On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping-a case study

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.