100米深孔液壓鉆機(jī)變速箱設(shè)計(jì)
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附錄1
快速原形機(jī)的軟件補(bǔ)償
摘要：這篇論文闡述了快速原形機(jī)在參數(shù)誤差成型法和軟件誤差補(bǔ)償方面的改進(jìn)。這種方法得到多年來在坐標(biāo)測量機(jī)和機(jī)械系統(tǒng)參數(shù)值發(fā)展的技術(shù)支持?？焖僭螜C(jī)所有的誤差結(jié)果將被輸入一個(gè)實(shí)際的參數(shù)誤差模型，普通的實(shí)體造型依賴于快速原形機(jī)和主要坐標(biāo)測量機(jī)的測量。測量結(jié)果用來顯示機(jī)械誤差函數(shù)和驅(qū)動(dòng)標(biāo)準(zhǔn)刀具的銼刀的誤差補(bǔ)償，對(duì)這個(gè)方法進(jìn)行了三次實(shí)驗(yàn)，結(jié)果顯示了它充分改善了標(biāo)準(zhǔn)件的精確度。
前言
快速原形機(jī)在刀具和輔助設(shè)計(jì)制造和隨后產(chǎn)生的商品化技術(shù)方面起著重要的作用。今天的工藝方法有許多種，例如：光敏液相固化法、熔絲沉積成型法、噴墨打印法、選區(qū)片層黏結(jié)法、選區(qū)激光燒結(jié)法。這些添加工藝應(yīng)用范圍很廣。如：概念成型技術(shù)，新產(chǎn)品開發(fā)、快速模具制造、生物學(xué)。將這種技術(shù)發(fā)展成產(chǎn)品技術(shù)是一個(gè)偶然的機(jī)遇，但它已得到了廣泛的應(yīng)用。但經(jīng)過CIRP的科學(xué)技術(shù)委員會(huì)調(diào)查確認(rèn)較差的工藝精度，將阻礙技術(shù)在機(jī)械制造方面的繼續(xù)滲透。
有兩種普通和快速原形一樣可以提高工藝精度。第一種解決這問題的方法是：通過“避免誤差”尋找誤差來源減小誤差。第二種是減小誤差產(chǎn)生的結(jié)果，被叫做“誤差補(bǔ)償”。
快速原形技術(shù)的研究使誤差避免和原形工藝的許多方面都有所提高。在這些技術(shù)中，最受關(guān)注的是原形工藝參數(shù)和基本定位完善。因?yàn)樵喂に囍校性S多工藝變量影響工件的精度，可以完善這些參數(shù)使工件達(dá)到最高的精度。復(fù)合表面處理技術(shù)是典型用來尋工件精度和制造工藝參數(shù)間關(guān)系的，一旦獲得復(fù)合表面，參數(shù)將控制完成更高的工件精度?；径ㄎ煌晟凭褪且粋€(gè)典型的參數(shù)應(yīng)用和延伸的例子。它被作為一種多元化標(biāo)準(zhǔn)完善問題，控制著表面精度和工件的壽命間的交換。決定變量是工藝參數(shù)設(shè)定和工件基本定位。復(fù)合表面符合參數(shù)問題是為了使產(chǎn)品優(yōu)化。除了工藝參數(shù)的實(shí)體成型和基本定位優(yōu)化，還有原形技術(shù)的工藝計(jì)劃。如：數(shù)據(jù)文件糾正、切削技術(shù)改善、結(jié)構(gòu)傳代、和路線計(jì)劃都在進(jìn)行詳細(xì)的研究，目的就是為了工藝改善。
好的工藝計(jì)劃，在一定范圍內(nèi)可以提高機(jī)械精度，但是隨著快速原形技術(shù)的發(fā)展，即使最好的TUNED系統(tǒng)生產(chǎn)的零件也只是為原形機(jī)進(jìn)行誤差補(bǔ)償。philpott和Gree提出為原形工藝進(jìn)行反復(fù)誤差補(bǔ)償?shù)牟呗?，它補(bǔ)償由于不懂得誤差創(chuàng)造機(jī)理而復(fù)制積累的誤差。然而因?yàn)闆]有機(jī)械誤差模型建立在這個(gè)策略上，所以反復(fù)工藝也許對(duì)每個(gè)新工件進(jìn)行反復(fù)補(bǔ)償。Nee et.al構(gòu)造了一個(gè)矯正的帶有n*n個(gè)格子的圖表來提高光敏液相固化工藝精度。這個(gè)矯正圖表是根據(jù) 系統(tǒng)的結(jié)構(gòu)來計(jì)算的，主要對(duì)激光束在平臺(tái)上定位的誤差補(bǔ)償。掃描緩沖器和定位標(biāo)準(zhǔn)可以利用這種圖表進(jìn)行補(bǔ)償。
