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英文原文Kinematic and dynamic synthesis of a parallel kinematic high speeddrilling machineAbstractTypically, the termhigh speed drilling is related to spindle capability of high cutting speeds. The suggested high speed drilling machine (HSDM) extends this term to include very fast and accurate point-to-point motions. The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs. The paper is focused on the kinematic and dynamic synthesis of this parallel kinematic machine (PKM). The kinematic synthesis introduces a new methodology of input motion planning for ideal drilling operation and accurate point-to-point positioning. The dynamic synthesis aims at reducing the input power of the PKM using a spring element. Keywords: Parallel kinematic machine; High speed drilling; Kinematic and dynamic synthesis1. IntroductionDuring the recent years, a large variety of PKMs were introduced by research institutes and by industries. Most, but not all, of these machines were based on the well-known Stewart platform 1 configuration. The advantages of these parallel structures are high nominal load to weight ratio, good positional accuracy and a rigid structure 2. The main disadvantages of Stewart type PKMs are the small workspace relative to the overall size of the machine and relatively slow operation speed 3,4. Workspace of a machine tool is defined as the volume where the tip of the tool can move and cut material. The design of a planar Stewart platform was mentioned in 5 as an affordable way of retrofitting non-CNC machines required for plastic moulds machining. The design of the PKM 5 allowed adjustable geometry that could have been optimally reconfigured for any prescribed path. Typically, changing the length of one or more links in a controlled sequence does the adjustment of PKM geometry.The application of the PKMs with constant-length links for the design of machine tools is less common than the type with varying-length links. An excellent example of a constant-length links type of machine is shown in 6. Renault-Automation Comau has built the machine named Urane SX. The HSDM described herein utilizes a parallel mechanism with constant-length links.Drilling operations are well introduced in the literature 7. An extensive experimental study of highspeed drilling operations for the automotive industry is reported in 8. Data was collected fromhundreds controlled drilling experiments in order to specify the parameters required for quality drilling. Ideal drilling motions and guidelines for performing high quality drilling were presented in 9 through theoretical and experimental studies. In the synthesis of the suggested PKM, we follow the suggestions in 9.The detailed mechanical structures of the proposed new PKM were introduced in 10,11. One possible configuration of the machine is shown in Fig. 1; it has large workspace, highspeed point-to-point motion and very high drilling speed. The parallel mechanism provides Y, and Z axes motions. The X axis motion is provided by the table. For achieving highspeed performance, two linear motors are used for driving the mechanism and a highspeed spindle is used for drilling. The purpose of this paper is to describe new kinematic and dynamic synthesis methods that are developed for improving the performance of the machine. Through input motion planning for drilling and point-to-point positioning, the machining error will be reduced and the quality of the finished holes can be greatly improved. By adding a well-tuned spring element to the PKM, the input power can be minimized so that the size the machine and the energy consumption can be reduced. Numerical simulations verify the correctness and effectiveness of the methods presented in this paper.2. Kinematic and dynamic equations of motion of the PKM moduleThe schematic diagram of the PKM module is shown in Fig. 2. In consistent with the machine tool conventions, the z-axis is along the direction of tool movement. The PKM module has two inputs (two linear motors) indicated as part 1 and part 6, and one output motion of the tool. The positioning and drilling motion of the PKM module in this application is characterized by (y axis motion for point-to-point positioning) and (z axis