Inverse kinematics

Inverse kinematics is the math a robot uses to figure out which joint angles will put its hand (or end-effector) exactly where you want it. It's the difference between "move forward" and "touch this specific point in 3D space."

The simple case: forward kinematics

If you know all the joint angles of a robotic arm, calculating where the hand ends up is straightforward — just add up the rotations and translations. That's forward kinematics: angles in, position out. Computers can do this in microseconds, even for arms with seven joints.

The hard case: inverse kinematics

But what we actually want is the opposite: given a target position for the hand, what joint angles get us there? That's inverse kinematics (IK). And it's hard, for three reasons:

1. Multiple solutions. For most arms, there's more than one way to reach the same point. Your own arm: you can touch your nose with your elbow up or your elbow down. Two valid solutions. A 7-joint arm has roughly infinite solutions for most targets.

2. No solutions. Some points are unreachable (out of arm length, or blocked by the body). IK has to detect this and give a sensible failure.

3. The math is nonlinear. Forward kinematics is a sequence of matrix multiplications. IK is solving for the angles inside those matrices — a much harder algebraic problem.

How robots actually solve it

Analytical IK — for simple arms (up to 6 joints, no funny geometry), there's a closed-form algebraic solution. Compute it in microseconds.

Numerical IK — for complex arms (7+ joints, humanoid arms, robots with redundancy), the solver iterates: start with a guess, see how far off the hand is, adjust the angles to reduce the error, repeat. Converges in milliseconds.

Neural IK — newer: train a neural network to learn the inverse mapping from data. Less accurate but much faster, and handles redundancy gracefully.

Why this matters for humanoids

A humanoid robot has roughly 30 joints. Getting its right hand to a specific point isn't "pick angles for the 6 joints in the arm" — it's "pick angles for 30 joints such that the whole body keeps its balance, the legs stay where they are, and the left hand doesn't bump into the chest". This is called whole-body IK. It's the math problem that Atlas, Optimus, and Figure 03 are solving 500-1,000 times per second.

The simplest place to see IK in action is a robotic arm with 6 joints. We'll build one in the Wire 06 course.