The simple chaotic model of passive dynamic walking

Recent findings on the dynamical analysis of human locomotion characteristics such as stride length signal have shown that this process is intrinsically a chaotic behavior. The passive walking has been defined as walking down a shallow slope without using any muscular contraction as an active controller. Based on this definition, some knee-less models have been proposed to present the simplest possible models of human gait. To maintain stability, these simple passive models are compelled to show a wide range of different dynamics from order to chaos. Unfortunately, based on simplifications, for many years the cyclic period-one behavior of these models has been considered as the only stable response. This assumption is not in line with the findings about the nature of walking. Thus, this paper proposes a novel model to demonstrate that the knee-less passive dynamic models also have the ability to model the chaotic behavior of human locomotion with some modifications. The presented novel model can show chaotic behavior as a stable and acceptable answer using a chaotic function in heel-strike condition. The represented chaotic model is also able to simulate different types of motor deficits such as Parkinson’s disease only by manipulating the value of chaotic parameter. Our model has extensively examined in complexity and chaotic behavior using different analytical methods such as fractal dimension, bifurcation and largest Lyapunov exponent, and it was compared with conventional passive models and the stride signal of healthy subjects and Parkinson patients.     

变刚度半被动双足机器人行走的仿人控制1)

研究了变刚度半被动双足机器人行走控制问题。采用仿人的行走控制策略,使用变刚度双足弹簧负载倒立摆模型,利用模型自稳定性,在双支撑阶段调整后腿刚度使机器人的能量保持在期望能量附近,在单支撑阶段调整摆动腿落地位置控制质点的高度和前向速度。仿真结果表明：本文采用的控制策略可以实现双足机器人在水平面上的稳定行走,无扰动时可以使机器人实现零输入的被动周期行走,有外部扰动时腿部变刚度控制可使机器人总能量恢复平衡并重新进入稳态行走,控制系统具有鲁棒性。     

HUMAN-LIKE CONTROL FOR SEMI-PASSIVE BIPEDAL ROBOT WALKING WITH VARIABLE LEG STIFFNESS1)

研究了变刚度半被动双足机器人行走控制问题。采用仿人的行走控制策略,使用变刚度双足弹簧负载倒立摆模型,利用模型自稳定性,在双支撑阶段调整后腿刚度使机器人的能量保持在期望能量附近,在单支撑阶段调整摆动腿落地位置控制质点的高度和前向速度。仿真结果表明：本文采用的控制策略可以实现双足机器人在水平面上的稳定行走,无扰动时可以使机器人实现零输入的被动周期行走,有外部扰动时腿部变刚度控制可使机器人总能量恢复平衡并重新进入稳态行走,控制系统具有鲁棒性。     

参数和直接激励下非线性压电俘能器性能分析的多尺度法

考虑几何非线性、阻尼非线性和梁的轴向不可伸长条件,利用Hamilton变分原理,建立了参数激励和直接激励下压电俘能器的非线性力电耦合的运动微分方程;利用Galerkin法,将所建立的动力学偏微分方程降阶为力电耦合的Mathieu-Duffing型方程;采用多尺度法获得了梁的位移和输出电压的解析表达式,给出了解的稳定性条件;利用解析表达式研究了单独参数激励以及参数激励和直接激励共同作用下阻尼系数对压电俘能器性能的影响。结果表明,在参数激励情况下,线性阻尼会显著影响超临界分岔点的位置,非线性二次阻尼不会影响超临界分岔点的位置。参数激励和直接激励的结合可以作为提升压电能量俘获器性能的解决方案。     

A passive-biped model with multiple routes to chaos

This paper presents a new passive-biped model consisting of a simplest walking model beneath an upper body, with no kinematic constraint. The upper body is attached to the legs with a linear torsional spring. The model is a passive dynamic walker, so it walks down a slope without energy input. The governing equations of motion are derived and simulated for the parameter analysis purposes. Simulation results reveal some different routes to chaos that have not been observed in previous models.     

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM 1)

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.     

一类三维非线性系统的复杂簇发振荡行为及其机理

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.      

