 Wrench (mechanics)

A wrench represents a system of loads on a rigid body by its resultant force and resultant torque about a stated reference point. In the moment-first convention, writeThe ${\mathbf{C}}_j$ are applied force couples. The six entries package two different physical units: newtons for force and newton-metres for torque. The primary robotics text and its author video use this representation to transform measured forces and moments between reference frames.^[ref]

An original example has a force $(0,10,0)\mathrm{N}$ applied at $(0.30,0,0)\mathrm{m}$, plus a free couple $(0,0,2)\mathrm{N}\mathrm{m}$. Its resultant torque about the origin is $(0,0,5)\mathrm{N}\mathrm{m}$. Shift the reference point to $P=(0.10,0,0)\mathrm{m}$ without rotating the axes. ThenThe physical loading has not changed, but its torque coordinates depend on the reference point. A rotation of axes additionally rotates the vector components.

The dual velocity object is a rigid-body twist, ordered as $(\mathbf{\omega },{\mathbf{v}}_O)$. Their pairing is mechanical power:Since ${\mathbf{v}}_P={\mathbf{v}}_O+\mathbf{\omega }\times {\mathbf{r}}_{OP}$, substituting the shifted torque cancels the extra scalar triple product and preserves power. This explains why changing the force-sensor frame must transform torque and force together.

A nonzero resultant force does not always replace the whole wrench by one force alone. Define wrench pitchThen ${\mathbf{r}}_0\times \mathbf{F}={\mathbf{\tau }}_O-h\mathbf{F}$. A force along the central line through ${\mathbf{r}}_0$ leaves the parallel residual couple $h\mathbf{F}$. For $\mathbf{F}=(0,0,10)\mathrm{N}$ and ${\mathbf{\tau }}_O=(2,0,3)\mathrm{N}\mathrm{m}$, ${\mathbf{r}}_0=(0,0.20,0)\mathrm{m}$ and $h=0.30\mathrm{m}$. The remaining axial couple is $(0,0,3)\mathrm{N}\mathrm{m}$. If the resultant force is zero, a nonzero pure couple must instead be retained directly.

A proposed check uses a calibrated force-torque sensor and known lever arms to compare the reported six components at two reference points. Agreement tests the transformation and loading model; it does not establish that two externally equivalent loads produce identical internal stress in a deformable part.