Rotation Representations

1
Rotation Representations
2
Rotations Differ from Translations

• Rotations are non-Euclidean
• like travelling on a globe vs. a grid
• Rotations are not commutative
• x-rotate, y-rotate is not equal y-rotate,
  x-rotate etc.
• Rotations are non-linear
• true of all parameterizations other than trivial
  SO(3)

3
Rotation Parameterization

• Represent rotation space in Euclidean R3
• e.g. Euler angles, exponential map
• Pros
• three parameters for three DOFs
• Cons
• singularities, potentially poor interpolation

4
Rotation Parameterization

• non-Euclidean space
• e.g. unit quaternions (S3)
• Pros
• singularity free
• Cons
• must take extra measures to stay in legal
  sub-space
• four parameters required for three DOFs

5
Euler angles (f,?,?)

• An Euler angle is a rotation about a single
  Cartesian axis
• Create multi-DOF rotations by concatenating
  Eulers
• R R? R? Rf
• 3 DOFs can be obtained by
  concatenating

Euler-X
Euler-Y
Euler-Z
6
X-Convention

• Most commonly used
• The rotation given by Euler angles (f,?,?), where
  the first rotation is by an angle f about the
  z-axis, the second is by an angle ? about the
  x-axis, and the third is by an angle ? about the
  z-axis (again).
• R R? R? Rf

7
Yaw-Pitch-Roll Convention
8
Singularities

• More than one sets of parameters can create the
  same rotation matrix.
• Gimbal lock - two or more axes align, results in
  loss of rotational DOFs
• For Yaw-Pitch-Roll Convention

9
Rotation Axis Angle

• Eulers Rotation Theorem
• all rotations can be expressed as axis/angle

10
Quaternions

• Traditional solution Use unit quaternions to
  represent rotations
• S3 has same topology as rotation space (a
  sphere), so no singularities
• A member of unit sphere in R4
• q(qx,qy,qz,qw)
• a rotation about unit axis v