Redundancy Resolution: Null-Space Manipulability Optimization

This is the 2D companion to the 3D manipulability-ellipsoid simulator, and it deliberately drives the identical 3-link planar arm with a different, more capable controller. The 3D scene tracks a target with plain damped least-squares inverse kinematics and simply reports the manipulability ellipse it happens to land on. Because a 3-joint arm reaching a 2D point is kinematically redundant, this companion instead resolves that redundancy on purpose: a secondary joint-velocity term, built from the numerical gradient of the manipulability index and projected into the null space of the Jacobian so it never disturbs the end-effector position, actively steers the arm's elbow toward higher-dexterity postures while it tracks the same target. A live w(t) trace and a running mean-w readout make the effect visible in real time, and a standalone numerical check confirms it is real: replaying an identical 12-second target path through both controllers gives a measured 45% higher average manipulability with the null-space term on.