Manipulability Ellipsoid & Singularities of a Robot Manipulator

This simulator drives a 3-link robotic arm with damped least-squares inverse kinematics toward a target you drag in 3D, and overlays the Jacobian manipulability ellipsoid at the end-effector on every frame. Watching the ellipse stretch, tilt and flatten as the arm reaches, folds and straightens makes visible the same dexterity measure — Yoshikawa's manipulability index w = √det(JJᵀ) — that real robot controllers use to detect and avoid kinematic singularities, where infinite joint speed would be needed to keep tracking a Cartesian path.