Jacobian Transpose Method

Uses the Jacobian's transpose instead of its inverse to iteratively solve IK — cheap to compute but slower to converge.

The Jacobian transpose method is a numerical approach to inverse kinematics, applied to IK in 1984 independently by Balestrino and colleagues and by Wolovich and Elliott. Each step sets Δθ = αJᵀe, where e is the position error between the end effector and the target, Jᵀ is the transpose of the Jacobian matrix, and α is a small step size. The reasoning comes from the statics relation τ = JᵀF: imagine a virtual spring pulling the end effector toward the target, producing a force F; converted into joint torques, that force is JᵀF, and moving the joints along it is guaranteed to reduce the error whenever the step is small enough, since ⟨JJᵀe, e⟩ = ‖Jᵀe‖² ≥ 0. It never requires a matrix inverse, so each step is very cheap and it never blows up numerically near a singularity — at the cost of slower, sometimes oscillatory, convergence. The same Jᵀ map is also the core of Cartesian impedance control: a virtual spring force at the end effector is converted to joint torques through Jᵀ.

ExampleMaking an animated character's or robot arm's hand reach for a target point: each iteration computes Δθ = αJᵀe to update the joint angles, and after a few dozen steps the hand gradually approaches the target. In Buss's comparison tests, the method worked adequately for a single end effector but performed noticeably worse than damped least squares on a Y-shaped branching structure with multiple end effectors.

Also called

Jᵀ Method, Transpose Jacobian Method

Related

Jacobian Matrix · Numerical Inverse Kinematics · Jacobian Pseudoinverse · Damped Least Squares · Cartesian Impedance Control · Principle of Virtual Work

Sources

Buss, Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods
Lynch & Park, Modern Robotics（2017）5.2 节：开链静力学 τ = JᵀF (Chinese)