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Euler/Cardan Angles for the Scapula - Replies

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  • Euler/Cardan Angles for the Scapula - Replies

    Last month I posted a message regarding Euler Angle sequences
    for describing 3D scapular motion. I asked if there is any FUNDAMENTAL
    motivation for the selection of a particular sequence that would make it
    the "best" sequence and how does this relate to the description of
    scapular rotations?

    In this follow-up posting, I have included a summary of the suggestions
    I received and the complete text of those responses. I appologize for
    length of this posting, but it was necessary for completeness.

    Thanks to all those who responded.

    - Andy

    __________________________________________________ ________

    Andrew Karduna, Ph.D.
    Assistant Professor, Department of Physical Therapy
    Biomechanics Laboratory, 219 N Broad St
    Allegheny University of the Health Sciences
    Philadelphia, PA 19107
    fax: (215) 762-6076 phone: (215) 762-5057
    __________________________________________________ _______


    There appears to be a consensus that there is no FUNDAMENTAL
    or CORRECT order of rotations and that it is difficult to use
    conventions from other bones because there is no "long axis"
    of the scapula.

    Here are some of the suggestions I have received:

    - Select the sequence that most clearly demonstrates a consistent
    difference between normal and a pathological movement.

    - Use the relationship between Euler and projected angles to help.

    - Instead of Euler Angles, use the attitude vector.

    - Use the standard proposed by Frans van der Helm at

    - Use the to Angulus Acromialis (posterior-lateral ridge of acromion)
    instead of the AC joint to avoid gimbal lock.


    1) Frances T. Sheehan -

    - Reply 1

    I have not worked on the shoulder joint at all, but have ran into
    similar difficulties when trying to describe the 3D movement of
    the knee. Basically, the orientation of 2 bodies is completely
    described by the orientation matrix (3x3 quantity) ONCE a set of
    reference frames have been fixed in both bodies. So, the first
    question to ask is:

    1) How are you defining your reference frames and are they
    consistent from experiment to experiment? Without consistent reference
    frames, you don't have a chance at comparing data.

    Next, there are numerous methods for "simplifying" the
    orientation matrix.

    If you are planning on using 3 orientation angles, there are 24 different
    geometric series you can use. Typically, the only constraint in choosing
    a sequence is mathematical (i.e., avoiding gimbal lock, as you noted).

    If you are trying to define a normal versus a pathological movement
    pattern, I would choose the sequence that most clearly demonstrates a
    consistent difference.

    - Reply 2

    It may sound post hoc to choose a sequence that best defines the
    pathology, but depending upon your aims, I think it can be very useful.
    First, the 3D orientation angles are not biological, they are a
    mathematical simplification of a complex 3D event (as your different
    results show). Thus, if you are trying to derive a test to define
    healthy versus pathological, it makes sense to process all the sequences
    and determine which one provides the largest statistical difference.
    Clinically, this is the standard method for processing data. For example,
    in investigating patellar tracking problems various 2D angles are typically
    analyzed and the researchers try to define which single quantity provides
    the clearest definition of the pathology.

    Of course, if you are attempting to derive a cohesive description of
    kinematics, regardless of the state of the joint, you are best of
    choosing just one orientation angle sequence and sticking with it.

    Frances T. Sheehan, Ph.D.
    126 Spencer Hall (Office 312)
    Mechanical Engineering Department
    University of Delaware
    Newark, Delaware 19716-3140
    (302) 831-1013 (phone)
    (302) 831-3619 (fax)

    2) Ingram Murray -

    I was quite amazed to see your posting on Biomech-L as I was
    about to post almost the same question myself. I am
    looking at motion of the humerus in relation to the scapula and have
    over the last couple of days trawled through the summary of the
    arguments in the archives, which though very entertaining failed to
    answer the question I had which was - why are some sequences better
    than others ? (specifically related to the JCS of Grood and Suntay (1983))
    I have recently carried out some work related to the movement at the
    shoulder and found like you that there is quite a discrepancy in
    results when using each of the six possible sequences. In my work
    I found the sequence suggested by Cole et al (1993) to give
    considerably better results than any of the others but have had
    difficulty convincing myself why this is the case.

