Investigation on effect of joint clearance on dynamics of four-bar mechanism

As a result of design, manufacturing and assembly processes or a wear effect, clearances are inevitable at the joints of mechanisms. In this study, dynamic response of mechanism having revolute joints with clearance is investigated. A four-bar mechanism having two joints with clearance is considered as a model mechanism. A neural network was used to model several characteristics of joint clearance. Kinematic and dynamic analyses were achieved using continuous contact mode between journal and bearing. A genetic algorithm was also used to determine the appropriate values of design variables for reducing the additional vibration effect due primarily to the joint clearance. The results show that the optimal adjusting of suitable design variables gives a certain decrease in shaking forces and their moments on the mechanism frame.     

Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation

The dynamic modeling and simulation of spatial rigid-multibody systems with lubricated spherical joints is the main purpose of the present work. This issue is of paramount importance in the analysis and design of realistic multibody mechanical systems undergoing spatial motion. When the spherical clearance joint is modeled as dry contact; i.e., when there is no lubricant between the mechanical elements which constitute the joint, a body-to-body (typically metal-to-metal) contact takes place. The joint reaction forces in this case are evaluated through a Hertzian-based contact law. A hysteretic damping factor is included in the dry contact force model to account for the energy dissipation during the contact process. The presence of a fluid lubricant avoids the direct metal-to-metal contact. In this situation, the squeeze film action, due to the relative approaching motion between the mechanical joint elements, is considered utilizing the lubrication theory associated with the spherical bearings. In both cases, the intra-joint reaction forces are evaluated as functions of the geometrical, kinematical, and physical characteristics of the spherical joint. These forces are then incorporated into a standard formulation of the system’s governing equations of motion as generalized external forces. A spatial four bar mechanism that includes a spherical clearance joint is considered here as an example. The computational simulations are carried out with and without the fluid lubricant, and the results are compared with those obtained when the system is modeled with perfect joints only. From the general results, it is observed that the system’s performance with lubricant effect presents fewer peaks in the kinematic and dynamic outputs, when compared with those from the dry contact joint model.     

Modeling joints with clearance and friction in multibody dynamic simulation of automotive differentials

Defects in kinematic joints can sometimes highly influence the simulation response of the whole multibody system within which these joints are included. For instance, the clearance, the friction, the lubrication and the flexibility affect the transient behaviour, reduce the component life and produce noise and vibration for classical joints such as prismatics, cylindrics or universal joints. In this work, a new 3D cylindrical joint model which accounts for the clearance, the misalignment and the friction is presented. This formulation has been used to represent the link between the planet gears and the planet carrier in an automotive differential model.     

Modeling 3D revolute joint with clearance and contact stiffness

The clearances in the kinematic joints are due to deformations, wear, and manufacturing errors; the accurate modeling of these effects in multibody analysis is a complex issue but in many practical applications, it is mandatory to take into them into account in order to understand the actual behavior of mechanical systems. In this paper, the authors present a general computer-aided model of a 3D revolute joint with clearance suitable for implementation in multibody dynamic solvers. While a perfect revolute joint imposes kinematic constraints, the proposed revolute joint with clearance leads to a force constraint. The revolute joint has been modeled by introducing a nonlinear equivalent force system, which takes into account the contact elastic deformations. The model depends on the structural and geometrical properties of materials in contact that have been investigated using finite element models. The purpose is to give a general approach to study the influence of actual joints on kinematic, dynamic, and structural behavior of mechanisms. The proposed model has been applied in dynamic simulations of a spatial slider-crank mechanism.     

