Lift the arm into 3D. A spatial manipulator maps its joint velocities to a 3D end-effector velocity through the Jacobian J. Feed every unit joint-motion through J and the reachable hand velocities trace an ellipsoid. Fat and round means the hand is nimble in every direction; flattened means it's strong one way and helpless another — and when the ellipsoid goes flat, the arm is at a singularity.
arm links — 3 revolute jointsellipsoid — reachable hand velocitiesThe velocity ellipsoid's axes are the singular values of J (equivalently, √ of the eigenvalues of JJT), oriented along its singular vectors — this is the SVD, computed live from the 3×3 Jacobian. The scalar w = √det(JJT) is Yoshikawa's manipulability measure, the ellipsoid's volume; maximizing it keeps a robot away from awkward, near-singular poses. As you drive the arm toward full extension one axis shrinks to zero — the ellipsoid flattens to a disc, then a line, w → 0, and the hand loses the ability to move in that direction no matter how hard the joints spin. This is exactly the map that resolved-rate and Jacobian-transpose controllers invert, and why singularity-robust (damped least-squares) inverses exist at all.