 Wrench pitch

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.