Jacobian Matrix

The matrix that converts joint speeds into end-effector velocity; it changes as the robot's posture changes.

In mathematics, a Jacobian matrix is the array of first-order partial derivatives of a multivariable function. In robotics, it describes the linear relationship between joint velocities and end-effector velocity: V = J(θ)·θ̇. Here θ̇ is the velocity of n joints; V is the end effector's 6-dimensional velocity (3D angular velocity plus 3D linear velocity, also called a twist); and J is a 6×n matrix whose i-th column shows how the end effector moves when only the i-th joint turns at unit speed. J depends on the current joint angles θ, so it has to be recomputed whenever the posture changes. It has three main uses: its inverse or pseudoinverse converts a desired end-effector velocity back into joint velocities, which is the basis of numerical inverse kinematics; checking whether J has lost rank reveals a singular configuration, where the end effector can't move in certain directions; and the static-force mapping τ = Jᵀ·F converts a desired end-effector force F into joint torques τ, which underlies force control and impedance control.

ExampleWhen a planar two-link arm is fully extended, no combination of joint rotations can move the end effector any farther outward along the arm's own direction — at that point J has lost rank, and the arm is in a singular configuration.

Also called

Spatial Jacobian, Body Jacobian

Related

Differential Kinematics · Jacobian Pseudoinverse · Singular Configuration (Kinematic Singularity) · Inverse Kinematics (IK) · Manipulability · Geometric vs. Analytical Jacobian

Sources

Modern Robotics 5.1.1: Space Jacobian (Northwestern)
Modern Robotics (Lynch & Park) preprint PDF, Ch. 5 Velocity Kinematics and Statics