Kinematics of a two-link arm

Forward kinematics turns joint angles into where the tip is: add each link’s vector, rotated by the angles before it. Inverse kinematics runs it backwards, from “put the gripper there” to joint angles. Drag the tip and the arm follows by inverse kinematics.

For two links the inverse is closed-form. The law of cosines gives the elbow angle up to a sign, and the two signs are the two arms that reach the same point: elbow up and elbow down. Both are drawn, the one you pick bold. Small arms, and many six-joint arms with a spherical wrist, have closed-form solutions; in general solving is numerical.

The tip can only reach a ring: out to the sum of the link lengths when the arm is straight, in to their difference when it is folded back. Outside the ring there is no solution, and the page pulls the target to the nearest point on the boundary and says so.

The Jacobian says how tip speed follows joint speed, and its determinant for this arm is the product of the link lengths times the sine of the elbow angle. It is zero when the arm is straight or folded back: there the tip cannot move along the arm, however the joints turn, and a small step along it needs huge joint motion. The red ring marks those boundaries. Statics uses the same matrix: the torques that hold a pose against a force are the Jacobian transposed times that force.

The link lengths here are illustrative, not any real arm’s.