# What is the difference between screw and wrench in rigid body motion?

A screw is defined by a six dimensional vector of forces and torques. It can represent any spatial movement of a rigid body (as written here). But I don't get the following distinction between screw and wrench:

The force and torque vectors that arise in applying Newton's laws to a rigid body can be assembled into a screw called a wrench.

It seems to be some kind of contextualisation but in what way?

I will try to make it as simple as possible. Imagine you have a SCREW, when you WRENCH it, it TWIST forward or backward.

The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line.

It means that any system can be described as those coordinate system is a screw. For example a normal screw has it axis and it pitch.

A WRENCH is one case of SCREW where the act (force or torque) have the axis and yield a movement (translation or rotation)

You can have a look at the slide of the Summer Screw Summer School for more detail.

• I haven't considered a screw to be a coordinate system. My way of thinking was always, that the screw enables a lower dimensional representation of displacement. Could you please explain in more detail? Jul 18, 2014 at 15:58
• What can be described in a system with an axis and the magnitudes vector along the axis is a SCREW. From the Summer Screw page 19 : a line l with a pitch h (a metric quantity) is a geometric element called a screw. A wrench is a system of forces (reduced at a point) with equivalent systems identified.
– Loan
Jul 21, 2014 at 15:37

Summarizing from Murray, Li, and Sastry (chapters 3 and 5) there are 3 related things:

• Twist: An element of se(3) (which is a bit like the derivative of an element of SE(3), which is the set of translations + rotations)
• Screw: A translation+rotation (i.e. and element of SE(3))
• Wrench: Generalized force (combination of linear force and torque)
• Thanks for the summary, that helps a lot! Could you please explain why it is possible to state a wrench as a combination of force and torque as well as a screw and a force magnitude? Jul 18, 2014 at 15:55
• Sorry, can you restate your question? I don't quite understand. Jul 18, 2014 at 15:59
• I refer to this paper: profs.sci.univr.it/~fiorini/corsi/robotica/icra2001notes.pdf page 28, where one can see that a wrench can either be expressed as a concatenation of force and moment or as a convolution of screw and force. For me it is still not clear, why this holds - but maybe this is no longer part of this thread Jul 18, 2014 at 16:25
• Sorry, but without studying it in more detail I'm not sure exactly what that paper is saying. I think it is saying that, similar to 3D motion, a wrench can be described by defining a line of action and a scalar times a pitch (i.e. similar to a screw describing a 3D motion). Jul 18, 2014 at 18:42

I strongly recommend a patient lecture of the chapters related to rigid body kinematics and dynamics of the book by Goldstein: classical mechanics. It will clear all our doubts.