Abstract
The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the nonlinear approximation of the local neighborhood-to-neighborhood mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear mapping, (1) the accuracy of the negative exponents calculated using the nonlinear mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.
Similar content being viewed by others
References
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physics D 16, 285–317 (1985)
Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos, an Introduction to Dynamical Systems. Springer, New York (1997)
Williams, G.P.: Chaos Theory Tamed. Joseph Henry Press, Washington, D.C. (1997)
Sekhavat, P., Sepehri, N., Wu, Q.: Calculation of Lyapunov exponents using nonstandard finite difference discretization scheme: a case study. J. Differ. Equ. Appl. 10(4), 369–378 (2004)
Müller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos Solitons Fractals 5, 1671–1681 (1995)
Oseledec, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197 (1968)
Asokanthan, S.F., Wang, X.H.: Characterization of torsional instability in a Hooke’s joint-driven system via maximal Lyapunov exponents. J. Sound Vib. 194(1), 83–91 (1996)
Gilat, R., Aboudi, J.: Parametric stability of non-linearly elastic composite plates by Lyapunov exponents. J. Sound Vib. 235(4), 627–637 (2000)
Zevin, A.A., Pinsky, M.A.: Absolute stability criteria for a generalized Lur’e problem with delay in the feedback. SIAM J. Control Optim. 43(6), 2000–2008 (2005)
Awrejcewicz, J., Kudra, G.: Stability analysis and Lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints. Nonlinear Anal. Theory Methods Appl. 63, 909–918 (2005)
Castillo, Rogelio, C., Gustavo, A., Javier, C.P.: Determination of limit cycles using both the slope of correlation integral and dominant Lyapunov methods. Nucl. Technol. 145(2), 139–149 (2004)
Wu, Q., Sekhavat, P., Sepehri, N., Peles, S.: On design of continuous Lyapunov’s feedback control. J. Franklin Inst. Eng. Appl. Math. 342(6), 702–723 (2005)
Sekhavat, P., Sepehri, N., Wu, Q.: Impact control in hydraulic actuators with friction: theory and experiments. IFAC J. Control Eng. Pract. 14(12), 1423–1433 (2006)
Yang, C., Wu, Q.: On stabilization of bipedal robots during disturbed standing using the concept of Lyapunov exponents. In: Proceedings of the 2006 American Control Conference, Minneapolis, MN, pp. 2516–2521, 14–16 June 2006
Yang, C., Wu, Q.: On stabilization of bipedal robots during disturbed standing using the concept of Lyapunov exponents. Robotica 24(5), 621–624 (2006)
Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from chaotic time series. Phys. Rev. Lett. 55, 1082 (1985)
Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)
Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2004)
Zeng, X., Pielke, R.A., Eykholt, R.: Extracting Lyapunov exponents from short time series of low precision. Mod. Phys. Lett. B 6(2), 55–75 (1992)
Rosenstein, M.T., Collins, J.J., DeLuca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D, Nonlinear Phenom. 65, 117–134 (1993)
Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series analysis: the TISEAN package. Chaos, Interdiscip. J. Nonlinear Sci. 9(2), 413–435 (1999)
Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185(1), 77–87 (1994)
Yasuaki, O., Muhammad, A., Akihiro, S., Kazuki, F., Hikaru, I., Ryoichi, N., Ichiro, T.: Assessment of walking stability of elderly by means of nonlinear time-series analysis and simple accelerometry. JSME Int. J. Ser. C, Mech. Syst. Mach. Elem. Manuf. 48(4), 607–612 (2006)
Burdet, E., Tee, K.P., Mareels, I., Milner, T.E., Chew, C.M., Franklin, D.W., Osu, R., Kawato, M.: Stability and motor adaptation in human arm movements. Biol. Cybern. 94(1), 20–32 (2006)
Dingwell, B.J., Cusumano, J.P.: Nonlinear time series analysis of normal and pathological human walking. Chaos 10(4), 848–863 (2000)
Dingwell, J.B., Cusumano, J.P., Sternad, D., Cavanagh, P.R.: Slower speeds in patients with diabetic neuropathy lead to improved local dynamic stability of continuous overground walking. J. Biomech. 33, 1269–1277 (2000)
Dingwell, J.B., Cusumano, J.P., Cavanagh, P.R., Sternad, D.: Local dynamic stability versus kinematic variability of continuous overground and treadmill walking. ASME J. Biomech. Eng. 123, 27–32 (2001)
Dingwell, J.B., Marin, L.C.: Kinematic variability and local dynamic stability of upper body motions when walking at different speeds. J. Biomech. 39, 444–452 (2006)
Dingwell, B.J.: Lyapunov exponents. In: Wiley Encyclopedia of Biomedical Engineering. Wiley, New York (2006)
Brown, R., Bryant, P., Abarbanel, H.D.: Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A 43, 2787–2806 (1991)
Yang, C., Wu, Q.: Effects of constraints on bipedal balance control. In: Proceedings of the 2006 American Control Conference, Minneapolis, MN, pp. 2510–2515, 14–16 June 2006
Haken, H.: At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys. Lett. A 94, 71–72 (1989)
Holzfuss, J., Lauterborn, W.: Lyapunov exponents from a time series of acoustic chaos. Phys. Rev. A 39, 2146–2152 (1989)
Vukobratović, M., Borovać, B.: Zero-moment point—thirty years of its life. Int. J. Humanoid Robot. 1(1), 157–173 (2004)
Abarbanel, H.D.I., Brown, R., Kennel, M.B.: Local Lyapunov exponents computed from observed data. J. Nonlinear Sci. 2, 343–365 (1992)
Takens, F.: Detect strange attractors in turbulence. In: Rand, D.A., Young, L.-S. (eds.) Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981)
Fraser, A.M.: Reconstructing attractors from scalar time series: a comparison of singular system analysis and redundancy criteria. Physica D 34, 391–404 (1989)
Broomhead, D.S., King, G.P.: Extracting qualitative dynamics from experimental data. Physica D 20, 217–236 (1986)
Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New York (1991)
Nusse, H., Yorke, J.A.: Dynamics: Numerical Exploration. Springer, New York (1998)
Parlitz, U.: Identification of true and spurious Lyapunov exponents from time series. Int. J. Bifurc. Chaos 2, 155–165 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, C., Wu, Q. On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study. Nonlinear Dyn 59, 239–257 (2010). https://doi.org/10.1007/s11071-009-9535-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9535-7