    Although I'm afraid all this falls into the 'General Reply' category
    and is not of specific help in your case, I would be very grateful if
    you could forward me any relevant replies than give a reasonable
    explanation of the different results obtained by applying different

    Ingram Murray
    Room G32
    Stephenson Building
    University of Newcastle upon Tyne
    Newcastle upon Tyne
    NE1 7RU

    3) Neil Crawford -

    - Reply 1

    Unlike other joints that have a clear plane of symmetry or a long axis,
    it is not so clear which is the appropriate Euler/Cardan sequence for
    the scapula. However, with your knowledge of scapular anatomy, you may
    be able to decide which is the appropriate projection angle to use in
    each plane, and then you can use the relationship between projection
    angles and Euler angles to choose the appropriate Euler sequence.

    In each plane, there are two possible projection angles. For example,
    looking down the x-axis at the y-z plane, it is clear that the
    projection angle about the x-axis (Px) can be described either by the
    projection of a vector j initially aligned with the y-axis (Pxj) or a
    vector k initially aligned with the z-axis (Pxk). There is usually an
    Euler/Cardan sequence that corresponds to the different projection
    sets. The relationship between projection angle sets and Euler angle
    sets is:

    Projection Euler
    Angle set Sequence
    -------------- --------------
    Pxj, Pyi, Pzi Rx->Ry->Rz
    Pxj, Pyi, Pzj Ry->Rx->Rz
    Pxj, Pyk, Pzi (none)
    Pxj, Pyk, Pzj Ry->Rz->Rx
    Pxk, Pyi, Pzi Rx->Rz->Ry
    Pxk, Pyi, Pzj (none)
    Pxk, Pyk, Pzi Rz->Rx->Ry
    Pxk, Pyk, Pzj Rz->Ry->Rx
    (no corresponding Euler sequence in the 2 cases where you use each axis
    once in the projection angles)

    You can choose the appropriate projection angles if you can
    identify a plane of symmetry for scapular movement and determine that
    motion about one of the three coordinate axes is different in nature
    from the other two (for instance, at a motion segment in the spine or
    at the wrist joint, there are two "bending" motions and one "twisting"
    motion). If two of the motions are similar in nature (like the 2
    bending motions), they should be described by the same vector projected
    in different planes (e.g., x-axis rotation and y-axis rotation can both
    be described by projecting the k vector, which was initially aligned with the
    z-axis: Pxk and Pyk). Also, motion describing deviation from a plane of
    symmetry should use a vector initially in that plane (e.g., if the y-z
    plane is a plane of symmetry, such as the midsagittal plane in the
    spine, Pyk and Pzj are preferred over Pyi and Pzi since the former two
    angles use vectors that were initially in the plane of symmetry while
    the latter two do not).

    I hope this explanation is not too convoluted. For a clearer
    presentation, see Crawford et al., "Methods for determining spinal
    flexion/extension, lateral bending, and axial rotation from marker
    coordinate data: Analysis and refinement", Human Movement Science
    15:55-78, 1996.

    If you cannot choose the appropriate projections using this
    method, I would suggest using whatever has most commonly been reported
    for scapular movement as long as the second Euler angle does not
    approach 90 degrees.

    - Reply 2

    Regarding the correspondence between Euler angles and projection angles,
    there is one Euler angle per sequence (the third rotation when rotations
    are taken about global fixed coordinates) that is mathematically
    identical to a certain projection angle. For the Euler sequence
    Rx->Ry->Rz, the third angle Rz is mathematically identical to the
    projection angle Pzi (the projection of the i vector on the plane
    perpendicular to the z-axis). The equation for both Rz in this sequence
    and Pzi is Rz=Pzi=atan2(iy,ix). (sorry, it's hard to write equations
    without subscripts, etc. iy and ix are the y and x components of the i
    vector, the unit vector that was initially aligned with the x-axis
    before the rotation).

