DYNAMICS MODELING,SIMULATION, AND CONTROL OF ROBOTS WITH FLEXIBLE JOINTS AND FLEXIBLE LINKS$^{\bf 1)}$

本文探究了铰柔性对机器人动力学响应和动力学控制的影响. 首先, 建立由$n$个柔性铰和$n$个柔性杆组成的空间机器人模型, 运用递推拉格朗日动力学方法, 得到柔性机器人系统的刚柔耦合动力学方程. 在动力学建模过程中, 除了考虑杆件的拉伸变形、弯曲变形、扭转变形以及非线性耦合变形对机器人系统动力学行为的影响, 还考虑了铰的柔性对机器人动力学响应和控制的影响. 其中, 柔性铰模型是基于Spong的柔性关节简化模型, 将柔性铰看成线性扭转弹簧, 不仅考虑了铰阻尼的存在, 还考虑了柔性铰的质量效应. 其次, 编写了空间柔性铰柔性杆机器人仿真程序, 研究铰的刚度系数和阻尼系数对系统动力学响应的影响. 研究表明: 随着柔性铰刚度系数的增大, 柔性机器人的动态响应幅值减小, 振动频率变大. 随着柔性铰阻尼系数的增大, 柔性机器人的动态响应幅值减小, 振动幅值的衰减速度变快. 可通过调节柔性铰的刚度和阻尼来减小柔性铰柔性杆机器人的振动, 因此铰阻尼的研究具有重要工程意义. 最后, 研究了铰柔性在机器人系统动力学控制中的影响. 在刚性铰机械臂和柔性铰机械臂完成相同圆周运动时, 通过逆动力学方法求解得到两种情况下的关节驱动力矩. 研究表明: 引入柔性铰会使控制所需的驱动力矩变小, 对机器人控制的影响显著.     

Vibration control for a flexible-link robot arm with deflection feedback

The use of flexible links in a robot inevitably causes the elastic deflection and vibration of the endpoint of the robot during high-speed operations. The deflection and vibration will tend to degrade the positioning performance of the robot. In this paper, an optical sensing system consisting of a laser diode and a position sensitive detector is introduced for the real-time measurement of the dynamic deflection. Utilising a non-linear, coupled and measurement-based dynamic system model, a Lyapunov-type controller based on the deflection feedback is then proposed to damp out the tip oscillations and regulate the endpoint of the flexible robot. Experimental tests are conducted for a flexible one-link robot arm with a payload mass at the tip. The results demonstrate the effectiveness of the proposed measuring and control schemes.     

Non-linear dynamic stability of damped Beck's column with variable cross-section

In this paper the non-linear dynamic stability of Beck's column with variable mass and stiffness properties in the presence of damping (both internal and external) is investigated using a complete non-linear dynamic analysis. This approach permits the examination of the global stability of the system in contrast to the static non-linear one, which, though more economical in computational cost, is associated only with the loss of local stability via flutter or divergence. The governing equations describing the dynamic response are derived in terms of the displacements taking also into account the axial deformation, which has a striking influence on the critical load. Since the cross-sectional properties of the beam vary along its axis, the resulting coupled non-linear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a FEM solution, which may experience lack of convergence. Interesting conclusions are drawn. The important, however, finding is that the inclusion of the axial deformation affects highly the critical load of Beck's column with varying cross-sectional properties, while it leaves it unaltered for Beck's column with uniform cross-section.     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

动态规划求解空间双臂机器人非完整运动最优控制问题

通过自适应动态规划研究自由漂浮空间双臂机器人运动的最优控制问题．针对空间双臂机器人的非完整性运动,采用自适应动态规划(Adaptive Dynamic Programming, ADP)方法求解其最优控制问题．根据多体动力学理论,推导了载体位置、姿态均无控制条件下,双臂空间机器人满足的系统动量守恒关系的非完整约束方程,并将其转化为控制系统的状态方程,从而将双臂空间机器人的非完整运动规划问题转化为对非线性系统的控制问题．文中根据自适应动态规划网络结构,利用神经网络来近似性能指标函数,进而用龙格库塔法求解状态变量．并给出了适合该类问题的一种效用函数具体表达式,保证了空间双臂机器人到达期望位置后不再继续运动．实现了对空间双臂机器人非完整运动规划的最优控制．数值仿真实验验证了ADP对求解空间双臂机器人非完整运动规划最优控制问题的有效性．     

考虑关节摩擦的空间机器人动力学建模与参数辨识

研究了考虑关节摩擦影响的空间机器人系统的动力学建模与参数辨识问题．采用单向递推组集方法和虚功率原理建立了含有关节摩擦的多体系统动力学方程,推导了关节摩擦对系统动力学方程的贡献,采用基于腕力传感器信号和最小二乘法的辨识方法进行了系统惯性参数的辨识．数值仿真结果验证了数学模型的正确性与辨识方法的有效性．     

Control of the spatial motions of a gantry manipulator with elastic links

A mathematical dynamic model is proposed for a controlled gantry robot with elastic compliance and inertia distributed along a two-link arm. The model includes a nonlinear system of hybrid differential equations. Kinematic and dynamic control problems for the robot are formulated. The dynamic characteristics of the robot are analyzed in comparison with an equivalent model of a robot manipulator with rigid links based on the Lagrangian formalism __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 121–128, February 2006.     

