wrench (force–torque pair)

A wrench is the way robotics bundles together everything you can do to push a solid object around: a straight push or pull (a force) and a twisting effort (a torque) packaged as one six-number object. The name is a small pun — it is not the hand tool, but it captures the same idea, since turning a real wrench applies both a push at your hand and a twist at the bolt. Three of the six numbers describe the force (how hard and in which direction it shoves), and the other three describe the torque (how hard and about which way it tries to spin the object). Carrying them together as a single quantity keeps the bookkeeping honest when a body is being shoved and twisted at once.

Why pair them rather than track force and torque separately? Because in the real world they almost never act alone, and they are deeply linked: a force applied off to the side of an object's center also makes it want to rotate, so a single off-center push secretly contains a torque. By writing the whole effect as one wrench tied to a chosen reference point, you can faithfully add up several pushes and twists, move the description from one frame of reference to another, and reason about balance — whether all the wrenches on a body cancel out, leaving it steady, or sum to something that will make it accelerate and turn.

Wrenches are the natural partner of twists, which package a body's linear and angular velocity into one six-number object the same way. This pairing is no accident: a twist says how a body is moving, a wrench says how it is being acted upon, and multiplying them gives the power flowing into the body. That symmetry is why grippers, force-controlled arms, and walking robots reason in wrenches — it lets a robot ask one clean question, what total force-and-twist am I applying or feeling, instead of juggling many separate numbers.