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I am wondering if pitch-roll-yaw is used to represent Eular angle? If not, what's the relationship between them?

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

So can anyone explain that in detail?


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up vote 3 down vote accepted

The 3 Euler angles $ ( \alpha, \beta, \gamma ) $ are often used 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. There are many ways of describing the orientation of an aircraft (or other rigid objection), some of which use all three -- Yaw some amount a, then Pitch some amount b, then Roll some amount c. 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 3x3 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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This is aerodynamics jargon. You can find an equivalence with Euler angles here.

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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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protected by Zev Chonoles Mar 3 at 7:04

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