inverse kinematics

Inverse kinematics is the reverse question, and the one a robot actually needs to answer in everyday work: I want the hand to be right here, pointing this way — so how much should each joint bend to make that happen? You start from the goal (a target position and orientation for the end-effector) and you have to solve backward for the joint values that achieve it. This is exactly what your own brain does, silently, every time you reach for a coffee mug: you decide where your hand should go, and your shoulder, elbow, and wrist angles sort themselves out.

Going backward is much harder than going forward, for two reasons. First, there can be several different joint settings that all put the hand in the same place — a human can grab the same mug with the elbow tucked low or flared out high — so the robot may face many valid answers and must pick one. Second, some targets have no solution at all because they simply lie beyond the arm's reach, and near awkward configurations the math can become touchy and unstable. Because of this, inverse kinematics is one of the central problems of robot motion: every time an arm is told to weld a seam, place a part, or follow a drawn path, something must be solving inverse kinematics, either with an exact algebraic formula or by an iterative numerical search.