這篇論文指出多年來CMM誤差參數(shù)值方面的技術(shù)發(fā)展使快速原形機(jī)的誤差補(bǔ)償更易理解，而且可以用實(shí)例參數(shù)誤差進(jìn)行誤差補(bǔ)償。典型CMM有三個(gè)線形托架，設(shè)計(jì)目的是為惡劣使X、Y、Z軸都獨(dú)立移動(dòng)，然而每個(gè)托架有六個(gè)自由度，而且硬件結(jié)構(gòu)不能完全消除不必要的移動(dòng)和轉(zhuǎn)動(dòng)。結(jié)果每個(gè)軸都有三個(gè)移動(dòng)誤差和三個(gè)垂直誤差。隨著三個(gè)垂直度的增加或軸間的空間誤差增加，一個(gè)三軸機(jī)器就會(huì)有21個(gè)參數(shù)誤差，假設(shè)此剛體運(yùn)動(dòng)，機(jī)械的空間誤差能寫出一個(gè)有21個(gè)參數(shù)的方程，這21個(gè)誤差和符號(hào)雜表格（1）中有所體現(xiàn)。
在快速原形工藝中，誤差預(yù)算除了軸的移動(dòng)誤差還有許多種，我們的方法是把所有討厭的誤差標(biāo)成21個(gè)實(shí)際參數(shù)誤差進(jìn)行機(jī)械容量誤差的全面測量，這有雙重的目的：第一：它將提供一個(gè)模型解決補(bǔ)償。第二：它將分離出有意義空間方向上的誤差預(yù)算，而且對(duì)硬件有一定的診斷作用，同時(shí)識(shí)別由于其它工藝特征引起的誤差源方向。和CMM、車床不同，快速原形機(jī)的21個(gè)“實(shí)際“參數(shù)不能在機(jī)器上直接測量，這個(gè)實(shí)物成型方法是間接對(duì)這些誤差進(jìn)行測量。也就是說，快速原形機(jī)是用來制造特殊設(shè)計(jì)的實(shí)物成型，而CMM具有大規(guī)模測量實(shí)物成型的特征，測量是為了推斷快速原形機(jī)的參數(shù)誤差，而且誤差補(bǔ)償規(guī)則應(yīng)用于工件工藝計(jì)劃的標(biāo)準(zhǔn)資料
這篇論文的剩余部分是結(jié)論，第二部分是解釋SLA機(jī)的誤差模型。第三部分介紹三維制圖和參數(shù)誤差方程的起源。第四部分是陳述了理解的方法，測試零件的補(bǔ)償結(jié)果。第五部分是總結(jié)工作和討論將來的發(fā)展。
2，SLA機(jī)的數(shù)學(xué)誤差模型
光敏液相固化法是用液態(tài)光敏樹脂做樣品，在一個(gè)槽內(nèi)裝滿光敏液態(tài)聚合物，在下方放有一個(gè)升降臺(tái)，把沒完成的部分放在樹脂表面下。計(jì)算機(jī)控制激光通過聚合物表面的方向選擇固化的表層，升降機(jī)下降一個(gè)距離重新覆蓋一層液態(tài)光敏樹脂，再進(jìn)行掃描固化，直到全部完畢，SLA250機(jī)用在研究層的厚度是0，004。
2．1 SLA機(jī)的誤差模型
機(jī)械數(shù)學(xué)模型根據(jù)機(jī)械類型不同實(shí)現(xiàn)剛體假設(shè)的多樣化。一般來說三軸機(jī)器可以根據(jù)探測器的移動(dòng)和工件分成XYFZ、XYZF、XFYZ或FYXZ幾類，F(xiàn)象征固定機(jī)器方程，當(dāng)其它字母出現(xiàn)在F的右側(cè)暗示探測器移動(dòng)。當(dāng)字母在F的左側(cè)暗示工件移動(dòng)。
在SLA 250 快速原形機(jī)內(nèi)激光焦點(diǎn)在刀頭上，它由鏡子控制方向，它能移動(dòng)二個(gè)正交水平方向，被分別定義為X、Y。這工作平臺(tái)延Z軸帶動(dòng)工件上下移動(dòng)，以垂直方向作為Z方向。根據(jù)這些特征SLA 250符合ZFXY典型機(jī)。這種類型的快速原形機(jī)的分類不能反映快速原形機(jī)的實(shí)際運(yùn)動(dòng)。但是能反映刀頭的運(yùn)動(dòng)結(jié)果。如：激光焦點(diǎn)。正如我們前面提到的，快速原形機(jī)參數(shù)誤差將由生產(chǎn)和測量一個(gè)系統(tǒng)部分來確定。而不是測量各個(gè)機(jī)械軸線移動(dòng)的精度誤差。這表示機(jī)械的實(shí)際參數(shù)誤差成型的選擇不代表精度的機(jī)械運(yùn)動(dòng)學(xué)。這就是它與交換方法的不同。每個(gè)軸線的六次移動(dòng)誤差就是以操作運(yùn)動(dòng)學(xué)鏈的每個(gè)同源改變基質(zhì)。當(dāng)用實(shí)際參數(shù)誤差成型時(shí)就必須決定是否這成型能成功地產(chǎn)生補(bǔ)償數(shù)據(jù)來提高機(jī)械的精度。討論補(bǔ)償?shù)慕Y(jié)果來顯示這成型能提供有用的補(bǔ)償數(shù)據(jù)的能力。