    The other two Euler angles for a given sequence are not identical to any
    particular projection angle, but for angles less than about 30 degrees
    (which is most of the range of motion of many joints), they are much
    closer to one of the two projections in a given plane than to the
    other. Rx in the sequence Rx->Ry->Rz is more closely approximated by
    Pxj than by Pxk. Ry in the sequence Rx->Ry->Rz is more closely
    approximated by Pyi than by Pyk. The reason can be understood by
    looking at the equations for the angles. Ry from Rx->Ry->Rz is
    Ry=atan2(-iz,ix cos Rz + iy sin Rz) while Pyi=atan2(-iz,ix) and
    Pyk=atan2(kx,kz). If Rz is small, then cos Rz approaches 1 and sin Rz
    approaches 0. Then, the equation for Ry reduces to Ry~=atan2(-iz,ix)
    is Rx=atan2(kx sin Rz - ky cos Rz, jy cos Rz - jx sin Rz) while
    Pxj=atan2(jz,jy) and Pxk=atan2(-ky,kz). If Rz is small,
    Rx~=atan2(-ky,jy) From my limited experience experimenting with these rotation sequences,having the largest rotation as the last rotation is the most confusing from a
    clinical perspective. I realize this "clinical perspective" isn't a
    very strong rationale. I think much of the difficulty arises because of our
    tendency (or at least mine) to want to interpret these rotations as the
    "path" of motion rather than a position description that needs to
    consider all of the rotations and the sequence used. As far as the potential
    singularity or "gimbal lock" problem with upward rotation as the second
    rotation, even using the AC joint in the axis definition, 90% of the
    subjects I have tested have had values below 70 degrees during abduction
    of the arm in the scapular plane. I do not test them at the extremes of
    their range of motion, however.

    - Reply 2

    I haven't seen Jurriaan de Groot's thesis but the statement makes sense.
    I would roughly estimate using the posterior acromion would decrease the
    upward rotation numbers maybe ten degrees or so. I actually digitized
    both the posterior lateral acromion and the AC joint in my latest study,
    but have presently calculated the data only using the AC joint. I like
    that point because I think the plane of the scapula is more correctly
    defined but it does increase the upward rotation values from the
    traditional clinical numbers. I collected both points to allow me to
    potentially experiment with this a bit and compare to the published data
    in both formats. I actually think these local coordinate system
    definitions are more critical than the rotation sequences for
    "standards" since it is more difficult to convert the data. Perhaps using
    the y coordinate of the AC joint and z coordinate of the posterior acromion
    would be ideal (assuming x lateral, y anterior, and z superior on the
    right shoulder) but I realize that would be somewhat bizarre...

    Paula Ludewig
    The University of Iowa

    7) Frans C.T. van der Helm -

    The subject of defining a standardized rotation sequence for rotations
    of the shoulder bones had my interest for some time. On request of the
    ISB, I wrote together with Jesus Dapena recommendations for a
    standardization effort of the ISB. No definite decision has been taken
    on that issue. Since then, we made a few adaptations, and I wrote a
    paper on standardization for the First Conference of the International
    Shoulder Group (Delft, The Netherlands, 1996):

    Frans C.T. van der Helm (1996). A standardized protocol for motion
    recordings of the shoulder. Proc. First Conf. of the International
    Shoulder Group (Eds: HEJ Veeger, FCT van der Helm, PM Rozing), Delft,
    The Netherlands, pp. 7-12.

    Euler/Cardan angles have been chosen to describe motions. For the
    scapula with respect to the thorax, the following choice has been made:

    A local coordinate systems has been defined with respect to bony
    landmarks as follows:

    Xs-axis: Along the scapular spine, from Trigonum Spinae (TS)
    to Angulus Acromialis (AA)
    Zs-axis: Perpendicular to the scapular plane, defined by TS, AA
    and Angulus Inferior (AI)
    Ys-axis: Perpendicular to Xs and Zs, hence in the scapular plane.

