2.3.2. Prismatic Joints

Prismatic Joint

A prismatic joint is a one-DOF joint that produces pure translation along a fixed axis (no rotation).

We choose a fixed translation direction v (typically unit) so that the joint displacement $\theta$ represents distance traveled along that direction.

Because pure translation moves every point of the rigid body with the same instantaneous velocity (independent of where the point is), the point velocity is:
$\dot{p}(t)=\dot{\theta }(t)v.$

Since there is no angular velocity component for a prismatic joint, the twist must have $\omega =0$, so:
$\xi =\left[\begin{array}{c}v\\ 0\end{array}\right].$

Because $\hat{\omega }=0$ in this case, the twist in $\mathfrak{se}(3)$ is:
$\hat{\xi }=\left[\begin{array}{cc}0& v\\ 0& 0\end{array}\right].$

Then, $\hat{\xi }$ is the infinitesimal generator of the motion in homogeneous coordinates. The instantaneous and finite kinematics satisfy:
$\left[\begin{array}{c}\dot{p}(\theta )\\ 0\end{array}\right]=\hat{\xi }\left[\begin{array}{c}p(\theta )\\ 1\end{array}\right],\left[\begin{array}{c}p(\theta )\\ 1\end{array}\right]=e^{\hat{\xi }\theta }\left[\begin{array}{c}p(0)\\ 1\end{array}\right].$

Exponentiating a pure-translation generator yields identity rotation and a translation of magnitude $\theta$ along $v$. Therefore, the transform is:
$e^{\hat{\xi }\theta }=\left[\begin{array}{cc}I& v\theta \\ 0& 1\end{array}\right].$

If $v$ is unit, then $\theta$ has units of length and is exactly the distance traveled; if $v$ is not unit, the actual translation is still $v\theta$.