Manipulability

An ellipsoid of available velocities

At a fixed robot configuration, the Jacobian maps a ball of joint velocities to an ellipsoidal set of task velocities. A long principal axis means more task velocity is available in that direction under the chosen joint-velocity bound. A short axis identifies a more restricted direction.

Modern Robotics derives the ellipsoid from J J^T. Its eigenvectors give the principal directions, and the square roots of its eigenvalues give the semi-axis lengths. At a kinematic singularity, the attainable velocity set loses a dimension rather than remaining a full-dimensional ellipsoid.

For a humanoid reaching towards a shelf, two postures may place the hand at the same pose yet leave different freedom for the next movement. Manipulability helps describe that local difference; it does not establish whether the route to either posture is collision-free.

A scalar needs a definition

There is no single interchangeable manipulability score. Modern Robotics presents both axis-ratio and volume-based measures. A large longest-to-shortest axis ratio indicates directional imbalance, whereas a larger ellipsoid volume describes a larger velocity set under the specified normalisation.

Linear and angular velocity have different units. The same source therefore treats their Jacobian components separately when constructing translational and rotational ellipsoids. Comparisons should state the task coordinates, units and weighting rather than treating every number called “manipulability” as comparable.

Kinematic freedom is not a safety certificate

The velocity ellipsoid describes a local kinematic mapping. It does not, by itself, impose collision avoidance, contact friction or the robot's actual torque limits. A favourable value is therefore not proof that a humanoid can safely lift an object or keep its balance.

For training and evaluation data, retain the robot configuration, model, selected task and metric formula alongside the score. Record any velocity normalisation and whether the quantity concerns translation, rotation or a weighted combination. This makes the score interpretable as a posture feature instead of an unexplained success label.