Skip to main content
SpringerLink
Log in

Synchronization of networked multibody systems using fundamental equation of mechanics

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

From the analytical dynamics point of view, this paper develops an optimal control framework to synchronize networked multibody systems using the fundamental equation of mechanics. A novel robust control law derived from the framework is then used to achieve complete synchronization of networked identical or non-identical multibody systems formulated with Lagrangian dynamics. A distinctive feature of the developed control strategy is the introduction of network structures into the control requirement. The control law consists of two components, the first describing the architecture of the network and the second denoting an active feedback control strategy. A corresponding stability analysis is performed by the algebraic graph theory. A representative network composed of ten identical or non-identical gyroscopes is used as an illustrative example. Numerical simulation of the systems with three kinds of network structures, including global coupling, nearest-neighbour, and small-world networks, is given to demonstrate effectiveness of the proposed control methodology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Murray, R. M. Recent research in cooperative control of multivehicle systems. Journal of Dynamic Systems Measurement and Control, 129(5), 571–583 (2007)

    Article  Google Scholar 

  2. Olfati-Saber, R., Fax, J. A., and Murray, R. M. Consensus and cooperation in networked multiagent systems. Proceedings of the IEEE, 95(1), 215–233 (2007)

    Article  Google Scholar 

  3. Ren, W. and Cao, Y. Distributed Coordination of Multiagent Networks: Emergent Problems, Models, and Issues, Springer-Verlag, London (2011)

    Book  MATH  Google Scholar 

  4. Liu, Y. C. and Chopra, N. Controlled synchronization of heterogeneous robotic manipulators in the task space. IEEE Transactions on Robotics, 28(1), 268–275 (2012)

    Article  Google Scholar 

  5. Chung, S. J. and Slotine, J. J. E. Cooperative robot control and concurrent synchronization of Lagrangian systems. IEEE Transactions on Robotics, 25(3), 686–700 (2009)

    Article  Google Scholar 

  6. Hokayem, P. F., Stipanovic, D. M., and Spong, M. W. Coordination and collision avoidance for Lagrangian systems with disturbances. Applied Mathematics and Computation, 217(3), 1085–1094 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nuño, E., Ortega, R., Basañez, L., and Hill, D. Synchronization of networks of non-identical Lagrange systems with uncertain parameters and communication delays. IEEE Transactions on Automatic Control, 56(4), 935–941 (2011)

    Article  Google Scholar 

  8. Wu, X., Zhou, J., Xiang, L., Lin, C., and Zhang, H. Impulsive synchronization motion in networked open-loop multibody systems. Multibody System Dynamics, 30, 37–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kovriguine, D. A. Synchronization and sommerfeld effect as typical resonant patterns. Archive of Applied Mechanics, 82(5), 591–604 (2012)

    Article  MATH  Google Scholar 

  10. Wen, G., Yu, W., Chen, M. Z. Q., Yu, X., and Chen, G. H8 pinning synchronization of directed networks with aperiodic sampled-data communications. IEEE Transactions on Circuits and Systems I: Regular Papers, 61(11), 3245–3255 (2014)

    Article  MathSciNet  Google Scholar 

  11. Liu, J., Ji, J., Zhou, J., Xiang, L., and Zhao, L. Adaptive group consensus in uncertain networked Euler-Lagrange systems under directed topology. Nonlinear Dynamics, 82(3), 1145–1157 (2015)

    Article  MathSciNet  Google Scholar 

  12. Zhou, J., Wu, X., and Liu, Z. Distributed coordinated adaptive tracking in networked redundant robotic systems with a dynamic leader. Science China Technological Sciences, 57(5), 905–913 (2014)

    Article  MathSciNet  Google Scholar 

  13. Scardovi, L. and Sepulchre, R. Synchronization in networks of identical linear systems. Automatica, 45(11), 2557–2562 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chopra, N., Stipanovic, D. M., and Spong, M. W. On synchronization and collision avoidance for mechanical systems. American Control Conference, IEEE, Seattle (2008)

    Book  Google Scholar 

  15. Nair, S. and Leonard, N. E. Stable synchronization of mechanical system networks. SIAM Journal on Control and Optimization, 47(2), 661–683 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao, Y., Duan, Z., and Wen, G. Distributed finite-time tracking of multiple Euler-Lagrange systems without velocity measurements. International Journal of Robust and Nonlinear Control, 25(11), 1688–1703 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Udwadia, F. E. and Kalaba, R. E. A new perspective on constrained motion. Proceedings of the Royal Society A, 439, 407–410 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Udwadia, F. E. and Kalaba, R. E. What is the general form of the explicit equations of motion for constrained mechanical systems? Journal of Dynamic Systems Measurement and Control, 69, 335–339 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Udwadia, F. E. A new perspective on the tracking control of nonlinear structural and mechanical systems. Proceedings of the Royal Society A, 459, 1783–1800 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Udwadia, F. E. and Han, B. Synchronization of multiple chaotic gyroscopes using the fundamental equation of mechanics. Journal of Applied Mechanics, 75, 021011 (2008)