其它原形機(jī)也做了同樣的分析，例如：熔司沉積成型機(jī)。這刀頭是由X、Y兩個(gè)方向移動(dòng)的托架驅(qū)動(dòng)的。沉積噴頭底座可以沿Z的方向上下移動(dòng)，因此熔絲沉積成型機(jī)也相當(dāng)于一個(gè)ZFYX典型機(jī)。圖FIG2是SLA 250機(jī)的誤差成型的運(yùn)動(dòng)學(xué)坐標(biāo)鏈?zhǔn)噶繄D。
用傳統(tǒng)的定義方式，0是坐標(biāo)原點(diǎn)，X、Y、Z是三個(gè)坐標(biāo)軸，它只包含平移和垂直誤差。在確定運(yùn)動(dòng)路線時(shí)旋轉(zhuǎn)誤差被認(rèn)為獨(dú)立存在。T是刀頭在拖架方向的補(bǔ)償。在SLA 250機(jī)上沒有真正的X、Y拖架，激光束直接聚焦在刀頭上。因?yàn)楣馐偸蔷劢褂谝后w表面，因此SLA 250 機(jī)的T=0。一般情況下，T包含在誤差成型的偏差中，但后來被省略了?？蚋裰械腤是有原點(diǎn)到激光聚焦點(diǎn)的矢量方向。它代表工件的真實(shí)尺寸。運(yùn)動(dòng)矢量鏈圖的內(nèi)容在下面已給出，它們的結(jié)果由機(jī)械誤差的理論知識(shí)是很容易理解的。從原點(diǎn)O到激光焦點(diǎn)有兩條路線：Z——W和X——Y——T。每個(gè)運(yùn)動(dòng)路線對(duì)一個(gè)軸線的理論誤差將影響軸線的實(shí)際誤差，因此軸線的移動(dòng)需要根據(jù)理論誤差來修改。X、Y、Z軸的旋轉(zhuǎn)可由旋轉(zhuǎn)半徑R（X）、R（Y）、R（Z）表示。讓U代表X、Y、Z，旋轉(zhuǎn)標(biāo)量和它們反向公式（2）在圖FIG2中Z和X沒有前身，W有前身軸Z、Y有前身軸X、T有兩個(gè)前身軸X和Y。因此這兩個(gè)從O到焦點(diǎn)的矢量平衡，可用下面的方程表示（3）。重新整理這方程，矢量W由其它矢量和標(biāo)量函數(shù)表示（4），用矢量機(jī)成型代替X、Y、Z、T、W和旋轉(zhuǎn)標(biāo)量。用坐標(biāo)系統(tǒng)XP、YP、ZP來表示在工件上激光焦點(diǎn)的坐標(biāo)（5）
TX、TY、TZ是不可重復(fù)的誤差，它是由不可重復(fù)的內(nèi)容產(chǎn)生的，因?yàn)镾LA 250 機(jī)沒有刀頭補(bǔ)償如：XT=YT=ZT=0所以數(shù)學(xué)模型可簡寫為（6）。EY（X）、EX（Y）、EY（Y）和EZ（Y）都沒有出現(xiàn)在模型中XT=YT=ZT=0就是ZFXY典型機(jī)的特征。因?yàn)樵谀Ｐ椭袇?shù)誤差函數(shù)是實(shí)際誤差，因此一些誤差不可能有可理解的物理意思。例如：激光束總是能消除桶表面的樹脂產(chǎn)生正確的誤差、不能表示來自樹脂表面激光束的偏差。
2．2勒讓德多項(xiàng)式約數(shù)
勒讓德多項(xiàng)式可以近似的表示每個(gè)參數(shù)誤差函數(shù)。每個(gè)平移和旋轉(zhuǎn)誤差都可以用勒讓德多項(xiàng)式方程的線性來近似表示。例如：是近似的系數(shù)，表示第個(gè)勒讓德多項(xiàng)式。要得到誤差近似值首先要確定勒讓德多項(xiàng)式的次序，勒讓德多項(xiàng)式的線性連接次序越高，剩余誤差平方的和越小。但是高次序的誤差函數(shù)可以近似制造工藝中的不可重復(fù)誤差，促使誤差最后得到補(bǔ)償，在CMM系統(tǒng)中，KRUTH ET。AL指出幾何誤差在系統(tǒng)中會(huì)慢慢改變，因此第三順序的勒讓德多項(xiàng)式是好的近似機(jī)會(huì)。高次序會(huì)減少剩余誤差，但會(huì)增大不可重復(fù)誤差的影響，第三次序的勒讓德多項(xiàng)式在研究中是最接近的。例如：（7）這個(gè)方程里有四個(gè)未知系數(shù)，通過設(shè)定絕對(duì)零點(diǎn)可使其減少到三個(gè)，一般將所有誤差都在軸線的起始點(diǎn)消除。例如:使 和 的關(guān)系就可表示為，用代替。則這個(gè)方程式就可重新寫為（8）為了方便看，把系數(shù)由代替，同樣表示的系數(shù)，表示系數(shù)，是軸、的平方誤差。在這樣的定義下，所有的參數(shù)誤差方程都可以用勒讓德方程來表示，但除了直接誤差方程（9）平方誤差和三個(gè)直接誤差、、存在一個(gè)特殊的關(guān)系，簡單說，直接誤差、、的線性關(guān)系分別是平方差、、接下來的例子將證實(shí)了這點(diǎn)。
