kinematic singularity
A kinematic singularity is a special pose where a robot arm suddenly loses the ability to move its hand freely in every direction — as if, in that one position, a door has quietly closed on one of the directions the hand could normally travel. The arm has not broken and nothing is jammed; it is simply folded into a shape where, no matter how you turn the joints, the hand just cannot budge along a certain line. The classic everyday picture is your own arm: stretch it out perfectly straight to reach a shelf, and you cannot push your hand any farther forward by straightening more — you have run out of reach in that direction. That fully-stretched pose is a singularity.
Near such a pose, things get even more troublesome than a plain dead-end. To make the hand inch in the "blocked" direction, the joints would have to whirl impossibly fast — the math demands enormous joint speeds for a tiny bit of hand motion. A controller that naively asks the arm to keep tracing a straight line through a singularity can suddenly command violent, lurching joint speeds. In the language of the Jacobian (the table linking joint speeds to hand motion), a singularity is exactly where that table "loses rank": it can no longer cover all the directions of hand motion, and trying to reverse it blows up.
Because of this, singularities are something robot designers and path planners actively map out and steer around. Routes are chosen to keep the arm comfortably away from these poses; and when one must be approached, the motion is slowed or reshaped so the joints never have to make their impossible sprint.
A six-joint arm is asked to drag its tool straight across a table, and its path happens to pass through the spot where the arm is fully stretched out — elbow straightened to its limit. Right at that spot the arm can no longer extend any farther, so the outward direction of hand motion vanishes; to keep tracing the straight line the controller demands a frantic joint spin to fake it, and the arm shudders. Replanning the path to keep the elbow slightly bent avoids the whole crisis.
At a singularity a direction of hand motion vanishes, and the joints would need impossible speeds to fake it.
"The Jacobian loses rank" is the formal way to say it: at a singularity the table linking joint speeds to hand motion can no longer reach in every direction, and reversing it (to find joint speeds from a desired hand motion) breaks down.