| The inverted pendulum system has the characteristics of instability,high-order and multivariable,and similar to the complex physical model with the centroid higher than the support point such as aircraft landing,rocket launch and robot joint,it can be simplified to a one-stage or even multi-stage inverted pendulum.Therefore,the system has been widely studied by many scholars.In particular,as a center of mass(COM)motion model,inverted pendulum is widely used in the field of robot passive walking to reveal the potential mechanism of human walking.Due to the saddle point characteristics in the dynamics of inverted pendulum system,the pendulum is easy to fall,which is closely related to the difficulty of maintaining the stability of passive walking gait.Therefore,the analysis of dynamic characteristics of inverted pendulum system has important guiding significance for the study of passive dynamic walking stability mechanism.Firstly,this thesis deduces the dynamic models of the inverted pendulum by using the Lagrange method,and introduces the dimensionless variables to make the dynamic equation dimensionless,and then transforms it into the state space equation required for numerical calculation and analysis,which is convenient for the calculation and analysis of the characteristics of system dynamics.Secondly,for the inverted pendulum systems,the main one-dimensional parameters are selected as the research object.The multi initial bifurcation,multi initial phase diagram and maximum Lyapunov exponent of single parameter are calculated by fourth-order Runge Kutta,and the bifurcation types and chaotic phenomena are analyzed.The dynamic behavior of the inverted pendulum system in two-dimensional parameter space is tracked by numerical calculation.Combined with single parameter bifurcation and Lyapunov exponent,the dynamic transition law of the system with the change of control parameters in two-dimensional parameter space is discussed.The mixed mode oscillation of double inverted pendulum is found by comparing the isospike stability diagrams of the plane,and the evolution law of overlapping adding double complex cascades in a single period of time response is analyzed.In order to further explain the difference of periodic stability phase diagram of inverted pendulum system under different initial values,the global stability of the system under the coexistence of attractors and the evolution of motion law before and after bifurcation point are studied and analyzed.The corresponding parameters are selected,the attraction region of inverted pendulum system with the change of parameters is calculated by cell mapping,and the global dynamic behavior of the system is analyzed.Then,the simulation model of the inverted pendulum system is designed with the help of three-dimensional software CATIA.The model is imported into ADAMS,and a reasonable simulation environment is set up to simulate the dynamics of the primary and secondary inverted pendulum models respectively.According to the simulation results,the evolution of the dynamic behavior of the primary inverted pendulum when the external excitation frequency and the length of the pendulum rod change,and the influence of the external excitation frequency and amplitude change on the motion characteristics of the secondary inverted pendulum are analyzed.By comparing the simulation results with the theoretical analysis,the bifurcation evolution process of the inverted pendulum system is verified.Finally,the experimental platform of double inverted pendulum is designed and built,and the experimental prototype of double inverted pendulum is made by 3D printing.In order to study the influence of the change of external excitation frequency on the motion characteristics of the double inverted pendulum system,several groups of experiments were carried out on this prototype platform.The motion state of the pendulum rod was accurately captured and recorded through the Qualisys track Manager(QTM)data acquisition system,and the experimental data were processed to obtain the time response diagram and phase diagram under different parameters.Through the comparative analysis of the experimental results,the correctness of the theoretical and simulation analysis was well verified.These results are helpful to recognize the motion mechanism of passive walking robot and promote the research and development of passive walker. |
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