Design and analysis of a tripod machine tool with an integrated Cartesian guiding and metrology mechanism

This paper introduces a novel parallel kinematic mechanism (PKM) machine called Cartesian-Guided Tripod (CGT) developed for high-speed machining of precise dies and molds. CGT has two kinds of functional independent legs, the driving functional leg and the integrated Cartesian guiding/metrology functional leg, parallel-linked to the platform. The guiding/metrology functional leg constraints and measures the platform motions in Cartesian coordinates. Because of the separation of metrology function from the drive mechanisms, the metrology loop of CGT is immunized to the geometric-, thermal-, and force-induced errors of the drive mechanisms. Another feature of this CGT is that the kinematic parameters can be auto-calibrated on line. CGT has explicit solutions of the inverse and forward kinematic analysis. This allows system specifications, such as the footprint/workspace ratio, maximum Cartesian speed/acceleration and even stiffness, be easily related to the machine component specifications. The CGT drive mechanism can be divided into two basic types: sliding leg and telescoping leg. The telescoping leg CGT has the higher edges in the kinematic categories than the sliding leg CGT. Conversely, the sliding leg CGT has better static rigidity and dynamic stiffness than the telescoping leg CGT.     

The influence of friction on contouring accuracy of a Cartesian guided tripod machine tool

This paper is focused on the friction-induced contouring errors of a three-axis parallel kinematic machine (PKM) called a Cartesian-guided tripod (CGT). A dynamic model of the CGT including the nonlinear dynamic, mechanical compliance and mechanical friction has been formulated. Servo control characteristics of the formulated CGT dynamic model are analyzed based on the conventional P-PI controller and feedforward friction compensation implemented in the joint level control scheme. Quantitative analysis and comparison of the various friction sources from the actuated joints (ballscrew/nut) and passive joints (balljoint and guideway) have been conducted. Analysis results show that the ballscrew/nut friction produces the most significant friction-induced contouring errors. The ball-joint friction and linear-guide friction induced contouring errors are two-order and one-order, respectively, lower than the errors by the ballscrew friction. The main reason is that the equivalent friction torques of these two friction sources coupled into the servo drive loop are reduced by the leverage effect of the driving leg length. It was also found that the conventional feedforward friction compensation can effectively reduce the ballscrew-friction-induced contouring errors. However, if not properly compensated, ballscrew friction could excite significant platform-leg structural vibration because the tripod-based PKM usually has a high compliance vibration mode in the horizontal platform-leg direction. The friction-excited structural vibration can be reduced by putting vibration absorbers on the guideways of the passive leg to damp out the platform-leg vibration. From the primary analysis results, it can be seen that the proposed tripod based PKM has high potential for high-speed 5-axis machining applications.     

Calibration of parallel kinematic machine tools using mobility constraint on the tool center point

In the application of parallel kinematic machine tools (PKM), because of errors in the geometric parameters, it is necessary to calibrate the PKM to improve the positioning accuracy. In existing self-calibration methods, either some redundant sensors on passive joints or some mobility constraints on the kinematic chains are used. However, the mobility constraints imposed on kinematic chains might apply large forces during the test on legs and passive joints. Also, these kinds of calibrating are applicable only on PKMs in which their actuated joints can be used as passive joints. To overcome weaknesses of existing methods, a novel approach to calibration based on imposing position constraints on the tool center point (TCP) is introduced. In this method, only the data of encoders installed on the actuated joints in some sets of configurations are required. In each set, the position of the TCP is fixed, but orientations of the tool are different. Simulations and experimental studies on a Hexaglide PKM built in Sharif University of Technology reveal the convenience and effectiveness of the proposed robot calibration method.     

