## Abstract

The center of mass (CoM) of a humanoid robot occupies a special place in its dynamics. As the location of its effective total mass, and consequently, the point of resultant action of gravity, the CoM is also the point where the robot’s aggregate linear momentum and angular momentum are naturally defined. The overarching purpose of this paper is to refocus our attention to *centroidal dynamics*: the dynamics of a humanoid robot projected at its CoM. In this paper we specifically study the properties, structure and computation schemes for the centroidal momentum matrix (CMM), which projects the generalized velocities of a humanoid robot to its spatial centroidal momentum. Through a *transformation diagram* we graphically show the relationship between this matrix and the well-known joint-space inertia matrix. We also introduce the new concept of “average spatial velocity” of the humanoid that encompasses both linear and angular components and results in a novel decomposition of the kinetic energy. Further, we develop a very efficient \(O(N)\) algorithm, expressed in a compact form using spatial notation, for computing the CMM, centroidal momentum, centroidal inertia, and average spatial velocity. Finally, as a practical use of centroidal dynamics we show that a momentum-based balance controller that directly employs the CMM can significantly reduce unnecessary trunk bending during balance maintenance against external disturbance.

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## Notes

The system Jacobian is not to be confused with the manipulator Jacobian in traditional fixed-based manipulators. The system Jacobian is an extension of the manipulator Jacobian and can contain it as one of its blocks if the corresponding coordinate frame is located at the task point.

The equations of motion for an \(n\)-dof robot can be expressed as:

$$\begin{aligned} \varvec{\tau }= \varvec{H}(\varvec{q}) \, \ddot{\varvec{q}}+ \varvec{C}(\varvec{q},\dot{\varvec{q}}) \, \dot{\varvec{q}}+ \varvec{\tau }_g(\varvec{q}) \, , \end{aligned}$$where \(\varvec{H}\) is the \(n\times n\) symmetric, positive-definite joint-space inertia matrix, \(\varvec{C}\) is an \(n\times n\) matrix such that \(\varvec{C}\,\dot{\varvec{q}}\) is the vector of Coriolis and centrifugal terms (collectively known as

*velocity product*terms), and \(\varvec{\tau }_g\) is the vector of gravity terms.For a stationary support foot, \(\ddot{\mathbf{q }}_S\) should satisfy the constraint equation following from the differentiation of Eq. 46,

*i.e.*,$$\begin{aligned} \ddot{\mathbf{q }}_S = -\mathbf L _S^{-1} \, (\mathbf L _P \, \ddot{\mathbf{q }}_P \, + \, \dot{\mathbf{L }}_P \, \dot{\mathbf{q }}_P + \, \dot{\mathbf{L }}_S \, \dot{\mathbf{q }}_S)\\ = -\mathbf L _S^{-1} \, (\dot{\mathbf{L }}_P \, \dot{\mathbf{q }}_P + \, \dot{\mathbf{L }}_S \, \dot{\mathbf{q }}_S) \, . \end{aligned}$$(Note that \(\dot{\mathbf{q }}_P\) and \(\dot{\mathbf{q }}_S\) are given by the system state.)

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## Acknowledgments

The authors gratefully thank Ghassan Bin Hammam for testing the recursive centroidal dynamics algorithm on a PC. Support for this work for David Orin was provided in part by Grant No. CNS-0960061 from NSF with a subaward to The Ohio State University. S.-H. Lee was partly supported by the Global Frontier R&D Program, NRF (NRF-2012M3A6A3055690).

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Orin, D.E., Goswami, A. & Lee, SH. Centroidal dynamics of a humanoid robot.
*Auton Robot* **35**, 161–176 (2013). https://doi.org/10.1007/s10514-013-9341-4

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DOI: https://doi.org/10.1007/s10514-013-9341-4