A proposed hybrid neural network for position control of a walking robot

The use of a proposed recurrent neural network control system to control a four-legged walking robot is presented in this paper. The control system consists of a neural controller, a standard PD controller, and the walking robot. The robot is a planar four-legged walking robot. The proposed Neural Network (NN) is employed as an inverse controller of the robot. The NN has three layers, which are input, hybrid hidden and output layers. In addition to feedforward connections from the input layer to the hidden layer and from the hidden layer to the output layer, there is also a feedback connection from the output layer to the hidden layer and from the hidden layer to itself. The reason to use a hybrid layer is that the robot’s dynamics consists of linear and nonlinear parts. The results show that the neural-network controller can efficiently control the prescribed positions of the stance and swing legs during the double stance phase of the gait cycle after sufficient training periods. The goal of the use of this proposed neural network is to increase the robustness of the control of the dynamic walking gait of this robot in the case of external disturbances. Also, the PD controller alone and Computed Torque Method (CTM) control system are used to control the walking robot’s position for comparison.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

弹簧刚度对被动步行的稳定性影响

由人类步行的生物力学研究得到启发,在被动双足步行机器人的髋关节处引入了扭簧,并通过仿真和试验研究了弹簧刚度对被动步行稳定性的影响.在仿真中,用胞映射方法计算被动步行机器人的吸引盆,并用吸引盆来衡量机器人的稳定性,研究了弹簧刚度对被动步行吸引盆大小的影响. 仿真结果表明, 吸引盆随着弹簧刚度的增大而增大. 在试验中,使机器人在各弹簧刚度参数下沿斜坡向下行走100次,记录下行走到头的次数作为稳定性的度量. 试验结果表明, 存在一个大小适中的弹簧刚度使机器人稳定性最大. 对弹簧提高机器人稳定性的原因进行了分析,对造成仿真与试验之间差异的原因进行了分析.      

Parametric resonance,stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model

This work examines dynamical behavior of a nonlinear oscillator with symmetric potential that models a quarter-car forced by the road profile under parametric excitation. The parametric resonance of a harmonically excited nonlinear quarter-car model with position and velocity time-delayed active control are investigated. We focus on the influence of delay and parametric excitation in the system. The influence of parametric excitation, time-delay and feedback gain parameters on the stability of the steady state response are investigated. By means of Melnikov's method, conditions for onset of chaos resulting from heteroclinic bifurcation is derived analytically and numerically.     

Treadmill walking of the pneumatic biped Lucy: Walking at different speeds and step-lengths

Actuators with adaptable compliance are gaining interest in the field of legged robotics due to their capability to store motion energy and to exploit the natural dynamics of the system to reduce energy consumption while walking and running. To perform research on compliant actuators we have built the planar biped Lucy. The robot has six actuated joints, the ankle, knee and hip of both legs with each joint powered by two pleated pneumatic artificial muscles in an antagonistic setup. This makes it possible to control both the torque and the stiffness of the joint. Such compliant actuators are used in passive walkers to overcome friction when walking over level ground and to improve stability. Typically, this kind of robots is only designed to walk with a constant walking speed and step-length, determined by the mechanical design of the mechanism and the properties of the ground. In this paper, we show that by an appropriate control, the robot Lucy is able to walk at different speeds and step-lengths and that adding and releasing weights does not affect the stability of the robot. To perform these experiments, an automated treadmill was built Published in Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 134–142, July 2008.     

Parametric random excitation of nonlinear coupled oscillators

The stochastic bifurcation and response statistics of nonlinear modal interaction under parametric random excitation are studied analytically, numerically and experimentally. Two basic definitions of stochastic bifurcation are first introduced. These are bifurcation in distribution and bifurcation in moments. bifurcation in moments is examined for the case of a coupled oscillator subjected to parametric filtered white noise. The center frequency of the excitation is selected to be close to either twice the first mode or second mode natural frequencies or the sum of the two. The stochastic bifurcation in moments is predicted using the Fokker-Planck equation together with gaussian and non-Gaussian closures and numerically using the Monte Carlo simulation. When one mode is parametrically excited it transfers energy to the other mode due to nonlinear modal interaction. The Gaussian closure solution gives close results to those predicted numerically only in regions well remote from bifurcation points. However, bifurcation points predicted by the non-Gaussian closure are in good agreement with those estimated by numerical simulation. Depending on the excitation level, the probability density of the excited mode is strongly non-Gaussian and exhibits multi-maxima as predicted by Monte Carlo simulation. Experimental tests are carried out at relatively low excitation levels. In the neighborhood of stochastic bifurcation in mean square the measured results exhibit different regimes of response characteristics including zero motion and occasional small random motion regimes. These two regimes are characterized by the phenomenon of on-off intermittency. Both regimes overlap and thus it is difficult to locate experimentally the bifurcation point.     