    Article  Google Scholar 

  21. Mylapilli, H. and Udwadia, F. E. A constrained motion approach to the synchronization of multiple coupled slave gyroscopes. Proceedings of the 13th ASCE Aerospace Division Conference and the 5th NASA/ASCE Workshop on Granular Materials in Space Exploration, American Society of Civil Engineers, Pasadena (2012)

    Google Scholar 

  22. Udwadia, F. E. and Schutte, A. D. A unified approach to rigid body rotational dynamics and control. Proceedings of the Royal Society A, 468, 395–414 (2012)

    Article  MathSciNet  Google Scholar 

  23. Schutte, A. D. and Udwadia, F. E. New approach to the modeling of complex multibody dynamical systems. Journal of Applied Mechanics, 78, 021018 (2011)

    Article  Google Scholar 

  24. Udwadia, F. E. Optimal tracking control of nonlinear dynamical systems. Proceedings of the Royal Society A, 464, 2341–2363 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peters, J., Mistry, M., Udwadia, F., Cory, R., Nakanishi, J., and Schaa, S. A unifying methodology for the control of robotic systems. IEEE/RSJ International Conference: Intelligent Robots and Systems, IEEE, Edmonton (2005)

    Google Scholar 

  26. Schutte, A. D., Udwadia, F. E., and Lam, T. Nonlinear dynamics and control of a dumbbell spacecraft system. Proceedings of the 11th Aerospace Division International Conference on Engineering, Science, Construction, and Operations in Challenging Environments, American Society of Civil Engineers, Long Beach (2008)

    Google Scholar 

  27. Udwadia, F. E. and Schutte, A. D. A unified approach to the modeling and control of multi-scale dynamical systems. Proceedings of the 13th ASCE Aerospace Division Conference on Engineering, Science, Construction, Environments and Operations in Challenging and the 5th NASA/ASCE Workshop on Granular Materials in Space Exploration, American Society of Civil Engineers, Pasadena (2012)

    Google Scholar 

  28. Udwadia, F. E. and Schutte, A. D. An alternative derivation of the quaternion equations of motion for rigid-body rotational dynamics. Journal of Applied Mechanics, 77, 044505 (2010)

    Article  Google Scholar 

  29. Udwadia, F. E. and Kalaba, R. E. Analytical Dynamics: A New Approach, Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  30. Zhang, H., Zhou, J., and Liu, Z. Synchronization of networked harmonic oscillators with communication delays under local instantaneous interaction. Journal of Dynamic Systems Measurement and Control, 134, 061009 (2012)

    Article  Google Scholar 

  31. Zhang, H. and Zhou, J. Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Systems and Control Letters, 61, 1277–1285 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, J., Zhang, H., Xiang, L., and Wu, X. Synchronization of coupled harmonic oscillators with local instantaneous interaction. Automatica, 48(8), 1715–1721 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chen, G., Wang, X., and Li, X. Introduction to Complex Networks: Models, Structures and Dynamics, Higher Education Press, Beijing (2012)

    Google Scholar 

  34. Zhou, J. and Chen, T. Synchronization in generral complex delayed dynamical networks. IEEE Transactions on Circuits and Systems I: Regular Papers, 53(3), 733–744 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Jun Liu & Jin Zhou

  2. Department of Mathematics, Jining University, Qufu, 273155, Shandong Province, China

    Jun Liu

  3. Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW, 2007, Australia

    Jinchen Ji

  4. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai, 200072, China

    Jun Liu & Jin Zhou

Authors
  1. Jun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinchen Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Zhou.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10972129 and 11272191), the Specialized Research Foundation for the Doctoral Program of Higher Education (No. 200802800015), the Science and Technology Project of High Schools of Shandong Province (No. J15LJ07), and the Shandong Provincial Natural Science Foundation (No.ZR2015FL026)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Ji, J. & Zhou, J. Synchronization of networked multibody systems using fundamental equation of mechanics. Appl. Math. Mech.-Engl. Ed. 37, 555–572 (2016). https://doi.org/10.1007/s10483-016-2071-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-016-2071-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation