Null-Space Control

Using spare degrees of freedom to accomplish a secondary goal without disturbing the primary task.

When a robot has more joints than the task strictly needs (a 7-DOF arm for a 6D end-effector pose, say), the same end-effector pose corresponds to infinitely many joint configurations — this is called kinematic redundancy. The Jacobian matrix J maps joint velocity q̇ to end-effector velocity; any joint motion satisfying J·q̇ = 0 forms J's null space — the joints move, but the end-effector doesn't, like an elbow tracing a circle while the hand stays put. Null-space control filters a secondary task's command through a projection matrix N = I − J⁺J (J⁺ the pseudoinverse of J) before adding it on top of the primary task, guaranteeing the secondary task never disturbs the primary one. Common secondary objectives include staying away from joint limits, avoiding singularities, keeping the elbow clear of obstacles, and holding a comfortable posture. At the torque level, Khatib's 1987 operational space control uses a ‘dynamically consistent’ pseudoinverse that accounts for the mass matrix — otherwise a secondary-task torque would still cause end-effector acceleration. Projecting several tasks layer by layer gives task priority and hierarchical quadratic programming.

ExampleFranka's Cartesian impedance control example: the primary task tracks the end-effector toward a target pose, while a secondary ‘spring-damper pulling back toward the startup joint posture’ torque is projected through (I − Jᵀ(Jᵀ)⁺) and added on top — pushing the elbow away lets it return on its own, without disturbing end-effector tracking.

Also called

Null-Space Projection, Redundancy Resolution

Related

Kinematic Redundancy · Null Space · Jacobian Pseudoinverse · Operational Space Control · Task Prioritization · Hierarchical Quadratic Programming

Sources

franka_ros cartesian_impedance_example_controller.cpp（零空间力矩实现） (Chinese)
StudyWolf: Robot control part 5 - Controlling in the null space
Implicit Null-space Manifold Generation for Redundant Robotic Systems (arXiv 2605.25770)