基于螺旋理论的卧式加工中心空间误差建模方法及其量化验证

针对现阶段机床空间误差建模过程中存在的繁琐性与非统一性问题,以及现有模型验证策略难以实现量化评价的局限性,以某卧式加工中心为研究对象,提出一套较完善的空间误差建模方法及其验证技术体系。以螺旋理论与串联机构运动学为理论基础,结合机床运动链拓扑分析,推导建立加工中心运动学模型。所提运动学建模方法可以有效避免传统方法中矩阵变换时潜在的奇异性问题,并且有利于简化机构运动学分析。通过系统讨论在不同参考系下定义运动误差对运动学模型的影响,提出名义运动矩阵与运动误差矩阵乘积关系的确定方法及原则,进而在此基础上构建卧式加工中心空间误差模型。为量化验证所建空间误差模型的准确性,提出基于空间矢量欧氏范数的量化验证策略与实施技术。数据对比显示,空间矢量欧氏范数偏差的预测值与实测值较吻合,最大相对误差为15.79%,表明所提空间误差建模方法可行且准确性较高,所提模型量化验证策略具有较好的直观性与有效性。     

球铰刚度计算模型及靠冗余支链实现并联机床刚度的改善

以Stewart机构为例提出一种基于解析法和有限元法的刚度分析新方法。应用有限元软件模拟球铰内的非线性接触变形,将计算结果抽象成一个特征参数带入机构的刚度解析模型中,然后得到机构的刚度矩阵,以解决在解析模型中无法处理铰链刚性的问题,并采用有限元软件证实该方法的可靠性。如果在解析模型中忽略球铰变形,则整个机构的刚度解析结果将产生较大误差。在此基础上,应用该方法定量研究采用主动冗余支链对提高并联机床刚性的效果。结果表明,在一定工作空间范围内,主动冗余支链可以显著提高并联机床的刚性。     

面向精度评价的并联机床参数辨识技术

基于并联机床外部标定,推导出辨识方程的残差近似等于标定后的位姿误差。为克服最小二乘法存在较大残差的缺陷,面向精度评价,提出最小最大优化的参数辨识技术,以残差最大绝对值最小为优化目标,直接控制误差范围。标定实例验证了最小二乘法缺陷的实际存在和最小最大优化抑制较大残差的效果。该参数辨识技术直接联系运动学标定和标定后的精度评价,简单有效的提高并联机床的整体精度。     

Computational model for the control, performance evaluation, and calibration of a parallel mechanism

This paper presents a computational model that defines a methodology for the control, performance evaluation, and calibration of a parallel mechanism by means of the kinematic model, the kinematic parameter identification, and the control of the system actuators and sensors. The developed software has special relevance to the calibration of parallel mechanisms, allowing us to perform the system control, performance evaluation, and mechanism calibration in a single application. Parallel computing techniques are used to overcome the high computational cost involved with solving complex nonlinear kinematic models and parameter identification, obtaining a significant computational cost saving. Finally, the developed procedure is validated, obtaining an important improvement in the accuracy of the mechanism. This methodology can be used in mechanisms for metrology applications and machine tool processes.     

Optimal design and development of a 2-DOF PKM-based machine tool

This paper presents multiobjective optimization of a typical 2-degree-of-freedom (DOF) parallel kinematic machine (PKM) tool that has only single DOF joints. Nondimensional indices, namely global stiffness index (GSI), global conditioning index (GCI), and workspace index, are considered as the objectives for optimization. The indices GSI and GCI depict the variation of stiffness and dexterity of PKM within the workspace. The leg length and distance between two rails on which actuators slide are treated as design variables as these greatly influence the characteristics of PKM. A multiobjective genetic algorithm (MOGA) approach is implemented in MATLAB to find an efficient solution to this complex optimization problem. Fitness function includes inverse kinematics equations, Jacobian and stiffness matrices to compute and optimize the nondimensional indices. First, the optimal value of each index is obtained by single-objective GA. To further improve the results, a hybrid function PATTERNSEARCH is used. This helps to select appropriate boundary conditions for MOGA. To obtain the optimal values of all the three indices, a multiobjective GA is carried out. The results are compared with a conventional exhaustive search method of optimization. The obtained results show that the use of MOGA enhances the quality of the optimization outcome. Secondly, a prototype has been designed and developed with the optimal dimensions. The actual workspace of the PKM and influence of leg collision on the workspace are studied. Finally, a preliminary experimentation was carried out. A comparison between PKM and the three-axis serial kinematic machine tool is presented.     

