Whole-Body Control: Redundant Arm Null-Space Simulator (2D)

This is the 2D companion to the 3D whole-body-control simulator, and it deliberately resolves the same kind of redundancy with a different secondary task. A 4-joint planar arm reaching a fixed 2D target has two spare degrees of freedom — a genuine null space of joint motions that never move the hand. Instead of dodging an obstacle, this version uses that spare freedom for joint-limit avoidance: a null-space-projected gradient term continuously pulls every joint back toward the centre of its own mechanical range, visualized live as a bar for each joint. Toggle the secondary task off and drag the target near full extension to watch the joints drift toward their limits with nothing to stop them; turn it back on and watch the same exact end-effector position get held while every joint bar recentres.