Wrench

• Also called: spatial force, force-torque vector
• Category: Foundations & spatial maths › Frames & transforms
• Tags: screw-theory, force-control, statics, spatial-math, contact

A six-dimensional vector combining the force and the moment (torque) acting on a rigid body, expressed at a chosen reference point and frame.

Definition

A wrench packs the force and the moment acting on a rigid body into a single six-dimensional vector: three components of force and three of moment, or torque. The name comes from screw theory, developed by Robert Ball, in which any system of forces on a rigid body is equivalent to a force along a line plus a moment about that same line. Wrenches are the force-side counterpart of twists, which combine angular and linear velocity in the same way.

A wrench is only meaningful together with the frame and point it is expressed at. The force part is the same wherever it is measured, but the moment part changes when the reference point moves: shifting the point by a vector r adds r × F to the moment. Transforming a wrench between frames therefore uses the transpose of the adjoint of the rigid-body transform, not just a rotation. Conventions on ordering also vary: some texts write [F; T], others, including Lynch and Park's Modern Robotics, put the moment first. Mixing conventions between libraries is a common source of bugs.

Wrenches connect directly to the rest of robot mechanics. The dot product of a twist and a wrench is mechanical power. The manipulator Jacobian maps joint velocities to an end-effector twist, and its transpose maps an end-effector wrench to the joint torques needed to apply or resist it: τ = Jᵀ F. Six-axis force-torque sensors measure a wrench in their own frame. In grasping, the set of wrenches that contacts can apply, built from their friction cones, decides whether a grasp can resist arbitrary disturbances.

Why it matters

Force control, grasp analysis, contact estimation and dynamics all work with wrenches. Treating force and moment as one object with a clear frame and reference point avoids a whole class of errors, and it is what allows a measured wrench at the wrist to be turned into contact locations, joint torques or tool forces.

Key applications

• Force and impedance control using wrist force-torque sensors
• Grasp analysis, force closure and wrench-space quality metrics
• Estimating contact location and in-hand object pose from measured forces
• Mapping end-effector loads to joint torques with the Jacobian transpose
• Rigid-body dynamics algorithms written with spatial vectors

In practice

A team fits a six-axis force-torque sensor between an arm's flange and a gripper. Before using the readings, they remove the sensor bias and subtract the gripper's own weight, computed as a gravity wrench from the tool's mass and centre of mass and the current orientation of the sensor. To command a pushing force at the fingertips, they transform the desired fingertip wrench to the sensor frame with the adjoint transform, compare it with the measured wrench, and map the error through the Jacobian transpose to joint torques. When the same sensor is used to estimate where an object touches the fingers, the ratio between the moment and force parts of the wrench constrains the line along which the contact force must act.

Common misconceptions

• A wrench is just a force — it includes the moment as well, and the moment changes with the point the wrench is expressed at
• Wrenches transform between frames by rotation alone — changing the reference point also changes the moment, so the full adjoint transform is needed
• Torque in a wrench and motor torque are the same thing — the wrench moment acts on the body in Cartesian space, while joint torques are related to it through the Jacobian transpose

Who should know this

• Manipulation and force-control engineers
• Grasp planning and dexterous manipulation researchers
• Engineers integrating force-torque sensors
• Robotics students studying spatial mechanics

Related technologies

• Six-Axis Force-Torque Sensor
• Twist
• Adjoint Transformation
• Spatial Vector Algebra
• Grasp Wrench Space

Related terms

• Force/Torque Sensor
• Screw Axis
• Jacobian
• Friction Cone
• SE(3)
• Impedance Control

Related projects

• Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design