Exechon并联模块的静刚度建模与分析*

为研究Exechon并联模块的刚度性能,采用子结构综合方法建立该模块的刚度解析模型。分别采用虚拟关节法和有限元法计入各关节和支链体的柔性,并通过推导动平台和支链装配体间的变形协调方程构建出系统的弹性静力学模型。由系统方程抽取出动平台刚度矩阵,据此预估Exechon并联模块在典型位姿和工作全域内的刚度性能,并将其与有限元仿真结果进行对比。结果表明,所建解析模型具有较高计算精度,可快速预估机构工作全域的刚度特性;Exechon并联模块在工作空间任一截面内的刚度关于x轴对称分布,且沿w方向的刚度受机构位形影响较小。     

Kinematic calibration of gantry hybrid machine tool based on estimation error and local measurement information

This paper presents the kinematic calibration of a four degrees-of-freedom (DOF) hybrid machine tool based on a novel planar 3-DOFs parallel mechanism and a long movement of the worktable. Closed-form solutions are developed for both the inverse and direct kinematics about the parallel mechanism. The error model is built and the mechanism accuracy is investigated. Two types of kinematic calibration method are proposed by a simple measurement device. The first type of calibration method is based on estimation error, and can easy improve the machine tool accuracy quickly by estimating the error trends. The second type of kinematic calibration method is based on local measurement information, which includes the position errors and does not include the pose errors of the machine tool. The calibration tests showed the effectiveness of the calibration methods, which can be useful for the similar types of parallel machine tool.     

Selection of optimal machining parameters for hexapod machine tool

Parallel kinematic machine tools (PKM) have the advantages of higher stiffness, higher payload capacity and lower inertia. Still their penetration into the machine tool industry is very less. One of the difficulties in using PKMs such as hexapod machine tools is that the stiffness continuously varies with configuration change at every instant. This makes location of work piece and selection of machining parameters difficult and complicated. A methodology is presented in this article to select optimal machining parameters for hexapod machine tools. Particle swarm optimization is used as a tool in the optimization process. A profile-milling example is also presented to demonstrate the selection of machining parameters.     

并联机床静刚度研究

常见的并联机床静刚度分析理论发展尚未完善。本文在研究并联机床静力学的基础上,使用能量法分析在外载荷作用下的并联机床变形量,从而获得在不同位姿下并联机床的静刚度。利用上述方法,针对一台五自由度的并联机床进行了分析,获得了其静刚度。为并联机床静刚度研究及其设计提供了理论依据和方法支持。     

3-RPS并联机构运动学标定方法的研究

针对六轴混联机床中因3-RPS并联机构结构参数误差引起的精度问题,分析了影响3-RPS并联机构几何精度的误差因素,给出了并联机构的误差模型;基于影响并联机构定平台运动精度较大的几何误差参数;建立了运动学标定模型.采用阻尼最小二乘法,经多次优化迭代实现了利用一组测量数据完成非线性超越矛盾标定方程组的求解.利用激光干涉仪完成了标定用数据的测量,通过3-RPS并联机构运动学逆解和各铰链的几何标定参数,得到动平台的实际位姿.通过对标定前后的Z轴定位精度的检测及实际零件加工试验,验证了3-RPS并联机构运动学标定模型和方法的正确性和有效性.     

A two-step calibration methodology of multi-actuated mechanical servo press with parallel topology

We present a two-step calibration methodology of multi-actuated mechanical press with parallel topology and illustrate the method with a case study of a dual-actuated servo press with parallel topology. The kinematic model of the servo press is established firstly. From the total differentials of the kinematic equations, the error sensitivity matrix is obtained to find out linear dependent parameters and the effects of kinematic parameters on the press accuracy are studied. It is found that the press mechanism has the advantage that the press accuracy at the working area is less sensitive to the kinematic parameter errors. The experiment is carried out to measure the motions of the moving platform and slider. By the kinematic error model, the kinematic parameters of the active and passive chains are identified, respectively. Experimental results show that the accuracy of the servo press improves by 82% after kinematic calibration.     

