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Centroidal dynamics of a humanoid robot

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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

  1. 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.

  2. 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.

  3. 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.)

References

  • Abdallah, M., Goswami, A. (2005). A biomechanically motivated two-phase strategy for biped robot upright balance control. In Proceedings of IEEE international conference on robotics and automation (ICRA), Barcelona, pp. 3707–3713.

  • Bin Hammam, G., Orin, D. E., & Dariush, B. (2010). Whole-body humanoid control from upper-body task specifications. In Proceedings of IEEE international conference on robotics and automation, Anchorage, pp. 398–3405.

  • De Luca, A., Albu-Schaffer, A., Haddadin, S., & Hirzinger, G. (2006). Collision detection and safe reaction with the DLR-III lightweight manipulator arm. In Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS), Beijing, pp. 1623–1630.

  • Essén, H. (1993). Average angular velocity. European Journal of Physics, 14, 201–205.

    Article  Google Scholar 

  • Fang, A. C., & Pollard, N. (2003). Efficient synthesis of physically valid human motion. ACM Transactions on Graphics ACM SIGGRAPH Proceedings, 22(3), 417–426.

    Article  Google Scholar 

  • Featherstone, R. (2008). Rigid body dynamics algorithms. New York: Springer.

    Book  MATH  Google Scholar 

  • Featherstone, R., & Orin, D. E. (2000). Robot dynamics: equations and algorithms. In Proceedings of IEEE international conference on robotics and automation, San Francisco, pp. 826–834.

  • Featherstone, R., & Orin, D. E. (2008). Dynamics, chapter 2. In B. Siciliano & O. Khatib (Eds.), Springer handbook of robotics. New York: Springer.

    Google Scholar 

  • Goswami, A., & Kallem, V. (2004). Rate of change of angular momentum and balance maintenance of biped robots. In Proceedings of IEEE international conference on robotics and automation (ICRA), New Orleans, pp. 3785–3790.

  • Herr, H., & Popovic, M. B. (2008). Angular momentum in human walking. The Journal of Experimental Biology, 211, 467–481.

    Article  Google Scholar 

  • Hofmann, A., Popovic, M., & Herr, H. (2009). Exploiting angular momentum to enhance bipedal center-of-mass control. In Proceedings of IEEE international conference on robotics and automation (ICRA), Kobe, pp. 4423–4429.

  • Kajita, S., Kanehiro, F., Kaneko, K., Fujiwara, K., Harada, K., Yokoi, K., & Hirukawa, H. (2003). Resolved momentum control: Humanoid motion planning based on the linear and angular momentum. In Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS), Las Vegas, pp. 1644–1650.

  • Khatib, O. (1987). A unified approach to motion and force control of robot manipulators: The operational space formulation. IEEE Transactions on Robotics and Automation, 3(1), 43–53.

    Article  Google Scholar 

  • Komura, T., Leung, H., Kudoh, S., & Kuffner, J. (2005). A feedback controller for biped humanoids that can counteract large perturbations during gait. In Proceedings of IEEE international conference on robotics and automation (ICRA), Barcelona, pp. 2001–2007.

  • Lee, S.-H., & Goswami, A. (2007). Reaction mass pendulum (RMP): An explicit model for centroidal angular momentum of humanoid robots. In Proceedimgs of IEEE international conference on robotics and automation, Rome, pp. 4667–4672.

  • Lee, S.-H., & Goswami, A. (November 2012). A momentum-based balance controller for humanoid robots on non-level and non-stationary ground. Autonomous Robots, 33(4), 399–414.

  • Macchietto, A., Zordan, V., & Shelton, C. R. (2009). Momentum control for balance. ACM Transactions on Graphics, 28(3), 80:1–80:8.

    Article  Google Scholar 

  • McMillan, S., & Orin, D. E. (1995). Efficient computation of articulated-body inertias using successive axial screws. IEEE Transactions on Robotics and Automation, 11(4), 606–611.

    Article  Google Scholar 

  • Mistry, M., Righetti, L. (2012). Operational space control of constrained and underactuated systems. In H. Durrant-Whyte (Ed.), Robotics: Science and systems VII (pp. 225–232). Cambridge: The MIT Press.

  • Mitobe, K., Capi, G., & Nasu, Y. (2004). A new control method for walking robots based on angular momentum. Mechatronics, 14, 163–174.

    Article  Google Scholar 

  • Morita, Y., & Ohnishi, K. (2003). Attitude control of hopping robot using angular momentum. In Proceedings of IEEE international conference on industrial technology, Vol. 1, pp. 173–178.

  • Nenchev, D., Umetani, Y., & Yoshida, K. (1992). Analysis of a redundant free-flying spacecraft/manipulator system. IEEE Transactions on Robotics and Automation, 8(1), 1–6.

    Article  Google Scholar 

  • Nakamura, Y., & Yamane, K. (2000). Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Transactions on Robotics and Automation, 16(2), 124–134.

    Google Scholar 

  • Naksuk, N., Mei, Y., & Lee, C. S. G. (2004). Humanoid trajectory generation: an iterative approach based on movement and angular momentum criteria. In Proceedings of the IEEE-RAS international conference on humanoid robots, Santa Monica, pp. 576–591.

  • Naksuk, N., Lee, C. S. G., & Rietdyk, S. (2005). Whole-body human to humanoid motion transfer. In Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS), Edmonton, pp. 104–109.

  • Naudet, J. (2005). Forward dynamics of multibody systems: A recursive Hamiltonian approach. PhD Thesis, Vrije Universiteit, Brussels.

  • Orin, D., & Goswami, A. (2008). Centroidal momentum matrix of a humanoid robot: Structure and properties. In Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS), Nice, pp. 653–659.

  • Nishiwaki, K., Kagami, S., Kuniyoshi, Y., Inaba, M., & Inoue, H. (2002). Online generation of humanoid walking motion based on a fast generation method of motion pattern that follows desired ZMP. In Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), pp. 2684–2689.

  • Papadopoulos, E. (1990). On the dynamics and control of space manipulators. PhD Thesis, Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge.

  • Popovic, M.B., Hofmann, A., & Herr, H. (2004). Zero spin angular momentum control: definition and applicability. In Proceedings of the IEEE-RAS international conference on humanoid robots, Santa Monica, pp. 478–493.

  • Popovic, M. B., Goswami, A., & Herr, H. (2005). Ground reference points in legged locomotion: Definitions, biological trajectories and control implications. International Journal of Robotics Research, 24(12), 1013–1032.

    Article  Google Scholar 

  • Roberson, R. E., & Schwertassek, R. (1988). Dynamics of multibody systems. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Rodriguez, G., Jain, A., & Kreutz-Delgado, K. (1991). A spatial operator algebra for manipulator modelling and control. International Journal of Robotics Research, 10(4), 371–381.

    Article  Google Scholar 

  • Sano, A., & Furusho, J. (1990). Realization of natural dynamic walking using the angular momentum information. In Proceedings of IEEE international conference on robotics and automation, Cincinnati, pp. 1476–1481.

  • Sciavicco, L., & Siciliano, B. (2005). Modeling and control of robot manipulators. Advanced textbooks in control and signal processing series. London: Springer London Limited.

  • Ugurlu, B., & Kawamura, A. (2010). Eulerian ZMP resolution based bipedal walking: Discussion on the intrinsic angular momentum rate change about center of mass. In Proceedings of IEEE international conference on robotics and automation, Alaska, pp. 4218–4223.

  • Walker, M. W., & Orin, D. (1982). Efficient dynamic computer simulation of robotic mechanisms in ASME. Journal of Dynamic Systems Measurement and Control, 104, 205–211.

    Article  MATH  Google Scholar 

  • Zordan, V. (2010). Angular momentum control in coordinated behaviors. Third annual international conference on motion in games. Zeist.

  • Wieber, P.-B. (2005). Holonomy and nonholonomy in the dynamics of articulated motion. Fast motions in biomechanics and robotics. Heidelberg: Springer.

    Google Scholar 

  • Wieber, P.-B. (2008). Viability and predictive control for safe locomotion. In Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Nice, pp. 1103–1108.

  • Wensing, P. M., & Orin, D. E. (2013). Generation of dynamic humanoid behaviors through task-space control with conic optimization. In Proceedings of IEEE international conference on robotics and automation (ICRA), Karlsruhe.

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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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Correspondence to Ambarish Goswami.

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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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