找一點(diǎn)P在二維測量機(jī)上測量沒有誤差，但在X0、Y0軸間卻有平方誤差，把非正交坐標(biāo)系中的P點(diǎn)（X0、Y0）可以平移到正交坐標(biāo)系中用、表示，則（10）得到第一個(gè)近似值。這是成立的。因?yàn)槭欠浅Ｐ∈荵0的函數(shù)，它是直接誤差##的第一個(gè)次序關(guān)系式，實(shí)際上，直接誤差的線性關(guān)系和平方誤差表達(dá)的是同一個(gè)誤差。它在Y軸的X方向是沒有必要移動(dòng)的，因?yàn)樗拇笮∨cY坐標(biāo)是成比例的。它是不可能分辨出這兩者，而且用同樣的誤差成型兩次是正確的。在成型中依賴假設(shè)事物制造出正交坐標(biāo)系統(tǒng)是一種方法。早期選擇這種理論這種機(jī)械就會(huì)被認(rèn)為是沒有正交坐標(biāo)系統(tǒng)。因此在成型中包括平方誤差在假設(shè)事物下，以上三種直接誤差的線性關(guān)系都為零。如：=0,=0, =0或者把機(jī)械假設(shè)為一個(gè)正交坐標(biāo)系例如：三個(gè)軸線彼此是正交的沒有平方誤差存在，但是直接誤差的線性關(guān)系最后誤差模型將有同樣的關(guān)系量，盡管一個(gè)誤差有不同的名字。接下來假設(shè)一個(gè)正交系統(tǒng)，平方誤差將不是單獨(dú)成型，在這個(gè)假設(shè)條件下，在多項(xiàng)式方程中所有的直接誤差都包括線性關(guān)系，SLA機(jī)成型可被表示為（11）
3.3D 實(shí)體造型和參數(shù)誤差方程
3.1 3D實(shí)體造型來估計(jì)誤差方程系數(shù)
每個(gè)誤差方程都需要確定三個(gè)系數(shù)，因?yàn)槠胀ㄝS正交系統(tǒng)18個(gè)參數(shù)誤差的已知系數(shù)的總量為54個(gè)，因?yàn)镾LA 250機(jī)正在研究中。因?yàn)樗?、、不能出現(xiàn)在模型中，因此未知系數(shù)總量變?yōu)?2，至少42個(gè)方程能解出所有未知系數(shù)，因?yàn)椴恢貜?fù)誤差的存在，所以確定方程容易一些，靠減少剩余誤差的總量才能解出未知系數(shù)。
用實(shí)體造型是零件關(guān)鍵特征的表面的位置（x.y.z）在CMM誤差值中，一個(gè)高精度的實(shí)物造型，如：用不同的定位和方向測量球棒和圓環(huán)來覆蓋，CMM全部的工作容量在快速原形中，快速原形機(jī)生產(chǎn)由CMM測量的普通實(shí)體造型。假設(shè)CMM的精度和重復(fù)能力比快速原形機(jī)高，用測量的特殊位置來表示誤差模型的函數(shù)去推斷，快速原形機(jī)的參數(shù)誤差參數(shù)，實(shí)體造型的機(jī)構(gòu)不是唯一的，因?yàn)樗凶銐虻牟煌恢玫狞c(diǎn)提供充足的方程來確定系數(shù)和誤差的最小值。
FIG4展示了這項(xiàng)研究的3D實(shí)體造型。它由169個(gè)圓柱和13層組成，而且由13個(gè)X層和13個(gè)Y層交叉形成。通過測量圓柱上表面的中心點(diǎn)可以寫出它的公稱位置和誤差系數(shù)的函數(shù)。圓柱高度排列成一條線，以至于盡可能X、Y、Z結(jié)合成獨(dú)立的方程。在測量時(shí)所有的圓柱表面很容易被CMM探測到，這部分可以提供163*3=507個(gè)方程?？繙p少不重復(fù)誤差來確定42個(gè)系數(shù)，這些方程是足夠用的。這個(gè)實(shí)體造型能研究范圍是200*200*100
用缺省的機(jī)械系數(shù)可以制定準(zhǔn)確的實(shí)體造型，這部分可以在平臺(tái)中自動(dòng)制成二次工藝后，3D實(shí)體造型在CarlZeiss ECLIPSE 550 CMM機(jī)上測量。把一部分坐標(biāo)系定義為基本表面：X-Y表面，3D實(shí)體造型中心作為X、Y的數(shù)據(jù)庫來確定中心點(diǎn)的坐標(biāo)（X、Y）。
3．2參數(shù)誤差函數(shù)
