Calculation of negative Lyapunov exponents using a time series for potentially stable robotic systems.

摘要

It is believed that Lyapunov Exponents can characterize the stability of nonlinear dynamic systems. Lyapunov Exponents can be calculated from the mathematic model or time series data of the system, which are independent of the initial conditions within the same stability region. Lyapunov Exponents have been mainly used for diagnosing chaotic systems, where at least one Lyapunov Exponent is positive. Little work has been done on calculating Lyapunov Exponents from a time series of a potentially stable system, where the largest Lyapunov Exponent is negative or zero. Most mechanical systems are complex, of which, the mathematical models are sketchy or even not available. For such systems, it is extremely difficult, even impossible, to derive a Lyapunov function for stability analysis. Therefore, an alternative method for stability analysis of nonlinear engineering systems is needed.;Lyapunov Exponents for each of the above systems are calculated using the mathematic models and the largest exponent is calculated from the time series. The results show that for the two-link position-controlled robotic system which has an isolated equilibrium point, the largest negative Lyapunov exponent calculated from the time series matches the one from the mathematic model very well. This indicates that Wolf's method has good potential for calculating largest negative Lyapunov exponent. However, for systems with a stable periodic motion, the stability should be studied using Lyapunov exponents calculated from mathematical model. For the pneumatic system, which has a set of infinite non-isolated equilibrium points, zero exponents are obtained from the mathematic model, which conflict with the conventional interpretation of the Lyapunov Exponents. However, the largest Lyapunov exponent calculated using a time series for the pneumatic system does not match the one from the mathematic model, and the cause is also explored. It is concluded based on the examples for systems with a set of infinite non-isolated equilibrium points, The largest Lyapunov Exponents can not be calculated using Wolf's method. Systems with infinite non-isolated equilibrium points occur naturally and frequently in mechanical engineering systems.;This work is the first step in applying the concept of Lyapunov Exponents for stable mechanical engineering systems. It enables us to understand the possibility and procedure for applying Wolf s method using time series for potentially stable robotic systems. More importantly, this work shows the limitations of the applications of Wolf's method to engineering systems.;The objective of this thesis is to explore the possibility and limitations of applying Wolf's method to calculate the largest Lyapunov exponent from a time series of potential stable systems. Two fundamentally different robotic systems are used as examples. One is a robotic arm with two rigid links moving in the horizontal plane. A position-controlled pneumatic actuator system is used as the second example. In addition to the different nature in their nonlinearity between the above two robotic systems, the pneumatic system has a set of infinite non-isolated equilibrium points, while the two-link robotic arm has one equilibrium point.