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Dynamics Analysis: Equations of Motion

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Humanoid Robotics: A Reference

Abstract

This chapter provides dynamics analyses of a humanoid robot as a mechanical system. It is modeled as a floating-base tree structure that exchanges forces with the environment through contacts. Lagrange’s equation of motion of the system is derived in order to focus on how to read it rather than how to compute it. The dominant dynamics of the robot embedded in the centroidal momentum is then discussed. Issues around the contact force are also addressed. Some useful ground references are introduced in order to intuitively associate the contact constraints with the whole-body motion. Finally, the equation of motion of a planar humanoid robot is explicitly derived in a different way from the above, which helps readers understand another mathematical structure of the dynamics.

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References

  1. C.G. Atkeson, C.H. An, J.M. Hollerbach, Estimation of inertial parameters of manipulator loads and links. Int. J. Robot. Res. 5(3), 101–119 (1986)

    Article  Google Scholar 

  2. R. Boulic, R. Mas, D. Thalmann, Inverse kinetics for center of mass position control and posture optimization, in Proceedings of European Workshop on Combined Real and Synthetic Image Processing for Broadcast and Video Production, 1994

    Google Scholar 

  3. Y. Fujimoto, A. Kawamura, Simulation of an autonomous biped walking robot including environmental force interaction. IEEE Robot. Autom. Mag. 5(2), 33–41 (1998)

    Article  Google Scholar 

  4. H. Hirukawa, S. Hattori, K. Harada, S. Kajita, K. Kaneko, F. Kanehiro, K. Fujiwara, M. Morisawa, A universal stability criterion of the foot contact of legged robots – adios ZMP, in Proceedings of the 2006 IEEE International Conference on Robotics & Automation, 2006, pp. 1976–1938

    Google Scholar 

  5. H. Hirukawa, S. Hattori, S. Kajita, K. Harada, K. Kaneko, F. Kanehiro, M. Morisawa, S. Nakaoka, A pattern generator of humanoid robots walking on a rough terrain, in Proceedings of the 2007 IEEE International Conference on Robotics & Automation, 2007, pp. 2181–2187

    Google Scholar 

  6. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, H. Hirukawa, Resolved momentum control: humanoid motion planning based on the linear and angular momentum, in Proceedings of the 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2003, pp. 1644–1650

    Google Scholar 

  7. H. Kawasaki, Trajectory control of manipulators by on-line parameter identification (in Japanese). Trans. Soc. Instrum. Control Eng. 25(9), 780–787 (1984)

    Article  Google Scholar 

  8. O. Khatib, A unified approach for motion and force control of robot manipulators: the operational space formulation. Int. J. Robot. Autom. RA-3(1), 43–53 (1987)

    Article  Google Scholar 

  9. H. Mayeda, K. Osuka, A. Kangawa, A new identification method for serial manipulator arm, in Proceedings of 9th IFAC World Congress, 1984, pp. 2429–2434

    Google Scholar 

  10. R.B. McGhee, A.A. Frank, On the stability properties of quadruped creeping gaits. Math. Biosci. 3, 331–351 (1968)

    Article  MATH  Google Scholar 

  11. K. Mitobe, G. Capi, Y. Nasu, Control of walking robots based on manipulation of the zero moment point. Robotica 18(6), 651–657 (2000)

    Article  Google Scholar 

  12. F. Miyazaki, S. Arimoto, A control theoretic study on dynamical biped locomotion. Trans. ASME J. Dyn. Syst. Meas. Control 102, 233–239 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Y. Nakamura, R. Mukherjee, Nonholonomic path planning of space robots via a bidirectional approach. IEEE Trans. Robot. Autom. 7(4), 500–514 (1991)

    Article  Google Scholar 

  14. Y. Nakamura, K. Yamane, Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Trans. Robot. Autom. 16(2), 124–134 (2000)

    Article  Google Scholar 

  15. D.E. Orin, A. Goswami, S.-H. Lee, Centroidal dynamics of a humanoid robot. Auton. Robot. 35, 161–176

    Google Scholar 

  16. K. Osuka, Adaptive control for nonlinear mechanical systems (in Japanese), in Proceedings of 14th Symposium on Control Theory, 1985, pp. 121–124

    Google Scholar 

  17. M. Popovic, A. Goswami, H.M. Herr, Ground reference points in legged locomotion: definitions, biological trajectories and control implications. Int. J. Robot. Res. 24(12), 1013–1032 (2005)

    Article  Google Scholar 

  18. J.J.E. Slotine, W. Li, On the adaptive control of robot manipulators. Int. J. Robot. Res. 6(3), 49–59 (1987)

    Article  Google Scholar 

  19. T. Sugihara, A study of the realtime generation of legged motions on a whole-body humanoid robot, Mc. thesis, University of Tokyo, 2001 (A brief English version is published as: T. Sugihara, Y. Nakamura, H. Inoue, Realtime humanoid motion generation through ZMP manipulation based on inverted pendulum control, in Proceedings of the 2002 IEEE International Conference on Robotics and Automation, 2002, pp. 1404–1409)

    Google Scholar 

  20. M. Takegaki, S. Arimoto, A new feedback method for dynamic control of manipulators. Trans. ASME J. Dyn. Syst. Meas. Control 102(2), 119–125 (1981)

    Article  MATH  Google Scholar 

  21. M. Vukobratović, J. Stepanenko, On the stability of anthropomorphic systems. Math. Biosci. 15, 1–37 (1972)

    Article  MATH  Google Scholar 

  22. K. Yoshida, D.N. Nenchev, A general formulation of under-actuated manipulator systems, in Proceedings of the Eighth International Symposium of Robotics Research, 1997, pp. 33–44

    Google Scholar 

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Authors and Affiliations

  1. Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University, 565-0871, 2-1 Yamadaoka, Suita, Osaka, Japan

    Tomomichi Sugihara

  2. Department of Electrical and Computer Engineering, Yokohama National University, 240-8501, 79-5 Tokiwadai, Hodogaya-ku, Yokohama, Japan

    Yasutaka Fujimoto

Authors
  1. Tomomichi Sugihara
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  2. Yasutaka Fujimoto
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Corresponding author

Correspondence to Yasutaka Fujimoto .

Editor information

Editors and Affiliations

  1. Intuitive Surgical , Sunnyvale, California, USA

    Ambarish Goswami

  2. Singapore, Singapur, Singapore

    Prahlad Vadakkepat

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Sugihara, T., Fujimoto, Y. (2017). Dynamics Analysis: Equations of Motion. In: Goswami, A., Vadakkepat, P. (eds) Humanoid Robotics: A Reference. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7194-9_3-1

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  • DOI: https://doi.org/10.1007/978-94-007-7194-9_3-1

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-7194-9

  • Online ISBN: 978-94-007-7194-9

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

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