motion for drilling). Motion equations for both rigid body and elastic body PKM module are developed. The rigid body equations are used for the synthesis of input motion planning of drilling and input power reduction. The elastic body equations are used for residual vibration control after point-to-point positioning of the tool.2.1. Equations of motion of the PKM module with rigid links Using complex-number representation of mechanisms 12, the kinematic equations of the tool unit (indicated as part 3 which includes the platform, the spindleand the tool) are developed as follows. The displacement of the tool is andwhere b is the distance between point B and point C, r is the length of link AB (the lengths of link AB, CD and CE are equal). The velocity of the tool iswhereThe acceleration of the tool iswhereThe dynamic equations of the PKM module are developed using Lagranges equation of the second kind 13 as shown in Eq. (7).echanism can be derived using the finite element method and take the form ofwhere M, C and K are system mass, damping and stiffness matrix, respectively; D is the set of generalized coordinates representing the translation and rotation deformations at each element node in global coordinate system; R is the set of generalized external forces corresponding to D; n is the number of the generalized coordinates (elastic degrees of freedom of the mechanism). In our FEA model, we use frame element shown in Fig. 3 in which EIe is the bending stiffness (E is the modulus of elasticity of the material, Ie is the moment of inertia), q is the material density, le isthe original length of the element. are nodal displacements expressed in local coordinate system(x, y). The mass matrix and stiffness matrix for the frame element will be 66 symmetric matrices which can be derived fromthe kinetic energy and strain energy expressions as Eqs. (12) and (13)where T is the kinetic energy and U is the strain energy of the element; are the linear 1 2 3 4 5 6 and angular deformations of the node at the element local coordinate system. Detailed derivations can be found in 14. Typically, a compliant mechanism is discretized into many elements as in finite element analysis. Each element is associated with a mass and a stiffness matrix. Each element has its own local coordinate system. We combine the element mass and stiffness matrices of all elements and perform coordinate transformations necessary to transform the element local coordinate systemto global coordinate system. This gives the systemmass M and stiffness K matrices. Capturing the damping characteristics in a compliant systemis not so straightforward. Even though, in many applications, damping may be small but its effect on the systemstability and dynamic response, especially in the resonance region, can be significant. The damping matrix C can be written as a linear combination of the mass and stiffness matrices 15 to form the proportional damping C which is expressed aswhere a and b are two positive coefficients which are usually determined by experiment. An alternate method 16 of representing the damping matrix is expressing CasThe element of C is defined as,where signKij=(Kij/|Kij|), Kij and Mij are the elements of K and M, is the damping ratio of the material.The generalized force in a frame element is defined as where Fj and Mj are the jth external force and moment including the inertia force and moment on the element acting at (xj ,yj), and m is the number of the externalforces acting on the element. The element generalized forces,are then combined to formthe systemgeneralized force R. The second order ordinary differential equations of motion of the system, Eq. (11), can be directly integrated with a numerical method such as Runge-Kutta method. For the PKM we studied, each link was discreted as 15 frame elements. Both Matlab and ADAMS software are used for programming and solving these equations.3. Input motion planning for drillingSuppose we know the ideal motion function of the drilling tool. How to determine the input motor motion so that the ideal tool motion can be realized is critical for high quality drillings. The created explicit input motion function also provides the necessary information for machine controls. According to the study done