Robot Jacobian

Each column describes one joint's contribution#

Hold the robot at a particular configuration and move one joint at unit speed while the others remain still. The resulting end-effector velocity forms that joint's Jacobian column. Combining the columns with the actual joint speeds gives the resulting task velocity.

Modern Robotics develops a spatial Jacobian that maps joint speeds to a twist expressed in the space frame. A body Jacobian expresses that motion in the body frame instead.

Choose the task and representation#

A position-only task has a different Jacobian from a full position-and-orientation task. A Jacobian for orientation-coordinate rates also differs from one for angular velocity. Frame conventions and component ordering must match the velocity being commanded.

In static force analysis, the transpose of a compatible Jacobian maps an end-effector wrench to joint forces and torques.

Local motion has limits#

The Jacobian is evaluated at the current configuration, so it changes as the robot moves. At a kinematic singularity, it loses rank relative to its maximum attainable rank.

MIT's differential IK discussion explains why a pseudoinverse near singularity can request very large joint velocities. Joint limits and other constraints require explicit handling; taking a matrix inverse is not a complete controller.