Wrench

A force can also produce a moment#

A downward load at a robot hand produces a force at the wrist and may also produce a turning moment, depending on its offset from the wrist. A wrench records both effects.

Modern Robotics illustrates this with an apple held by a hand fitted with a force-torque sensor. The same gravitational load has zero moment about the apple's center of mass but a nonzero moment about an offset sensor frame.

The reference point matters#

Forces and moments must be expressed in a declared frame. Moving the reference point changes the moment through the force's moment arm, even when the physical loading stays the same. Moment and force also have different units, conventionally newton-metres and newtons.

A wrench and a twist, expressed with compatible conventions, have a dot product equal to instantaneous mechanical power. That power is independent of the coordinate frame.

Connecting contact loads to joints#

The Jacobian-transpose relationship maps an end-effector wrench to corresponding joint forces and torques under the stated static convention. It is useful in force control and load analysis. During acceleration, inertial and other dynamic terms also matter; a wrench transformation alone is not a complete dynamics calculation.