in 9, the drilling process can be divided into three phases: entrance phase, middle phase, and exit phase. In order to increase the productivity and quality of the drilling, many operation constraints such as minimum tool life constraint, hole location error constraint, exit burr constraint, drill torsion breakage constraint, etc. must be considered and satisfied. Under these conditions, the feed velocity of the tool should be slow at the entrance phase to reduce the hole location errors. The tool velocity should also be slow at the exit phase to reduce the exit burr. At the middle phase, the tool drilling velocity should be fast and kept constant. The retraction of the tool after finishing the drilling should be done as quickly as possible to increase the productivity. Based on these considerations, we assume that the ideal drilling and retracting velocities of the tool are given by Eq. (17).where vT1 is the maximum drilling velocity, T1, T2,and T3 are the times corresponding to the entrance phase, the middle phase and the exit phase. vT2 is the maximum retracting velocity. T4, T5, and T6 are corresponding to accelerating, constant velocity, and decelerating times for retracting operation. is the cycle time for a single drilling. As a numerical example, suppose we drill a 25.4 mm (1 in) deep hole with Tc=0.4s, 0.3s for drilling, 0.1s for retracting. Set T1=T3 0.06s, T4=T6=0.03s. Under these con-ditions, vT1=106(mm/s), vT2=-363(mm/s). The graphical expression of the ideal tool motion is shown in Fig. 4. If the link length in PKM r=500 mm, the angle=53 at the starting point of drilling, the corresponding input motor velocity relative to the idealtool motion is shown in Fig. 5. Generally, the curve fitting method can be used to create the input motion function. But according to the shape of the curve shown in Fig. 5, we create the linear motor velocity function manually section by section as shown in Eq. (18).where vB=143.48mm/s, vC=165.77mm/s, vE=-557.36mm/s, vF=-499.44mm/s. When plotting the velocity curve with Eq. (18), no visual difference can be found with the curve shown in Fig. 5. Eq. (18) is composed of six parts with four cycloidal functions and two linear functions. If we control the two linear motors to have the same motion as described in Eq. (18), the drilling and retracting velocity of the tool will be almost the same as shown in Fig. 4. The absolute errors between the ideal and real tool velocity are shown in Fig. 6, in which the maximum error is less than 8 mm/s, the relative error is less than 1.5%. At the start and the end positions of the drilling, the errors are zero. These small absolute and relative errors illustrate the created input motion and are quite acceptable. The derived function is simple enough to be integrated into the control algorithmof the PKM.4. Input motion planning for point-to-point positioningIn order to achieve fast and accurate positioning operation in the whole drilling process, the input motion should be appropriately planned so that the residual vibration of the tool tip can be minimized. Conventionally the constant acceleration motion function is commonly used for driving the axes motions in machine tools. Although this kind of motion function is simple to be controlled, it may excite the elastic vibration of the systemdue to the sudden changes in acceleration. Take the same PKM module used in previous for example. A FEA model is built using ADMAS with frame elements. The positioning motion is the y-axis motion, which isrealized by the two linear motors moving in the same direction. Suppose the positioning distance between the two holes is 75mm, the constant acceleration is 3g(approximated as 30m/s here). The input motion of the linear motors with constant acceleration and deceleration is shown in Fig. 7, in which the maximum velocity is 1500 mm/s, the positioning time is 0.1 s. Assuming the material damping ratio as 0.01, the residual vibration of the tool tip is shown in Fig. 8. In order to reduce the residual vibration and make the positioning motion smoother, a six order polynomial input motion function is built as Eq. (19)where the coeffcients ci are the design variables which have to be determined by minimizing the residual vibration of the tool tip. Selecting the boundary conditions as that when t=0, sin=0, vin=0, ain=0;and when t=Tp, sin=h, vin=0, ain=0, where Tp is the point-to-point positioning time, the first six coeffcients are resulted:Logically, set the optimization objective aswhere c6 is the independent design variable; is the maximum fluctuation of residual vibrations of the tool tip after the point-to-point positioning. Set and start the calculation from c6=0. The optimization results in c6=-10mm/s . Consequently, c5=7.510mm/s , c4 =-1.42510mm/s , c3=8.510mm/s , c2=c1=c0=0. It can be seen that the optimization calculation brought the design variable c6 to the boundary. If further loosing the limit for c6, the objective will continue reduce in value, but the maximum value of acceleration of the input motion will become too big. The optimal input motions after the optimization are shown in Fig. 9. The corresponding residual vibration of the tool tip is shown in Fig. 10. It is seen from comparing Fig. 8 and Fig. 10 that the amplitude and tool tip residual vibration was reduced by 30 times after optimization. Smaller residual vibration will be very useful for increasing the positioning accuracy. It should be mentioned that only link elasticity is included in above calculation. The residual vibration after optimization will still be very small if the compliance from other sources such as bearings and drive systems caused it 10 times higher than the result shown in Fig. 10.5. Input power reduction by adding spring elementsReducing the input power is one of many considerations in machine tool design. For the PKM we studied, two linear motors are the input units which drive the PKM module to perform drilling and positioning operations. One factor to be considered in selecting a linear motor is its maximum required power. The input power of the PKM module is determined by the input forces multiplying the input velocities of the two linear motors. Omitting the friction in the joints, the input forces are determined from balancing the drilling force and inertia forces of the links and the spindle unit. Adding an energy storage element such as a spring to the PKM may be possible to reduce the input power if the stiffness and the initial (free) length of the spring are selected properly. The reduction of the maximum input power results in smaller linear motors to drive the PKM module. This will in turn reduce the energy consumption and the size of the machine structure. A linear spring can be added in the middle of the two links as shown in Fig. 11(a). Or two torsional springs can be added at points B and C as shown in Fig. 11(b). The synthesis process is the same for the linear or torsional springs. We will take the linear spring as an example to illustrate the design process. The generalized force in Eq. (10) has the form ofwhere l0 and k are the initial length and the stiffness of the linear spring. The input power of the linear motors is determined byIn order to reduce the input power, we set the optimization objective as follows:where v is a vector of design variables including the length and the stiffness of the spring, . For the PKM module we studied, the mass properties are listed in Table 1. The initial values of the design variables are set as . The domains for design variables are set as lmin;lmax=400, 500 mm, kmin; kmax=1,20 N/mm. The PKM module is driven by the input motion function described as Eq. (18). Through minimizing objective (24), the optimal spring parameters are obtained as and k=14.99 N/mm. The input powers of the linear motors with and without the optimized spring are shown in Fig. 12, in which the solid lines represents the input power without spring, the dotted lines represents the input power with the optimal spring. It can be seen from the result that the maximum input power of the right linear motor is reduced from 122.37 to 70.43 W. A 42.45% reduction is achieved. For the left linear motor, the maximum input power is reduced from 114.44 to 62.72 W. A 45.19% reduction is achieved. The effectiveness of the presented method by adding a spring element to reduce the input power of the machine is verified. Torsional springs may be sued to reduce the inertial effect and the size of the spring attachment.6. ConclusionsThe paper