Damped Least-Squares IK

Key Insight

Inverse kinematics runs forward kinematics backward — given a desired hand pose, find the joint angles that achieve it — and the workhorse method repeatedly inverts the Jacobian to step the joints toward the target. The catch is that near a singularity the Jacobian loses rank, and a plain inverse then demands impossibly fast joint motion; damped least-squares cures this by adding a small penalty (the "damping") that trades a little tracking accuracy for bounded, stable joint velocities. Watching the arm ease through a wrist singularity instead of flailing is the clearest lesson in why this one regularization trick ships inside real industrial controllers.