Inverse kinematics

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Find joint angles that place the hand at a target: solve $f(q) = p^\ast$ . Usually iteratively with the Jacobian — Newton/Gauss–Newton steps $\Delta q = J^+(p^\ast - f(q))$ , or damped least squares near singularities. Game engines use it to plant feet and aim hands.

Formulas

$\Delta q = J^{\mathsf T}\big(JJ^{\mathsf T} + \lambda^2 I\big)^{-1}\big(p^\ast - f(q)\big)$
damped least squares (Levenberg–Marquardt)

The mathematics behind it

Jacobian matrix★★★★★fundamental
IK solvers iterate $\Delta q = J^{+}\Delta p$ (pseudo-inverse) or damped least squares.
Newton's method★★★★★frequent
Solving for joint angles that reach a target is a Newton (Gauss–Newton) iteration with the robot Jacobian.
Intermediate value theorem★★★★★indirect
Existence arguments of the type "some joint angle reaches this position" are intermediate-value arguments.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.