多体系统动力学中关节效应模型的研究进展

在一般的多体系统动力学研究中认为运动关节是理想运动副. 然而,实际中的运动关节不仅含有间隙与摩擦,还有间隙引起的关节元素之间的接触碰撞、局部变形和磨损. 多体系统动力学中的关节效应不仅引起了系统的振动和噪声,减小了系统的可靠性和寿命,而且损失了系统的精度和稳定性. 为此,对近十几年多体系统动力学中关节效应的研究进行了详细分析,总结了关节效应中间隙运动学模型、接触力模型与磨损模型在多体系统动力学中的建模过程. 其中,着重分析了多体系统动力学中关节磨损效应的研究进展,并对常用的Reye'shypothesis 和Archard 磨损模型进行了比较,详细地分析了Archard 磨损模型的演变形式以及主要磨损参数(接触应力,接触面积和滑移距离),特别分析了关键磨损参数接触应力的建模方法,解释了基于Winkler 弹性基础理论在求解接触应力时遇到的困难. 另外,介绍了4 种间隙运动副(转动副、移动副、圆柱副和球面副) 的运动学模型. 分析了考虑关节磨损多体系统动力学模型的一般建模方法,并以平面五杆机构为例说明了其建模过程.最后,简要地展望了多体系统动力学中关节效应模型的发展趋势以及应用前景.      

Modeling and analysis of sliding joints with clearances in flexible multibody systems

The modeling of the sliding joint with clearance between a flexible beam and a rigid hole is investigated in this paper. The flexible beam is discretized using the three-dimensional curved Euler–Bernoulli beam element of the Absolute Nodal Coordinate Formulation, while the motion of the rigid hole is described by the Cartesian coordinates. Moreover, the cross sections of both the flexible beam and the rigid hole are assumed to be circular. The existing joints with clearances are mainly rigid joints with small clearances, and the contact detection algorithm adopted can solve only one pair of potential contact points within one section. In order to model the contact problem in the sliding joint with clearance, a new contact detection method based on the intersection of the rigid hole’s cross section and the flexible beam is proposed, which yields a two-dimensional contact detection problem. Based on the common-normal concept, the ellipse–circle contact detection problem within the hole’s cross section can be solved. The potential contact point on the hole’s cross section will be determined, and the closest point projection on the beam’s neutral axis can be defined further. The proposed contact detection method can deal with the sliding joint with large clearance and the multiple-point contact problem within one section. In addition, the penalty method is adopted to model the frictionless contact between the flexible beam and the rigid hole. Finally, two numerical examples about sliding joints with clearances, one with an initially curved beam under gravity and the other with a straight beam under zero gravity, are presented to demonstrate the influence of the clearance of sliding joint on the dynamic performance of flexible multibody systems.     

Optimal free-defect synthesis of four-bar linkage with joint clearance using PSO algorithm

This paper presents the design of planar four-bar linkages free of order, branch and circuit defects, for the purpose of path generation, having clearances at one, two, three or all of its joints. Joint clearance is treated as a massless virtual link and its direction is known by the direction of the joint force. A Particle Swarm Optimization based algorithm is given here to solve this highly nonlinear optimization problem with some constraints, namely; the Grashof’s and free of the foregoing defects conditions. An example is included in which the optimal problem is solved for different cases; namely planar four-bar linkage having clearances at one, two, three, all of the joints and without clearance. For all the designs, the generated paths, the errors and the directions of the virtual links are plotted and are compared. Finally, we compare the optimal designs with reality.     

A validated model for a pin-slot clearance joint

The numerical modeling of joints with a certain amount of clearance and a subsequent validation of the model are important for accurate multibody simulations. For such validated modeling, not only the kinematic constraints, but also the contact models, are important. If a joint has no clearance, it is assumed to be ideal. However, in real applications, there is frequently some clearance in the joints. Adding clearance and kinematic conditions to a pin-slot joint significantly increases the number of kinematic and contact parameters. Consequently, the resulting kinematics and the contact forces can vary significantly with regard to the selection of those parameters. This research covers the development of a validated model for a pin-slot clearance joint. Different kinematic constraints and contact models are discussed. The presented model is an experimentally validated one for a pin-slot clearance joint that is commonly used in safety-critical applications like electrical circuit breakers. Special attention is given to the Hertz, Kelvin–Voigt, Johnson, and Lankarani–Nikravesh contact models. When comparing different contact models within numerical approaches and comparing the results with experimental data, significant differences in the results were observed. With a validated model of a pin-slot clearance joint, a physically consistent numerical simulation was obtained.     