Nonlinear dynamic analysis for coupled vehicle-bridge system with harmonic excitation

In this paper, a multi-degree-of-freedom lumped parameter coupled vehicle-bridge dynamic model is proposed considering the nonlinearities of suspension and tire stiffness/damping and the nonlinear foundation of bridge. In terms of modelling, the continuous expressions of the kinetic energy, potential energy and the dissipation function are constructed. The dynamic equations of the coupled vehicle-bridge system (CVBS) are derived and discretized using Galerkin’s scheme, which yield a set of second-order nonlinear ordinary differential equations with coupled terms. The numerical simulations are conducted by using the Newmark-β integration method to perform a parametric study of the effects on excitation amplitude, suspension stiffness and position relation. The bifurcation diagram, 3-D frequency spectrum and largest Lyapunov exponent are demonstrated in order to better understand the vibration properties and interaction between the vehicle and bridge with the key system parameters. It can be found that the nonlinear dynamic characteristics such as parametric resonance, jump phenomena, periodic, quasi-periodic and chaotic motions are strongly attributed to the interaction between vehicle and bridge. Significantly, under the combined internal and external excitations, the vibration amplitudes of the CVBS have a certain degree of dependence on the external excitation. Suspension stiffness could lead to complex dynamics such as the higher-order bifurcations increase and the chaotic regions broaden. The increasing of distance could effectively control the nonlinear vibration of CVBS. The application of the proposed nonlinear coupled vehicle-bridge model would bring higher computational accuracy and make it possible to design the vehicle and bridge simultaneously.     

随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

变长度弹性伸缩腿双足机器人动力学与控制

研究了半被动双足机器人的平面稳定行走的控制问题．基于弹簧质点模型，采用拉格朗日方法分别得到双足机器人单支撑阶段与双支撑阶段的动力学方程，对机器人系统的动力学方程求得周期解．应用非线性系统状态反馈线性化理论，在双足机器人的单支撑阶段和双支撑阶段中，通过控制双足机器人的腿长度，实现稳定的周期行走．在理论分析的基础上，对控制算法进行了仿真与研究．结果表明：在周期行走过程中，文中采用的变长度控制算法可以使双足机器人克服外界的干扰，并具有较强的抗干扰性．     

Bifurcation and chaos of a simple walking model driven by a rhythmic signal

The walk of animals is achieved by the interaction between the dynamics of their mechanical system and the central pattern generator (CPG). In this paper, we analyze dynamic properties of a simple walking model of a biped robot driven by a rhythmic signal from an oscillator. In particular, we examine the long-term global behavior and the bifurcation of the motion that leads to chaotic motion, depending on the model parameter values. The simple model consists of a hip and two legs connected at the hip through a rotational joint. The joint is driven by a rhythmic signal from an oscillator, which is an open loop. In order to analyze the bifurcation, we first obtained approximate solutions of the walking motion and then constructed discrete dynamics using the Poincaré map. As a result, we found that consecutive period-doubling bifurcations occur as the model parameter values change, and that the walking motion leads to chaotic motion over the critical value of the model parameters. Moreover, we approximately obtained the period-doubling solutions and the critical value by employing a Newton-Raphson method. Our analytical results were verified by the numerical simulations.     

Random excitation of coupled oscillators with single and simultaneous internal resonances

The primary objective of this paper is to examine the random response characteristics of coupled nonlinear oscillators in the presence of single and simultaneous internal resonances. A model of two coupled beams with nonlinear inertia interaction is considered. The primary beam is directly excited by a random support motion, while the coupled beam is indirectly excited through autoparametric coupling and parametric excitation. For a single one-to-two internal resonance, we used Gaussian and non-Gaussian closures, Monte Carlo simulation, and experimental testing to predict and measure response statistics and stochastic bifurcation in the mean square. The mean square stability boundaries of the coupled beam equilibrium position are obtained by a Gaussian closure scheme. The stochastic bifurcation of the coupled beam is predicted theoretically and experimentally. The stochastic bifurcation predicted by non-Gaussian closure is found to take place at a lower excitation level than the one predicted by Gaussian closure and Monte Carlo simulation. It is also found that above a certain excitation level, the solution obtained by non-Gaussian closure reveals numerical instability at much lower excitation levels than those obtained by Gaussian and Monte Carlo approaches. The experimental observations reveal that the coupled beam does not reach a stationary state, as reflected by the time evolution of the mean square response. For the case of simultaneous internal resonances, both Gaussian and non-Gaussian closures fail to predict useful results, and attention is focused on Monte Carlo simulation and experimental testing. The effects of nonlinear coupling parameters, internal detuning ratios, and excitation spectral density level are considered in both investigations. It is found that both studies reveal common nonlinear features such as bifurcations in the mean square responses of the coupled beam and modal interaction in the neighborhood of internal resonances. Furthermore, there is an upper limit for the excitation level above which the system experiences unbounded response in the neighborhood of simultaneous internal resonances.     

Principal resonance of a nonlinear system with two-frequency parametric and self-excitations

The principal resonance of a single-degree-of-freedom system with two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. Qualitative analyses are employed to study the behaviour of steady state responses, limit cycle responses and 2-torus responses, including their stability and bifurcation. The effects of damping, detuning, and magnitudes of self-excitation and parametric excitations are analyzed. The theoretical analyses are verified by numerical integration results of the governing equation and the modulation equations.     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。