## Methods of Representing Solid Angles

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell

 Representation Definition Problems Advantages Notes/Conversions 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. near gimbal lock when second  rotation = 90 or 270° , i.e. first & third axes  parallel (e.g. earth at poles for Fick, shoulder abducted to 90°, false torsion in eye movements in Fick & Helmholz) 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,  Xzx, Xyx, Yxy Sequence Dependence Gimbal lock when second  rotation is 0 or 180°, i.e. first & third axes parallel 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) q2 = ey sin (f/2) q3 = ez sin (f/2) q4 = cos(f/2) where: e = unit vector along axis of rotation f = total rotation angle 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: q1 ~ roll / 2 q2 ~ pitch / 2 q3 ~ yaw / 2 q4 ~ 1 Conversion from Quaternion to Euler tan(yaw) = 2(q1q2+q4q3) / (q42 + q12 - q22 - q32) sin(pitch) = -2(q1q3-q4q2) tan(roll)  =  2(q4q1+q2q3) / (q42 - q12 - q22 + q32)  [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

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Please let me know if you spot any errors, or have any additions!
Dr. Chris Kirtley 