{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T03:22:49Z","timestamp":1777778569722,"version":"3.51.4"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,10]],"date-time":"2018-11-10T00:00:00Z","timestamp":1541808000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["MOST 106-2628-E-006-010-MY3 and 107-2218-E-194-002"],"award-info":[{"award-number":["MOST 106-2628-E-006-010-MY3 and 107-2218-E-194-002"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This study designs and characterizes a novel precise measurement system for simultaneously measuring six-degree-of-freedom geometric motion errors of a long linear stage of a machine tool. The proposed measurement system is based on a method combined with the geometrical optics method and laser interferometer method. In contrast to conventional laser interferometers using only the interferometer method, the proposed measurement system can simultaneously measure six-degree-of-freedom geometric motion errors of a long linear stage with lower cost and faster operational time. The proposed measurement system is characterized numerically using commercial software ZEMAX and mathematical modeling established by using a skew-ray tracing method, a homogeneous transformation matrix, and a first-order Taylor series expansion. The proposed measurement system is then verified experimentally using a laboratory-built prototype. The experimental results show that, compared to conventional laser interferometers, the proposed measurement system better achieves the ability to simultaneously measure six-degree-of-freedom geometric errors of a long linear stage (a traveling range of 250 mm).<\/jats:p>","DOI":"10.3390\/s18113875","type":"journal-article","created":{"date-parts":[[2018,11,14]],"date-time":"2018-11-14T10:58:22Z","timestamp":1542193102000},"page":"3875","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["Design of a Measurement System for Simultaneously Measuring Six-Degree-Of-Freedom Geometric Errors of a Long Linear Stage"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3537-5828","authenticated-orcid":false,"given":"Chien-Sheng","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, National Cheng Kung University, Tainan City 70101, Taiwan"},{"name":"Department of Mechanical Engineering, National Chung Cheng University, Chiayi County 62102, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-Fan","family":"Pu","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National Chung Cheng University, Chiayi County 62102, Taiwan"},{"name":"Advanced Institute of Manufacturing with High-tech Innovations, National Chung Cheng University, Chiayi County 62102, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-Ta","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National Chung Cheng University, Chiayi County 62102, Taiwan"},{"name":"Advanced Institute of Manufacturing with High-tech Innovations, National Chung Cheng University, Chiayi County 62102, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong-Tai","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National Chung Cheng University, Chiayi County 62102, Taiwan"},{"name":"Advanced Institute of Manufacturing with High-tech Innovations, National Chung Cheng University, Chiayi County 62102, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1007\/s12541-012-0199-8","article-title":"Uncertainty investigation of grating interferometry in six degree-of-freedom motion error measurements","volume":"13","author":"Lee","year":"2012","journal-title":"Int. J. Precis. Eng. Manuf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s12206-012-1217-6","article-title":"Multi-degree-of-freedom motion error measurement in an ultraprecision machine using laser encoder \u2013 Review","volume":"27","author":"Lee","year":"2013","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gao, W. (2010). Precision Nanometrology: Sensors and Measuring Systems for Nanomanufacturing, Springer.","DOI":"10.1007\/978-1-84996-254-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1243\/09544054JEM1620","article-title":"Development of a straightness measuring system and compensation technique using multiple corner cubes for precision stages","volume":"224","author":"Liu","year":"2010","journal-title":"Proc. IMechE Part B J. Eng. Manuf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1243\/09544054JEM880","article-title":"Development of a three-dimensional contouring measuring system and error compensation method for a CNC machine tool","volume":"221","author":"Jywe","year":"2007","journal-title":"Proc. IMechE Part B J. Eng. Manufact."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"055110","DOI":"10.1063\/1.1915520","article-title":"Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage","volume":"76","author":"Liu","year":"2005","journal-title":"Rev. Sci. Instrum."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.optlaseng.2016.10.026","article-title":"Design and experimental verification of novel six-degree-of freedom geometric error measurement system for linear stage","volume":"92","author":"Chen","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/S0141-6359(99)00021-5","article-title":"A 6-degree-of-freedom measurement system for the accuracy of X-Y stages","volume":"24","author":"Fan","year":"2000","journal-title":"Precis. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6735","DOI":"10.1364\/OE.24.006735","article-title":"System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser","volume":"24","author":"Cui","year":"2016","journal-title":"Opt. Express"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1115\/1.2035692","article-title":"Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool","volume":"127","author":"Lee","year":"2005","journal-title":"J. Manuf. Sci. Eng.-Trans. ASME"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"25805","DOI":"10.1364\/OE.21.025805","article-title":"Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide","volume":"21","author":"Feng","year":"2013","journal-title":"Opt. Express"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1007\/s00170-008-1743-7","article-title":"Development of an optical measuring system for integrated geometric errors of a three-axis miniaturized machine tool","volume":"43","author":"Wang","year":"2009","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.ast.2014.11.012","article-title":"Real-time estimation of helicopter blade kinematics using integrated linear displacement sensors","volume":"42","author":"Allred","year":"2015","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1016\/j.mechatronics.2011.09.001","article-title":"Six d.o.f. displacement measuring device based on a modified Stewart platform","volume":"21","author":"Mura","year":"2011","journal-title":"Mechatronics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/s00542-012-1470-8","article-title":"Multi-dofs MEMS displacement sensors based on the Stewart platform theory","volume":"18","author":"Mura","year":"2012","journal-title":"Microsyst. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1177\/0954406213482071","article-title":"Sensitivity analysis of a six degrees of freedom displacement measuring device","volume":"228","author":"Mura","year":"2014","journal-title":"Proc. Inst. Mech. Eng. C"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"065109","DOI":"10.1063\/1.4953335","article-title":"Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology","volume":"87","author":"Yu","year":"2016","journal-title":"Rev. Sci. Instrum."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1080\/23080477.2016.1207407","article-title":"High precision optical sensors for real-time on-line measurement of straightness and angular errors for smart manufacturing","volume":"4","author":"Lee","year":"2016","journal-title":"Smart Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2451","DOI":"10.1364\/OE.23.002451","article-title":"Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements","volume":"23","author":"Hsieh","year":"2015","journal-title":"Opt. Express"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.sna.2005.05.022","article-title":"A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module","volume":"125","author":"Kuang","year":"2005","journal-title":"Sens. Actuators A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"095105","DOI":"10.1063\/1.2786272","article-title":"A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques","volume":"78","author":"Kuang","year":"2007","journal-title":"Rev. Sci. Instrum."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.precisioneng.2005.06.003","article-title":"Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage","volume":"30","author":"Gao","year":"2006","journal-title":"Precis. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1016\/j.precisioneng.2013.03.005","article-title":"A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage","volume":"37","author":"Li","year":"2013","journal-title":"Precis. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"9052","DOI":"10.1364\/OE.23.009052","article-title":"Laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters","volume":"23","author":"Chen","year":"2015","journal-title":"Opt. Express"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"066103","DOI":"10.1063\/1.2743165","article-title":"Development of a three-degree-of-freedom laser linear encoder for error measurement of a high precision stage","volume":"78","author":"Huang","year":"2007","journal-title":"Rev. Sci. Instrum."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2767","DOI":"10.1364\/AO.48.002767","article-title":"Five-degrees-of-freedom diffractive laser encoder","volume":"48","author":"Liu","year":"2009","journal-title":"Appl. Opt."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"054005","DOI":"10.1088\/0957-0233\/21\/5\/054005","article-title":"Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder","volume":"21","author":"Kimura","year":"2010","journal-title":"Meas. Sci. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3122","DOI":"10.1364\/AO.54.003122","article-title":"Compensation for straightness measurement systematic errors in six degree-of-freedom motion error simultaneous measurement system","volume":"54","author":"Cui","year":"2015","journal-title":"Appl. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1364\/AO.54.000458","article-title":"Errors crosstalk analysis and compensation in the simultaneous measuring system for five-degree-of-freedom geometric error","volume":"54","author":"Gao","year":"2015","journal-title":"Appl. Opt."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6805","DOI":"10.1364\/OE.25.006805","article-title":"Laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology","volume":"25","author":"Lou","year":"2017","journal-title":"Opt. Express"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"20993","DOI":"10.1364\/OE.25.020993","article-title":"Measurement system and model for simultaneously measuring 6DOF geometric errors","volume":"25","author":"Zhao","year":"2017","journal-title":"Opt. Express"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"105101","DOI":"10.1088\/0957-0233\/24\/10\/105101","article-title":"A novel laser displacement sensor with improved robustness toward geometrical fluctuations of the laser beam","volume":"24","author":"Liu","year":"2013","journal-title":"Meas. Sci. Technol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1016\/j.optlaseng.2014.10.004","article-title":"Precise autofocusing microscope with rapid response","volume":"66","author":"Liu","year":"2015","journal-title":"Opt. Lasers Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1717","DOI":"10.1007\/s00542-013-1883-z","article-title":"Design and characterization of precise laser-based autofocusing microscope with reduced geometrical fluctuations","volume":"19","author":"Liu","year":"2015","journal-title":"Microsyst. Technol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"122408","DOI":"10.1117\/1.OE.53.12.122408","article-title":"Numerical and experimental characterization of reducing geometrical fluctuations of laser beam based on rotating optical diffuser","volume":"53","author":"Liu","year":"2014","journal-title":"Opt. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2272","DOI":"10.1364\/JOSAA.29.002272","article-title":"Jacobian and Hessian matrices of optical path length for computing the wave front shape, irradiance, and caustics in optical systems","volume":"29","author":"Liu","year":"2012","journal-title":"J. Opt. Soc. Am. A-Opt. Image Sci. Vis."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Chen, Y.T., Huang, Y.S., and Liu, C.S. (2016). An optical sensor for measuring the position and slanting direction of flat surfaces. Sensors, 16.","DOI":"10.3390\/s16071061"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1364\/JOSAA.33.000785","article-title":"Free-form surface design method for a collimator TIR lens","volume":"33","author":"Tsai","year":"2016","journal-title":"J. Opt. Soc. Am. A-Opt. Image Sci. Vis."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lin, P.D. (2013). New Computation Methods for Geometrical Optics, Springer.","DOI":"10.1007\/978-981-4451-79-6"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Rodr\u00edguez-Navarro, D., L\u00e1zaro-Galilea, J.L., Bravo-Mu\u00f1oz, I., Gardel-Vicente, A., and Tsirigotis, G. (2016). Analysis and calibration of sources of electronic error in PSD sensor response. Sensors, 16.","DOI":"10.3390\/s16050619"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3875\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:29:03Z","timestamp":1760196543000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3875"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,10]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["s18113875"],"URL":"https:\/\/doi.org\/10.3390\/s18113875","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,10]]}}}