用一自然非線性程序問題去解算系數(shù)，它的目標(biāo)函數(shù)是縮小剩余非重復(fù)性誤差的平方總和。（12）用LinGo編一個(gè)優(yōu)化程序可以解出42個(gè)系數(shù)。每個(gè)多項(xiàng)式誤差函數(shù)的結(jié)果在表2中已列出。
三個(gè)軸的系數(shù)和參數(shù)誤差函數(shù)已在FIG5中列出，得到以下結(jié)論：
1.數(shù)誤差不總是線性函數(shù)，也不總是X=0或Y=0對(duì)稱的。這表示參數(shù)誤差補(bǔ)償對(duì)模型補(bǔ)償是比應(yīng)用簡單同類收縮率因素補(bǔ)償更精確。
2.在許多多項(xiàng)式中，Z軸的標(biāo)準(zhǔn)誤差，ZTZ是最大的變換誤差，這很容易理解，因?yàn)檫@部分在Z方向上制成一層一層的，這層與上層的連接處會(huì)比X、Y方向產(chǎn)生更多的誤差。
4用機(jī)械誤差模型進(jìn)行補(bǔ)償
根據(jù)我們的假設(shè)，可以在非補(bǔ)償部分參數(shù)誤差函數(shù)，預(yù)測點(diǎn)的位置可以提前對(duì)部件模型應(yīng)用補(bǔ)償來提高其精度。預(yù)測和補(bǔ)償?shù)慕Y(jié)果可以用估算誤差模型的精度。在這部分首先介紹補(bǔ)償?shù)姆椒ǎ缓蠼榻B他在不同部件上用3D實(shí)體造型估算正確的誤差模型方面應(yīng)用。
4.1當(dāng)用CAD設(shè)計(jì)完成原形時(shí)，可有好幾種文件格式表示：CAD成型、Pro/E成型、三角測量后的STL文件、二進(jìn)制格式或AS格式。切削文件由快速原形機(jī)的切削軟件制作，闡述誤差補(bǔ)償?shù)哪繕?biāo)是有必要的?？焖僭螜C(jī)的誤差模型不是同類時(shí)，補(bǔ)償部分的同一多項(xiàng)式當(dāng)然也將改變，也就是說平面將不是平面，球面將不是球面。假如可能用CAD系統(tǒng)進(jìn)行補(bǔ)償，但也是很難的，因此用CAD模型補(bǔ)償是不實(shí)際的。STL文件在快速原形工業(yè)中是defacto格式，應(yīng)用在快速原形工藝上有兩種典型的格式：二進(jìn)制文件和ASC11文件、二進(jìn)制格式很常用，因?yàn)槿萘啃?，但它的格式?jīng)]有ASC11格式易讀易改。另一方面ASC11 STL文件有以下格式：
除了第一行和最后一行，這文件可以分為幾個(gè)小單元，每七行一個(gè)單元，每個(gè)單元都是以facet normal 開始以 end facet結(jié)束。每個(gè)單元都由記錄的三個(gè)垂直度的坐標(biāo)來描述一個(gè)面，每個(gè)面是一個(gè)普通的單元矢量。在FIG6有一個(gè)圓柱，在兩個(gè)圓環(huán)邊界上有三條線相互垂直，補(bǔ)償可以應(yīng)用在這些邊界環(huán)的垂直點(diǎn)上，這暗示了當(dāng)創(chuàng)造層建立部件的標(biāo)準(zhǔn)工藝時(shí)，圓柱體只能接受來自上下底面邊界移位留下的補(bǔ)償，這是粗糙補(bǔ)償，另一個(gè)精致補(bǔ)償代替了每層的輪廓線或切片。當(dāng)STL文件成為切片后，被顯示在FIG7中，每個(gè)輪廓由線組成，最后連成環(huán)成為層的邊界，而且在層形成時(shí)建立了標(biāo)準(zhǔn)刀具。對(duì)切片進(jìn)行補(bǔ)償，其補(bǔ)償方法和機(jī)械分層方法效果是一樣的，然而切片格式有時(shí)是專用的，而且不易理解的。ASC11 STL文件應(yīng)用誤差補(bǔ)償更容易被接受，并且得以證實(shí)。
假設(shè)一機(jī)械誤差模型在STL文件中每個(gè)頂點(diǎn)的實(shí)際位置都可被預(yù)測實(shí)現(xiàn)位置和虛擬位置間的誤差可以補(bǔ)償。相反，意義不同的是提前STL文件中添加虛擬頂點(diǎn)的坐標(biāo)，每個(gè)單元矢量要用補(bǔ)償垂直度為每個(gè)面進(jìn)行反復(fù)計(jì)算。一個(gè)FORTRAN程序可以對(duì)STL文件進(jìn)行系統(tǒng)的修改。
下面這段是用補(bǔ)償程序提高三個(gè)不同例子部件的精度，為了證明補(bǔ)償程序?qū)μ岣咛厥馕恢镁龋瑐?cè)面精度和厚度精度的能力。
4.2用補(bǔ)償來提高特殊位置的精度
