A 3D Model of One-Legged Hopping Machine

The paper models the spatial motion of a hopping machine with allowance for the inertial properties of its leg. A control system is synthesized and a program trajectory is determined in view of the collision processes accompanying touchdowns. The synthesis procedure is based on optimization of periodic systems. The results from numerical simulation of the hopper's nonstationary motion are demonstrative of the high efficiency of the control system     

A Control Algorithm for a 3D Hopping Machine

The paper addresses several control problems for a three-dimensional hopping machine. The machine consists of a ponderable torso and an imponderable leg. The leg is a kinematic chain with joints allowing rotation and translation. The leg is attached at the torso's center of mass. The force in the leg is developed by a linear spring. A control system is synthesized within the framework of a periodic linear quadratic problem. The efficiency of the controller is exemplified by a case where the hopper changes the direction of its motion     

单足机器人周期跳跃控制的虚拟约束方法

研究单足机器人周期跳跃控制问题.弹簧支撑倒立摆模型可以比较准确地描述动物的跳跃行为,但无控制的自然跳跃抗干扰能力较差,一般采用轨迹跟踪控制方法实现单足机器人周期跳跃.当系统存在比较大的误差时,传统的时间轨迹跟踪控制方法存在明显的不足.引入虚拟约束技术,采用基于空间路径跟踪的控制方法可以克服时间轨迹跟踪的不足.采用点足机器人模型,并通过控制腿伸缩的方式为系统提供动力,将跳跃过程分为地面摆动和腾空飞行两个阶段,并通过起飞和着陆两个事件完成两个阶段之间的转换,整个系统模型属于欠驱动非光滑动力学系统.根据简化的动力学方程获得系统的虚拟约束解析表达式,并采用部分反馈线性化方法结合PD控制设计系统的控制律.分析了系统的混合零动力学方程,并证明了闭环系统的临界稳定性.仿真结果表明,提出的控制方法可以实现单足机器人的周期跳跃控制,并且对外部干扰具有较强的鲁棒性.     

High-accuracy algorithms for solving the discrete-time periodic Riccati equation

The paper proposes an algorithm for solving the discrete-time periodic Riccati equation, which arises, for instance, in designing a stabilization system for a hopping robot. This algorithm does not have a number of constraints on the matrix coefficients peculiar to other algorithms. It is significant that variable-precision arithmetic can be used in numerical implementation of the algorithm. This makes it possible to solve discrete-time periodic Riccati equations with high accuracy. Some examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 98–105, September 2007.     

A Note on a Walking Machine Model

The paper continues studies intended to find out whether it is possible to create a prototype walking machine with relatively simple components. In this connection, the control problem is solved for a two-dimensional model of biped machine. It has a torso and two telescopic legs. Each leg includes a ponderable section of constant length and an imponderable section of variable length. The machine, regarded as a system with variable constraints, implements a single-stance gait (one stance leg at a time) with a step of constant duration. The contact of the swing leg with the ground is analyzed within the framework of Carnot's theorem (perfectly inelastic impact). It is assumed that the force developed in the stance leg is due to the deformation of the leg's spring and that this deformation can be controlled. An algorithm is proposed to synthesize a control system that takes into account collisions occurring at reverse of the roles of the legs. This algorithm is based on methods of optimizing periodic systems. The algorithm is compared with approaches used by other authors     

Hopping numerical approximations of the hyperbolic equation

Polynomial functions can be used to derive numerical schemes for an approximate solution of hyperbolic equations. A conventional derivation technique requires a polynomial to pass through every node values of a continuous computational stencil, leading to severe manifestation of the Gibbs phenomenon and strict time‐step limitation. To overcome the problem, this paper introduces polynomials that skip regularly (‘hop’ over) one or more nodes from the computational grid. Polynomials hopping over odd and even nodes yield a series of explicit numerical schemes of a required accuracy, with Lax–Friedrichs method being a particular simplest case. The schemes have two times wider stability interval compared to conventional continuous‐stencil explicit methods. Convex combinations of odd‐ and even‐node‐based updates improve further accuracy and stability of the method. Out of considered combinations (up to third‐order accuracy), derived odd‐order methods are stable for the Courant number ranging from 0 to 3, and even‐order ones from 0 to 5. A 2‐D extension of the hopping polynomial method exhibits similar properties. Copyright © 2007 John Wiley & Sons, Ltd.     

