Editing Screw theory (section)

== Work of forces acting on a rigid body ==
Consider the set of forces  '''F'''<sub>1</sub>, '''F'''<sub>2</sub> ... '''F'''<sub>''n''</sub> act on the points '''X'''<sub>1</sub>, '''X'''<sub>2</sub> ... '''X'''<sub>''n''</sub> in a rigid body. The trajectories of '''X'''<sub>''i''</sub>, ''i''&nbsp;=&nbsp;1,...,''n''  are defined by the movement of the rigid body with rotation [''A''(''t'')] and the translation '''d'''(''t'') of a reference point in the body, given by 
: <math> \mathbf{X}_i(t)= [A(t)]\mathbf{x}_i + \mathbf{d}(t)\quad i=1,\ldots, n, </math>
where '''x'''<sub>''i''</sub> are coordinates in the moving body.

The velocity of each point '''X'''<sub>i</sub> is
: <math>\mathbf{V}_i = \vec{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \mathbf{v},</math>
where '''ω''' is the angular velocity vector and '''v''' is the derivative of '''d'''(''t'').

The work by the forces over the displacement ''δ'''''r'''<sub>''i''</sub>='''v'''<sub>''i''</sub>''δt'' of each point is given by
: <math> \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \cdots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t.</math>
Define the velocities of each point in terms of the twist of the moving body to obtain
: <math> \delta W =  \sum_{i=1}^n \mathbf{F}_i\cdot (\vec{\omega}\times(\mathbf{X}_i -\mathbf{d}) + \mathbf{v})\delta t. </math>

Expand this equation and collect coefficients of ω and '''v''' to obtain
: <math>
\begin{align}
\delta W & = \left(\sum_{i=1}^n \mathbf{F}_i\right) \cdot\mathbf{d}\times \vec{\omega}\delta t+ \left(\sum_{i=1}^n \mathbf{F}_i\right)\cdot\mathbf{v}\delta t + \left(\sum_{i=1}^n \mathbf{X}_i \times\mathbf{F}_i\right) \cdot \vec{\omega}\delta t \\[4pt]
& = \left(\sum_{i=1}^n \mathbf{F}_i\right) \cdot(\mathbf{v}+\mathbf{d}\times \vec{\omega}) \delta t + \left(\sum_{i=1}^n \mathbf{X}_i \times\mathbf{F}_i\right) \cdot\vec{\omega}\delta t.
\end{align}
</math>

Introduce the twist of the moving body and the wrench acting on it given by
: <math> \mathsf{T} = (\vec{\omega},\mathbf{d}\times \vec{\omega} +\mathbf{v})=(\mathbf{T},\mathbf{T}^\circ),\quad\mathsf{W} = \left(\sum_{i=1}^n \mathbf{F}_i, \sum_{i=1}^n \mathbf{X}_i \times\mathbf{F}_i\right) = (\mathbf{W},\mathbf{W}^\circ), </math>
then work takes the form
: <math>\delta W = (\mathbf{W}\cdot\mathbf{T}^\circ +  \mathbf{W}^\circ \cdot\mathbf{T})\delta t.</math>

The 6×6 matrix [Π] is used to simplify the calculation of work using screws, so that
: <math>\delta W = (\mathbf{W}\cdot\mathbf{T}^\circ +  \mathbf{W}^\circ \cdot\mathbf{T})\delta t = \mathsf{W}[\Pi]\mathsf{T}\delta t,</math>
where
: <math> [\Pi] =\begin{bmatrix} 0 & I \\ I & 0 \end{bmatrix},</math>
and [I] is the 3×3 identity matrix.

=== Reciprocal screws ===
If the virtual work of a wrench on a twist is zero, then the forces and torque of the wrench are constraint forces relative to the twist.  The wrench and twist are said to be ''reciprocal,'' that is if
: <math>\delta W =\mathsf{W}[\Pi]\mathsf{T}\delta t = 0,</math>
then the screws ''W'' and ''T'' are reciprocal.

=== Twists in robotics ===
In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6×6 matrix [Π]  in the calculation of work.<ref name="McCarthy"/>  In this case the twist is defined to be
: <math>\check{\mathsf{T}} =  (\mathbf{d}\times \vec{\omega} +\mathbf{v},\vec{\omega}),</math>
so the calculation of work takes the form
: <math>\delta W =\mathsf{W}\cdot\check{\mathsf{T}}\delta t.</math>

In this case, if
: <math>\delta W =\mathsf{W}\cdot\check{\mathsf{T}}\delta t= 0,</math>
then the wrench W is reciprocal to the twist T.