Damped Least Squares

Adds a damping term to the Jacobian inverse in numerical IK so the arm doesn't move wildly near singular poses.

Damped least squares is an iterative method for solving inverse kinematics, also called the Levenberg-Marquardt method; Wampler, and separately Nakamura and Hanafusa, applied it to inverse kinematics in 1986. Each step computes the joint increment Δθ = Jᵀ(JJᵀ + λ²I)⁻¹e, where J is the Jacobian matrix (mapping joint velocity to end-effector velocity), e is the error between the end effector's current pose and the target, λ is a damping coefficient, and I is the identity matrix. This is equivalent to minimizing ‖JΔθ − e‖² + λ²‖Δθ‖² — reducing the error while also keeping the joint step from being too large. With the plain pseudoinverse, approaching a singular pose (where the end effector loses the ability to move in some direction) produces enormous joint velocities; adding λ keeps the denominator from going to zero, so the motion stays smooth. The tradeoff is that too large a λ slows convergence, so λ is often adjusted dynamically based on how close the arm is to a singularity.

ExampleAn arm fully extended is near a singularity, where the plain pseudoinverse might demand that some joint instantly spin through many revolutions; setting λ to around 0.05 makes convergence along that direction slower but keeps joint velocities within a normal range.

Also called

DLS, Levenberg-Marquardt Inverse, Singularity-Robust Inverse

Related

Inverse Kinematics (IK) · Numerical Inverse Kinematics · Jacobian Matrix · Jacobian Pseudoinverse · Singular Configuration (Kinematic Singularity) · Jacobian Transpose Method

Sources

Buss: Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods
Wikipedia: Levenberg–Marquardt algorithm