Editing Screw theory (section)

== Coordinate transformation of screws ==
The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of  line, which in turn are obtained from the transformations of the coordinate of points on the line.

Let the displacement of a body be defined by ''D''&nbsp;=&nbsp;([''A''],&nbsp;'''d'''), where [''A''] is the rotation matrix and '''d''' is the translation vector.  Consider the line in the body defined by the two points '''p''' and '''q''', which has the [[Plücker coordinates]],
: <math> \mathsf{q}=(\mathbf{q}-\mathbf{p}, \mathbf{p}\times\mathbf{q}),</math>
then in the fixed frame we have the transformed point coordinates '''P'''&nbsp;=&nbsp;[''A'']'''p'''&nbsp;+&nbsp;'''d''' and '''Q'''&nbsp;=&nbsp;[''A'']'''q'''&nbsp;+&nbsp;'''d''', which yield.
:<math>\mathsf{Q}=(\mathbf{Q}-\mathbf{P}, \mathbf{P}\times\mathbf{Q}) = ([A](\mathbf{q}-\mathbf{p}), [A](\mathbf{p}\times\mathbf{q}) + \mathbf{d}\times[A](\mathbf{q}-\mathbf{p}))</math>

Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by
: <math>
\begin{Bmatrix} \mathbf{Q}-\mathbf{P} \\ \mathbf{P}\times\mathbf{Q} \end{Bmatrix}
= \begin{bmatrix} A & 0 \\ DA & A \end{bmatrix}
\begin{Bmatrix} \mathbf{q}-\mathbf{p} \\ \mathbf{p}\times\mathbf{q} \end{Bmatrix}.
</math>
The matrix [''D''] is the skew-symmetric matrix that performs the cross product operation, that is [''D'']'''y'''&nbsp;=&nbsp;'''d'''&nbsp;×&nbsp;'''y'''.

The 6&times;6 matrix obtained from the spatial displacement ''D''&nbsp;=&nbsp;([''A''],&nbsp;'''d''') can be assembled into the dual matrix
: <math>[\hat{\mathsf{A}}] =([A], [DA]),</math>
which operates on a screw ''s''&nbsp;=&nbsp;('''s'''.'''v''') to obtain,
: <math>\mathsf{S} = [\hat{\mathsf{A}}]\mathsf{s}, \quad (\mathbf{S}, \mathbf{V}) = ([A], [DA])(\mathbf{s}, \mathbf{v})  = ([A]\mathbf{s}, [A]\mathbf{v}+[DA]\mathbf{s}).</math>

The dual matrix [Â]&nbsp;=&nbsp;([''A''],&nbsp;[''DA'']) has determinant 1 and is called a ''dual orthogonal matrix''.
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== Screw theory and robotics ==
The angular velocity and velocity of the end-effector of a robot are assembled into the pair of vectors called a twist.  The twists used in robotics are transposed versions of the twists used in screw theory.  This transposition facilitates the calculation of virtual work.

=== Twists ===
[[Image:Velocity twist.jpg|thumb|A change in the reference point of the moving frame changes the twist.]]
The twist includes the velocity of the reference point of the moving frame.  To change the reference point in the moving from A to B, one must account for the rotation of the body.  Introduce the notation:
* <math>\vec v_A</math> denotes the linear velocity at point ''A''
* <math>\vec v_B</math> denotes the linear velocity at point ''B''
* <math>\vec \omega</math> denotes the angular velocity of the rigid body
* <math>[\vec r_{AB}]_\times </math> denotes the 3×3 [[Cross product#Conversion_to_matrix_multiplication|cross product matrix]]

In screw notation velocity twists transform with a 6×6 transformation matrix
: <math>\hat{v_A} = \begin{Bmatrix}  \vec v_A\\ \vec \omega \end{Bmatrix} = \begin{bmatrix} 1 & [\vec r_{AB}]_\times \\ 0 & 1 \end{bmatrix}\begin{Bmatrix} \vec v_B \\ \vec \omega \end{Bmatrix}. </math>
Notice that in this formulation the velocity of the reference point is identified as the first vector of the twist, while the angular velocity is the second vector.

=== Wrenches ===
[[Image:Force wrench.jpg|thumb|A change in the reference point of the moving frame changes the wrench.]]
Similarly the equipolent moments expressed at each location within a [[rigid body]] define a helical field called the force wrench. To move representation from point ''B'' to point ''A'', introduce the notation:
* <math>\vec \tau_A</math> denotes the [[equipollent]] (link: [http://en.wikibooks.org/wiki/Statics/Resultants_of_Force_Systems_(contents) wikibooks.org] ) moment at point ''A''
* <math>\vec \tau_B</math> denotes the [[equipollent]] (link: [http://en.wikibooks.org/wiki/Statics/Resultants_of_Force_Systems_(contents) wikibooks.org] ) moment at point ''B''
* <math>\vec F</math> denotes the total force applied to the rigid body
* <math>[\vec r_{AB}]_\times </math> denotes the 3×3 [[Cross product#Conversion_to_matrix_multiplication|cross product matrix]]

In screw notation force wrenches transform with a 6x6 transformation matrix,
:<math>\hat {\tau}_A = \begin{Bmatrix} \vec F \\ \vec {\tau}_A \end{Bmatrix} = \begin{bmatrix}  1 & 0 \\  [ \vec r_{AB} ]_\times  &  1 \end{bmatrix} \begin{Bmatrix} \vec F  \\ \vec {\tau}_B  \end{Bmatrix} . </math>
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