Difference between revisions of "Realization of transformations"

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{{Navigation|before=[[Composition of rotations]]|overview=[[Quaternions]]|next=[[???]]}}
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{{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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q' = e\ q\ e^*+ p
 
q' = e\ q\ e^*+ p
 
</math>
 
</math>
 +
 +
----
 +
 +
Assume a pure translation is to be described using the above notation. Thus the rotation quaternion <math>e_t</math> of a pure translation describes a rotation with an angle <math>\phi=0^\circ</math> which is actually no rotation at all. According to [[Rotations using quaternions]] <math>e</math> is defined as:
 +
:<math>
 +
e_t(\vec{\mathbf{u}},0^\circ)=\cos{0^\circ} + \sin{0^ \circ}\vec{\mathbf{u}} = 1+0\cdot\vec{\mathbf{u}} = 1
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</math>
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The rotation axis <math>\vec{\mathbf{u}}</math> can be any arbitrary unit vector as the sine of <math>0^\circ</math> is zero. So <math>e_t</math> of a pure translation is always just <math>1</math>.
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The translation quaternion <math>p_t</math> of a pure translation has the scalar part <math>0</math> and the translation vector <math>\vec{\mathbf{p}}</math> as vector part:
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:<math>
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p_t=0\oplus\left[\begin{array}{c}p_x\\p_y\\p_z\end{array}\right] =  p_xi+p_yj+p_zk
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</math>
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So a pure translation by vector <math>[p_x\ p_y\ p_z]^T</math> is defined as:
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:<math>
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\left\{e_t,p_t\right\}=\left\{1,\ p_xi+p_yj+p_zk\right\}
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</math>
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{{Example
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|Title=Translation
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|Contents=
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A translation by <math>2</math> on the <math>y</math>-axis for example would be
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:<math>
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\mathbf{Trans}(y,2) \equiv \left\{1,\ 2j\right\}
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</math>
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}}
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 +
----
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For a pure rotation there is no translation. Thus the translation quaternion <math>p_r</math> of a pure rotation is a zero quaternion:
 +
:<math>
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p_r=0 \oplus \left[\begin{array}{c}0\\0\\0\end{array}\right] = 0
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</math>
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The rotation quaternion <math>e_r</math> of a pure rotation around an axis defined by a unit vector <math>\vec{\mathbf{u}}</math> by the angle <math>\phi</math> is:
 +
:<math>
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e_r(\vec{\mathbf{u}},\phi)=\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}}
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</math>
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So a pure rotation is defined as:
 +
:<math>
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\left\{e_r,p_r\right\}=\left\{\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}},\ 0\right\}
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</math>
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{{Example
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|Title=Rotation
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|Contents=
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A rotation by <math>60^\circ</math> around the <math>z</math>-axis for example would be
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:<math>
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\mathbf{Rot}(z,60^\circ) \equiv \left\{\cos{30^\circ} + \sin{30^\circ}\left[\begin{array}{c}0\\0\\1\end{array}\right],\ 0\right\} = \left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 0\right\}
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</math>
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}}
  
 
===Combination of transformations===
 
===Combination of transformations===
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\end{align}
 
\end{align}
 
</math>
 
</math>
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But how can the two quaternions <math>e_c</math> and <math>p_c</math> of the quaternion notation be calculated based on the quaternions of individual transformations? The first transformation leads to  
+
But how can the two quaternions <math>e_c</math> and <math>p_c</math> of the quaternion notation be calculated based on the quaternions of the individual transformations? The first transformation leads to  
 
:<math>
 
:<math>
 
q' = e_1\ q\ e_1^*+ p_1
 
q' = e_1\ q\ e_1^*+ p_1
 
</math>
 
</math>
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Now the second transformation is applied on <math>q'</math>:
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Now the second transformation is applied on <math>q'</math>. The resulting equation can be solved using the [[Multiplication of quaternions|distributive law for quaternions]] to determine <math>e_c</math> and <math>p_c</math>:
 
:<math>
 
:<math>
 
\begin{align}
 
\begin{align}
 
q'' &= e_2\ q'\ e_2^*+ p_2 \\
 
q'' &= e_2\ q'\ e_2^*+ p_2 \\
−
&= e_2\Big(e_1\ q\ e_1^*+ p_1\Big)\ e_2^*+ p_2
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&= e_2\Big(e_1qe_1^*+ p_1\Big)e_2^*+ p_2 \\
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&= \Big(e_2e_1qe_1^*+ e_2p_1\Big)e_2^*+ p_2 \\
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&= \underbrace{e_2e_1}_{e_c}q\underbrace{e_1^*e_2^*}_{e_c^*}+\underbrace{e_2p_1e_2^*+ p_2}_{p_c} \\
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&= e_c\ q\ e_c^*+p_c \\
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\end{align}
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</math>
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Thus the combination of two transformations can be denoted in quaternion notation as
 +
:<math>
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\left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}=\left\{e_2e_1\ , \ e_2p_1e_2^*+p_2\right\}
 +
</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
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|Title=Combination of rotation and translation
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|Contents=
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The two examples for rotation and rotation are now to be combined. The combined transform is defined as:
 +
:<math>
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\mathbf{T}_c = \mathbf{Trans}(y,2)\mathbf{Rot}(z,60^\circ) \quad \equiv \quad \left\{e_c,p_c\right\}
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</math>
 +
Using the explanations above this leads to:
 +
:<math>
 +
\begin{align}
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\left\{e_c,p_c\right\}&=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}\\
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&= \left\{1,\ 2j\right\}\left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 0\right\}\\
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&= \left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 2j\right\}
 
