Redundancy and Null-Space Motion

A three-link arm placing a point in the plane has one more joint than it needs. That extra freedom means the joints can move together in a way that leaves the hand exactly where it is. Those self-motions live in the null space of the Jacobian.

Self-motion speed Show past poses

Drag the red target to move the hand.

Two parts of the motion

The joint velocity is split into a part that moves the hand and a part that does not:

$$\dot{\boldsymbol\theta} = J^{+}\,\dot{\mathbf{x}}_{\text{task}} + \left(I - J^{+}J\right)\mathbf{z}$$

The matrix $I - J^{+}J$ projects any vector $\mathbf{z}$ into the null space, the joint motions with $J\dot{\boldsymbol\theta} = 0$. Here the null space is a single direction, and the slider sets how fast the arm moves along it.

What to notice

The hand stays pinned to the target while the elbow and wrist swing through a whole family of poses. Robots use this freedom for a second goal without disturbing the first one: keeping the elbow away from an obstacle, staying far from joint limits, or keeping away from singularities. A 7-joint arm reaching a 6-number pose has exactly one such extra freedom, which is why many research arms have seven joints.