Inverse kinematics

Forward kinematics is easy: set each joint's angle and follow the chain out to the tip. The inverse problem — "where should the joints sit so the foot lands there?" — often has several solutions or none at all, and that is where the difficulty lies. The technique comes from robotics and has become indispensable in games for planting feet on uneven ground, letting a hand grab an object or aiming a gaze at a target.

For short chains an analytic two-bone solution suffices. For longer ones there are iterative methods: CCD (cyclic coordinate descent), the Jacobian-based family, and FABRIK (2011), which is fast and gives natural poses without heavy matrix work. IK sits on top of skeletal animation and corrects the hand-made pose against the world in real time; orientations are often handled with quaternions to avoid gimbal lock.