Wrench pitch ($h=\mathbf{F}\cdot \mathbf{\tau }/\Vert \mathbf{F}{\Vert }^2$)

For a wrench with nonzero resultant force $\mathbf{F}$ and moment ${\mathbf{\tau }}_O$ about a stated point $O$, its wrench pitch isIt has units of length: torque divided by force. It measures the signed moment remaining parallel to the resultant, per unit force. The force/moment representation and its frame transformations are developed in the primary robotics text’s wrench supplement.^[ref]

The wrench pitch does not depend on the moment origin. Moving from $O$ to $P$, with displacement $\mathbf{r}=\overrightarrow{OP}$ expressed in the same axes, gives ${\mathbf{\tau }}_P={\mathbf{\tau }}_O-\mathbf{r}\times \mathbf{F}$. Taking the dot product with $\mathbf{F}$ removes the added term because $\mathbf{F}\cdot (\mathbf{r}\times \mathbf{F})=0$. A common rotation of both vectors also preserves the ratio. Scaling every applied load by the same nonzero scalar leaves $h$ unchanged.

The central axis of a wrench makes the meaning visible. DefineThe vector triple-product identity givesEvery point on the line ${\mathbf{r}}_0+s\hat{\mathbf{F}}$ has that same parallel residual moment. This line also minimizes the moment magnitude over all choices of origin: the perpendicular part can be removed, while the invariant parallel part cannot. The minimum is $|h|\Vert \mathbf{F}\Vert$.

For an original example, let $\mathbf{F}=(0,0,20)\mathrm{N}$ and ${\mathbf{\tau }}_O=(4,-2,6)\mathrm{N}\mathrm{m}$. Then $h=0.30\mathrm{m}$ and ${\mathbf{r}}_0=(0.10,0.20,0)\mathrm{m}$. Translating the moment origin to that point leaves $(0,0,6)\mathrm{N}\mathrm{m}$: a $20\mathrm{N}$ axial force together with a $6\mathrm{N}\mathrm{m}$ axial force couple. No relocation of a single force can eliminate this remaining couple.

If $h=0$ with $\mathbf{F}\ne 0$, the central-axis moment vanishes and one force on that line reproduces the external load system. If $\mathbf{F}=0$ and the moment is nonzero, the load is a pure force couple; the formula for $h$ is undefined. An infinite-pitch description is a limiting screw-theory convention, not a finite value supplied by this quotient. Both zero force and zero moment give the zero wrench and no distinguished axis.

A proposed load-sensor check combines a known force on an offset line with a known parallel couple. Recompute moments about several origins and compare the resulting $h$ values. Errors in origin translation can corrupt the moment components while the exact invariant remains fixed; near zero resultant force, the division by $\Vert \mathbf{F}{\Vert }^2$ makes pitch sensitive to measurement error. This geometric load parameter is not automatically a physical screw’s thread spacing.

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.