設(shè)計(jì)一個(gè)部件與3D實(shí)體造型相似的幾何圖形拉進(jìn)行研究，這部件有49個(gè)直徑相同的圓柱，但位置無序而且高度不同，用同一個(gè)SLA機(jī)和一樣的參數(shù)背景復(fù)制了兩個(gè)部件，但一個(gè)用補(bǔ)償、一個(gè)沒有補(bǔ)償。容量誤差如：計(jì)算每個(gè)圓柱上表面中心的實(shí)際位置和虛擬位置的距離，并且作為誤差圖繪制如圖FIG8數(shù)據(jù)顯示，在補(bǔ)償部分的容量誤差急劇下降。
計(jì)算誤差減少量，計(jì)算每點(diǎn)補(bǔ)償后的容量和補(bǔ)償前的容量誤差比率。49個(gè)點(diǎn)的比率的一部分繪制在圖FIG9中。平均容量誤差比原來的值減少大約30%，這意味著誤差補(bǔ)償后實(shí)際點(diǎn)與虛擬位置更接近。然而由于在快速原形工藝中無重要重復(fù)因素，并且Z值量是分層制造使Z方向誤差增大。所以說這比率與統(tǒng)計(jì)分配有關(guān)。大部分?jǐn)?shù)據(jù)下降在10——60%之間，當(dāng)一個(gè)點(diǎn)的數(shù)值降副大于1，則Z值將由下段來闡述。
4．3用補(bǔ)償來提高側(cè)面精度
在前面已經(jīng)證明補(bǔ)償可以提高個(gè)別點(diǎn)的位置精度，在這段設(shè)計(jì)了一個(gè)半徑為45.72mm的半圓形去研究怎樣的補(bǔ)償才能提高連續(xù)表面?zhèn)让娴木?，如圖FIG10在半圓形表面選擇90個(gè)點(diǎn)進(jìn)行測量，這些點(diǎn)覆蓋半圓表面，它可以靠減小偏差的平方總數(shù)來確定一個(gè)完美的球面。在這些點(diǎn)中有一個(gè)最大偏差值和最小偏差值點(diǎn)，這兩個(gè)點(diǎn)可以確定兩個(gè)同心圓，所有的餓點(diǎn)都包括在內(nèi)，用這兩個(gè)同心圓的范圍作為這表面的餓側(cè)面精度的近似值。在表3中列出了補(bǔ)償前和補(bǔ)償后的計(jì)算結(jié)果。
附錄2
Software compensation of rapid prototyping machines
Kun Tong, E. Amine Lehtihet, Sanjay Joshi
The Harold and Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University,358 Leonhard Building, University Park, PA 16802, USA Received 22 January 2003 ; received in revised form 6 October 2003 ;accepted 6 November 2003.
Abstract:This paper addresses accuracy improvement of rapid prototyping (RP) machines by parametric error modeling and software error compensation. This approach is inspired by the techniques developed over the years for the parametric evaluation of coordinate measuring machines (CMM) and machine tool systems. The confounded effects of all errors in a RP machine are mapped into a “virtual” parametric machine error model. A generic artifact is built on the RP machine and measured by a master CMM. Measurement results are then used to develop a machine error function and error compensation is applied to the files which drive the build tool. The method is applied to three test parts and the results show a significant improvement in dimensional accuracy of built parts.
