Difference between revisions of "Realization of transformations"
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− | {{Navigation|before=[[Composition of rotations]]|overview=[[Quaternions]]|next=[[ | + | {{Navigation|before=[[Composition of rotations]]|overview=[[Quaternions]]|next=[[Denavit-Hartenberg Convention]]}} |
===Quaternion notation for general transformations=== | ===Quaternion notation for general transformations=== | ||
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\left\{e_t,p_t\right\}=\left\{1,\ p_xi+p_yj+p_zk\right\} | \left\{e_t,p_t\right\}=\left\{1,\ p_xi+p_yj+p_zk\right\} | ||
</math> | </math> | ||
+ | |||
{{Example | {{Example | ||
|Title=Translation | |Title=Translation | ||
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\left\{e_r,p_r\right\}=\left\{\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}},\ 0\right\} | \left\{e_r,p_r\right\}=\left\{\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}},\ 0\right\} | ||
</math> | </math> | ||
+ | |||
{{Example | {{Example | ||
|Title=Rotation | |Title=Rotation | ||
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</math> | </math> | ||
Using the knowledge about [[Addition of quaternions|addition of quaternions]] and [[Rotations using quaternions|rotations]] and [[Composition of rotations|composition of rotations]] using quaternions this can directly be determined regarding the homogeneous transformation matrix <math>\mathbf{T}_c</math> and its rotational and translational components <math>\mathbf{R}_c</math> and <math>\vec{\mathbf{p}}_c</math>. | Using the knowledge about [[Addition of quaternions|addition of quaternions]] and [[Rotations using quaternions|rotations]] and [[Composition of rotations|composition of rotations]] using quaternions this can directly be determined regarding the homogeneous transformation matrix <math>\mathbf{T}_c</math> and its rotational and translational components <math>\mathbf{R}_c</math> and <math>\vec{\mathbf{p}}_c</math>. | ||
+ | |||
{{Example | {{Example | ||
|Title=Combination of rotation and translation | |Title=Combination of rotation and translation | ||
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The two examples for rotation and rotation are now to be combined. The combined transform is defined as: | The two examples for rotation and rotation are now to be combined. The combined transform is defined as: | ||
:<math> | :<math> | ||
− | \mathbf{T}_c = \mathbf{Trans}(y,2)\mathbf{Rot}(z,60^\circ) | + | \mathbf{T}_c = \mathbf{Trans}(y,2)\mathbf{Rot}(z,60^\circ) \quad \equiv \quad \left\{e_c,p_c\right\} |
+ | </math> | ||
+ | Using the explanations above this leads to: | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | \left\{e_c,p_c\right\}&=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}\\ | ||
+ | &= \left\{1,\ 2j\right\}\left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 0\right\}\\ | ||
+ | &= \left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 2j\right\} | ||
+ | \end{align} | ||
</math> | </math> | ||
}} | }} | ||
+ | |||
+ | [[Category:Article]] | ||
+ | [[Category:Quaternion]] |
Latest revision as of 18:18, 13 November 2015
← Back: Composition of rotations | Overview: Quaternions | Next: Denavit-Hartenberg Convention → |
Quaternion notation for general transformations
Up to now transformations have been defined by homogeneous matrices combining a rotation matrix and a translation vector
. Now a new notation is introduced to represent a transformation using two quaternions
and
:
The quaternion is equivalent to
and describes the rotation while
is defined as
and so equivalent to the translation.
Applying such a transformation to a quaternion is done by first rotating
with
corresponding to the rotation equation and then adding
:
Assume a pure translation is to be described using the above notation. Thus the rotation quaternion of a pure translation describes a rotation with an angle
which is actually no rotation at all. According to Rotations using quaternions
is defined as:
The rotation axis can be any arbitrary unit vector as the sine of
is zero. So
of a pure translation is always just
.
The translation quaternion of a pure translation has the scalar part
and the translation vector
as vector part:
So a pure translation by vector is defined as:
![]() A translation by |
For a pure rotation there is no translation. Thus the translation quaternion of a pure rotation is a zero quaternion:
The rotation quaternion of a pure rotation around an axis defined by a unit vector
by the angle
is:
So a pure rotation is defined as:
![]() A rotation by |
Combination of transformations
It is known that a combination of transformations is defined as:
But how can the two quaternions and
of the quaternion notation be calculated based on the quaternions of the individual transformations? The first transformation leads to
Now the second transformation is applied on . The resulting equation can be solved using the distributive law for quaternions to determine
and
:
Thus the combination of two transformations can be denoted in quaternion notation as
Using the knowledge about addition of quaternions and rotations and composition of rotations using quaternions this can directly be determined regarding the homogeneous transformation matrix and its rotational and translational components
and
.
![]() The two examples for rotation and rotation are now to be combined. The combined transform is defined as: Using the explanations above this leads to: |