請用此 Handle URI 來引用此文件: http://hdl.handle.net/11455/18591
標題: 建立雷射追蹤儀量測工具機靜態幾何誤差之數學模式
A Mathematical Model for Measuring the Static Geometry Errors of a Machine Tool with a Laser Tracker System
作者: 何瑞樑
Her, Jui-Liang
關鍵字: triangular plate
三角鋁板
imagined pyramid
static geometry errors
laser tracker
sensor target ball
geometry errors on a circle path
angular errors of a rotary table
table flatness
假想三角錐體
靜態幾何誤差
雷射追蹤儀
感應標的球
圓形路徑之幾何誤差
旋轉台角度誤差
床台平面度
出版社: 應用數學系
摘要: 本文建立一套三角鋁板與假想三角錐體的數學模式,利用該模式於活動空間之移動翻轉變化情形,求出工具機之靜態幾何誤差,包含九個線性誤差δ j (i)與九個角度誤差ε j (i)。所謂線性誤差δ j (i)是指當機器主軸沿著i軸方向移動時在j軸方向所產生的位置偏移量;而角度誤差ε j (i) 則是指當機器主軸沿著i軸方向移動時在繞著j軸方向所產生的角度誤差。 文中提出特製三角鋁板一塊,於該板面銑三個凹穴,以置入感應標的球。量測時,將此三角鋁板固定在機器主軸上,使其得以隨機器主軸於活動空間中運動。每當鋁板運動至欲量測位置而停止時,即將感應標的球輪流置入三個凹穴中,於感應球移動過程,雷射追蹤儀的雷射頭將隨感應球慢速轉動至定位,從而測出凹穴之空間座標。本數學模式乃建構在雷射追蹤儀之量測模式上,而將鋁板上三點設計成三個凹穴以置入感應球,倘若有更適合的儀器可量測三角鋁板上既定三點座標,則亦適用於本數學模式。文中於其中一個凹穴建立一假想三角錐體,利用鋁板移動前後凹穴之座標變化與剛體運動特性,配合該假想三角錐體,可計算出鋁板之移動量與轉動量,從而求出此機器在該位置的線性誤差與角度誤差。其中包括位置誤差、傾角誤差、翻轉誤差與偏擺誤差等。根據ASME B5.54 標準第5.9.2節所提,藉由沿活動空間對角線位移精度的量測,可快速得知該體積的精度。乃因對角線的量測對所有誤差都很敏感,諸如線性位置誤差、角度誤差及直角度誤差等,故本文採對角線的量測方法以完成靜態幾何誤差之量測;由活動空間某一下方角落,將三角鋁板逐步朝對角線方向移動至上方對角處,真正移動路徑為先沿平行x軸移動至一固定距離後停止,隨即量測三個凹穴之座標,接著平行y軸移動另一固定距離,同樣量測出凹穴的座標後,再平行z軸移動回到對角線上,並量測之,至此完成第一個量測循環。依此方式逐步完成每一量測循環,直到三角鋁板到達活動空間之上方對角處時,即完成整個量測工作。 本文以電腦模擬方式來驗證三角鋁板與假想三角錐之數學模式的正確性。另提出雷射追蹤儀配合三角鋁板之量測模式,以量測空間任何平面上圓形路徑之幾何誤差、旋轉台角度誤差及床台或垂直牆之平面度等。 未來研究方向,可分為三方面:其一為實際量測,在實測過程,可探討實驗誤差、可靠度分析等等;其二是量測三個座標軸之直角度誤差;其三則為發展線上或離線之誤差補償方法。
This dissertation proposes a mathematical model for measuring the static geometry errors of a machine tool. The mathematical model consists of a triangular plate with three caves on it and a pyramid is imagined on one of the three caves. The model is based on the measurement behavior of a laser tracker system. Three caves are then needed on the plate for putting sensor ball on the three specific points there. If there exits another suitable instrument that can measure the three specific points on the plate, then the caves must be changed to fit the new requirement. The static geometry errors include nine linear errors and nine angular ones. The linear error δ j (i) is defined as the linear error happened in j-axis direction due to the step movement in i-axis direction. And the angular error ε j (i) is defined as the angle error rotated about j-axis due to the step movement in i-axis direction. The static geometry errors of a machine tool can be calculated with the turning over rigid body motion of the plate and the imagined pyramid. The plate is fixed on the machine spindle and moves with it steps by steps toward the diagonal of the working space. After setup, the plate is moved to the place where the location is wanted and then stops there. Then a sensor ball is put into the three caves on the plate sequentially and slowly. When the sensor ball moves from one cave to the other, the laser tracker will rotate automatically with it and then detect the coordinates of each cave. There is an imagined pyramid created on one of the three caves. The movement and rotation of the plate can be calculated by means of the coordinates of the three caves and the imagined pyramids before and after the plate's rigid body motion. Then the eighteen static geometry errors including linear positional errors and angular pitch, roll, yaw errors of the machine tool at that position can be calibrated. The body diagonal measurement has been recommended for a quick check of the volumetric accuracy in the ASME B5.54 standard [1], section 5.9.2. This is because the body diagonal measurement is sensitive to all the errors such as the position errors, straightness errors, squareness errors, and the angular errors. Thus, it is a good way to check the volumetric accuracy. The measurements in this dissertation is just completed by moving the machine spindle from one lower corner of the working space to the opposite upper corner along the diagonal line steps by steps. The real paths of movements are moving in x, y, z-axis direction, respectively. The simulation proves the correctness of the mathematical model in a CAD/CAM system named CATIA. The method proposed here has the merits of easy setup, quick measurement and accuracy. The model of laser tracker system and triangular plate is also proposed to measure the geometry errors on a circle path and the angular errors of a rotary index table. The flatness situation of a machine table is also studied. The further work of experiment is expected to be finished recently. In the experimental study, some topics will be discussed, including measurement errors and uncertainty analysis, etc. The method is expected to measure the squareness errors in the future. And the on-line or off-line compensation can also be considered together in the final system.
URI: http://hdl.handle.net/11455/18591
顯示於類別:應用數學系所

文件中的檔案:
沒有與此文件相關的檔案。


在 DSpace 系統中的文件,除了特別指名其著作權條款之外,均受到著作權保護,並且保留所有的權利。