Rigid Body Motion: Twists and Screws

Chasles' theorem says any rigid body motion can be done as one rotation about an axis plus a slide along that same axis. In the plane, that becomes a rotation about a single fixed point, or a pure translation if there is no rotation at all.

End x End y End rotation Motion progress Play the motion Also show straight-line interpolation

The pivot point

A planar motion with rotation $R$ and translation $\mathbf{t}$ leaves exactly one point fixed, the point $\mathbf{c}$ that does not move:

$$R\,\mathbf{c} + \mathbf{t} = \mathbf{c} \quad\Longrightarrow\quad \mathbf{c} = (I - R)^{-1}\,\mathbf{t}$$

The constant twist that produces the motion rotates at rate $\omega$ about $\mathbf{c}$. Its linear part is the velocity of the point at the origin:

$$\boldsymbol{\xi} = (\omega,\ v_x,\ v_y), \qquad \mathbf{v} = -\,\boldsymbol{\omega} \times \mathbf{c} = (\omega\, c_y,\ -\omega\, c_x)$$

What to notice

Every corner of the body moves on a circle around the same red pivot. Set the end rotation close to zero and the pivot runs off to infinity, since the motion becomes a pure slide. The gray ghost interpolates x, y, and angle separately, which takes a different and less natural path.

In 3D the pivot becomes a screw axis, and the body may also slide along it, like a nut on a bolt. This is the foundation of the product of exponentials formula for robot arms.