The Fundamental Problem of Robotics
Every time a robot arm reaches for an object, it must solve inverse kinematics: given a desired hand position, compute the joint angles that achieve it. Forward kinematics is easy — plug angles into trigonometry and get the hand position. Inverse kinematics reverses this process, and it's far more challenging because multiple solutions (or no solution) may exist.
The 2-Link Planar Arm
The 2-link planar arm is the simplest non-trivial IK problem. Two rigid links of lengths L₁ and L₂, connected by revolute joints, must reach a target (x, y). The law of cosines gives θ₂ (elbow angle), then θ₁ (shoulder angle) follows from geometry. The ± sign on θ₂ yields two valid solutions: elbow-up and elbow-down. This elegant closed-form solution exists because the 2-link problem has exactly 2 DOF matching the 2D target.
Workspace and Singularities
The reachable workspace of a 2-link arm is an annular ring: the target must be at least |L₁-L₂| and at most L₁+L₂ from the base. At the outer boundary (arm fully extended), the arm hits a kinematic singularity — it can only move tangentially, not radially. Near singularities, tiny target movements demand huge joint velocities, making control difficult. Industrial robots are programmed to avoid singularity regions.
Scaling to Real Robots
Industrial robots typically have 6 degrees of freedom (3 for position, 3 for orientation). Closed-form IK solutions exist for most 6-DOF industrial arm geometries. For arbitrary kinematic chains, numerical methods like the Jacobian transpose, damped least squares, or iterative algorithms (CCD, FABRIK) are used. Redundant robots with 7+ DOF have infinitely many IK solutions, offering flexibility to avoid obstacles or singularities.