\end{align}
 
\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 \mathbf{R} and a translation vector \vec{\mathbf{p}}. Now a new notation is introduced to represent a transformation using two quaternions e and p:

\mathbf{T} = \left[\begin{array}{cccc} & & & \\ & \mathbf{R} & & \vec{\mathbf{p}} \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \quad \equiv \quad \left\{e,p\right\}

The quaternion e is equivalent to \mathbf{R} and describes the rotation while p is defined as 0 \oplus \vec{\mathbf{p}} and so equivalent to the translation.

e \equiv \mathbf{R} \qquad \quad p = 0 \oplus \vec{\mathbf{p}}

Applying such a transformation to a quaternion q is done by first rotating q with e corresponding to the rotation equation and then adding p:

q' = e\ q\ e^*+ p

Assume a pure translation is to be described using the above notation. Thus the rotation quaternion e_t of a pure translation describes a rotation with an angle \phi=0^\circ which is actually no rotation at all. According to Rotations using quaternions e is defined as:

e_t(\vec{\mathbf{u}},0^\circ)=\cos{0^\circ} + \sin{0^ \circ}\vec{\mathbf{u}} = 1+0\cdot\vec{\mathbf{u}} = 1

The rotation axis \vec{\mathbf{u}} can be any arbitrary unit vector as the sine of 0^\circ is zero. So e_t of a pure translation is always just 1.

The translation quaternion p_t of a pure translation has the scalar part 0 and the translation vector \vec{\mathbf{p}} as vector part:

p_t=0\oplus\left[\begin{array}{c}p_x\\p_y\\p_z\end{array}\right] = p_xi+p_yj+p_zk

So a pure translation by vector [p_x\ p_y\ p_z]^T is defined as:

\left\{e_t,p_t\right\}=\left\{1,\ p_xi+p_yj+p_zk\right\}
Example: Translation

A translation by 2 on the y-axis for example would be

\mathbf{Trans}(y,2) \equiv \left\{1,\ 2j\right\}

For a pure rotation there is no translation. Thus the translation quaternion p_r of a pure rotation is a zero quaternion:

p_r=0 \oplus \left[\begin{array}{c}0\\0\\0\end{array}\right] = 0

The rotation quaternion e_r of a pure rotation around an axis defined by a unit vector \vec{\mathbf{u}} by the angle \phi is:

e_r(\vec{\mathbf{u}},\phi)=\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}}

So a pure rotation is defined as:

\left\{e_r,p_r\right\}=\left\{\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}},\ 0\right\}
Example: Rotation

A rotation by 60^\circ around the z-axis for example would be

\mathbf{Rot}(z,60^\circ) \equiv \left\{\cos{30^\circ} + \sin{30^\circ}\left[\begin{array}{c}0\\0\\1\end{array}\right],\ 0\right\} = \left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 0\right\}

Combination of transformations

It is known that a combination of transformations is defined as:

\begin{align} \mathbf{T}_c = \mathbf{T}_2\mathbf{T}_1 = \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_2\mathbf{R}_1 & & \vec{\mathbf{p}}_2+\mathbf{R}_2\vec{\mathbf{p}}_1 \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] = \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_c & & \vec{\mathbf{p}}_c \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \quad \equiv \quad \left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\} \end{align}

But how can the two quaternions e_c and p_c of the quaternion notation be calculated based on the quaternions of the individual transformations? The first transformation leads to

q' = e_1\ q\ e_1^*+ p_1

Now the second transformation is applied on q'. The resulting equation can be solved using the distributive law for quaternions to determine e_c and p_c:

\begin{align} q'' &= e_2\ q'\ e_2^*+ p_2 \\ &= e_2\Big(e_1qe_1^*+ p_1\Big)e_2^*+ p_2 \\ &= \Big(e_2e_1qe_1^*+ e_2p_1\Big)e_2^*+ p_2 \\ &= \underbrace{e_2e_1}_{e_c}q\underbrace{e_1^*e_2^*}_{e_c^*}+\underbrace{e_2p_1e_2^*+ p_2}_{p_c} \\ &= e_c\ q\ e_c^*+p_c \\ \end{align}

Thus the combination of two transformations can be denoted in quaternion notation as

\left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}=\left\{e_2e_1\ , \ e_2p_1e_2^*+p_2\right\}

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 \mathbf{T}_c and its rotational and translational components \mathbf{R}_c and \vec{\mathbf{p}}_c.

Example: Combination of rotation and translation

The two examples for rotation and rotation are now to be combined. The combined transform is defined as:

\mathbf{T}_c = \mathbf{Trans}(y,2)\mathbf{Rot}(z,60^\circ) \quad \equiv \quad \left\{e_c,p_c\right\}

Using the explanations above this leads to:

\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}
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