Inverse kinematics

Inverse kinematics finds joint coordinates that place an end effector at a requested position or configuration. It reverses the mapping specified by forward kinematics, but need not have a unique solution: a target can be unreachable, attainable through several configurations, or attainable through a continuous family when the mechanism has redundant degrees of freedom.^[ref]

For a two-link planar robot arm, the law of cosines givesWith unit link lengths and target $(1,1)$, $\cos {\theta }_2=0$. Two position solutions are $({\theta }_1,{\theta }_2)=(0,\pi /2)$ and $(\pi /2,-\pi /2)$. They have different arm shapes and tool orientations. A target at $(3,0)$ is unreachable because it lies beyond the total link length of two. Joint limits, obstacles and a specified tool orientation can exclude solutions even when the position is geometrically reachable.