Delta并联机构精度标定方法研究

以Delta并联机构为对象,研究一类含平行四边形支链的3自由度并联机构误差建模技术,所建模型可有效分离出影响末端姿态误差的几何误差源。在此基础上提出一种精度标定方法,该方法利用并联机构操作空间与关节空间非线性映射的性质,仅需检测末端沿z向的位置误差、以及在初始位形下的姿态误差便可识别出几何参数,并可通过修改系统输入实现末端位置误差补偿。给出算例以验证该方法的有效性。     

Multi-objective optimization of a 3-DOF translational parallel kinematic machine

In this paper, stiffness modelling and analysis of a typical 3-DOF parallel kinematic machine (PKM) that provides translational motion along X, Y and Z axes is presented. The mechanism consists of three limbs each having an arm and a forearm with prismatic-revoluterevolute-revolute joints (PRRR). The joint arrangement is in such a way that the moving or tool platform maintains same orientation in the entire workspace. Through inverse kinematics, the joint angles for a given position of tool platform necessary for the stiffness modelling and analysis are obtained. The stiffness modelling is based on the compliance matrices of arm and forearm of each limb. Typical non-dimensional performance indices, namely, workspace volume index (WVI), global translational stiffness index (GTSI), and global rotational stiffness index (GRSI), are introduced and used to study the influence of dimensions. Attempts are also made to find the optimal dimensions of the translational PKM using multi-objective optimization based on the genetic algorithms (MOGA) in MATLAB. The methodology presented and the results obtained are useful for predicting the performance capability of the PKM under study.     

三自由度虚拟轴机床静力学及动力学的若干研究

针对我国特殊钢的钢坯局部修磨问题,开发出一种三自由度虚拟轴机床。建立静力学和动力学方程,并对静力学和动力学特性进行了研究。由于机床的外力和外力矩分别由驱动杆和平行机构承受,其力学模型和位置控制都变得简单易行。通过结构柔度系数矩阵对机床刚度特性进行考虑,表明在工作空间的横截面内,越接近中心,机床刚度越高。动力学方程的仿真表明各杆驱动力有相似性,且运动平台的质量对各杆驱动力的影响较大。     

仿人型机器人腿的探讨

提出一种用气动人工肌肉驱动的仿人型机器人腿。首先描述了气动人工肌肉的结构与工作原理，人工肌肉具有变刚度特性，它的刚度可通过控制橡胶管内的气压实现，调节管内气压的大小就可改变肌肉的刚度。它的刚度大小决定肌肉的驱动力。然后提出了用气动人工肌肉驱动的机器人腿的结构形式以及机器人腿的摆动控制方式，腿的驱动力和行走运动全由工控机控制实现。     

Enhanced stiffness modeling of manipulators with passive joints

The paper presents a methodology to enhance the stiffness analysis of serial and parallel manipulators with passive joints. It directly takes into account the influence of external and internal loadings on the manipulator configuration and, consequently, on its Jacobians and Hessians. The main contributions of this paper are the introduction of a non-linear stiffness model for the manipulators with passive joints, a relevant numerical technique for computing the Cartesian stiffness and stability criteria for configurations of the kinematic chains. Within the developed technique, the manipulator elements are presented as pseudo-rigid bodies separated by multidimensional virtual springs and perfect passive joints. Simulation examples are presented that deal with parallel manipulators of the Ortholide family and demonstrate the ability of the developed methodology to describe non-linear behavior of the manipulator structure such as a sudden change of the elastic instability properties (buckling).     

车铣复合加工中心综合误差检测及补偿策略

根据五轴车铣复合加工中心的结构及其运动链构型特点,设计了综合误差检测方案。检测包括车主轴床身至铣主轴运动链的空间误差检测以及车主轴的热误差检测两部分。由于检测方案使刀具-工件之间构成了完整的运动链,解决了单纯的空间误差检测方法未考虑车主轴运动链误差影响的问题。文中同时提出了车铣复合加工中心综合误差补偿策略以及运用神经网络算法的几何误差和热误差综合补偿模型。采用分步体对角空间误差检测后,实施了空间误差补偿。补偿后四条体对角线的空间误差都明显减小,减小幅度从15.24 μm到50.83 μm,误差补偿效果从39.10%提高到78.06%。本文提出的方法极大地改善了空间误差补偿精度。