On Applying Approximate Entropy To Analyzing The Chaotic Motion Of Underactuated Mechanism

Chaos refers to a phenomenon in nonlinear dynamic system which is internally irregular and externally deterministic, and is very sensitive to the initial conditions of the system. In recent years, the study of chaos has been emerging from various disciplines and fields, and some great progress and achievements have been made. For its great worth of research and prospects of application in engineering, chaos or chaotic motion in mechanism design has attracted lots of attention from experts and researchers.Accuracy is an important feature of classical mechanism design. It is pursued to represent the motion of a mechanism accurately. However, for the chaotic motion of a mechanism one can’t be able to explain or express it by some kind of precise quantity or value. As a measure of random and uncertainty, entropy should be referenced to the study of chaotic motion. Although entropy has been applied to the study of chaos arising from some other discrete or/and continuous dynamic systems, its applications in mechanism design is rare, and need to be explored systematically and extensively.To this end, taking typical open-chain and closed-chain underactuated mechanisms as examples of chaotic motion, this paper is devoted to putting forwards the application of entropy in the study of chaotic motion of mechanism, with the aim of finding out parameters of mechanism at the edge of chaos and laying some foundations for controlling of chaotic motion of mechanism.The main work is the followings:1. Based on the theory of entropy and edge of chaos, the definitions and algorithms for Shannon entropy, symbolic entropy, Kolmogorov entropy and approximate entropy are studied in detail. Symbol entropy and approximate entropy are calculated for two dynamic systems, namely Logistic map and Lorenz system. The variation charts of entropy with respect to the changes of the system’s parameters are obtained, by which the edges of chaos for two systems are identified. The method to determine the threshold value for calculation of approximate entropy is modified. All these research work will provide some references for applying entropy to study of chaotic motion of mechanism.2. Inverted pendulum on a cart is a typical open-chain underactuated mechanism. It is taken as the first subject to be studied. Using MATLAB to simulate the system, the edge of chaos with one system’s parameter changing is found by approximate entropy. The resultsSponsored by National Science Foundation of China(Grant No:51175437) show that the approximate entropies undergo a change from equal to zero into some positive numbers at the edge of chaos, and according to the definition of entropy, the dynamic behavior of the system undergoes a change from regular motion into chaotic motion. This procedure forms an approach to find the edge of chaos. In addition, the threshold values used to calculate the approximate entropy for the motions of inverted pendulum on a cart are discussed, and the appropriate threshold value rang is fixed.3. The second subject to be studied is planar closed-chain underactuated five-bar mechanism. Firstly, SimMechanics is used to model the mechanism, and then approximate entropy is employed to find the edge of chaos. By extracting the time series with each changing rod length, the entropy is calculated and the variation chart of entropy is plotted to find the range of rod length as the system changes from periodic motion to chaotic motion; at the same time, the conclusion drawn from variation of entropy is verified by the Lyapunov exponents and the phase diagrams. Finally, the appropriate threshold value rang is fixed.