A linear force $f$ acting on a rigid body at point $p$ creates a torque (moment) $m$ that can be represented in a reference frame $\{a\}$ as:

With:

• $m_a \in \mathbb{R}^3$: Torque or moment
• $r_a \in \mathbb{R}^3$: Point of action (of rigid body)
• $f_a \in \mathbb{R}^3$: Force acting on the rigid body

Wrench

The moment and the force can be combined in a 6-dimensional vector (like with twists) called wrench (or spacial force).

• $F_a \in \mathbb{R}^6$: Wrench (spacial force)
• $m_a \in \mathbb{R}^3$: Torque or moment on rigid body
• $f_a \in \mathbb{R}^3$: Force acting on rigid body

A wrench that has a zero linear component is a pure moment

Sum of Wrenches

If multiple wreches act on a rigid body, the total wrench is the sum of all wrench vectors. The wrenches need to be represented in the same frame.

Conversion between Frame Representations

• $F_s$: spacial wrech
• $F_b$: body wrech

From Body Frame to Space Frame

From Space Frame to Body Frame

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017