singularities

Connect joint velocities to end-effector velocity through the Jacobian matrix. Learn how to build J, spot singularities where det J = 0, invert J with the pseudoinverse for velocity control, and measure manipulability — with NumPy code for the 2-link arm.

Flip the FK problem: given a desired end-effector pose, find the joint angles that achieve it. Master analytical closed-form IK for the 2-link arm, the elbow-up/elbow-down duality, atan2 arithmetic, and the basics of numerical IK via the Jacobian.