Centroidal momentum matrix

Collect momentum from the whole mechanism#

Every link contributes linear and angular momentum. The centroidal momentum matrix combines those contributions into a configuration-dependent map from floating-base and joint velocities to the robot's total momentum about its centre of mass. It has six output rows in spatial motion: three for linear momentum and three for angular momentum.

This compact map is useful when a controller cares about the net motion of the entire mechanism rather than every link separately. A humanoid can regulate angular momentum while moving its arms, or plan contact forces that change linear momentum without specifying all joint accelerations at the same level.

A bridge to estimation and whole-body control#

Khorshidi and colleagues use the centroidal momentum matrix with constrained whole-body dynamics to relate measured joint torques to centroidal-state dynamics in a legged-robot estimator. Their experiments are on a quadruped and are evidence for that estimator, not for every use of the matrix.

In whole-body control, the matrix can express momentum tasks alongside contact and posture tasks. In centroidal dynamics, it links the reduced momentum variables back to the full robot configuration and velocity.

The map depends on a model and configuration#

The matrix changes as the robot moves and depends on link masses, inertias, joint geometry, and coordinate conventions. Model errors therefore become momentum errors. Angular momentum must also be stated about a point, usually the centre of mass, because changing that point changes the value.

The matrix describes a kinematic momentum relationship at an instant. It does not by itself determine feasible contact forces, prevent foot slip, or enforce actuator limits. A controller still needs contact models and dynamics, while an estimator still needs enough measurements to resolve uncertain states and disturbances.