




Direction cosine (transformation, rotation) matrices  a_{11}, a_{12}, a_{13}
a_{21}, a_{22}, a_{23} a_{31}, a_{32}, a_{33} 
Redundant, Interpretation  No gimbal lock  Most complete representation 
Cardan (TaitBryant)  roll, pitch, yaw
(somersault, tilt, twist) no axis repeats cyclic: (Xyz (Earth Fixed  longitude/latitude, Fick), Yzx (Helmholz), Zxy or anticyclic: Zyx, Xzy,Yxz 
Sequence Dependence
Increasingly sensitive to measuring errors.

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,

Sequence Dependence
Gimbal lock when second

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 = q_{4} + iq_{1} + jq_{2} +
kq_{3}
where: i^{2} = j^{2} = k^{2} = ijk = 1 q_{1} = e_{x} sin (f/2)

Interpretation in terms of anatomical motion  Sequenceindependent
Insensitive to roundoff errors 
Conversion from Euler Angles
q_{1} = cos(yaw/2) cos(pitch/2) sin(roll/2)  sin(yaw/2) sin(pitch/2) cos(roll/2) q_{2} = cos(yaw/2) sin(pitch/2) cos(roll/2) + sin(yaw/2) cos(pitch/2) sin(roll/2) q_{3} = sin(yaw/2) cos(pitch/2) cos(roll/2)  cos(yaw/2) sin(pitch/2) sin(roll/2) q_{4} = 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] 
Angleaxis  q_{1} = U_{x}
q_{2} = Uy q_{3} = Uz q_{4 }= f 
Conversion from Euler Angles  
Rotation Vector (Benati, RodriguezHamilton)  q_{1} = e_{x} tan (f/2)
q_{2} = e_{y} tan (f/2) q_{3} = e_{z} tan (f/2) 
Gimbal lock when 180°  
Helical angles (finite helical axis, screw theory, Woltring)  h_{1} = A*U_{x}
h_{2} = A*U_{y} h_{3} = A*U_{z} 
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
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