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Volume 6 Issue 1
Jan.  2019

IEEE/CAA Journal of Automatica Sinica

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Zeguo Wang, Leonid B. Freidovich and Honghua Zhang, "Periodic Motion Planning and Control for Double Rotary Pendulum via Virtual Holonomic Constraints," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 291-298, Jan. 2019. doi: 10.1109/JAS.2017.7510712
Citation: Zeguo Wang, Leonid B. Freidovich and Honghua Zhang, "Periodic Motion Planning and Control for Double Rotary Pendulum via Virtual Holonomic Constraints," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 291-298, Jan. 2019. doi: 10.1109/JAS.2017.7510712

Periodic Motion Planning and Control for Double Rotary Pendulum via Virtual Holonomic Constraints

doi: 10.1109/JAS.2017.7510712
Funds:  This work was supported by China Scholarship Council (201504980073) for Zeguo Wang to visit Umeå University
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  • Periodic motion planning for an under-actuated system is rather difficult due to differential dynamic constraints imposed by passive dynamics, and it becomes more difficult for a system with higher underactuation degree, that is with a higher difference between the number of degrees of freedom and the number of independent control inputs. However, from another point of view, these constraints also mean some relation between state variables and could be used in the motion planning.We consider a double rotary pendulum, which has an underactuation degree 2. A novel periodic motion planning is presented based on an optimization search. A necessary condition for existence of the whole periodic trajectory is given because of the higher underactuation degree of the system. Moreover this condition is given to make virtual holonomic constraint (VHC) based control design feasible. Therefore, an initial guess for the optimization of planning a feasible periodic motion is based on this necessary condition. Then, VHCs are used for the system transformation and transverse linearization is used to design a static state feedback controller with periodic matrix function gain. The controller gain is found through another optimization procedure. The effectiveness of initial guess and performance of the closed-loop system are illustrated through numerical simulations.


