Wrench

Packs a 3D force and a 3D torque into one 6-dimensional vector describing the full load acting on a rigid body.

A wrench is the 6-dimensional vector F = (m, f): f is the 3D force and m is the 3D torque, both expressed in the same frame. It's the dual of the twist — their dot product VᵀF equals power, and because power doesn't depend on the choice of reference frame, the rule for transforming a wrench between frames follows directly from that fact. Robotics uses it anywhere the question is 'how much force is being applied': a wrist-mounted six-axis force-torque sensor reads out a wrench directly; the statics equation τ = JᵀF converts an end-effector wrench into joint torques; and grasp analysis sums the contact forces at each fingertip into the total wrench acting on the object. Component ordering isn't standardized across sources: Lynch and Park's textbook writes it as (torque, force), while ROS's geometry_msgs/Wrench and Franka's libfranka both list force before torque.

ExampleAn example from Modern Robotics (taking g = 10 m/s²): a 0.5 kg robot hand holds a 0.1 kg apple, and the wrist's six-axis force-torque sensor reads not just the 6 N of weight but also 0.75 N·m of torque, because the centers of mass of both the hand and the apple sit offset from the sensor.

Also called

Force-Torque, Six-Dimensional Force/Torque

Related

Twist · Six-Axis Force/Torque Sensor · Statics · Torque · Grasp Matrix · Adjoint Representation

Sources

Modern Robotics 3.4: Wrenches
ROS 2 geometry_msgs/Wrench.msg
Modern Robotics: Mechanics, Planning, and Control (Lynch & Park, free preprint)