Null Space

Every input that a matrix maps to zero; for a redundant arm, the joint motions that don't change the end-effector pose at all.

Null space is originally a linear-algebra concept: the null space of a matrix A is every vector x satisfying Ax = 0. In robotics, it usually refers to the null space of the Jacobian matrix J: since J converts joint velocities into end-effector velocity, any joint velocity that falls in its null space produces no motion or rotation of the end effector at all. A kinematically redundant arm — one with more joints than the task strictly needs, like a 7-axis arm performing a 6-dimensional pose task — always has this kind of motion available, called self-motion; a non-redundant arm only gets it at a singular configuration. Null-space control exploits exactly this: the primary task keeps the end effector tracking its target, while secondary objectives — staying away from joint limits, avoiding obstacles, steering clear of singularities — get executed by projecting them into the null space with the projection matrix (I − J⁺J), where J⁺ is the Jacobian pseudoinverse, so they never interfere with the primary task. Task-priority schemes in humanoid whole-body control are also built from layered null-space projections like this.

ExampleHolding a 7-axis arm's end effector fixed directly above a cup, the elbow can still trace a circle around the shoulder-wrist line (changing the arm angle) — a one-dimensional self-motion, which can be used to steer the elbow away from a nearby obstacle.

Also called

Kernel, Jacobian Null Space, Self-Motion

Related

Kinematic Redundancy · Null-Space Control · Jacobian Pseudoinverse · Task Prioritization · Swivel Angle · Jacobian Matrix

Sources

Wikipedia: Kernel (linear algebra)
StudyWolf: Robot control part 5 – Controlling in the null space