representation is used?

Under many clinical circumstances, gait/ motion information is nearly

unhelpful, and such

mathematical delicacies are ignored completely.

I don't dispute the academic validity of this type of argument, and

perhaps its mathematical

fascination. Detailed discussion of these concepts would be important

for the purposes of

computer simulations and calculations of movements, say in VR

applications or other

computerised processes, where dealing with the errors is not only a

calculation but a

mathematical flirtation. However, when trying to understand the gait

recovery of a 57 year old

stroke patient and decide where to focus the physiotherapy efforts, all

these seem like "star

trek".

I would like to encourage contributions from all colleagues who have the

pleasure of having to

deal with the pragmatic world of health care. At least from a clinical

point of view, it is

important to stimulate discussions about gait data analysis that aims to

enhance patient care,

or at least make it clear that how the discussion could be potentially

relevant to "shop

floor".

Dr. H Rassoulian BSc, MSc, PhD, MIPEM, SRCS

Head of Clinical Bioengineering Group

Dept. Medical Physics & Bioengineering

Southampton University Hospitals NHS Trust

Southampton SO16 6YD

United Kingdom

Tel: 023 80796945

Fax: 023 80794117

email1: Hamid.Rassoulian@suht.swest.nhs.uk

email2: hamidR@soton.ac.uk

--

Dr. Chris Kirtley MD PhD

Associate Professor

HomeCare Technologies for the 21st Century (Whitaker Foundation)

NIDRR Rehabilitation Engineering Research Center on TeleRehabilitation

Dept. of Biomedical Engineering, Pangborn 105B

Catholic University of America

620 Michigan Ave NE

Washington, DC 20064

Tel. 202-319-6247, fax 202-319-4287

Email: kirtley@cua.edu

http://engineering.cua.edu/biomedical

Clinical Gait Analysis: http://guardian.curtin.edu.au/cga

Send subscribe/unsubscribe to listproc@info.curtin.edu.au

--

Dr. Chris Kirtley MD PhD

Associate Professor

HomeCare Technologies for the 21st Century (Whitaker Foundation)

NIDRR Rehabilitation Engineering Research Center on TeleRehabilitation

Dept. of Biomedical Engineering, Pangborn 105B

Catholic University of America

620 Michigan Ave NE

Washington, DC 20064

Tel. 202-319-6247, fax 202-319-4287

Email: kirtley@cua.edu

http://engineering.cua.edu/biomedical

Clinical Gait Analysis: http://guardian.curtin.edu.au/cga

Send subscribe/unsubscribe to listproc@info.curtin.edu.au

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