Dr. Chris Kirtley
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| Direction cosine (transformation, rotation) matrices | a11, a12, a13
a21, a22, a23 a31, a32, a33 |
Redundant, Interpretation | No gimbal lock | Most complete representation |
| Cardan (Tait-Bryant) | roll, pitch, yaw
(somersault, tilt, twist) no axis repeats cyclic: (Xyz (Earth Fixed - longitude/latitude, Fick), Yzx (Helmholz), Zxy or anti-cyclic: Zyx, Xzy,Yxz |
Sequence Dependence
Increasingly sensitive to measuring errors.
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Interpretaton in terms of anatomical motion (sagittal, frontal, transverse) | Sometimes also grouped with Euler angles
[Grood & Suntay, 1983] |
| Euler | As for Cardan but first and last rotations about same axis:
Xyx, Yzy, Zxz,
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Sequence Dependence
Gimbal lock when second
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Interpretaton in terms of anatomical motion
No false torsion in eye movements for Xzx |
[Craig, 1989 has conversion matrices between all Euler sequence combinations] |
| Euler parameters (Quaternions, Euler's axis and angle, finite rotational axis) | q = q4 + iq1 + jq2 +
kq3
where: i2 = j2 = k2 = ijk = -1 q1 = ex sin (f/2)
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Interpretation in terms of anatomical motion | Sequence-independent
Insensitive to round-off errors |
Conversion from Euler Angles
q1 = cos(yaw/2) cos(pitch/2) sin(roll/2) - sin(yaw/2) sin(pitch/2) cos(roll/2) q2 = cos(yaw/2) sin(pitch/2) cos(roll/2) + sin(yaw/2) cos(pitch/2) sin(roll/2) q3 = sin(yaw/2) cos(pitch/2) cos(roll/2) - cos(yaw/2) sin(pitch/2) sin(roll/2) q4 = cos(yaw/2) cos(pitch/2) cos(roll/2) + sin(yaw/2) sin(pitch/2) sin(roll/2) For small angles:
Conversion
from Quaternion to Euler
[Haug, 1989; Kuipers, 1999] |
| Angle-axis | q1 = Ux
q2 = Uy q3 = Uz q4 = f |
Conversion from Euler Angles | ||
| Rotation Vector (Benati, Rodriguez-Hamilton) | q1 = ex tan (f/2)
q2 = ey tan (f/2) q3 = ez tan (f/2) |
Gimbal lock when 180° | ||
| Helical angles (finite helical axis, screw theory, Woltring) | h1 = A*Ux
h2 = A*Uy h3 = A*Uz |
sensitive to
measurement error & noise |
combines description of location & attitude (absolute or relative)
or location & attitude
displacements |
[Ball, 1900; Woltring 1994] |
The current rotations must respect the previous rotations to avoid gimbal lock to prevent any alignment of rotation axis which causes this problem. This problem doesn't depend on the mathematical way you choose to express rotations, it depends on the way to combine rotations. The problem comes from Euler's classical transform: R=Rx.Ry.Rz
Implicitely or explicitly you use it with matrices, quaternions, Euler or Cardan angles, ...
You should correct this transform and it works whatever you used to express your rotations:
R=R[R[R[Ox,a]Oy,b]R[Ox,a]Oz,c] . R[R[Ox,a]Oy,b] .R[Ox,a].
Contributed by Julien Gouesse
Andrews Letter to the Editor, Journal of Biomechanics 17(1984)2, 155-158.
Ball, R.S. (1900) A Treatise on the Theory of Screws, Cambridge University Press
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Blankevoort, L., Huiskes, R., Lange, A. de (1988) The Envelope of passive knee joint motion. J. Biomechanics 21, 705-720.
Bortz A John E. New Mathematical Formulation for Strapdown Inertial Navigation IEEE Transactions on Aerospace and Electronic Systems January 1971, Vol. AES-7, No. 1, pp 61-66
Bottema & Roth "Theoretical Kinematics", Chapter 3 (Elsevier, Amsterdam 1979)
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Goeran Selvik, Roentgen Stereophotogrammetric Analysis - A Review Article, Acta Radiologica 31(1990), Fasc. 2.
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