Inverse Kinematics (IK)

Working backward from a target end-effector position and orientation to find the joint angles that reach it.

Inverse kinematics is the reverse of forward kinematics: given a target end-effector pose, find the joint angles that achieve it. It's considerably harder. The equations are nonlinear and can have multiple solutions — an elbow-up or elbow-down configuration might reach the same point — a redundant arm with 7 degrees of freedom has infinitely many solutions, a target outside the workspace has no solution at all, and the numerics get unstable near singular configurations (joint poses where the end effector can't move in certain directions). Solving methods fall into two camps: analytical (closed-form) solutions compute the answer directly from a formula, which is fast but only works for specific arm geometries; numerical solutions iterate using the Jacobian matrix (the mapping from joint velocities to end-effector velocity), which is general-purpose but depends on a good starting guess and may fail to converge. Whenever a VLA policy outputs an end-effector pose, IK is what turns it into joint commands; dragging a character's hand in animation software, with the shoulder and elbow following automatically, is also IK.

ExampleTo move the gripper to 10 centimeters directly above a cup, jaws facing down, IK solves for the joint angles that achieve it. In MATLAB's Robotics System Toolbox, the inverseKinematics function solves numerically using BFGS gradient projection by default, and can return only an approximate answer if the initial guess is poor.

Also called

IK

Related

Forward Kinematics (FK) · Analytical Inverse Kinematics · Numerical Inverse Kinematics · Jacobian Matrix · Singular Configuration (Kinematic Singularity) · Kinematic Redundancy

Sources

Wikipedia: Inverse kinematics
MathWorks: Inverse Kinematics Algorithms