The Effect of the Elastic Compliance of Actuator Components on the Dynamics of a Robot

Kinematic and dynamic control problems for a pedestal-mounted robot with a multilink arm are formulated. The robot is considered a system of perfectly rigid bodies controlled by a combined actuating system. The mathematical model of robot dynamics accounts for the elastic properties of actuator components based on the formalism of Lagrange equations of the second kind. The effect of the elastic compliance of the actuator components on the dynamics of manipulator links and actuator motors is discussed. A robot with a two-link arm is considered as an example     

BUFFER AND COMPLIANT DYNAMIC SURFACE CONTROL OF SPACE ROBOT CAPTURING SATELLITE BASED ON COMPLIANT MECHANISM$^{\bf 1)}$

研究了空间机器人在轨捕获非合作卫星过程避免关节受碰撞冲击破坏的缓冲从顺控制问题, 为此在机械臂与关节电机之间配置了一种柔性机构, 其作用在于: (1)在接触、碰撞阶段可通过其内置弹簧的变形来吸收被捕获卫星对空间机器人关节产生的冲击力矩; (2)在镇定运动阶段, 结合与之配合的缓冲从顺控制策略来适时开、关关节电机, 以保证关节受到的冲击力矩受限在安全范围. 首先, 利用多刚体系统理论获得配置柔性机构空间机器人及目标卫星分体系统动力学方程; 之后, 结合整个系统动量守恒关系, 捕获操作后系统运动几何关系及力的传递规律, 建立了两者形成联合体系统的动力学方程, 并计算了碰撞过程的冲击效应与冲击力. 为了实现失稳联合体系统的镇定控制, 提出了一种基于动态面的缓冲从顺控制方案. 上述控制方案可在实现吸收捕获操作产生的冲击力矩的同时, 还能在冲击力矩过大时适时开启、关闭关节电机, 以避免关节电机发生破坏; 此外, 动态面的引入避免了反演法存在的计算膨胀问题, 有效减少了计算量. 基于Lyapunov函数法证明了系统的稳定性, 并通过系统数值仿真结果验证了上述缓冲从顺控制策略的正确性.     

3-PRS并联机器人惯量耦合特性研究

惯量是影响机器人动态性能的主要因素,并联机器人因其多支链耦合的结构特点,关节空间各驱动轴出现惯量耦合的动力学特性,在高速、高加速度运动时易引起控制超调、振动等现象,破坏机器人的动态性能,因此研究并联机器人惯量耦合特性具有重要意义. 以3-PRS 并联机器人为例,通过虚功原理求得惯量矩阵,提出惯量耦合指标,该耦合指标表征了并联机器人在工作空间不同位姿时各驱动轴的耦合惯量大小,并给出了该耦合指标在机器人工作空间内的分布规律. 进一步在一台3-PRS 并联机器人样机上进行了实验验证,结果表明耦合惯量会改变驱动轴负载,负载的改变将最终影响动态性能. 同时各驱动轴的负载变化量随着惯量耦合指标的变大而变大,与理论分析有较好的一致性. 研究成果可帮助评价并联机器人的动力学耦合特性,并可用于并联机器人的结构参数优化及伺服参数调试以提高机器人的动态性能.      

双臂空间机器人相对载体运动的增广自适应控制方法

本文讨论了载体位置、姿态均不受控制情况下,漂浮基双臂空间机器人末端爪手基联坐标系内轨迹跟踪的自适应控制问题.系统运动学、动力学分析结果表明,结合系统动量守恒关系得到的双臂空间机器人系统动力学方程,可以表示为一组新的适当选择的系统组合惯性参数的线性函数;从而克服了惯常的空间机器入系统动力学方程,关于系统惯性参数呈非线性函数关系的难点.以此为基础,并借助于增广变量法,针对双臂空间机器人末端爪手所持载荷参数未知的情况,设计了漂浮基双臂空间机器人末端爪手基联坐标系内轨迹跟踪的增广自适应控制方案.提出的控制方案具有不需要反馈、测量漂浮基的位置、移动速度及移动加速度的显著优点.系统数值仿真,证实了方法的有效性.     