Stable Periodic Motion of a System Using Echo for Position Control

We study a system of equations which is based on Newton's law and models automatic position control by echo. Rewritten as a delay differential equation with state-dependent delay the model defines a semiflow on a submanifold in the space of continuously differentiable maps [-h, 0] ². All time-t maps and the restriction of the semiflow given by t > h are continuously differentiable. For simple nonlinearities and suitable parameters we find a set of initial data to which the solution curves return, after an excursion into the ambient manifold. The associated return map is semiconjugate to an interval map. Estimates of derivatives yield a unique, attracting fixed point of the latter, which can be lifted to the return map. The proof that the resulting periodic orbit is stable and exponentially attracting with asymptotic phase involves a Poincaré return map and a discussion of derivatives of its iterates.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

Stability of Nonstationary Automatic-Control Systems of Variable Structure in Forced Motion

Conditions of stability in sliding mode are obtained for nonstationary dynamic automatic-control processes of variable structure. The nonstationary parameters of a control process subjected to external perturbations vary in given ranges, whereas the parameters of the switching plane remain constant for the appropriately chosen parameters of discontinuous logic control laws. The corresponding system of differential equations contains coefficients that vary stepwise in time together with the parameters of the controlled process. The general stability criteria for the solutions of variable-structure systems are used to analyze the stability of motion in sliding mode of an automatic variable-structure control system of the second order     

结构试验机的神经网络内模自适应跟踪控制系统

针对结构试验系统的非线性和不确定性特性，研究了一种基于神经网络的非线性内模自适应加载控制方法。引入的神经网络内模可跟踪学习对象的时变动力学，控制器的设计较少依赖于对象的先验知识，控制参数的调整是基于被控过程的测量信息，利用导出的神经网络算法来实现的。实验结果证明该系统具有良好的控制效果。     

Technical Stability of Forced Motion in Nonstationary Automatic-Control Systems of Variable Structure

Sufficient conditions of technical stability are established for nonstationary automatic-control systems of variable structure. The case is considered where for appropriate controls and for all admissible initial perturbations from a predefined measurable set of initial states, the nonstationary parameters of a process under external actions vary within given ranges. The control process depends on the mismatch signal and its derivatives up to an order that is less by one than the order of the equations of motion. The associated system of differential equations contains discontinuous time-dependent coefficients. A relationship is established between the eigenvalues of the quadratic forms of the given Lyapunov functions and the technical-stability criteria. It is shown that the technical-stability conditions do not depend on the sliding-mode conditions for variable-structure control processes     

Longitudinal free vibration of inhomogeneous elastic straight strut with a variable cross section

The scope of the present article, motivated by the case of the composite wooden propeller of an airplane, is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar with a more general mathematical treatment.In this analysis, we have assigned to the modulus of elasticity, the bar cross section as well as the mass per unit length of the bar an exponential function variation, and then found a general solution, wherein three parameters were considered as the main factors to affect the longitudinal free vibration of the inhomogeneous elastic straight bar with a variable cross section.     

The solution for the generalized riccati algebraic equations of linear equality constraint system

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper. Project supported by the National Natural Science Youth Foundation of China     

Damping the vibrations of a rectangular plate with piezoelectric actuators

The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM (Ⅰ)

Based on the mathematical model of a kind of complicated financial system, all possible things that the model shows in the operation of our country’s macro-financial system were analyzed, such as balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcations,and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior were analyzed. This study will deepen people’s understanding of the lever function of all kinds of financial policies.     

碰摩裂纹转子轴承系统的周期运动稳定性及实验研究

根据碰摩裂纹耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶法，研究了系统周期运动的稳定性。研究发现，小偏心量下系统周期运动发生Hopf分岔，大偏心量下系统周期运动发生倍周期分岔，偏心量的加大使周期解的稳定性明显降低；系统碰摩间隙变小，碰摩影响了油膜涡动的形成，使失稳转速有所提高；裂纹深度的加大降低了系统周期运动的稳定性。本文的研究为转子轴承系统的安全稳定运行提供了理论参考。     

Unsteady Motion of a Maxwellian Fluid Droplet in a Maxwellian Medium under the Action of Monotonic and Periodic Forces

Unsteady motion of a Maxwellian fluid droplet, which arises in a quiescent Maxwellian medium under the action of monotonic and periodic forces, is considered. In the initial period of time smaller than the relaxation time, the droplet is affected by elastic forces on the part of the fluid; moreover, the droplet itself is a viscoelastic material. A solution of the problem in the first approximation is found. The dependence of the amplitude of droplet velocity and the shift of the phase of oscillations on the relaxation time of the external and internal media and also on the frequency of oscillations of the driving force is analyzed. The passage to the limit in terms of density and viscosity of the internal medium is performed.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 55–65, July–August, 2005.     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅱ)

Based on the work discussed on the former study, this article first starts from the mathematical model of a kind of complicated financial system, and analyses all possible things that the model shows in the operation of our country’s macro-financial system: balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcation, and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior have been analyzed. This study will deepen people’s understanding of the lever function of all kinds of financial policies.     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

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Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b₀|≠1 instead of the condtion |b₀|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption. Foundation item: the Natural Science Foundation of Yunnan Province, China (97A012G)