Central axis of a wrench ($\mathbf{r}=\mathbf{F}\times {\mathbf{\tau }}_O/\Vert \mathbf{F}{\Vert }^2+s\hat{\mathbf{F}}$)

The central axis of a wrench is the line of reference points about which the resultant torque is parallel to the nonzero resultant force. It converts a general wrench into a force along one line plus a parallel force couple. A nonzero parallel couple cannot be removed merely by moving the force’s application point.

Let $\mathbf{F}\ne 0$ and ${\mathbf{\tau }}_O$ be the force and moment about origin $O$. DefineThen the central axis, expressed relative to $O$, isHere $h$ is the wrench pitch. Since ${\mathbf{r}}_0\cdot \mathbf{F}=0$, ${\mathbf{r}}_0$ is the point on the axis closest to $O$. The primary robotics wrench representation supplies the force/moment translation law used in this construction.^[ref]

For an original derivation, write the moment at a point displaced by $\mathbf{r}$ as ${\mathbf{\tau }}_{\mathbf{r}}={\mathbf{\tau }}_O-\mathbf{r}\times \mathbf{F}$. The triple-product identity gives ${\mathbf{r}}_0\times \mathbf{F}={\mathbf{\tau }}_O-h\mathbf{F}$, henceThe two terms are perpendicular. ThereforeThis proves both the minimum-moment property and the uniqueness of the minimizing line: the minimum magnitude is $|h|\Vert \mathbf{F}\Vert$, attained exactly on $\mathcal{L}$. It also quantifies the moment penalty for a perpendicular displacement from that line.

For a numerical example, take $\mathbf{F}=(10,0,0)\mathrm{N}$ and ${\mathbf{\tau }}_O=(2,3,-4)\mathrm{N}\mathrm{m}$. Then $h=0.20\mathrm{m}$, ${\mathbf{r}}_0=(0,0.40,0.30)\mathrm{m}$, and the axis is parallel to the $x$ direction. The closest-axis distance from the original origin is $0.50\mathrm{m}$. On the axis the moment is $(2,0,0)\mathrm{N}\mathrm{m}$; at a point $0.10\mathrm{m}$ perpendicularly from it, its magnitude is $\sqrt{2^2+1^2}\approx 2.236\mathrm{N}\mathrm{m}$. At $O$, it is $\sqrt{29}\mathrm{N}\mathrm{m}$.

If $h=0$, a single force on the central axis represents the wrench. If $\mathbf{F}=0$ and the moment is nonzero, every origin has the same pure couple and this construction has no unique finite axis. With both zero, no direction is selected at all. Near zero force, the denominator also makes a reconstructed axis sensitive to measurement uncertainty; an enormous calculated offset need not be a well-resolved physical location.

A proposed six-component load-sensor test applies a known force on an offset line and a known parallel couple, then computes the axis from measurements about several reference points. Correctly translated axes should describe the same geometric line even though their coordinate vectors differ. Subtract tare loads and retain axis conventions before comparing the resulting moments; minimum moment does not mean zero moment unless the wrench pitch is zero.