Difference between revisions of "Euler angles"

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[[File:euler-main.gif|right|350px]]
 
[[File:euler-main.gif|right|350px]]
  
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The roll, pitch and yaw angles are three angles defined in regard of [[Three-Angle Representations|absolute transformation]] to describe the orientation of an object, generally vehicles, in three-dimensional space. In the following the common convention will be used, so the three angles can be described as follows (in the order they are applied):
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The Euler angles are three angles defined in regard of [[Three-Angle Representations|relative transformation]] to describe the orientation of an object in three-dimensional space. So unlike the [[Roll-Pitch-Yaw|roll, pitch and yaw]] angles which are applied to static axes, Euler angles are always applied to the new coordinate axes.  
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* '''Yaw''': Rotation around the vertical axis of the object or vehicle, respectively
 
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* '''Pitch''': Rotation around the lateral axis
 
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* '''Roll''': Rotation around the longitudinal axis (what is generally the movement axis of a vehicle)
 
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There are different notations to define the axes of an object. Usually and in recent publications the vertical axis is the z-axis, the longitudal axis is x and then the lateral axis is the y-axis and directed to the left.
 
  
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So the roll-pitch-yaw transformation matrix of the orientation is defined as follows:
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There are different conventions for Euler angles which differ in the definition of the three rotation axes. Two pure Euler conventions are ''ZXZ'' and ''ZYZ''. In the ''ZXZ'' convention the first rotation is around the z-axis, then a rotation around the new x-axis follows and another rotation around the current z-axis. The ''ZYZ'' convention implies, as the name already says, a rotation around the z-axis, then a rotation around the new y-axis followed by a rotation around the new z-axis.
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So the ''ZYZ'' transformation matrix of the orientation is defined as follows:
 
:<math>
 
:<math>
 
\begin{align}
 
\begin{align}
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RPY(\phi,\theta,\psi)&=Rot(x,\phi)Rot(y,\theta)Rot(z,\psi) \\
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Euler(\phi,\theta,\psi)&=Rot(z,\phi)Rot(y,\theta)Rot(z,\psi) \\
 
&=
 
&=
 
\left[\begin{array}{cccc}
 
\left[\begin{array}{cccc}
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\cos{\psi}\cos{\theta} & -\cos{\theta}\sin{\phi} & \sin{\theta} & 0\\
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\cos{\phi}\cos{\psi}\cos{\theta}-\sin{\phi}\sin{\psi} & -\cos{\phi}\sin{\psi}-\cos{\psi}\cos{\theta}\sin{\phi} & \cos{\psi}\sin{\theta} & 0\\
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\cos{\psi}\sin{\phi}+\cos{\phi}\sin{\psi}\sin{\theta} & \cos{\phi}\cos{\psi}-\sin{\phi}\sin{\psi}\sin{\theta} & -\cos{\theta}sin{\psi} & 0\\
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\cos{\psi}\sin{\phi}+\cos{\phi}\sin{\psi} & \cos{\phi}\cos{\psi}-\cos{\theta}\sin{\phi}\sin{\psi} & \sin{\psi}sin{\theta} & 0\\
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\sin{\phi}\sin{\psi}-\cos{\phi}\cos{\psi}\sin{\theta} & \cos{\psi}\sin{\phi}\sin{\theta} & \cos{\psi}\cos{\theta} & 0\\
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-\cos{\phi}\sin{\theta} & \sin{\phi}\sin{\theta} & \cos{\theta} & 0\\
 
0 & 0 & 0 & 1
 
0 & 0 & 0 & 1
 
\end{array}\right]
 
\end{array}\right]
 
\end{align}
 
\end{align}
 
</math>
 
</math>

Revision as of 15:56, 7 May 2015

← Back: Roll-Pitch-Yaw Overview: Three-Angle Representations Next: Quaternions →
Euler-main.gif

The Euler angles are three angles defined in regard of relative transformation to describe the orientation of an object in three-dimensional space. So unlike the roll, pitch and yaw angles which are applied to static axes, Euler angles are always applied to the new coordinate axes.

There are different conventions for Euler angles which differ in the definition of the three rotation axes. Two pure Euler conventions are ZXZ and ZYZ. In the ZXZ convention the first rotation is around the z-axis, then a rotation around the new x-axis follows and another rotation around the current z-axis. The ZYZ convention implies, as the name already says, a rotation around the z-axis, then a rotation around the new y-axis followed by a rotation around the new z-axis.

So the ZYZ transformation matrix of the orientation is defined as follows:

\begin{align} Euler(\phi,\theta,\psi)&=Rot(z,\phi)Rot(y,\theta)Rot(z,\psi) \\ &= \left[\begin{array}{cccc} \cos{\phi}\cos{\psi}\cos{\theta}-\sin{\phi}\sin{\psi} & -\cos{\phi}\sin{\psi}-\cos{\psi}\cos{\theta}\sin{\phi} & \cos{\psi}\sin{\theta} & 0\\ \cos{\psi}\sin{\phi}+\cos{\phi}\sin{\psi} & \cos{\phi}\cos{\psi}-\cos{\theta}\sin{\phi}\sin{\psi} & \sin{\psi}sin{\theta} & 0\\ -\cos{\phi}\sin{\theta} & \sin{\phi}\sin{\theta} & \cos{\theta} & 0\\ 0 & 0 & 0 & 1 \end{array}\right] \end{align}
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