Math & Kinematics

Differential Inverse Kinematics

Differential Inverse Kinematics is the velocity-level resolution of inverse kinematics, mapping desired end-effector twists to joint velocities through the manipulator Jacobian. Practical solvers use the pseudoinverse or damped least squares to remain stable near singularities, and quadratic-programming formulations add joint limits, collision constraints, and secondary objectives such as posture optimization for redundant arms. It underlies resolved-rate control and most real-time Cartesian servoing.

Why it matters for physical AI

Learned policies frequently output end-effector velocities or pose targets, delegating joint-space execution to differential IK. Its conditioning near singularities and constraint handling determine whether those commands translate into safe, smooth hardware motion.