Difference between revisions of "Rotations using quaternions"
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− | + | Assume an object is shifted 1 unit on the x-axis. So its position vector and the corresponding quaternion are defined as follows: | |
+ | :<math> | ||
+ | \vec{\mathbf{p}} = \left[\begin{array}{c}1\\0\\0\end{array}\right], \qquad | ||
+ | p = 0 \ \oplus \ \vec{\mathbf{p}} = (0,1,0,0) | ||
+ | </math> | ||
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Revision as of 14:09, 9 September 2015
← Back: Multiplication of quaternions | Overview: Quaternions | Next: Realization of transformations → |
Usually rotations are defined by 3 angles, either Euler or Roll-Pitch-Yaw angles. So three successive rotations around three different axes lead to a combined rotation, that can be described by a single rotation matrix. Such a combined rotation is equal to a rotation around a certain axis in three-dimensional space about a certain angle. This is shown in the figure on the right. The vector is rotated such that it results in . This could be done by rotating the vector around the z-, the y- and the x-axis successively or by just rotating it around the rotation axis by .
Mathematical description
First the rotation axis has to be defined by a unit vector. So it holds:
Then a rotation about by the angle can be represented by a quaternion like follows:
So the scalar part is defined as the cosine of and the vector part consisting of , and equals multiplied by the sine of .
That the resulting quaternion is a unit quaternion can be proven as follows:
So if a quaternion is given, the angle and the rotation axis can be computed as follows:
Mathematical solving of a quaternion rotation
Assume a quaternion describing a rotation and a vector that should be rotated. To be able to apply the rotation, has to be presented as a quaternion. So the pure quaternion is defined with scalar part equal to 0 and as vector part:
The same is done for the resulting vector of the rotation:
The rotation is then applied by the following quaternion multiplication:
Following the rules for quaternion multiplication, this equation leads to
This shows, that the result of the multiplication is again a pure quaternion describing the rotated vector . The corresponding rotation matrix describing the rotation obtained by the quaternion can easily be computed using the equation for .
Example: Quaternion rotation and rotation matrices
Assume an object is shifted 1 unit on the x-axis. So its position vector and the corresponding quaternion are defined as follows: ... ... |
Applet
The following three-dimensional applet helps you to understand the relation between Roll-Pitch-Yaw angles and a quaternion. The initial position of the object can be set using the sliders for x, y and z. Then the object can be rotated by defining the roll, pitch and yaw angles. The most intuitive way is to start with the yaw angle, because this one is applied first. Then the object is rotated aroud the y-axis by the pitch angle followed by a rotation around the x-axis by the roll angle. The quaternion describing the same rotation is shown dynamically and the corresponding angle and the rotation axis are presented. The rotation axis and the rotational path are visualized on the left side. After pressing the Show Quaternion Rotation button, the rotation of the object around gets animated.