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長春工業(yè)大學(xué)本科畢業(yè)設(shè)計(jì)論文
Dimensional Tolerance Allocation for New Type of Parallel Machine Tools
LI Bing, WANG Zhi xing, HU Ying (Dept. of Mechanical Engineering, Harbin Institute of Technology, Harbin 150001, China) ?
Abstract：Presents an inverse error estimation method for deriving errors of links from cutter head errors to overcome the difficulty in obtaining analystic solutions for new type of machine tools due to the multi solution of direct kinematics so that the total dimension chain tolerance of machine tool links can be determined by Monte Carlo random simulation for a certain spindle dimension of the machine tool and the precision requirement specified, and the method of dimensional tolerance allocation developed by analysing the elements of machine tool links and the combination of machine tool precision with actual dimensional tolerance for justification of tolerance design for this kind of machine tools.
Key Words： parallel machine tool Monte Carlo simulation tolerance allocation
1 INTRODUCTIONS
The prototype of a new type of parallel machine tools is a 6-6 Stewart platform mechanism. Compared to the traditional machine tool, a parallel machine tool possesses many advantages, such as high precision, high structural rigidity and high machining force/torque capacity, etc. Currently, many universities have research on this kind of machine tools. As shown in Fig.1, the parallel machine tool comprises the following four parts: a fixed platform, six in-parallel actuators, the length variation of each actuator is implemented by ball-screws and the ball-screws are driven by servo motors. The ball joints connect the fixed and moving platforms with ball-screws; a spindle is installed on moving platform bottom. When machining work is being carried out the variation of the moving platform allows a parallel machine tool to machine complicated curved work piece such as die, impeller etc.
The developing tendency of modern NC machine tools is high speed and high precision, and the machine tool errors have direct effect on machine tool precision. In order to meet the precision specified, a scientific dimensional tolerance allocation is urgently required in the design of this kind of machine tool. The errors of the machine tool include the following parts: 1)transmission error in transmission chains, i.e. ,pitch cumulative error of the ball-screw in each actuator and dead domain error during starting or reversing operation;2)length transform error of the ball-screw due to the influence of ambient temperature;3)pares clearance error that connect fixed and moving platform;4)other errors, i.e. , positioning error caused by transmission rigidity, dynamic error caused by moving parts mass and velocity damping. The above errors will finally lead to error at the cutter head of the machine tool. This paper will carry out research on dimensional tolerance allocation based on a kind of statistical experiment method, Monte-Carlo method.
2 SOLUTION OF INVERSE KINEMATICS FOR MACHINE TOOL
The kinematics solution for this parallel machine tool can be classified into two types. When the known quantity is the lengths of the six actuators and the orientation of the cutter head, this is called direct kinematics; On the contrary, when the known quantity is the position and the orientation of the cutter head and the unknown quantity is the lengths of the six actuators, this is called inverse kinematics. The solution of the direct kinematics of the machine tool is used to monitor while the solution of the inverse kinematics of the machine tool is used to control. As shown in Fig.2, a fixed coordinate frame O-XYZ is attached to the fixed platform and a moving coordinate frame O’-X’Y’Z’ is attached to the moving platform. The center of the joint connecting the itch actuator to the fixed platform will be denoted as whereas the center of the joint connecting the same leg to the moving platform will be denoted as , the Y-axis of the fixed coordinate frame is selected along the line which bisects the angle and the Y’-axis of the moving coordinate frame is selected along the line which bisects the angle . Let the position of point O’ with respect to the origin of the fixed coordinate frame be denoted by vector , vectors will be defined as the position vectors of the moving coordinate frame, thus we can write the position vectors of the moving platform joints in fixed coordinate frame as :
Where matrix is the rotation matrix describing the orientation of the moving platform with respect to the fixed platform, the elements of matrix are shown by RPY expression. Let be angles that the moving platform rotates about X-axis, Y-axis, Z-axis of the fixed coordinate frame, and then we can obtain:
Let vectors be the position vectors of the platform joints with respect to the fixed coordinate frame, then the length of each actuator can be written as:
3 DIMENSIONAL TOLERANCE ALLOCATIONS WITH MONTE-CARLO METHOD
Determination of Total Dimensional Chain Error of the Machine Tool Links
In order to carry out dimension tolerance allocation, the variation feature between the errors of the in-parallel links and orientation error of the cutter head (or the center point of the moving platform) need to know. The direct kinematics is used to derive the cutter head error from the link errors of the machine tool. It is difficult to obtain analytical solution by direct kinematics. So an inverse error estimation method by which the link errors can be derived from the position and orientation errors of the cutter head is adopted.
