manipulator Jacobian
The manipulator Jacobian is a table of numbers that translates how fast you turn a robot arm's joints into how fast — and in which direction — its hand actually moves through space. Picture an arm with several motors, each spinning its joint at some speed. The hand at the tip ends up gliding, lifting, and twisting all at once. The Jacobian is the bookkeeping device that adds up every joint's individual contribution and tells you the total result: the tip's straight-line speed (how fast it travels left-right, up-down, forward-back) together with its turning speed (how fast it rolls and tilts).
Why a table and not a single number? Because each joint affects the hand differently depending on the arm's current pose, and there are many joints feeding into many directions of motion at once. So the Jacobian is a grid: read down a column to see what one joint does to the hand right now; read across a row to see how all the joints together contribute to one particular direction of hand motion. Crucially, this grid is not fixed — it changes as the arm folds and unfolds, because the same joint that mostly lifts the hand when the arm is stretched out may mostly swing it sideways when the arm is curled up. The Jacobian is the single tool that turns joint speeds into hand motion, and (run in reverse) hand motion back into the joint speeds needed to produce it.
An arm is welding along a straight seam. The controller wants the torch tip to move at a steady 5 centimeters per second. It uses the Jacobian to work out, moment by moment, exactly how fast each of the six joints must turn so that all their motions add up to that one smooth, straight glide.
The Jacobian converts a desired hand motion into the joint speeds that produce it.
The full manipulator Jacobian has six rows — three for straight-line speed and three for turning speed — and one column per joint, so its width tells you how many joints the arm has.