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英文原文 Kinematic and dynamic synthesis of a parallel kinematic high speed drilling machine Abstract Typically, the term‘‘high 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 synthesis 1. Introduction During 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 module The 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 spindle and the tool) are developed as follows. The displacement of the tool is and where 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 is where The acceleration of the tool is where The dynamic equations of the PKM module are developed using Lagrange’s equation of the second kind [13] as shown in Eq. (7). ·echanism can be derived using the finite element method and take the form of where [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 is the 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 as where a and b are two positive coefficients which are usually determined by experiment. An alternate method [16] of representing the damping matrix is expressing [C]as The element of [C’] is defined as,where signKij=(Kij/|Kij|), Kij and Mij are the elements of [K] and [M], ζis 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 drilling Suppose 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 angleβ=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 positioning In 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 is realized 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/s2 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 as where 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=7.5×10mm/s , c4 =-1.425×10mm/s , c3=8.5×10mm/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 elements Reducing 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 of where l0 and k are the initial length and the stiffness of the linear spring. The input power of the linear motors is determined by In 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. Conclusions The 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. Acknowledgements The 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 Center’s industrial partners. 中文翻譯 高速鉆床的動力學(xué)分析 摘要 通常情況下，術(shù)語“高速鉆床”就是指具有較高切削速率的鉆床。高速鉆床(HSDM)也是指具有非?？斓暮驼_的點到點運動的鉆床。新的HSDM是由帶有兩個直線電動機的平面并聯(lián)機構(gòu)組成。本文主要就是對并聯(lián)機器(PKM)的動力學(xué)分析。運動合成是為了介紹一種新方法，它能夠完善鉆孔操作和點到點定位的準(zhǔn)確性。動態(tài)合成旨在減少因使用彈簧機械時PKM的輸入功率。 關(guān)鍵詞: 并聯(lián)運動機床; 高速鉆床; 動力學(xué)的合成 1.介紹 在最近的幾年里，研究所和工業(yè)協(xié)會介紹了各式各樣的PKM。其中大部分（但不是所有）,以眾所周知的斯圖爾特月臺[1]為基礎(chǔ)結(jié)構(gòu)。這一做法的好處是高公稱的負載重量比，良好的位置精度和結(jié)構(gòu)剛性[2]。斯圖爾特式PKM的主要缺點是相對小的工作空間和相對慢的操作速度 [3,4]。機床刀具的工作空間是指刀尖能夠移動和切削材料所需要的容積。平面的斯圖爾特月臺的設(shè)計在[5]中被提到，像是對無CNC機器作翻新改進的方法需要塑料的鑄模機制一樣。PKM[5]的設(shè)計允許可以調(diào)整幾何學(xué)已經(jīng)被規(guī)定了的最佳的再配置的任何路徑。 一般的，改變一根或較多連桿的長度是以PKM受約束的順序來做幾何學(xué)的調(diào)整。 在機床設(shè)計中，“定長度連桿”的PKM應(yīng)用比“不定長度連桿”的共同點要少的多。一個優(yōu)秀“定長度連桿”型的機器例子被顯示在[6]。Renault-Automation Comau已經(jīng)建造叫做“Urane SX”的機器。在此HSDM被描述成是一個采用“定長度連桿”組成的并聯(lián)機械裝置。 鉆床操作在文學(xué)[7]中被很好的介紹了。汽車工業(yè)中，一項關(guān)于高速鉆孔的操作的廣泛的實驗研究在[8]中被報告。數(shù)據(jù)從數(shù)百個鉆床控制實驗上收集起來，是為了具體指定鉆床質(zhì)量所必須的參數(shù)。理想的鉆床運動和制造高質(zhì)量鉆床的指導(dǎo)方針通過理論和實驗的研究被呈現(xiàn)在[9]中。在被建議的PKM綜合中，我們遵循[9]中的結(jié)論。 新推出的PKM的詳細機械結(jié)構(gòu)在[10,11]被介紹，機器的大致結(jié)構(gòu)顯示在圖1中；它有很大的工作空間，點到點的高速運動和非常高的鉆速。并聯(lián)的機械裝置提供給了Y和Z軸的動作，X軸動作是由工作臺提供的。為了達成高速的運轉(zhuǎn)，用了兩個線性馬達來驅(qū)駛機械裝置和用一個高速的主軸來鉆孔。這篇文章的目的就是描述新的運動學(xué)的和動力學(xué)合成的方法的發(fā)展，為了改良機器的運轉(zhuǎn)。通過輸入運動，規(guī)劃鉆井和點對點定位，機器的誤差將會被減少，而且完成孔的質(zhì)量能被極大的提高。通過增加一個彈簧機械要素到PKM，輸入動力就能被最小，以便機器的尺寸和能量損耗降低。數(shù)字模擬的正確查證和