Null space

Joint motion invisible to a task#

A redundant manipulator may move its elbow while keeping its hand still. At the current configuration, the associated joint-velocity vector lies in the hand task's Jacobian null space.

The word is task-specific. If the task constrains only hand position, null-space motion may change hand orientation. Adding orientation constraints can remove that freedom. Modern Robotics illustrates how redundancy depends on the selected velocity task.

Combining primary and secondary objectives#

MIT's differential IK treatment projects a preferred joint motion into the Jacobian null space to help return an arm toward a nominal posture while tracking an end-effector velocity.

This supplies a useful building block for task prioritization: the primary task uses visible motion directions, while a secondary objective uses remaining directions.

An instantaneous result needs continued updating#

The robot Jacobian changes as the configuration changes. A vector in its null space now is not necessarily in the null space after a finite step. Maintaining the task requires recalculating the relationship as the robot moves.

Null-space projection also does not automatically satisfy joint or contact constraints. MIT notes that constraints can make primary and secondary objectives conflict, so a weighted or hierarchical optimization must handle their interaction explicitly.