Fu Q, Wang D H, Xu L, Yuan G
Precision and Intelligence Laboratory, Department of Optoelectronic Engineering, Chongqing University, Chongqing, 400044, China.
Key Laboratory of Optoelectronic Technology and Systems of the Ministry of Education of China, Chongqing University, Chongqing, 400044, China.
Biol Cybern. 2018 Jun;112(3):227-235. doi: 10.1007/s00422-018-0746-1. Epub 2018 Jan 13.
Nonlinear oscillators are usually utilized by bionic scientists for establishing central pattern generator models for imitating rhythmic motions by bionic scientists. In the natural word, many rhythmic motions possess asymmetric time ratios, which means that the forward and the backward motions of an oscillating process sustain different times within one period. In order to model rhythmic motions with asymmetric time ratios, nonlinear oscillators with asymmetric forward and backward trajectories within one period should be studied. In this paper, based on the property of the invariant set, a method to design the closed curve in the phase plane of a dynamic system as its limit cycle is proposed. Utilizing the proposed method and considering that a cardioid curve is a kind of asymmetrical closed curves, a cardioid oscillator with asymmetric time ratios is proposed and realized. Through making the derivation of the closed curve in the phase plane of a dynamic system equal to zero, the closed curve is designed as its limit cycle. Utilizing the proposed limit cycle design method and according to the global invariant set theory, a cardioid oscillator applying a cardioid curve as its limit cycle is achieved. On these bases, the numerical simulations are conducted for analyzing the behaviors of the cardioid oscillator. The example utilizing the established cardioid oscillator to simulate rhythmic motions of the hip joint of a human body in the sagittal plane is presented. The results of the numerical simulations indicate that, whatever the initial condition is and without any outside input, the proposed cardioid oscillator possesses the following properties: (1) The proposed cardioid oscillator is able to generate a series of periodic and anti-interference self-exciting trajectories, (2) the generated trajectories possess an asymmetric time ratio, and (3) the time ratio can be regulated by adjusting the oscillator's parameters. Furthermore, the comparison between the simulated trajectories by the established cardioid oscillator and the measured angle trajectories of the hip angle of a human body show that the proposed cardioid oscillator is fit for imitating the rhythmic motions of the hip of a human body with asymmetric time ratios.
仿生科学家通常利用非线性振荡器来建立中枢模式发生器模型，以模仿节律性运动。在自然界中，许多节律性运动具有不对称的时间比，这意味着振荡过程的向前和向后运动在一个周期内持续的时间不同。为了对具有不对称时间比的节律性运动进行建模，需要研究在一个周期内具有不对称向前和向后轨迹的非线性振荡器。本文基于不变集的性质，提出了一种在动态系统的相平面中设计封闭曲线作为其极限环的方法。利用该方法并考虑到心形曲线是一种不对称封闭曲线，提出并实现了一种具有不对称时间比的心形振荡器。通过使动态系统相平面中的封闭曲线的导数等于零，将该封闭曲线设计为其极限环。利用所提出的极限环设计方法并根据全局不变集理论，实现了一种以心形曲线作为极限环的心形振荡器。在此基础上，进行了数值模拟以分析心形振荡器的行为。给出了利用所建立的心形振荡器模拟人体髋关节在矢状面内节律性运动的示例。数值模拟结果表明，无论初始条件如何且无任何外部输入，所提出的心形振荡器具有以下特性：（1）能够产生一系列周期性且抗干扰的自激轨迹；（2）所产生的轨迹具有不对称的时间比；（3）时间比可通过调整振荡器参数进行调节。此外，所建立的心形振荡器模拟轨迹与人体髋关节角度测量轨迹的比较表明，所提出的心形振荡器适合于模拟具有不对称时间比的人体髋关节节律性运动。
