Screw Theory - Twist

In order to define the twist of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, D(t)=(,d(f)), where is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body to trace a curve P(t) in the fixed frame given by,

\mathbf{P}(t) = \mathbf{p} + \mathbf{d}(t).

The velocity of P is

\mathbf{V}_P(t) = \left\mathbf{p} + \mathbf{v}(t),

where v is velocity of the origin of the moving frame, that is dd/dt. Now substitute p= (P-d) into this equation to obtain,

\mathbf{V}_P(t) = \mathbf{P} + \mathbf{v} - \mathbf{d}\quad\mbox{or}\quad\mathbf{V}_P(t) = \mathbf{\omega}\times\mathbf{P} + \mathbf{v} + \mathbf{d}\times\mathbf{\omega},

where = is the angular velocity matrix and ω is the angular velocity vector.

The screw

is the twist of the moving body. The vector V=v + d×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=0, then the twist is a pure rotation about a line, then the twist is

and (ii) when =0, that is the body does not rotate but only slides in the direction v, then the twist is a pure slide given by

Read more about this topic:  Screw Theory

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