Numerical Inverse Kinematics

An inverse-kinematics method that starts from an initial guess and repeatedly refines the joint angles to approach a target pose.

Inverse kinematics has to find joint angles that produce a desired end-effector pose. Analytical IK derives a closed-form formula and only works for arms with special geometry; numerical IK is more general-purpose: starting from an initial set of joint angles, it uses forward kinematics to compute the current error between the end effector and the target, converts that error into a joint-angle correction using the Jacobian matrix, and repeats until the error is small enough. The most basic version is the Newton-Raphson method with the Jacobian pseudoinverse; common refinements include damped least squares (which keeps joint angles from swinging wildly near a singularity), the Jacobian transpose method, and formulating the whole problem as an optimization with joint limits and obstacle avoidance written in as constraints. The downside is that it only converges to whichever solution happens to be near the starting guess, and it can fail to converge or get stuck in a local optimum. In real-time control, using the previous time step's joint angles as the initial guess usually means convergence in just a few iterations. Libraries like KDL, TRAC-IK, and mink all provide numerical IK.

ExampleDuring VR teleoperation, each frame takes the headset controller's pose as the arm's target end-effector pose, and runs a few iterations of numerical IK starting from the previous frame's joint angles to get real-time joint angles.

Also called

Numerical IK, Iterative IK

Related

Inverse Kinematics (IK) · Analytical Inverse Kinematics · Jacobian Pseudoinverse · Damped Least Squares · Jacobian Transpose Method · TRAC-IK

Sources

Modern Robotics 6.2: Numerical Inverse Kinematics (Part 1 of 2)
Modern Robotics 6.2: Numerical Inverse Kinematics (Part 2 of 2)
Wikipedia: Inverse kinematics