基于干扰观测器的柔性关节空间机器人退步自适应控制

研究了载体位置、姿态均不受控的情况下,系统参数不确定的柔性关节空间机器人轨迹跟踪的控制问题.结合系统动量、动量矩守恒关系,利用拉格朗日法推导出系统的动力学模型.为减小系统柔性关节对系统控制精度的影响,采用关节柔性补偿器来等效降低系统关节的柔性.再借助奇异摄动法,针对系统参数不确定的情况,设计了柔性关节空间机器人基于干扰观测器的退步自适应滑模控制方案.该方案不需要对系统惯性参数进行线性化处理,控制器结构简单,且实现了空间机器人期望轨迹的精确跟踪控制.通过平面两杆空间机器人的数值仿真证明了该方法的有效性.     

柔性关节-柔性臂空间机器人的神经网络自适应反演控制及双重柔性振动抑制

空间机器人系统的柔性主要体现在空间机器人的臂杆和连接各臂杆之间的铰关节。由于空间机器人系统结构的复杂性,以往研究人员对同时具有柔性关节和柔性臂的系统关注不够。为此探讨了参数未知柔性关节-柔性臂空间机器人系统的动力学模拟、轨迹跟踪控制算法设计和关节、臂杆双重柔性振动的主动抑制问题。首先,采用多体动力学建模方法并结合漂浮基空间机器人固有的线动量和角动量守恒动力学特性,推导了系统的动力学方程。以此为基础,考虑到空间机器人实际应用中各关节铰具有较强柔性的情况,引入一种关节柔性补偿控制器解决了传统奇异摄动法应用受关节柔性限制问题,导出了适用于控制系统算法设计的数学模型。然后,利用该模型,基于反演思想在慢时标子系统中设计神经网络自适应控制算法来补偿系统参数未知和柔性关节引起的转动误差,实现系统运动轨迹跟踪性能;针对快时标子系统,设计了鲁棒最优控制算法抑制因柔性关节及柔性臂引起的系统双重弹性振动,保证系统的稳定性。最后,通过仿真对比实验验证了所设计控制算法的有效性。     

应用Winkler弹性基础模型的间隙铰接副磨损预测

间隙铰接副磨损与机构动力学之间存在交互耦合作用.通过Winkler弹性基础模型既可表达铰接副共形接触反力用于构建多体力学方程,又可得到界面接触压力分布用于磨损计算,从而可获得较Hertz理论对共形接触问题更好的计算准确度,且避免有限元方法计算接触压力分布导致的计算耗时性.以曲柄滑块机构为例的分析和实验结果表明:虽然有限元方法可得到更高的磨损预测精度,但Winkler模型对微间隙铰接副磨损预测具有可接受的精度和更好的计算效率,从而可为含间隙铰接副机械系统摩擦学设计提供简便的算法.     

Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches

The main objective of this work is to present a computational and experimental study on the contact forces developed in revolute clearance joints. For this purpose, a well-known slider-crank mechanism with a revolute clearance joint between the connecting rod and slider is utilized. The intra-joint contact forces that are generated at these clearance joints are computed by considering several different elastic and dissipative approaches, namely those based on the Hertz contact theory and the ESDU tribology-based cylindrical contacts, along with a hysteresis-type dissipative damping. The normal contact force is augmented with the dry Coulomb’s friction force. In addition, an experimental apparatus is used to obtained some experimental data in order to verify and validate the computational models. From the outcomes reported in this paper, it is concluded that the selection of the appropriate contact force model with proper dissipative damping plays a significant role in the dynamic response of mechanical systems involving contact events at low or moderate impact velocities.     

