Euler angles

Terminology

• Euler axis
  • 회전의 기준이 되는 축

 

• Euler pole
  • Euler axis가 가르키는 방향

 

• Euler angle
  • Euler axis를 기준으로 한 rotation의 변화량

 

 

 

Sign convention

• clock-wise

 

• counter clock-wise (Usally used)

 

 

 

Composition convention

• As matrix multiplication does not commute, order or rotation axes will affect the result

 

• Two successive rotations must be performed with different axes.

 

• There are 12 conventions for the combination of these three axes which can be divided into 2 groups
  • 6 - symmetric conventions involving the same axes
    • XYX, XZX, YXY, YZY, ZXZ, ZYZ
  • 6 - asymmetric conventions
    • XYZ, XZY, YXZ, YZX, ZXY, ZYX

 

 

 

Rotation matrix properties

• Positive orthogonal matrix
  • $R_i^T(\theta) = R_i^{-1}(\theta) \quad (i = x, y, z)$
  • $R_i(-\theta) = R_i^T(\theta) \quad (i = x, y, z)$

 

• Det = +1

 

• Additive in same axis
  • $R_i(\alpha) + R_i(\beta) = R_i(\alpha + \beta) \quad (i = x, y, z)$

 

 

 

Passive rotations

 

 

 

Active rotations

• Active rotation transforms a vector in same fixed coordinate system

 

 

 

Extrinsic rotation & Intrinsic rotation

• Extrinsic rotation
  • Extrinsic rotation is a rotation with fixed axis

 

• Intrinsic rotation
  • Intrinsic rotation is a rotation with sequentially changing axis

 

• Any extrinsic rotation is equivalent to a intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa.

 

• For instance, the intrinsic rotations x-y’-z″ by angles α, β, γ are equivalent to the extrinsic rotations z-y-x by angles γ, β, α.