3-Link Robotic Arm – Inverse Kinematics

A 3-link planar arm has three rotational joints $(\theta_1, \theta_2, \theta_3)$ and can control both position $(x, y)$ and orientation $\phi$ of the end-effector. The extra degree of freedom means there are now infinitely many solutions for a position-only target — the system is redundant — unless you also fix the desired wrist angle.

Wrist decoupling splits the 3-DOF problem into two simpler parts:

1. Subtract the contribution of link 3 to find the wrist position: $x_w = x - l_3\cos\phi, \quad y_w = y - l_3\sin\phi$

2. Solve the remaining 2-link IK for $(x_w, y_w)$ using the law of cosines: $\cos\theta_2 = \frac{x_w^2 + y_w^2 - l_1^2 - l_2^2}{2\,l_1\,l_2}$

3. Recover $\theta_3$ from the orientation constraint: $\theta_3 = \phi - \theta_1 - \theta_2$

1. Choose a desired end-effector orientation $\phi$.
2. Compute the wrist point $(x_w, y_w)$ by "walking back" along link 3.
3. Run standard 2-link IK on the wrist point to get $\theta_1$ and $\theta_2$.
4. Set $\theta_3 = \phi - \theta_1 - \theta_2$.
5. Repeat for each elbow sign ($\pm$) to enumerate all configurations.

Six-DOF industrial arms (KUKA, UR) use wrist decoupling to separate the "where" (position, solved by the first three joints) from the "how" (orientation, solved by the wrist joints). This trick makes real-time IK computationally feasible for complex robot arms used in car assembly and electronics manufacturing.

1. If $l_1 = l_2 = l_3 = 1\,\text{m}$ and the target is at $(2, 0)$ with $\phi = 0$, what are $\theta_1$, $\theta_2$, and $\theta_3$?
2. Why does fixing the desired orientation $\phi$ remove the redundancy of the 3-link arm?