A nonlinear contact pressure distribution model for wear calculation of planar revolute joint with clearance

As various errors result from manufacture and assembly processes or wear effect, clearance joint widely exists in mechanical system as a base component. The coupling analysis of tribology and dynamics of clearance joint is important to the reliability of mechanical system. A nonlinear contact pressure distribution mode (NLCP) is proposed to combine dynamics analysis with wear calculation together in this paper. The discrete thought of Winkler model is adopted to deal with contact problem with a high conformal rate. The contact relationship in a local microcontact area can be regarded as the contact between cylinder and plane. And the local contact pressure is acquired based on Hertz contact theory. The NLCP model has not only described the nonlinear relationship between contact pressure and penetration depth, but also avoided the complexity in contact pressure computation. The performance of NLCP model is demonstrated in comparison with asymmetric Winkler model, revealing that NLCP model has enhanced the calculation accuracy with a good efficiency. A comprehensive experimental study on the wear calculation of a slider–crank mechanism with clearance joint is presented and discussed to provide an experimental verification for NLCP model. The paper’s work has solved the contact problem with a high conformal rate and has described the nonlinear relationship between contact pressure and penetration depth. It has great value to the wear analysis of clearance joint.     

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.     

全柔性空间机器人运动振动一体化输入受限重复学习控制

探究基座、臂、关节全柔性影响下空间机器人动力学模拟、运动控制及基座、臂、关节三重柔性振动主动抑制的问题, 设计了不基于系统模型信息的运动振动一体化输入受限重复学习控制算法. 将柔性基座与关节等效为线性弹簧与扭转弹簧, 柔性臂视为欧拉-伯努利梁模型, 利用拉格朗日方程与假设模态法建立动力学模型, 然后, 用奇异摄动理论将模型分解为包含刚性变量与臂柔性振动的慢变子系统, 包含基座、关节柔性振动的快变子系统, 并分别设计相应的子控制器, 构成了带关节柔性补偿的一体化控制算法. 针对慢变子系统, 提出输入受限重复学习控制算法, 由双曲正切函数, 饱和函数与重复学习项构成, 双曲正切函数与饱和函数实现输入力矩受限要求, 重复学习项补偿周期性系统误差, 以完成对基座姿态、关节铰周期轨迹的渐进稳定追踪. 然而, 为了同时抑制慢变子系统臂的柔性振动, 运用虚拟力的概念, 构造同时反映臂柔性振动与系统刚性运动的混合轨迹, 提出了基于虚拟力概念的输入受限重复学习控制器, 保证基座、关节轨迹精确追踪的同时, 对臂的柔性振动主动抑制. 针对快变子系统, 采用线性二次最优控制算法抑制基座与关节的柔性振动. 仿真结果表明: 控制器适用于一般柔性非线性系统, 满足输入力矩受限要求, 实现对周期信号的高精度追踪, 有效抑制基座、臂、关节的柔性振动, 证实算法的可行性.      

Design and kinematic optimization of a novel underactuated robotic hand exoskeleton

This study presents the design and the kinematic optimization of a novel, underactuated, linkage-based robotic hand exoskeleton to assist users performing grasping tasks. The device has been designed to apply only normal forces to the finger phalanges during flexion/extension of the fingers, while providing automatic adaptability for different finger sizes. Thus, the easiness of the attachment to the user’s fingers and better comfort have been ensured. The analyses of the device kinematic pose, statics and stability of grasp have been performed. These analyses have been used to optimize the link lengths of the mechanism, ensuring that a reasonable range of motion is satisfied while maximizing the force transmission on the finger joints. Finally, the usability of a prototype with multiple fingers has been tested during grasping tasks with different objects.     

Sensitivity analysis of the equal angle divider mechanism kinematics with the synthesis of the joint gap tolerances

Tolerance analysis of mechanical components and assemblies is a key element in the industry for improving product quality and reducing overall cost. This paper is an attempt to propose a methodology to formulate and compute the kinematic assembly variations with respect to the critical dimensions (joint clearances) by calculating the homogeneous matrix of loops with considering the transforming and rotation matrix, the tolerance sensitivities, the geometrical responses to assembly variations, and the tolerance sensitivity of mechanism for clearance in the joints in different motion phases. The analysis is performed by the systematic methodology which is extended for kinematic sensitivity and positional error analysis with published results.     

