Key takeaways
- A null space contains joint motions that create no instantaneous change in the selected end-effector task.
- A redundant arm can realize the same hand pose with multiple joint configurations.
- A controller solves the primary task first, then projects obstacle, joint-limit or singularity objectives into the remaining directions.
- The Jacobian and its projector are local models, so long motions require repeated computation and sensor feedback.
- Damping, saturation, model error and delay mean the primary task is not perfectly protected in a real robot.
One hand pose can have more than one joint posture
Null-space control uses the extra degrees of freedom of a redundant robot to pursue another objective without intentionally changing its primary end-effector task. A person can hold a cup at one pose while moving the elbow inward or outward. Likewise, a seven-joint arm that controls six components of a hand pose can, in suitable configurations, retain at least one remaining direction of joint motion.
Null does not mean that the robot is inactive. It means that the effect is zero in the task coordinates selected by the controller. If position and orientation of the tool are the primary task, a combination of joint velocities belongs to the null space when it produces no instantaneous tool velocity. That freedom can move an elbow away from a post while the tool stays on an assembly point.
The Jacobian reveals directions with no task effect
The local velocity relation is commonly written as x-dot equals J times q-dot. Here J is the Jacobian that maps joint velocity into task-space velocity at the current configuration. If a nonzero q-dot satisfies J q-dot equals zero, that direction is in the task’s null space. Because J changes with posture, the null space is not a permanent corridor; it is a local set of directions recomputed each control cycle.
One primary-task solution can be written q-dot-one equals J-sharp times x-dot, where J-sharp is a pseudoinverse. The projector N equals I minus J-sharp J removes components that affect the primary task. A final command q-dot equals J-sharp x-dot plus N z combines the requested hand velocity with an arbitrary secondary command z. The choice of pseudoinverse, weighting and damping changes both the selected posture and numerical behavior.
A projector places secondary goals behind the primary task
Consider a hypothetical assembly cell. A robot holds a driver over a bolt while its elbow approaches a safety post. The primary task keeps the driver position and orientation fixed. A secondary objective asks the joints to increase the elbow-to-post distance. The controller forms an unconstrained avoidance direction from the gradient of a distance function, then multiplies it by N to remove the component that would move the tool.
One projection does not complete a long avoidance maneuver. The robot takes a small step and then recomputes hand error, obstacle distance and the Jacobian from the new state. The avoidance command shrinks when clearance is sufficient, while a growing hand error increases the primary correction. This sensor-and-model feedback loop turns an instantaneous velocity relation into a physical trajectory.
An elbow-avoidance step in practice
Secondary objectives extend beyond obstacles. A controller can bias joints toward the middle of their ranges, reduce a heavily loaded joint’s speed or torque, or reorient an arm-mounted camera for a better view. It can also prefer configurations with higher manipulability to move away from a singular posture. Each behavior is expressed as a direction or cost that uses the freedom left by the primary task.
Multiple objectives need explicit priorities. If hand pose is first, human clearance second and joint centering third, each lower task should operate only inside the null space left by the tasks above it. Hierarchical controllers chain projections so a lower task cannot simply undo a higher command. A weighted sum can be useful, but an unfortunate weight can trade away a safety-related objective without making the decision obvious.
Null space is useful for more than obstacle avoidance
The amount and quality of null-space freedom are not constant. Near a singular configuration, the Jacobian becomes ill-conditioned and a small hand request may demand very large joint velocity. A damped pseudoinverse limits that growth. Damping, however, changes the ideal projector, so some secondary motion can leak into the hand task. It is a deliberate trade between numerical robustness and exact task separation.
Velocity, acceleration and torque limits also matter. When a computed command exceeds an actuator limit, saturation or an optimizer changes the result, potentially altering both objectives. If the stopping distance is larger than the remaining clearance, a null-space command alone cannot prevent collision. Independent speed limits, protective stops and collision monitoring are still required; null-space control is not a safety certification.
Singularities and model error can weaken the hierarchy
A projector is only as accurate as its state and robot model. Incorrect link dimensions, tool calibration or joint estimates mean that a mathematically zero task effect is not zero on the hardware. Backlash, structural compliance, communication delay and filtering can add visible hand motion. Tight insertion and human-proximate work therefore need position and force feedback plus execution monitoring, not projection alone.
A serious evaluation should show more than an elbow moving around an object. Compare end-effector position and orientation error, minimum obstacle clearance, joint-limit margin, peak joint speed and torque on the same timeline with the secondary objective off and on. Repeat the test near singularities and under sensor delay, model error and sudden obstacle changes to reveal how well the primary task survives difficult conditions.
Evaluate hand error and the secondary goal together
The statement ‘seven axes make the arm flexible’ is not a performance guarantee. Review which task coordinates were constrained, the actual null-space dimension, who selected the secondary cost and priority, and what is sacrificed when saturation occurs. Identical hardware can produce very different elbow paths and hand errors depending on the projector, damping and constraint handling.
A practical pilot can start with one fixed-tool task and one simple obstacle. Set an acceptable hand-error bound first, then measure whether enabling the null-space objective improves clearance and joint margin. Add singularity, saturation and emergency-stop cases step by step. The value of null-space control is its use of remaining freedom, but it becomes an operational feature only when tests show that the freedom does not compromise the primary task or safety.
Sources reviewed
- Oussama Khatib: A Unified Approach for Motion and Force Control of Robot Manipulators (1987)
- MIT Robotic Manipulation: Manipulator Control
- libfranka official examples
- libfranka Cartesian impedance control example source
- libfranka Robot API: stiffness frame and Cartesian impedance
Sources reviewed on September 23, 2026.
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Sean Woo — I have spent more than 15 years shaping robotics technology and business direction. Drawing on public technical documents, research papers and company announcements, I explain changes in robotics and AI in accessible terms. The interpretations in this publication do not represent the official position of any company or organization.