在最近製造業和工業4.0大規模定制的趨勢中，機械手臂是用於以更有效的方式操作工程任務以響應消費者需求。機械手臂是可靠的，因為它能夠以高精度（即高精度的可重複性）執行重複任務。然而，分配任務的運動軌跡通常由人來定義。手動分配運動軌跡可能需要一些時間，並且可能降低質量效率定制生產線。本文介紹如何使用拉線編碼器來測量不同負載條件下機械手臂的末端效應器絕對位置的三維誤差使用Kriging參數化模擬末端效應器工作空間中的誤差修正。每根拉線編碼器用於測量其與末端效應器的距離。三邊測量被用來計算末端執行器的三維座標定位。目標座標之間的絕對誤差測量坐標計算並通過Kriging進行參數化建模。裝載條件也被認為是克里金模型。隨著基於建立的克里金模型的修正，末端效應器的絕對精度可以從1毫米提高到小於0.5毫米。這項研究旨在為製造商提供有效的測量和克利金方法來校準末端效應器定位在工作空間中，更有效地執行大規模定制任務。

In the recent trend of mass customization in manufacturing and Industry 4.0, industrial robots are used to operate engineering tasks in a more efficient manner to react to consumer demand. Robot is reliable because it is capable of performing repeated tasks with high accuracy (i.e. high accuracy of repeatability). However, the motion trajectories of the assign tasks are often defined by human. Manuel assignment of the motion trajectories may take some time and it could reduce the efficiency of mass customization manufacturing line. This paper presents how to use cable encoders to measure the three-dimensional errors of the end effector absolute positions under different loading conditions and parametrically model the error fields in the working space of the end effector using Kriging. Each cable encoder was used to measure its distance from the end effector. Trilateration was used to calculate the three-dimensional coordinate of the end effector. The absolute error between the target coordinate the measured coordinate was computed and parametrically modeled by Kriging. The loading conditions were also considered in the Kriging models. With the correction based on the built Kriging models, the end effector absolute accuracy can be improved from millimeters to less than 0.5 mm. This research aims to provide manufacturers an effective measurement and Kriging modeling method to calibrate the end effector positioning in the working space and more efficiently perform mass-customization tasks.

R. Mautz, "Overview of current indoor positioning systems," Geodezija ir kartografija, vol. 35, no. 1, pp. 18-22, 2009.

阮柏鈞, "使用拉線編碼器及克利金模型的三邊量測法進行三維定位," 中原大學機械工程研究所學位論文, pp. 1-243, 2017.

B. Cook, G. Buckberry, I. Scowcroft, J. Mitchell, and T. Allen, "Indoor location using trilateration characteristics," in Proc. london communications symposium, 2005, pp. 147-150: Citeseer.

L. Peneda, A. Azenha, and A. Carvalho, "Trilateration for indoors positioning within the framework of wireless communications," in 2009 35th Annual Conference of IEEE Industrial Electronics, 2009, pp. 2732-2737: IEEE.

S. Ye, Y. Chen, and T. Hu, "Evolutionary algorithmic deployment of radio beacons for indoor positioning," in 2016 IEEE Congress on Evolutionary Computation (CEC), 2016, pp. 2829-2835: IEEE.

S. Subedi, G.-R. Kwon, S. Shin, S.-s. Hwang, and J.-Y. Pyun, "Beacon based indoor positioning system using weighted centroid localization approach," in 2016 Eighth International Conference on Ubiquitous and Future Networks (ICUFN), 2016, pp. 1016-1019: IEEE.

L. Angrisani, P. Arpaia, and D. Gatti, "Fast beacon recognition for accurate ultrasonic indoor positioning," in 2017 IEEE International Workshop on Measurement and Networking (M&N), 2017, pp. 1-5: IEEE.

"ASM. Available: https://www.asm-sensor.com/en."

"Linear Encoder. Available: https://www.asm-sensor.com/en/produkt-detail-import-en.html?page=120&prod=6."

"Arduino UNO. Available: https://store.arduino.cc/usa/arduino-uno-rev3."

格雷碼 (Gray Code). Available: https://en.wikipedia.org/wiki/Gray_code

"Bell Labs. Available: https://en.wikipedia.org/wiki/Bell_Labs."

J. Denavit, "A kinematic notation for lower-pair mechanisms based on matrices," ASME J. Appl. Mech., pp. 215-221, 1955.

"HIWIN_RA605. Available: http://www.hiwin.tw/Products/Products_robot.aspx?type=robot&subtype=robot_six_axis."

M. Raghavan and B. Roth, "Inverse kinematics of the general 6R manipulator and related linkages," Journal of Mechanical Design, vol. 115, no. 3, pp. 502-508, 1993.

J. Zhao and N. I. Badler, "Inverse kinematics positioning using nonlinear programming for highly articulated figures," ACM Transactions on Graphics (TOG), vol. 13, no. 4, pp. 313-336, 1994.

D. Tolani, A. Goswami, and N. I. Badler, "Real-time inverse kinematics techniques for anthropomorphic limbs," Graphical models, vol. 62, no. 5, pp. 353-388, 2000.

D. Krige, Two-dimensional weighted moving average trend surfaces for ore evaluation. South African Institute of Mining and Metallurgy Johannesburg, 1966.

D. Krige and H. Ueckermann, "Value contours and improved regression techniques for ore reserve valuations," Journal of the South African Institute of Mining and Metallurgy, vol. 63, no. 10, pp. 429-452, 1963.

D. Krige, "On the departure of ore value distributions from the lognormal model in South African gold mines," Journal of the Southern African Institute of Mining and Metallurgy, vol. 61, no. 4, pp. 231-244, 1960.

D. G. Krige and E. J. Magri, "Studies of the effects of outliers and data transformation on variogram estimates for a base metal and a gold ore body," Mathematical Geology, vol. 14, no. 6, pp. 557-564, 1982.

M. Armstrong, "Problems with universal kriging," Mathematical Geology, vol. 16, no. 1, pp. 101-108, 1984.

D. J. Brus and G. B. Heuvelink, "Optimization of sample patterns for universal kriging of environmental variables," Geoderma, vol. 138, no. 1-2, pp. 86-95, 2007.

D. Zimmerman, C. Pavlik, A. Ruggles, and M. P. Armstrong, "An experimental comparison of ordinary and universal kriging and inverse distance weighting," Mathematical Geology, vol. 31, no. 4, pp. 375-390, 1999.

C. V. Deutsch, "Correcting for negative weights in ordinary kriging," Computers & Geosciences, vol. 22, no. 7, pp. 765-773, 1996.

D. L. Zimmerman and M. B. Zimmerman, "A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors," Technometrics, vol. 33, no. 1, pp. 77-91, 1991.

X. Emery, "Simple and ordinary multigaussian kriging for estimating recoverable reserves," Mathematical Geology, vol. 37, no. 3, pp. 295-319, 2005.

N. Cressie, "The origins of kriging," Mathematical geology, vol. 22, no. 3, pp. 239-252, 1990.

Po-Chun Juan, Wei-Hao Lu, Shu-Ping Lin, and P. T. Lin, "Accuracy Improvement of Cyber-Physical Robots (CPR)," presented at the 12th World Congress of Structural and Multidisciplinary Optimization (WCSMO 2017), Braunschweig, Germany.

Wei-Liang Lian, Shu-Ping Lin, Po-Chun Juan, Wei-Hao Lu, and P. T. Lin, "A Kriging-based Absolute Error Correction in Robot Control," presented at the Asian Congress of Structural and Multidisciplinary Optimization, ACSMO 2016, Paper Number 190, Nagasaki, Japan.