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I am wondering if pitch, roll and yaw are used to represent Euler angles? If not, what's the relationship between them?

From wiki, I know that Euler angles are used to represent the rotation from three axes independently, which seems like pitch, roll and yaw. But from this wiki, it seems that they are two different things.

So, can anyone explain that in detail?

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  • $\begingroup$ This is explained at length in the first wiki link. Read carefully. $\endgroup$ – Yves Daoust May 18 '17 at 19:43
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The $3$ Euler angles (usually denoted by $\alpha, \beta$ and $\gamma$) are often used to represent the current orientation of an aircraft.

Starting from the "parked on the ground with nose pointed North" orientation of the aircraft, we can apply rotations in the Z-X'-Z'' order:

  1. Yaw around the aircraft's Z axis by $ \alpha $
  2. Roll around the aircraft's new X' axis by $ \beta $
  3. Yaw (again) around the aircraft's new Z'' axis by $ \gamma $

to get the current orientation of the aircraft represented by the $3$ Euler angles $(\alpha, \beta, \gamma)$.

You may have noticed that we yaw twice, and we never use pitch. In fact, there are many ways of describing the orientation of an aircraft (or other rigid objection), some of which use all three: some amount $a$ of yaw, then some amount $b$ of pitch, then some amount $c$ of roll.

There exist standard formulas for converting between different ways of describing some given orientation: orientation in $(a, b, c)$ format; orientation in Euler angle $ ( \alpha, \beta, \gamma ) $ format; orientation as described by a $3 \times 3$ rotation matrix, etc.


I'm assuming one popular coordinate system in flight dynamics which associates:

  • X-axis is positive forward, through the nose of the aircraft
  • Y-axis is positive out the right wing
  • Z-axis is positive down.
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  • 1
    $\begingroup$ Just to give the non-Euler angles a name, they are called Tait–Bryan angles. $\endgroup$ – Ruslan Jul 22 '19 at 12:47
  • $\begingroup$ and there is a useful table of formulae slightly further down in that Wikipedia article. $\endgroup$ – almagest Jan 27 at 9:04
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You can say that Euler angle representation (rep) is ZYX representation where as roll-pith-yaw is XYZ representation.
If $\alpha$ is rotation about Z axis, $\beta$ is rotation about Y axis, and $\gamma$ is rotation about X axis:

  • Transformation matrix for Euler angles rep would be $R^{\alpha}_Z$*$R^{\beta}_Y$*$R^{\gamma}_X$.
  • Transformation matrix for Roll-pitch-yaw rep would be $R^{\gamma}_X$*$R^{\beta}_Y$*$R^{\alpha}_Z$.

Hence, if you consider rotation about X was done first, then about Y and then about Z. Then, rotations in Euler angles rep are w.r.t. global reference frame and those in roll-pitch-yaw are w.r.t. local frames.

You can refer to 1 for better understanding.

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