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  • [1]
    M. Yamakita, K. Nonaka, and K. Furuta, "Swing up control of a double pendulum, " in Proc. American Control Conf., San Francisco, USA, 1993, pp. 2229-2233.
    M. W. Spong, "Partial feedback linearization of underactuated mechanical systems, " in Proc. IEEE/RSJ/GI Int. Conf. Intelligent Robots and Systems, Munich, Germany, 1994, pp. 314-321. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=407375
    M. Reyhanoglu, A. van der Schaft, N. H. McClamroch, and I. Kolmanovsky, "Dynamics and control of a class of underactuated mechanical systems, " IEEE Trans. Automat. Control, vol. 44, no. 9, pp. 1663-1671, Sep. 1999. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=788533
    R. Ortega, M. W. Spong, F. Gomez-Estern, and G. Blankenstein, "Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, " IEEE Trans. Automat. Control, vol. 47, no. 8, pp. 1218-1233, Aug. 2002. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=1024334
    N. Sun, Y. C. Fang, H. Chen, and B. Lu, "Amplitude-saturated nonlinear output feedback antiswing control for underactuated cranes with doublependulum cargo dynamics, " IEEE Trans. Ind. Electron., vol. 64, no. 3, pp. 2135-2146, Mar. 2017. http://ieeexplore.ieee.org/document/7725990/
    N. Sun, Y. M. Wu, Y. C. Fang, H. Chen, and B. Lu, "Nonlinear continuous global stabilization control for underactuated RTAC systems: design, analysis, and experimentation, " IEEE/ASME Trans. Mechatronics, vol. 22, no. 3, pp. 1104-1115, Apr. 2017. http://ieeexplore.ieee.org/document/7752953/
    Y. Liu and H. N. Yu, "A survey of underactuated mechanical systems, " IET Control Theory Appl., vol. 7, no. 7, pp. 921-935, May 2013. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6562926
    L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, "Generating selfexcited oscillations via two-relay controller, " IEEE Trans. Automat. Control, vol. 54, no. 2, pp. 416-420, Feb. 2009. http://ieeexplore.ieee.org/document/4782016/
    L. Freidovich, A. Shiriaev, F. Gordillo, F. Gomez-Estern, and J. Aracil, "Partial-energy-shaping control for orbital stabilization of highfrequency oscillations of the furuta pendulum, " IEEE Trans. Control Syst. Technol., vol. 17, no. 4, pp. 853-858, Jul. 2009. http://ieeexplore.ieee.org/document/4837870/
    A. S. Shiriaev, L. B. Freidovich, A. Robertsson, R. Johansson, and A. Sandberg, "Virtual-holonomic-constraints-based design of stable oscillations of Furuta pendulum: theory and experiments, " IEEE Trans. Robot., vol. 23, no. 4, pp. 827-832, Aug. 2007. http://www.mendeley.com/research/virtualholonomicconstraintsbased-design-stable-oscillations-furuta-pendulum-theory-experiments/
    L. Freidovich, A. Robertsson, A. Shiriaev, and R. Johansson, "Periodic motions of the pendubot via virtual holonomic constraints: theory and experiments, " Automatica, vol. 44, no. 3, pp. 785-791, Mar. 2008. http://www.sciencedirect.com/science/article/pii/S0005109807003676
    A. S. Shiriaev, L. B. Freidovich, and S. V. Gusev, "Transverse linearization for controlled mechanical systems with several passive degrees of freedom, " IEEE Trans. Automat. Control, vol. 55, no. 4, pp. 893-906, Apr. 2010. http://ieeexplore.ieee.org/document/5404418/
    P. X. M. La Hera, A. S. Shiriaev, L. B. Freidovich, U. Mettin, and S. V. Gusev, "Stable walking gaits for a three-link planar biped robot with one actuator, " IEEE Trans. Robot., vol. 29, no. 3, pp. 589-601, June. 2013. http://ieeexplore.ieee.org/document/6423288/
    M. Maggiore and L. Consolini, "Virtual holonomic constraints for EulerLagrange systems, " IEEE Trans. Automat. Control, vol. 58, no. 4, pp. 1001-1008, Apr. 2013. http://ieeexplore.ieee.org/document/6286994/
    J. W. Grizzle, G. Abba, and F. Plestan, "Asymptotically stable walking for biped robots: analysis via systems with impulse effects, " IEEE Trans. Automat. Control, vol. 46, no. 1, pp. 51-64, Jan. 2001. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=898695
    E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. H. Choi, and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. Boca Raton: CRC Press, 2007.
    L. Consolini and M. Maggiore, "Control of a bicycle using virtual holonomic constraints, " Automatica, vol. 49, no. 9, pp. 2831-2839, Sep. 2013. https://linkinghub.elsevier.com/retrieve/pii/S000510981300304X
    A. P. Aguiar, J. P. Hespanha, and P. V. Kokotovic, "Path-following for nonminimum phase systems removes performance limitations, " IEEE Trans. Automat. Control, vol. 50, no. 2, pp. 234-239, Feb. 2005. http://ieeexplore.ieee.org/document/1393141
    C. Nielsen, C. Fulford, and M. Maggiore, "Path following using transverse feedback linearization: application to a maglev positioning system, " Automatica, vol. 46, no. 3, pp. 585-590, Mar. 2010. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5159998
    A. Shiriaev, J. W. Perram, and C. Canudas-de-Wit, "Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach, " IEEE Trans. Automat. Control, vol. 50, no. 8, pp. 1164-1176, Aug. 2005. http://ieeexplore.ieee.org/document/1492559/
    S. V. Gusev, A. S. Shiriaev, and L. B. Freidovich, "SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations, " Int. J. Control, vol. 89, no. 7, pp. 1396-1405, Jul. 2016. doi: 10.1080/00207179.2015.1131850
    A. S. Shiriaev, L. B. Freidovich, and M. W. Spong, "Controlled invariants and trajectory planning for underactuated mechanical systems, " IEEE Trans. Automat. Control, vol. 59, no. 9, pp. 2555-2561, Sep. 2014. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6760115
    W. J. Rugh, Linear System Theory. 2nd ed. Upper Saddle River, USA:Prentice Hall, 1996.
    H. K. Khalil, Nonlinear Systems. 3rd ed. Upper Saddle River, New Jersey, USA:Prentice Hall, 2002.


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