Keywords: Rapid prototyping; Parametric modeling; Software error compensation; STL files
1. Introduction
Rapid prototyping (RP) machines are now an important part of the vast array of tools and techniques used to assist in the design, manufacture and subsequent commercialization of a product [1,2]. Today’s commercial machines offer a variety of processes such as the stereo lithography apparatus (SLA), fused deposition modeling (FDM), ink jet printing (IJP), laminated object manufacturing (LOM) and selective laser sintering (SLS). These additive processes are used in a wide range of applications such as concept modeling, new product marketing, rapid tooling, and biomedical. Many of the challenges encountered to develop this technology into a production technique with a wide range of applications have been overcome. However, the survey by the CIRP’s Scientific Technical Committee [3] identifies the inferior dimensional accuracy of these processes as the one remaining obstacle which prevents this technology from greater penetration of manufacturing activities. There are two general approaches which can be used to improve the accuracy of a process such as RP. The first approach attacks the problem through “error avoidance” and seeks to eliminate the source of an error. The second approach strives to cancel the effect of an error without removing the error source and is known as “error compensation” [4–6].
Most studies on RP accuracy improvement to date fall within the error avoidance category and have focused on different aspects of the RP process. Among these, RP process parameters tuning [7–10] and build orientation optimization [11–14] have drawn the most attention. For any RP process, there are many process variables that affect part accuracy. The setting of these parameters could be optimized to achieve the best part accuracy. Response surface methodology (RSM) is typically used to find the relationship between part accuracy and manufacturing process parameters [8]. Once a response surface is obtained, parametric tuning is then conducted to achieve better part accuracy. Optimization of build orientation is a typical example of an application and extension of parameter tuning. It is usually formulated as a multi-criteria optimization problem, managing the trade off between surface finish, part accuracy and part build time. Decision variables are the process parameter settings and part building orientation. Response surfaces fitted in parameter tuning problem are then used in optimization. Besides process parameters tuning and build orientation optimization, other aspects of process planning of RP techniques such as data file correction, slicing technique improvement, support structure generation and path planning have also been studied in detail for process improvement [15–20]. Good process planning can improve machine accuracy to some extent, but with the current RP technology, even the best tuned system will still produce parts with considerable systematic errors. Error compensation can be used to further reduce errors. However, very little work, if any, has been done on error compensation for RP machines. Philpott and Green [21] present an iterative error compensation strategy for one RP process, which compensates for cumulative error build-up during replication without knowledge of the error creating mechanism. However, since no machine error models were built in this strategy, the iterative process needs to be repeated for every new part. Nee et al. [14] constructed a correction table with n × n lattice points to improve the stereo lithography process accuracy. This correction table is calculated according to the configuration of the Galvano-mirror system and is used mainly to compensate for the error in positioning the laser beam on the platform. This table is uploaded to the scanner buffer and positioning values are compensated by values in the table. This paper presents a more comprehensive method for RP machines error evaluation and error compensation using” virtual ” parametric errors, inspired by the technique developed over the years for parametric evaluation of CMM errors [4–6,22–29]. A typical CMM has three linear carriages, designed to move independently along X, Y,or Z axis. How- ever, each carriage has six degrees of freedom and hardware construction usually cannot completely suppress the undesired translational or rotational movement. As a result, each axis has three translational errors and three rotational errors. With the addition of three perpendicularity or squareness errors between the axes, a three-axis machine has a total of 21 parametric errors [4–6,23,27]. Assuming rigid body kinematics, the volumetric error of the machine can be written as a function of the 21 parametric errors. These 21 errors and the notation used for representation are summarized in Table 1.
In the RP processes, the error budget is quite large and includes many items other than axes motion errors [30–32]. Our approach is to map all these confounded errors into 21“virtual” parametric errors as a global measure of machine volumetric accuracy. This will serve a dual purpose: first, it will provide a model with sufficient resolution for compensation; second, it will partition the error budget along meaningful spatial directions and serve as a diagnostic tool for intervention on the hardware (drives, controller) as well as a diagnostic tool for the identification of direction dependent error sources due to other process characteristics. Unlike the case of CMM’s and machine tools, the 21 “virtual” parametric errors of a RP machine cannot be measured directly on the machine, and the artifact method is thus used for an indirect measurement of these errors. That is, the RP machine is used to manufacture a specially designed artifact, and a CMM is used as a master scale to measure artifact characteristics. Measurements are then used to infer the parametric errors of the RP machine, and an error compensation routine can then be applied to the build files of any part scheduled for processing by the machine. The rest of the paper is organized as follows. Section 2 explains the SLA machine error model. Section 3 introduces the 3D artifact and the parametric error functions derived from it. Section 4 presents the compensation method, test parts compensation results and Section 5 summarizes the work and discusses possible future work.