    Rotations are interpreted with respect to a starting position of the
    scapula aligned with the Local coordinate system of the thorax:

    Yt-axis: (IJ + C7)/2 - (PX + T8)/2 (and normalized. )
    (IJ: Incisura Jugularis; C7: proc. spinosus 7th Cervical Vertebra;
    PX: Processus Xiphoideus; T8: proc. spinosus 8th Thoracic Vertebra)
    Xt-axis: Yt x (C7 - IJ) (vector product)
    Zt-axis: Xt x Yt

    Rotation order:
    Pro/retraction about the Ys-axis, defines the angle of the scapular
    spine w.r.t. the frontal plane.
    Lateral/medial rotation about the Zs'-axis (moved by the previous
    rotation), i.e. rotation in the scapular plane
    Tipping forward/backward about the Xs"-axis (scapular spine, moved by
    the previous rotations).

    The first two rotations define the orientation of the scapular spine in
    space, the last rotation is a rotation about the scapular spine. It is
    preferred to have the last rotation about a longitudinal axis of the
    bone, however for the scapula there is no unique longitudinal axis. The
    second rotation can result in gimbal lock problems (if the rotation
    angle approaches 90 degrees), therefore AA has been chosen instead of
    the acromioclavicular joint (AC), as in previous recommendations of our
    group. The order of rotations is close to interpretation of medical
    terminology. Note that the zero-zero-zero orientation (starting
    position) is not a feasible orientation of the scapula, due to the
    motion constraints of clavicle and thorax!

    In my opinion, there is no FUNDAMENTAL argument to choose for the one or
    other rotation order. Interpretation of rotation in combination with
    avoidance of gimbal lock should be the first rationale, and
    interpretation is certainly subjective. It is important to define local
    coordinate systems with respect to bony landmarks. If one uses the same
    bony landmarks, every definition can be reconstructed. However, it is
    preferred to use a standardized definition in order to improve the
    communication between researchers and clinicians.

    Frans C.T. van der Helm, PhD
    Man-Machine Systems & Control group
    Dept. of Mechanical Engineering
    Delft University of Technology
    Mekelweg 2
    2628 CD Delft
    The Netherlands

    tel. (+31)-15-2785616
    fax. (+31)-15-2784717

    8) Jurriaan de Groot -

    - Reply 1

    I agree with you that there should be an agreement on the definition of
    the rotation axes, rotation order and defined initial position for the
    description of motions of the shoulder bones.

    As a part of my Ph.D. thesis I looked at the accuracy of the Cardan
    parameters by which the scapular Cardan angles can be expressed. I must
    confess that the choice of rotation axes and rotation order where
    initially based on the traditions within our group. The arguments for
    these definitions however were rather convincing, based on the fact that
    the rotations should be interpretable and accurate.

    Your questions were:
    1) Is there any FUNDAMENTAL (mechanical, clinical or mathematical)
    motivation for the selection of a particular sequence that would make
    it the "best" sequence? and
    2) How does this relate to the description scapular rotations?

    I think that the 'best sequence' is not unique. A mathematical best sequence
    will e.g. be based on the most accurate description, the
    clinical best will be based in the interpretation of the angles.
    To my opinion the best sequence for the scapula will be one that is well
    interpretable (e.g. in the clinic) and in which the variability introduced
    by the definition is negligibly small with respect to the variability of
    intra- and interindividual shoulder motions.

    The best sequence depends on:
    1 - the axis definition with respect to the recordable landmarks.
    2 - the definition of the initial reference position.
    3 - the need of interpretation of the rotations.
    4 - the wish to describe bony motions or joint motions

    Van der Helm and Pronk 1995 (J.Biom.Eng 117: 27-40) define the
    orientation of the scapula with respect to a 'vertical' thorax. The
    rotation axes are based on palpable bony landmarks: the
    acromioclavicular joint [AC], the angulus acromialis [AA], the trigonum
    spinae [TS] and the angulus inferior [AI].

    Three perpendicular axes were defined:
    - A medial-lateral axis defined by the landmarks TS and AC (Sx-axis).
    - An anterior-posterior axis, perpendicular to the 'scapular plane'
    defined by AC,TS and AI (Sz axis).
    - A caudal-cranial Sy-axis perpendicular to the Sz and Sx axis.