柔性铰柔性杆机器人动力学建模、仿真和控制

本文探究了铰柔性对机器人动力学响应和动力学控制的影响. 首先, 建立由$n$个柔性铰和$n$个柔性杆组成的空间机器人模型, 运用递推拉格朗日动力学方法, 得到柔性机器人系统的刚柔耦合动力学方程. 在动力学建模过程中, 除了考虑杆件的拉伸变形、弯曲变形、扭转变形以及非线性耦合变形对机器人系统动力学行为的影响, 还考虑了铰的柔性对机器人动力学响应和控制的影响. 其中, 柔性铰模型是基于Spong的柔性关节简化模型, 将柔性铰看成线性扭转弹簧, 不仅考虑了铰阻尼的存在, 还考虑了柔性铰的质量效应. 其次, 编写了空间柔性铰柔性杆机器人仿真程序, 研究铰的刚度系数和阻尼系数对系统动力学响应的影响. 研究表明: 随着柔性铰刚度系数的增大, 柔性机器人的动态响应幅值减小, 振动频率变大. 随着柔性铰阻尼系数的增大, 柔性机器人的动态响应幅值减小, 振动幅值的衰减速度变快. 可通过调节柔性铰的刚度和阻尼来减小柔性铰柔性杆机器人的振动, 因此铰阻尼的研究具有重要工程意义. 最后, 研究了铰柔性在机器人系统动力学控制中的影响. 在刚性铰机械臂和柔性铰机械臂完成相同圆周运动时, 通过逆动力学方法求解得到两种情况下的关节驱动力矩. 研究表明: 引入柔性铰会使控制所需的驱动力矩变小, 对机器人控制的影响显著.      

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

Modal test and analysis of cantilever beam with tip mass

The phenomenon of dynamic stiffening is a research field of general interest for flexible multi-body systems. In fact, there are not only dynamic stiffening but also dynamic softening phenomenon in the flexible multi-body systems. In this paper, a non-linear dynamic model and its linearization characteristic equations of a cantilever beam with tip mass in the centrifugal field are established by adopting the general Hamilton Variational Principle. Then, the problems of the dynamic stiffening and the dynamic softening are studied by using numerical simulations. Meanwhile, the modal test is carried out on our centrifuge. The numerical results show that the system stiffness will be strengthened when the centrifugal tension force acts on the beam (i.e. the dynamic stiffening). However, the system stiffness will be weakened when the centrifugal compression force acts on the beam (i.e. the dynamic softening). Furthermore, the equilibrium position of the system will lose its stability when the inertial force reaches a critical value. Through theoretical analysis, we find that this phenomenon comes from the effect of dynamic softening resulting from the centrifugal compression force. Our test results verify the above conclusions and confirm that both dynamic stiffening and softening phenomena exist in flexible multi-body systems. The project supported by the National Natural Science Foundation of China (19972002) and the Doctoral Programme from The State Education Commission China (20010001011)     

Dynamics of a Robot Manipulator with Elastic Links

The problems of dynamic and kinematic control of a multilink robot manipulator are formulated. The distributed elasticity and inertia of the manipulator links are taken into account on the basis of the Timoshenko-beam model on the assumption that each of its elements moves complexly. The results obtained here are compared with data on the dynamic characteristics of a manipulator with rigid links and a manipulator with elastic links, whose elastic properties were simulated on the basis of the Euler–Bernoulli-beam model     

Input torque sensitivity to uncertain parameters in biped robot

Input torque is the main power to maintain bipedal walking of robot, and can be calculated from trajectory planning and dynamic modeling on biped robot. During bipedal walking, the input torque is usually required to be adjusted due to some uncertain parameters arising from objective or subjective factors in the dynamical model to maintain the pre-planned stable trajectory. Here, a planar 5-link biped robot is used as an illustrating example to investigate the effects of uncertain parameters on the input torques. Kine-matic equations of the biped robot are firstly established by the third-order spline curves based on the trajectory planning method, and the dynamic modeling is accomplished by taking both the certain and uncertain parameters into account. Next, several evaluation indices on input torques are intro-duced to perform sensitivity analysis of the input torque with respect to the uncertain parameters. Finally, based on the Monte Carlo simulation, the values of evaluation indices on input torques are presented, from which all the robot param-eters are classified into three categories, i.e., strongly sensi-tive, sensitive and almost insensitive parameters.     

A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems

This paper presents a dynamic behaviour study of non-linear friction systems subject to uncertain friction laws. The main aspects are the analysis of the stability and the associated non-linear amplitude around the steady-state equilibrium. As friction systems are highly sensitive to the dispersion of friction laws, it is necessary to take into account the uncertainty of the friction coefficient to obtain stability intervals and to estimate the extreme magnitudes of oscillations. Intrusive and non-intrusive methods based on the polynomial chaos theory are proposed to tackle these problems. The efficiency of these methods is investigated in a two degree of freedom system representing a drum brake system. The proposed methods prove to be interesting alternatives to the classic methods such as parametric studies and Monte Carlo based techniques.     

旋转倒立双摆摆起控制量的动态设计变量优化

倒立摆的摆起过程是复杂的非线性力学与控制的综合问题,当力学模型和参数确定时,寻求有效方法计算控制律函数是实现摆起的关键。本文以倒立双摆模型为研究对象,将动态设计变量优化方法引入对摆起开环控制律的分析,编制计算机程序进行数值仿真,确定了控制律函数与状态量随时间变化的状态,为复杂强非线性系统的控制提出了一种有效方法。     

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．     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.