Push and Twist: Force and Torque
A force is a push or a pull — measured in newtons (N), it is what changes how fast a thing moves in a straight line. When a robot's finger presses on a table or a wheel pushes against the ground, that is a force. But most of a robot's moving parts do not slide in straight lines; they *rotate* around joints. For rotation, the quantity that matters is torque — a twisting effort, measured in newton-metres (N·m).
Torque is force multiplied by a lever arm: how hard you push, times how far from the pivot you push. Loosening a bolt with a long wrench is easy because the long handle gives a big lever arm; the same hand force near the bolt barely budges it. Inside a robot's revolute joint, the motor delivers exactly this kind of twisting effort — the joint torque — about the joint's rotation axis.
Newton's Bargain: Effort Buys Acceleration
Newton's second law is the deal at the heart of all motion: force equals mass times acceleration, written F = m·a. Read it as a price tag. To make a heavy thing accelerate as fast as a light thing, you must pay more force. Mass is the resistance to being sped up — its *inertia*.
Rotation has a twin of this law. Torque equals *rotational inertia* times *angular acceleration* — often written τ = I·α. Here I (the moment of inertia) plays the role mass played before: it measures how stubbornly something resists being spun up. A long, heavy arm stretched straight out has a large I and accelerates slowly for a given torque; the same arm folded in close spins up easily.
Straight line: F = m * a (force buys linear acceleration) Rotation: tau = I * alpha (torque buys angular acceleration) Lever link: tau = F * r (force F at distance r from axis = torque tau)
This is why dynamics matters in robotics: if a controller knows the inertia of each link, it can predict the exact torque needed to produce a wanted acceleration. That prediction is the seed of the full robot equation of motion, which later chapters assemble link by link.
Friction: The Cut Off the Top
In a perfect world, every newton-metre the motor produces would go straight into accelerating the arm. Real joints are not perfect: bearings, gears, and seals rub. That rubbing is friction, and it always opposes the motion you are trying to make. Joint friction is the tax that the physical world collects before the rest of the torque does any useful work.
- Static (stiction): the joint is parked and a small torque arrives. Below a threshold, nothing moves at all — friction silently cancels the whole command until you push hard enough to break free.
- Coulomb (dry) friction: once moving, a roughly constant opposing torque remains, no matter the speed — a flat toll on every motion.
- Viscous friction: a drag that grows with speed, like stirring honey. The faster the joint turns, the bigger the bite.
Friction is why a robot's commands are never quite obeyed. Ask a joint to nudge by a hair's width and stiction may swallow the request entirely; ask it to track a fast curve and viscous drag drains torque you budgeted for acceleration. Good controllers therefore *model* friction and add a little extra torque to pay it off — a small but essential part of the equation of motion.
What the Motor Must Win
Put the pieces together and you can see the job a joint's motor actually faces. The actuation force (or torque) it commands must simultaneously: (1) overcome gravity holding the limb down, (2) pay the friction tax, and (3) leave enough left over to produce the acceleration the task demands. Only the leftover does the useful accelerating; the rest just keeps the robot from losing ground.
Holding an arm still, stretched out horizontally, takes real torque even though nothing moves — pure gravity load. Controllers handle this with gravity compensation: a baseline torque computed to exactly cancel weight, so the operator only ever fights the *extra* effort of moving. Add the friction model on top, and the motor's command starts to look like a small, honest budget.
Working out that full budget — gravity, friction, inertia, and the wanted acceleration, summed for every joint — is exactly the problem of inverse dynamics: given the motion you want, solve for the torques to command. The quantities in this chapter are the line items; the chapters ahead show how to total them up.