2. Mathematical error model of SLA machine
Stereo lithography apparatus (SLA) creates a prototype by photo curing a liquid resin. A vat is filled with photo-curable liquid polymer with an elevator platform carrying the unfinished part set below the surface of the resin. The computer controlled optical scanning system directs a laser beam across the top of the polymer, which selectively hardens the surface layer. The machine then lowers slightly to cover the top surface of the unfinished part with another layer of the liquid resin, continuing to harden layer by layer until the complete part is built. Layer thickness of the SLA 250 machines used in this study is set to 0.004 in. (0.1016 mm) (Fig. 1).
2.1. SLA machine error model
Mathematical models of machines based on rigid body assumptions vary according to machine types. In general, three-axes machines can be classified according to the motion of the probe and work piece as XYFZ, XYZF, XFYZ or FYXZ. In this representation, F designates the fixed machine Twenty-one parametric errors of a three axes system foundation while letters appearing to the right of F indicate work piece motion [23]
In the SLA 250 rapid prototyping machine, the focus of the laser beam corresponds to the tool tip in machine tools. It is directed by mirrors and can move in two orthogonal horizontal directions, which are defined as X and Y, respectively. The work platform carrying the work piece can move up and down along the Z-axis, with the vertical up as positive Z-direction. According to these features, SLA 250 corresponds to a ZFXY type machine. This classification of the RP machine type does not reflect the actual kinematics of the RP machine, but reflects the resulting motion of the tool tip, i.e., the laser focus in this case. As mentioned previously, the RP machine parametric errors will be determined by producing and measuring282 b
Fig. 1. The SLA machine.
an artifact part, rather than measuring the actual error of the motions of the individual axis of the machine. This allows for the selection of a “virtual” parametric error model of the machine which does not represent the actual machine kinematics. This is different from the traditional approach where the six error motions of each axis are directly measured to populate the individual homogeneous transformation matrices in the kinematics chain. When using “virtual” parametric error model, it has to be determined if the model can successfully produce compensation data to improve the machine accuracy. Compensation results discussed later in the paper will demonstrate the ability of this model to provide useful compensation data.
The same analysis can be applied to other RP machines. For example, in a FDM machine, the tool tip is a deposition nozzle driven by two carriages in X and Y directions and the table moves up and down in Z direction. A FDM machine thus corresponds to a ZFYX type machine. The kinematic axes chain vector diagram in Fig. 2 is used to build the error model for SLA 250.
Fig.2.The SLA250 machine axes chain vector diagram.
Under the conventional definition method [29,33],O is the fixed origin. X, Y and Z are the three axes vectors, including only the translational and squareness errors. The rotational errors will be considered separately later when writing the kinematical paths. T is the tool tip offsets xt , yt and zt with respect to the carriage to which the tool is attached. In the SLA 250 machine, there are no real X and Y carriages and the laser beam focus itself is considered the tool tip. Since the beam focus is always on the liquid surface, thus for the SLA 250 machine T = 0. For generality, T will be included in the error model derivations but dropped afterwards. W is the vector directed from the part origin to the laser focus in the part frame, which represents the actual size of the part. The components of the kinematical vector chain diagram are given below. Their structure is easily understood by rationalizing the effects of the machine errors on positioning ability [29]:
(1)
There are two kinematic paths from origin O to laser beam focus: Z → W and X → Y → T. In each kinematic path, the rotational error of the predecessors to an axis will affect the actual movement of that axis, thus the axis movement needs to be modified by the rotational error of its predecessors. Rotation of X, Y and Z axes can be represented by infinitesimal rotation matrices R(X), R(Y), and R (Z). Letting u represent X,Y or Z, the rotation matrix and its inverse are respectively:
There are two kinematic paths from origin O to laser beam focus: Z → W and X → Y → T. In each kinematic path, the rotational error of the predecessors to an axis will affect the actual movement of that axis, thus the axis movement needs to be modified by the rotational error of its predecessors. Rotation of X, Y and Z axes can be represented by infinitesimal rotation matrices R(X), R(Y), and R(Z). Letting u epresent X,Y or Z, the rotation matrix and its inverse are respectively:
In Fig. 2, Z and X have no predecessor. W has a predecessor axis Z, Y has a predecessor axis X, and T has two predecessor axes: X and Y. Thus, the two equivalent vector chains from origin O to laser beam focus can be expressed by the following equation:
After rearranging the terms, vector W is explicitly written as a function of all other vectors and matrices:
Substituting X, Y, Z, T, W and rotational matrixes into the vector machine model, the expressions for the coordinates of the laser focus in the work piece coordinates s
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