forward kinematics

Forward kinematics is the question: if I know exactly how much every joint of a robot arm is bent or extended, where does the hand end up, and which way is it pointing? You feed in the joint values — say, shoulder rotated 30 degrees, elbow bent 90 degrees, wrist twisted 45 degrees — and forward kinematics tells you the precise position and orientation of the tip (the gripper, tool, or whatever the arm carries, called the end-effector). It is like folding a carpenter's ruler: once you fix the angle at each hinge, the location of the far end is completely determined.

It works by chaining the links together one at a time. Each joint has a small piece of geometry that says how the next link is shifted and turned relative to the current one; mathematically each piece is written as a transform (a compact recipe for a rotation plus a shift in space). You multiply these transforms in order from the fixed base out to the tip, and the combined result places the end-effector in the world. This direction is the easy one: for any given set of joint values there is always exactly one answer, with no ambiguity and no guessing — which is why forward kinematics is the dependable foundation that the harder, backward problem is built on top of.