A.: Centroidal Momentum Matrix of a humanoid robot: Structure and Properties

Abstract—The centroidal momentum of a humanoid robot is the sum of the individual link momenta, after projecting each to the robot’s Center of Mass (CoM). Centroidal momentum is a linear function of the robot’s generalized velocities and the Centroidal Momentum Matrix is the matrix form of this function. This matrix has been called both a Jacobian matrix and an inertia matrix by others. We show that it is actually a product of a Jacobian and an inertia matrix. We establish the relationship between the Centroidal Mo-mentum Matrix and the well-known joint-space inertia matrix. We present a Transformation Diagram that graphically captures the inter-relationships of the matrix operators and motion and momentum variables in Joint Space, CoM Space as well as the System Space. The Centroidal Momentum Matrix is a local scaling function that maps the joint rates to the centroidal momentum. Follow-ing the concept of the manipulability ellipsoid, we propose the centroidal momentum ellipsoid that quantifies the momentum generation ability of the robot. We present a simulation plot showing the evolution of the singular values of the Centroidal Momentum Matrix during the walking motion of a humanoid. Index Terms—momentum matrix, inertia matrix, angular momentum, linear momentum, matrix properties, centroidal momentum ellipsoid. I