Euler angles

Parents: rotations-so3-group-index , orientation-parametrisations

Source: Phil’s Lab

• Three angles that describe the orientation of an object w.r.t. a fixed coordinate system
• Roll $\phi$, Pitch $\theta$, Yaw $\psi$

Source:  http://en.wikipedia.org/wiki/Euler_angles

Possible representations

• Proper Euler angles (e.g. $zxz$) vs Tait-Bryan (e.g. $xyz$, $zyx$)
• Intrinsic vs. extrinsic rotations
  • Extrinsic rotations (around fixed CS $xyz$)
  • Intrinsic rotations (around body CS $XYZ = x''' y''' z'''$)

As a rotation matrix

$$R = X(\alpha) Y(\beta) Z(\gamma)$$

This means either: (s.  Intrinsic vs extrinsic rotations )

• extrinsic rotations about z -> y -> x / yaw pitch roll
• intrinsic rotations about x -> y' -> z'' = Z = z'''

Note:  Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. [ Source ]

Intrinsic rotations $x-y’-z″$ by angles $\alpha, \beta, \gamma$ are equal to extrinsic rotations $z-y-x$ by angles $\gamma, \beta, \alpha$.

Definition of X(alp), Y(beta), Z(gamma) [elemental rotation matrix] depends on convention chosen

Table of Euler rotation matrices (RH, active, intrinsic):

Proper Euler	Tait-Bryan
	s. derivation here:  bryan-tait-kardanwinkel
	intrinsic yaw pitch roll

Source: Markley 2014

Table of Euler rotation matrices (RH, passive, intrinsic): 1: phi, 2:theta, 3:psi

Proper Euler	Tait-Bryan
	(transpose of the active version)
	(transpose of the active version)

Euler to quaternion conversions (note: these are all passive transformations??) 1: phi, 2:theta, 3:psi