The key to the dimensional tolerance allocation is to determine the total dimensional chain error of the machine tool links. Monte-Carlo method is a numeral method to solve mathematics based on random sampling. In this parallel an evenly-distributed Monte-Carlo random simulation method on a certain position and orientation precision of the cutter head is used. By Esq. (1) the lengths of the machine tool links under the sample volume can be obtained. The maximum length error of each link can be obtained in different positions in the machine tool workspace. Take the minimum length errors as the total error for dimensional tolerance allocation of each link. Before random simulation by Monte-Carlo method, the machine tool precision should be estimated and the variation of the moving platform’s position and orientation should be determined. Let the position and orientation of the center point of the moving platform be three transforming quantities and three rotating quantities. A simplified model of spindle system for parallel machine too is shown in Fig.3. From the figure we can see that point a and b represent cone bearings, segment bc represents cutter holder and segment cd represents the cutter, the cutter head point d bears machining force P.
4 CONCLUSIONS
The presented dimensional tolerance allocation method is the combination between machine tool precision and the actual dimension tolerances. The factors that influence on the cutter head errors of the parallel machine tool are analyzed first. Due to the multi-solution and the difficulty in obtainment of the analytical solution of the direct kinematics, the derivation of the cutter head error from the error of machine tool links is difficult. So a inverse error estimation method to derive errors of links from cutter head error is presented in this paper.
To a certain spindle dimension of the parallel machine tool, the total dimensional tolerance of the machine tool link can be determined by Monte-Carlo random simulation method. The dimensional tolerance allocation for adjustable loop of the link is developed. If the allocated dimension tolerance can not meet the design or manufacturing requirement, the adjustments towards precision grade of the ball screw or the spindle dimension are desired for tolerance re-allocation. Example shows the presented tolerance allocation method is reasonable. This work provides a basis in the design stage of the parallel machine tool.
新型并聯(lián)機(jī)床的三維公差分配
李冰，王知行，胡穎（哈爾濱工業(yè)大學(xué)機(jī)械工程系，中國哈爾濱150001）
摘要：提出了一種逆誤差估計(jì)的方法以克服刀盤刀具連接中存在錯(cuò)誤和誤差這一難題，獲取新型機(jī)床由于直接運(yùn)動(dòng)學(xué)多解的解決方案，通過蒙特卡羅隨機(jī)模擬的方法直接確定機(jī)床環(huán)節(jié)對機(jī)床的主軸尺寸和精度要求，對尺寸公差分配和組合機(jī)床機(jī)床精度與實(shí)際的尺寸公差的分析，可采用該方法的公差分配來設(shè)計(jì)機(jī)床。
關(guān)鍵詞：并聯(lián)機(jī)床 蒙特卡洛模擬 公差分配
1介紹