Design and analysis of a compliant parallel pan-tilt platform

In combination of the advantages of both parallel mechanisms and compliant mechanisms, a compliant parallel mechanism with two rotational degrees of freedom is designed to meet the requirement of a lightweight and compact pan-tilt platform. Firstly, two commonly-used design methods i.e. direct substitution and Freedom and Constraint Topology are applied to design the configuration of the pan-tilt system, and similarities and differences of the two design alternatives are compared. Then inverse kinematic analysis of the candidate mechanism is implemented by using the pseudo-rigid-body model, and the Jacobian related to its differential kinematics is further derived to help designer realize dynamic analysis of the 8R compliant mechanism. In addition, the mechanism’s maximum stress existing within its workspace is tested by finite element analysis. Finally, a method to determine joint damping of the flexure hinge is presented, which aims at exploring the effect of joint damping on actuator selection and real-time control. To the authors’ knowledge, almost no existing literature concerns with this issue.     

Dynamics analysis of 2-DOF complex planar mechanical system with joint clearance and flexible links

Joint clearance and flexible links are two important factors that affect the dynamic behaviors of planar mechanical system. Traditional dynamics studies of planar mechanism rarely take into account both influence of revolute clearance and flexible links, which results in lower accuracy. And many dynamics studies mainly focus on simple mechanism with clearance, the study of complex mechanism with clearance is a few. In order to study dynamic behaviors of two-degree-of-freedom (DOF) complex planar mechanical system more precisely, the dynamic analyses of the mechanical system with joint clearance and flexibility of links are studied in this work. Nonlinear dynamic model of the 2-DOF nine-bar mechanical system with revolute clearance and flexible links is built by Lagrange and finite element method (FEM). Normal and tangential force of the clearance joint is built by the Lankarani–Nikravesh and modified Coulomb’s friction models. The influences of clearance value and driving velocity of the crank on the dynamic behaviors are researched, including motion responses of slider, contact force, driving torques of cranks, penetration depth, shaft center trajectory, Phase diagram, Lyapunov exponents and Poincaré map of clearance joint and slider are both analyzed, respectively. Bifurcation diagrams under different clearance values and different driving velocities of cranks are also investigated. The results show that clearance joint and flexibility of links have a certain impact on dynamic behavior of mechanism, and flexible links can partly decrease dynamic response of the mechanical system with clearance relative to rigid mechanical system with clearance.     

机器人关节非线性摩擦模型关键参数反求

机器人关节非线性摩擦的准确描述对提高机器人轨迹精度、定位精度及其可靠性等具有重要理论意义和科学价值. 然而, 机器人关节通常包含电机、减速器、驱动器和传感器, 是一个复杂的机电耦合系统, 随服役时间及工况的变化, 机器人关节的摩擦参数也存在显著时变效应, 难以准确描述, 造成轨迹精度下降, 为机器人后期精度维护造成巨大困难. 因此, 本文定量评价了摩擦参数对机器人输出力矩的影响, 提出考虑时变效应的机器人关节非线性摩擦参数反求方法. 首先, 建立机器人关节一般非线性摩擦模型. 设计机器人关节恒速跟踪实验, 通过卡尔曼滤波对实验采集的数据进行处理, 进而建立关节速度和驱动电机电流之间的关系, 完成关节一般非线性摩擦模型建立. 其次, 择取非线性摩擦模型关键参数. 建立包含非线性摩擦的机器人动力学模型, 基于激励轨迹计算各关节力矩, 并对其开展灵敏度分析, 择取对关节力矩灵敏性较高的摩擦参数. 再次, 建立关节输出力矩和摩擦参数一一对应的数据集. 基于实际工况构建摩擦参数取值空间, 采用最优拉丁超立方法对摩擦参数采样, 并将其代入机器人动力学模型计算出相应的力矩, 从而求得关节输出力矩和摩擦参数一一对应的数据集. 最后, 建立反问题神经网络并对其进行训练, 实现非线性摩擦模型关键参数反求, 并进行验证. 研究结果表明关节非线性摩擦的准确描述减小了机器人低速运动换向时摩擦力矩突变对机器人轨迹的影响, 显著提升了机器人轨迹精度.