velocity kinematics
Velocity kinematics is the study of speeds rather than positions: given how fast each of a robot arm's joints is turning right now, how fast and in what direction is its hand actually moving? It is the moving-picture companion to ordinary position kinematics, which only asks where the hand ends up for a given set of joint angles. Velocity kinematics instead asks how the hand is sliding and rotating at this very instant.
The tool that links the two is the Jacobian — the table that maps joint speeds to hand speed. Feed in the joint turning-rates and the Jacobian hands back the hand's linear speed (its travel left-right, up-down, forward-back) and angular speed (its rolling and tilting). Run the same relationship backward and you solve the everyday control problem: "I want the hand to glide this way at this speed — how fast should each joint turn?" Because the link is so direct, velocity kinematics is what lets a robot trace smooth, controlled paths in real time, rather than just hopping from one fixed pose to the next.
There is a catch worth knowing early: the relationship holds only for the arm's pose at this instant, and it shifts as the arm moves. So a controller cannot solve it once and forget it; it keeps recomputing the link, tick after tick, as the arm flexes. This constant refreshing is exactly what makes the motion come out smooth instead of jerky.
A camera-arm follows a runner along the track. The operator just dials in "pan right at this speed, lift slightly"; velocity kinematics continuously converts that desired hand motion into the exact turning speed for each joint, so the shot stays smooth even as the arm changes shape.
Smooth real-time tracking is velocity kinematics solved backward, instant by instant.
Velocity kinematics is often called "first-order" kinematics because it deals with rates of change (speeds), one step beyond plain position.