新類型的并聯(lián)機(jī)床的原型是一個(gè)6-6斯圖爾特平臺(tái)機(jī)構(gòu)。與普通機(jī)床相比，并聯(lián)機(jī)床有許多優(yōu)點(diǎn)，比如高的精度，高的結(jié)構(gòu)剛度和高機(jī)制力量/轉(zhuǎn)力矩能力等?，F(xiàn)在，許多大學(xué)都對這種并聯(lián)機(jī)床都加以研究。如圖1所示。并聯(lián)機(jī)床包括四個(gè)部分：一個(gè)固定的平臺(tái)，六個(gè)平行的主動(dòng)件，每個(gè)主動(dòng)件的長度變化由球鉸鏈來控制，而球鉸鏈由伺服馬達(dá)驅(qū)使。球關(guān)節(jié)用鉸鏈連接固定部分和可動(dòng)部分，轉(zhuǎn)軸安裝在移動(dòng)平臺(tái)底部。當(dāng)機(jī)加工進(jìn)行時(shí)，移動(dòng)平臺(tái)使并聯(lián)機(jī)床能加工復(fù)雜的工件，如鋼模的復(fù)雜彎工作塊及其他的東西。
現(xiàn)代的控制母機(jī)的發(fā)展趨向是高速度和高精度，并且機(jī)床的誤差應(yīng)能直接影響機(jī)床的精度。為了解決這種精密的需要，一種尺寸公差分配方法需要應(yīng)用于機(jī)床的設(shè)計(jì)中。機(jī)床的誤差包括下列各項(xiàng)部分：（1）傳輸鏈的傳輸誤差，也就是，在開始或顛倒操作的時(shí)候在每個(gè)主動(dòng)件和死區(qū)中球鉸鏈的累積誤差（2）周圍溫度變化導(dǎo)致的球鉸鏈長度變化誤差（3）剝連固定和可動(dòng)工作臺(tái)的清除誤差（4）其它的誤差，也就是，由移動(dòng)部分塊和速度降低引起的動(dòng)態(tài)誤差。上述的誤差最終會(huì)導(dǎo)致在工作母機(jī)的切削頭上產(chǎn)生的誤差。這篇論文將對一種基于統(tǒng)計(jì)的尺寸公差分配的實(shí)驗(yàn)方法——蒙地卡羅方法加以研究。
2 機(jī)床的逆向運(yùn)動(dòng)學(xué)的解決
運(yùn)動(dòng)學(xué)解決這種并聯(lián)機(jī)床的運(yùn)動(dòng)學(xué)解決方法可以分為兩類。當(dāng)已知量是六個(gè)主動(dòng)件的長度和切削頭的方向的時(shí)候，這叫做正向的運(yùn)動(dòng)學(xué)。另一方面，當(dāng)已知量是位置和切削頭的方向而未知量是六個(gè)主動(dòng)件的長度的時(shí)候，這叫做逆向的運(yùn)動(dòng)學(xué)。并聯(lián)機(jī)床的正向運(yùn)動(dòng)學(xué)用于解決檢測問題，而逆向運(yùn)動(dòng)學(xué)用于解決控制問題。如圖2所示，固定的坐標(biāo)系附在固定的平臺(tái)上，而移動(dòng)的坐標(biāo)系附在移動(dòng)的平臺(tái)上。連接主動(dòng)件到固定的平臺(tái)關(guān)節(jié)的中心被表示為，同樣，連接相同的腿到那個(gè)移動(dòng)平臺(tái)的關(guān)節(jié)中心被表示為。固定坐標(biāo)系的Y軸是按沿著角挑選的，而移動(dòng)坐標(biāo)系的Y`軸是沿著角。讓有關(guān)固定的坐標(biāo)系的起源用矢量表示，而移動(dòng)坐標(biāo)系的起源用矢量表示。如此，我們把移動(dòng)平臺(tái)的位置在固定平臺(tái)中用位置矢量表示為：
點(diǎn)陣式是描述移動(dòng)平臺(tái)相對于固定平臺(tái)的旋轉(zhuǎn)點(diǎn)陣式，點(diǎn)陣式元素用RPY表示。讓為移動(dòng)平臺(tái)與X軸，Y軸，Z軸之間的旋轉(zhuǎn)夾角，然后，我們能獲得：
讓矢量是有關(guān)于坐標(biāo)體格平臺(tái)位置的矢量。然后每個(gè)主動(dòng)件的長度可以寫作：
3 蒙地卡羅尺寸公差分配方法
為了要實(shí)現(xiàn)尺寸公差分配，必須弄清楚聯(lián)編尺寸誤差和切削頭的定位誤差之間的變化特征。正向運(yùn)動(dòng)學(xué)用于來制來自機(jī)床的聯(lián)編誤差的切削頭誤差。直接運(yùn)動(dòng)學(xué)的獲得分析是十分困難的。如此一個(gè)可能起源于位置的逆向誤差和切削頭定方位的判斷方法可以被采用。
尺寸公差分配的關(guān)鍵是要決定機(jī)床的聯(lián)編總體尺寸公差。蒙地卡羅方法是基于隨意抽取樣品的解決尺寸公差分配的一種數(shù)學(xué)方法。平均地隨意分配在一個(gè)某位置和切削頭方向的蒙地卡羅模擬方法被采用在并聯(lián)機(jī)床上。樣本容量下的機(jī)床的聯(lián)編尺寸的長度可能被獲得。聯(lián)編的最大長度誤差可能在工作母機(jī)工作空間的不同位置獲得，尺寸公差分配為拿聯(lián)編最小量長度誤差作為總誤差。在蒙地卡羅隨意模擬之前，工作母機(jī)的精度應(yīng)被估計(jì)，并且移動(dòng)平臺(tái)的位置變化和方位應(yīng)被考慮。那個(gè)移動(dòng)平臺(tái)的中心點(diǎn)位置和方向?yàn)槿D(zhuǎn)換量和三替換量。并聯(lián)機(jī)床的轉(zhuǎn)軸的簡化模型也被顯示在圖片3中。從這個(gè)圖片，我們可以看出a和b代表圓錐體，弦bc代表切削者，而弦cd切削頭，切削頭點(diǎn)d產(chǎn)生機(jī)制力量P。
3 結(jié)論
尺寸公差分配方法是機(jī)床精確度和真實(shí)尺寸公差之間的組合。我們應(yīng)首先分析影響并聯(lián)機(jī)床的切削頭的誤差。由于直接運(yùn)動(dòng)學(xué)的分析解決獲得困難和形式多樣，源自機(jī)床聯(lián)編誤差的切削頭誤差的引出很困難。因此，本論文中闡述了源自切削頭誤差的聯(lián)編誤差的倒轉(zhuǎn)誤差判斷方法。
對于并聯(lián)機(jī)床轉(zhuǎn)軸來說，機(jī)床聯(lián)編總體尺寸公差分配可能被蒙地卡羅隨意模擬方法確定。調(diào)整性的尺寸公差分配正被發(fā)展。如果尺寸公差分配不能滿足機(jī)械制造業(yè)和設(shè)計(jì)的需要，那么，能調(diào)整的精密的球螺釘或轉(zhuǎn)軸被需要出現(xiàn)。事例證明這個(gè)尺寸公差分配方法是合理可行的。這一工作為并聯(lián)機(jī)床的設(shè)計(jì)階段提供了基礎(chǔ)。
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