manipulability ellipsoid
A manipulability ellipsoid is a little squashed balloon, drawn at the robot arm's hand, that shows how easily the hand can move in each direction from where it is right now. The directions where the balloon is long and stretched are the directions the hand can move fast and freely; the directions where the balloon is thin and pinched are the directions where moving is sluggish and hard, because the joints would have to spin furiously just to nudge the hand a little.
It comes from a simple idea: imagine spinning the robot's joints in every possible combination, but always keeping the total joint speed the same modest amount, and ask where the hand ends up able to go. Collect all those reachable hand-velocities and they trace out an ellipsoid — an egg or pancake shape. A fat, round, ball-like shape means the arm is in a comfortable, agile pose that handles motion in any direction about equally well. A flat, cigar-like shape that is collapsing toward a flat disc or a line is a warning: the arm is getting awkward, and in the pinched direction it has almost lost the ability to move at all.
Engineers use this shape as a quick health check on a pose. When planning how an arm should reach for something, they often steer it toward poses with rounder ellipsoids, so the hand stays nimble in every direction and avoids the cramped, near-frozen configurations where fine control falls apart.
Stretch your own arm straight out and try to push your hand farther forward — it barely moves, because the elbow is locked out. But it swings up, down, and sideways easily. Your hand's manipulability ellipsoid right then is a thin pancake: nearly flat in the forward direction, wide in all the others.
A fully extended arm is poor at moving along its own length but fine sideways — a flattened ellipsoid.
When the ellipsoid collapses toward a flat disc or a line, the arm is at or near a kinematic singularity — a pose where one direction of motion is lost.