Inverse Kinematics for Robotic Arms

Forward kinematics asks: given joint angles, where is the end-effector? Inverse kinematics (IK) flips the question: given a desired end-effector position $(x, y)$, what joint angles $\theta_1$ and $\theta_2$ produce it? IK is what lets a robot arm reach out and grab an object at a known location.

For a 2-link planar arm with link lengths $l_1$ and $l_2$:

Step 1 — find $\theta_2$ using the law of cosines:

$\cos\theta_2 = \frac{x^2 + y^2 - l_1^2 - l_2^2}{2\,l_1\,l_2}$

$\theta_2 = \pm\arccos\!\left(\frac{x^2 + y^2 - l_1^2 - l_2^2}{2\,l_1\,l_2}\right)$

The $\pm$ gives two solutions: elbow-up and elbow-down.

Step 2 — find $\theta_1$:

$\theta_1 = \operatorname{atan2}(y,\,x) - \operatorname{atan2}\!\left(l_2\sin\theta_2,\; l_1 + l_2\cos\theta_2\right)$

1. Measure the straight-line distance from base to target: $r = \sqrt{x^2 + y^2}$.
2. Check reachability: the target must satisfy $|l_1 - l_2| \le r \le l_1 + l_2$.
3. Apply the cosine rule to find $\theta_2$ (two solutions).
4. Substitute each $\theta_2$ into the atan2 formula to get the matching $\theta_1$.
5. Choose the solution (elbow-up or elbow-down) based on workspace constraints.

Every industrial robot welding a car frame, every surgical robot making a precise incision, and every animated video-game character reaching for an object uses IK. In robot arms like the KUKA or UR5, IK runs hundreds of times per second so the end-effector tracks a moving target smoothly.

1. A 2-link arm has $l_1 = l_2 = 1\,\text{m}$. What is $\theta_2$ when the target is at $(1, 0)$? (Hint: substitute into the cosine formula.)
2. Why does IK sometimes have two solutions, and why might a robot prefer one over the other?