differential kinematics
Differential kinematics looks at tiny changes: if you nudge each of a robot arm's joints by a hair, exactly how much, and in which direction, does its hand shift? Instead of asking about full speeds over a second, it asks about the smallest imaginable steps — a whisker of joint rotation and the matching whisker of hand movement. It is the "zoomed-in" view of how joint motion and hand motion are tied together.
Working in tiny steps turns out to be a wonderful simplification. Over a full, large motion the link between joints and hand is tangled and curved; but if the step is small enough, that link becomes very nearly a simple, straight proportion — and that proportion is captured by the Jacobian, the table that maps a small twist of the joints into a small shift of the hand. This is why so much of robot control is built on tiny increments: stack up thousands of these small, well-behaved steps and you get a complete, smooth, controllable motion, without ever wrestling with the full curved relationship all at once.
Differential kinematics and velocity kinematics are two faces of the same coin. Velocity kinematics phrases the link as speeds (motion per second); differential kinematics phrases it as increments (motion per tiny step). Both lean on the same Jacobian, and a digital controller — which updates in tiny ticks anyway — naturally thinks in these small increments.
To draw a circle, a robot does not solve the whole circle at once. It breaks the loop into hundreds of tiny straight hops; for each hop it uses differential kinematics to find the small joint nudge that moves the pen by that little step. Strung together, the tiny straight hops trace a smooth circle.
Big curved motions are built from many tiny straight steps, each one a differential-kinematics solve.
"Differential" here just means "about very small differences" — the same idea behind calculus, where curved relationships are tamed by looking at them one tiny step at a time.