presents a new type of high speed drilling machine based on a planar PKM module. The study introduces synthesis technology for planning the desirable motion functions of the PKM. The method allows both the point-to-point positioning motion and the up-and-down motion required for drilling operations. The result has shown that it is possible to reduce substantially the residual vibration of the tool tip by optimizing a polynomial motion function. Reducing residual vibration is critical when tool positioning requirement for the HSDM is in the range of several microns. By adding a well-tuned optimal spring to the structure, it was possible to reduce the required input power for driving the linear motors. The simulation has demonstrated that more than 40% reduction in the required input power is achieved relative to the structure without the spring. The reduction of required input power may allow choosing smaller motors and as a result reducing costs of hardware and operations.In order to better understand the properties of the HSDM and to complete its design, further study is required. It will include error analysis of the machine as well as the control strategies and control design of the system. 7. AcknowledgementsThe authors gratefully acknowledge the financial support of the NSF Engineering Research Center for Reconfigurable Machining Systems (US NSF Grant EEC95-92125) at the University of Michigan and the valuable input fromthe Centers industrial partners.中文翻译高速钻床的动力学分析摘要通常情况下，术语“高速钻床”就是指具有较高切削速率的钻床。高速钻床(HSDM)也是指具有非常快的和正确的点到点运动的钻床。新的HSDM是由带有两个直线电动机的平面并联机构组成。本文主要就是对并联机器(PKM)的动力学分析。运动合成是为了介绍一种新方法，它能够完善钻孔操作和点到点定位的准确性。动态合成旨在减少因使用弹簧机械时PKM的输入功率。关键词: 并联运动机床; 高速钻床; 动力学的合成1.介绍在最近的几年里，研究所和工业协会介绍了各式各样的PKM。其中大部分（但不是所有）,以众所周知的斯图尔特月台1为基础结构。这一做法的好处是高公称的负载重量比，良好的位置精度和结构刚性2。斯图尔特式PKM的主要缺点是相对小的工作空间和相对慢的操作速度 3,4。机床刀具的工作空间是指刀尖能够移动和切削材料所需要的容积。平面的斯图尔特月台的设计在5中被提到，像是对无CNC机器作翻新改进的方法需要塑料的铸模机制一样。PKM5的设计允许可以调整几何学已经被规定了的最佳的再配置的任何路径。 一般的，改变一根或较多连杆的长度是以PKM受约束的顺序来做几何学的调整。在机床设计中，“定长度连杆”的PKM应用比“不定长度连杆”的共同点要少的多。一个优秀“定长度连杆”型的机器例子被显示在6。Renault-Automation Comau已经建造叫做“Urane SX”的机器。在此HSDM被描述成是一个采用“定长度连杆”组成的并联机械装置。钻床操作在文学7中被很好的介绍了。汽车工业中，一项关于高速钻孔的操作的广泛的实验研究在8中被报告。数据从数百个钻床控制实验上收集起来，是为了具体指定钻床质量所必须的参数。理想的钻床运动和制造高质量钻床的指导方针通过理论和实验的研究被呈现在9中。在被建议的PKM综合中，我们遵循9中的结论。新推出的PKM的详细机械结构在10,11被介绍，机器的大致结构显示在图1中；它有很大的工作空间，点到点的高速运动和非常高的钻速。并联的机械装置提供给了Y和Z轴的动作，X轴动作是由工作台提供的。为了达成高速的运转，用了两个线性马达来驱驶机械装置和用一个高速的主轴来钻孔。这篇文章的目的就是描述新的运动学的和动力学合成的方法的发展，为了改良机器的运转。通过输入运动，规划钻井和点对点定位，机器的误差将会被减少，而且完成孔的质量能被极大的提高。通过增加一个弹簧机械要素到PKM，输入动力就能被最小，以便机器的尺寸和能量损耗降低。数字模拟的正确查证和热交换率的方法呈现在这篇文章中。2.PKM模型的运动学和动力学的运动方程式PKM模型的概要线图在图2中被显示。由于机床刀具库的一致，Z轴是沿着工具运动的方向的。PKM模型有部分1和部分6二个输入指示(二个线性电机)，和一个刀具的输出动作。在PKM模型应用中，定位和钻孔运动分别通过 ( y 轴动作相对点到点的定位)和 (z轴动作相对钻孔)表示。刚体和柔性体的PKM模型运动方程式都被发展了。刚体方程式被用于合成输入钻床的动作计划和输入力量还原。柔性体方程式被用来在刀具点到点定位之后的剩余振动控制。2.1.刚性连杆的PKM模型的运动方程式机械装置12的特点是使用了数字集成,刀具设备(含工作台，主轴和刀具3部份)。它的运动学方程式的发展依下列各项。刀具的变位是且 其中b是点B和点C之间的距离,r是连杆AB的长度(连杆AB、CD和CE的长度是相等的)。刀具的速度是 其中 刀具的加速度是 其中 PKM模型的动力学方程式的发展如方程（7）所示，使用了拉格朗日的第二个类型的方程式13。 其中t是系统的总动能；和是总坐标值和速度值;是总力对应到的的值。k是坐标系中总的独立数目。在这里,k=2，q1= y1和q2=y6，引出之后，公式(7)可被表达成其中n是移动连杆的数目；是连杆i的大量惯性矩；是连杆i的质量中心坐标；是PKM模型中连杆i的旋转角。总力的值通过（9）决定 其中V是势能， 是没有势能的力。为了对PKM模型的钻孔操作，我们有 其中是切削力, F1和F6是线性马达在PKM上输入的力。情绪商数。公式(1)到公式(10)构成了刚性连杆PKM模型的运动学和动力学方程式。2.2.柔性连杆的PKM模型的动作方程式顺从的机械装置的动微分方程式能用有限的机械要素方法和以下的公式得到其中M、C和K分别是系统质量，阻尼和刚性母体；D是在全球同等坐标系中的每个机械要素平移和旋转变形表现的总坐标值；R是总外力值，与D保持一致；n是坐标的总数目值(机械装置的柔性自由度)。在我们的FEA模型中，我们使用在图3中被显示的机械要素结构，其中EIe是弯曲刚性(E是材料的柔性系数,Ie是惯性矩),是物质的密度,le是机械要素的最初长度。是(x,y)坐标系统中表现的结点变位。机械要素的大众基地和刚性基地将会是66个对称的矩阵，能从动能和应变能中得到，表达在公式（12）和（13）中 其中t是动能，U是机械要素的应变能；是机械要素基本坐标系中线性的123456和角变形节。详细的推论能在14被发现。典型地，在有限的机械要素分析中，一个顺从的机械装置是被离散成许多个机械要素的。每个机械要素与一个质量和一个刚性母体有关。每个机械要素有它自己的基本坐标系。我们结合机械要素质量和所有机械要素的刚性矩阵运行坐标转换时，必须把机械要素的基本坐标系转换成世界坐标系，这就提供了系统质量M和刚性K矩阵。在一个顺从的系统中捕获阻尼特性不是这么顺利的。即使, 在许多应用中，阻尼可能很小，但是它能作用在系统安全性和动力的频率响应中，尤其在共振区域中,可能是重要的。阻尼基地C能被写做一种质量和刚性矩阵15的线性结合，构成比例阻尼C如下式表达所示其中和是二个通常由实验决定的正系数。一个表现阻尼基地的交互方法16 表达成C如下机械要素C被定义为,其中,和是K和M的机械要素, 是材料的阻尼比。机械要素结构中的总力被定义为 其中和是的外力和力矩，包括在上动作的机械要素的惯性力和力矩,m 是在机械要素上动作的外力数目。机械要素的总力,组合构成了系统总力R。系统动作的第二次序普通微分方程式，如公式(11), 用一个数字能直接被整合的方法,就像是Runge- Kutta的方法那样。对于我们研究的PKM,每个连杆被分离成15个机械要素结构。Matlab和ADAMS软件都被用来规划和解决这些方程式。3.为钻床输入动作计划假如我们知道钻床理想的动作功能。高质量钻床的关键是如何决定输入电动机动作以便刀具的理想动作能被了解。创建明白的输入动作功能时也为机器控制提供了必需的数据。依照研究在9中所做的,钻孔的过程能分为三个时期: 入口期,中间期和出口期。为了增加生产能力和钻孔的质量，许多操作限制，例如最小刀具的寿命限制，孔位置误差限制,退出毛边限