    The missing longitudinal axis of the scapula could be referred to as the
    axis between the AC-joint and the GH-joint, but the rotations would not
    be very interpretable.

    In the initial 'reference' position all axes of the scapula were
    aligned with the axes of the thorax. Otherwise an pre-defined
    thorax-arm posture should be agreed in order to be able to compare
    recorded data.

    The chosen rotation order was Sy-Sz-Sx, which agrees to the definition
    mentioned in the original posting: protraction-elevation-tilt.

    The order is motivated by the fact that the second rotation should be
    the largest rotation of a clear structure (the scapular spine),
    preceeded by a rotation that defines the plane of the second rotation.
    In case of the scapula, the second rotation is the elevation of the
    scapular spine, preceeded by the initial rotation to align the scapula
    to the 'tangential plane' of the curved thorax. As such the scapular
    elevation coincides best with the elevation in the scapular plane

    The choice of rotation order is not based on mechanical or mathematical
    motivation but purely on interpretation of the resulting angles

    However, if the second rotation is in the order of 90 degrees one must
    be aware of the problems of the indetermined gimbal-lock position. In a
    study in which the total variation the recorded motions was
    identified, it was shown that the scapula position based on the AC, TS
    and AI landmarks, was liable to this gimbal-lock inaccuracy (Groot 1998,
    Clin. Biomech. in press]. Using the AA-landmark instead of the AC-landmark
    hardly changes the interpretation of the recorded angles, and reduces the
    gimbal-lock to a negligible influence.

    Concluding: the chosen rotation order of Van der Helm and Pronk (1995)
    combined with the coordinate system based on AA, TS and AI (Groot 1998)
    and referred to an initial scapular reference position which is aligned
    with the thorax coordinate system, results in an accurate and
    interpretable description of the scapular motions.

    An advantage of this description is, that the global orientation of the
    scapula can be recorded directly by means of a scapula locator, used by
    Johnson et al. 1993 (Clin.Biom.5:123-128)

    - Reply 2

    To my opinion it is best to define all the landmarks that you can record
    in a local coordinate system. In this way e.g. the GH-rotation centre
    can be approximated. In any case every body can compare his/her data
    with the data presented. The AC-landmark is absolutely a better landmark
    in order to approximate the rotations in the actual AC-joint.

    However from the point of accurate description of the motion of the
    scapula, which was your major concern I would advice to use the
    AA-landmark instead of the AC-landmark. The position of the AC-TS-AI
    triangle ('scapular plane') can always accurately be reconstructed.
    The scapular ridge of major importance is the medial border, wich is
    assumed to make contact to the thoracic-gliding plane under normal
    circumstances. This is defined by both AC-TS-AI and AA-TS-AI triangles
    (but is not congruent with either Ys-axes however).

    Jurriaan de Groot (Ph.D) |Tel : -31-(0)15-2782156
    Lab. of Measurement and Control |Fax : -31-(0)15-2784747
    Fac. Mechanical Engineering |E-mail :
    Delft University of Technology |
    | |____/(O----------------------
    Mekelweg 2 | |:::::/
    2628 CD Delft | |:::/ DUTCH SHOULDER GROUP
    The Netherlands | |:/

    9) Carolyn Anglin -

    As you have described, the descriptions used for the long bones cannot
    be easily adapted to the scapula. Van der Helm's group in the
    Netherlands has done a lot of work and thinking on this topic. I would
    therefore support using their method in order to standardize the
    reporting of data, which greatly simplifies comparing studies.

    The technique is described in:
    Van der Helm FCT & Pronk GM (1995) Three-dimensional recording
    and description of motions of the shoulder mechanism. J. Biomechanical
    Engineering 117:27-40.

    Another reference for your interest is:
    Peterson B (1994) On a model of the upper extremity. Advances
    in the Biomechanics of the Hand and Wrist. Schuind F [ed.]. Plenum
    Press, New York.

    Carolyn Anglin
    Sulzer Orthopedics Ltd.
    P.O. Box 65
    CH 8404 Winterthur

    Tel: +41 (52) 262 68 32
    Fax: +41 (52) 262 01 87

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