2.2. Twists and Screws

Twists and Screws

Chasles’ theorem states that any rigid body displacement in three-dimensional space can be represented as a rotation about some axis combined with a translation along that same axis. In other words, every rigid motion is a screw motion.

This geometric fact motivates the structure of the Lie algebra of $SE(3)$. The infinitesimal counterpart of $SE(3)$ is its Lie algebra $\mathfrak{se}(3)$, whose elements are called twists. A twist represents the instantaneous version of a screw motion and is written as a six-dimensional vector
$\xi =\left[\begin{array}{c}v\\ \omega \end{array}\right],$
where $\omega \in {\mathbb{R}}^3$ represents angular velocity and $v\in {\mathbb{R}}^3$ represents linear velocity. These two components together describe a screw motion, meaning that at any instant a rigid body’s motion can be viewed as rotation about some axis combined with translation along that axis.

Every twist can be converted into a $4\times 4$ matrix using the “hat” operator,
$\hat{\xi }=\left[\begin{array}{cc}[\omega {]}_{\times }& v\\ 0& 0\end{array}\right]\in se(3),$
where $[\omega {]}_{\times }$ is the skew-symmetric matrix representing the cross product with $\omega$. The matrix exponential of this object produces a finite rigid motion:
$g(\theta )=\exp (\hat{\xi }\theta )=\left[\begin{array}{cc}R(\theta )& p(\theta )\\ 0& 1\end{array}\right]\in SE(3),$
where $\theta$ is a scalar displacement parameter (an angle, distance, or just a scalar depending on the type of joint). Thus, the six numbers in $\xi$ define the direction of motion, while $\theta$ specifies how far to move along that direction.

Geometrically, a twist may correspond to pure rotation (zero pitch), pure translation (infinite pitch), or a general screw motion with finite pitch, defined as the translation per unit rotation. Because rigid motions are fundamentally screw-like, twists provide a unified way to describe all joint motions in robotics, which naturally leads to the classification of joint types.

Twists are not absolute objects: the same physical instantaneous motion has different coordinate representations depending on the reference frame. If a rigid transform $g\in SE(3)$ relates two frames, then the twist coordinates change by the adjoint representation of SE(3):

${\xi }^{\prime }={\mathrm{Ad}}_g\xi ,{\mathrm{Ad}}_g=\left[\begin{array}{cc}R& [p{]}_{\times }R\\ 0& R\end{array}\right],$
which mixes the linear and angular components because translating the reference frame changes the linear velocity induced by a given angular velocity. This distinction also motivates the two standard conventions for expressing twists: a spatial twist is the instantaneous velocity of the body written in the fixed (space) frame, whereas a body twist is the same instantaneous velocity written in the moving (body) frame. If g(t) is the body pose, these are defined by
${\hat{\xi }}_s=\dot{g}{g}^{-1},{\hat{\xi }}_b=g^{-1}\dot{g},$
and they are related by the same adjoint map,
${\xi }_s={\mathrm{Ad}}_g{\xi }_b.$
Choosing spatial versus body coordinates does not change the underlying motion; it changes only the frame in which the 6D velocity is reported, and the adjoint provides the exact conversion between the two.

Intuition

A twist is the infinitesimal generator of a screw motion, just as a skew-symmetric matrix $\hat{\omega }\in \mathfrak{so}(3)$ is the infinitesimal generator of a rotation. The six components of $\xi =(v,\omega )$ encode two things: the direction of the screw axis (determined by $\omega$ and $v$) and the type of motion (rotation, translation, or helical). The scalar $\theta$ says how far along that screw to move.

The classification by pitch $h={\omega }^{T}v/\Vert \omega {\Vert }^2$ captures the ratio of translation to rotation along the screw axis. A door hinge has $h=0$ (pure rotation); a sliding drawer has $h=\infty$ (pure translation); a corkscrew or bolt has finite $h$ (helical motion). Every possible rigid body motion falls into one of these three categories (Chasles’ theorem), so twists provide a universal description of joint motions.

Properties

Twist Classification

(Revolute Twist) Pitch $h=0$, $\Vert \omega \Vert =1$. The twist has the form $\xi =\left[\begin{array}{c}-\omega \times q\\ \omega \end{array}\right]$ where $q$ is any point on the rotation axis. The linear component $v=-\omega \times q$ encodes the axis location, not a translational velocity.

(Prismatic Twist) $\omega =0$, $\Vert v\Vert =1$. The twist $\xi =\left[\begin{array}{c}v\\ 0\end{array}\right]$ represents pure translation along $v$. The exponential is $e^{\hat{\xi }\theta }=\left[\begin{array}{cc}I& v\theta \\ 0& 1\end{array}\right]$.

(Helical Twist) Pitch $h={\omega }^{T}v/\Vert \omega {\Vert }^2\ne 0$. The motion combines rotation by $\theta$ about an axis with translation by $h\theta$ along that axis.

Key Identities

(Twist Exponential Conjugation) For $g\in SE(3)$ and twist $\xi$:

$g{e}^{\hat{\xi }\theta }{g}^{-1}=e^{\widehat{{\mathrm{Ad}}_g\xi }\theta }.$
This says that conjugating a screw motion by a rigid transform is the same as transforming the twist by the adjoint. It is the $SE(3)$ analogue of the hat conjugation identity for $SO(3)$.

(Adjoint Properties) The adjoint satisfies: ${\mathrm{Ad}}_{g_1{g}_2}={\mathrm{Ad}}_{g_1}{\mathrm{Ad}}_{g_2}$ (composition), ${\mathrm{Ad}}_{g^{-1}}=({\mathrm{Ad}}_g{)}^{-1}$ (inverse), and $\det ({\mathrm{Ad}}_g)=1$ (volume-preserving). See adjoint map for the explicit formula and its role in velocity kinematics.

(Recovering the Axis) For a revolute twist $\xi =(-\omega \times q,\omega )$, the axis direction is $\omega$ and a point on the axis is $q=\omega \times v/\Vert \omega {\Vert }^2$ (when ${\omega }^{T}v=0$, i.e., zero pitch).

Examples

Constructing a Revolute Twist

A joint rotates about the $z$-axis through the point $q=(a,0,0)$:

Twist computation $\omega =(0,0,1)$, $v=-\omega \times q=-(0,0,1)\times (a,0,0)=(0,a,0)$.

So $\xi =\left[\begin{array}{c}0\\ a\\ 0\\ 0\\ 0\\ 1\end{array}\right]$. Exponentiating: $e^{\hat{\xi }\theta }$ gives a rotation by $\theta$ about the $z$-axis through $(a,0,0)$.

If instead the axis passes through the origin ($q=0$), then $v=0$ and $\xi =\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]$—a pure rotation about $z$.