Editing Screw theory (section)

== Twist ==
In order to define the ''twist'' of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, {{nowrap|1=D(''t'') = ([A(''t'')], '''d'''(''t''))}}, where [A] is a rotation matrix and '''d''' is a translation vector.  This causes a point '''p''' that is fixed in moving body coordinates to trace a curve '''P'''(t) in the fixed frame given by
: <math>
\mathbf{P}(t) = [A(t)]\mathbf{p} + \mathbf{d}(t).
</math>

The velocity of '''P''' is
: <math>
\mathbf{V}_P(t) = \left[\frac{dA(t)}{dt}\right]\mathbf{p} + \mathbf{v}(t),
</math>
where '''v''' is velocity of the origin of the moving frame, that is d'''d'''/dt.  Now substitute '''p'''&nbsp;=&nbsp; [''A''<sup>T</sup>]('''P'''&nbsp;−&nbsp;'''d''') into this equation to obtain,
: <math>
\mathbf{V}_P(t) = [\Omega]\mathbf{P} + \mathbf{v} - [\Omega]\mathbf{d}\quad\text{or}\quad\mathbf{V}_P(t) = \mathbf{\omega}\times\mathbf{P} + \mathbf{v} + \mathbf{d}\times\mathbf{\omega},
</math>
where [Ω]&nbsp;=&nbsp;[d''A''/d''t''][''A''<sup>T</sup>] is the angular velocity matrix and ω is the angular velocity vector.

The screw 
: <math> \mathsf{T}=(\vec{\omega}, \mathbf{v} + \mathbf{d}\times \vec{\omega}),\!</math>
is the ''twist'' of the moving body.  The vector '''V'''&nbsp;=&nbsp;'''v'''&nbsp;+&nbsp;'''d'''&nbsp;×&nbsp;''ω'' is the velocity of the point in the body that corresponds with the origin of the fixed frame.

There are two important special cases: (i) when '''d''' is constant, that is '''v'''&nbsp;=&nbsp;0,  then the twist is a pure rotation about a line, then the twist is
: <math>\mathsf{L}=(\omega, \mathbf{d}\times\omega),</math> 
and (ii) when [Ω]&nbsp;=&nbsp;0, that is the body does not rotate but only slides in the direction '''v''', then the twist is a pure slide given by
: <math> \mathsf{T}=(0, \mathbf{v}).</math>

=== Revolute joints ===
For a [[revolute joint]], let the axis of rotation pass through the point '''q''' and be directed along the vector '''''ω''''', then the twist for the joint is given by,
: <math> \xi =  \begin{Bmatrix} \boldsymbol{\omega} \\ q \times \boldsymbol{\omega} \end{Bmatrix}.</math>

=== Prismatic joints ===
For a [[prismatic joint]], let the vector '''v''' pointing define the direction of the slide, then the twist for the joint is given by,
: <math> \xi =  \begin{Bmatrix} 0\\v \end{Bmatrix}.</math>