Jacobian Pseudoinverse

The stand-in for a Jacobian inverse when it doesn't exist, giving the solution with the smallest error and least joint motion.

The Jacobian matrix J maps joint velocity θ̇ to end-effector velocity ẋ: ẋ = Jθ̇. Solving for joint velocity requires inverting J, but J is often not square (a 7-joint arm's J is 6×7), or it becomes non-invertible near a singular pose (an orientation where the end effector loses the ability to move in some direction). The usual fix is the Moore-Penrose pseudoinverse J⁺ (proposed independently by Moore in 1920, Bjerhammar in 1951, and Penrose in 1955): when an exact solution exists, it gives the one with the least joint motion (smallest norm) among all solutions; when no exact solution exists, it gives the least-squares solution with the smallest error. When J has full rank and there are more joints than task dimensions, J⁺ = Jᵀ(JJᵀ)⁻¹. It's a basic tool of numerical inverse kinematics, but solutions blow up near a singularity, so damped least squares is often used instead in practice.

ExampleA 7-DoF arm solves for joint velocity using θ̇ = J⁺ẋ + (I − J⁺J)φ: the first term makes the end effector track the desired velocity ẋ, while the second projects an arbitrary vector φ into the null space, adjusting the elbow and other posture without affecting the end effector — useful for avoiding joint limits, a technique introduced by Liégeois in 1977.

Also called

Moore-Penrose Pseudoinverse, Generalized Inverse, J⁺, J†

Related

Jacobian Matrix · Numerical Inverse Kinematics · Damped Least Squares · Null Space · Singular Configuration (Kinematic Singularity) · Kinematic Redundancy

Sources

Buss, Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods
Wikipedia: Moore–Penrose inverse
Lynch & Park, Modern Robotics（2017）6.2 节：数值逆运动学 (Chinese)