{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T18:42:33Z","timestamp":1780512153002,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2013,5,17]],"date-time":"2013-05-17T00:00:00Z","timestamp":1368748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Strain distributions are crucial criteria of cross-beams six-axis force\/torque sensors. The conventional method for calculating the criteria is to utilize Finite Element Analysis (FEA) to get numerical solutions. This paper aims to obtain analytical solutions of strains under the effect of external force\/torque in each dimension. Genetic mechanical models for cross-beams six-axis force\/torque sensors are proposed, in which deformable cross elastic beams and compliant beams are modeled as quasi-static Timoshenko beam. A detailed description of model assumptions, model idealizations, application scope and model establishment is presented. The results are validated by both numerical FEA simulations and calibration experiments, and test results are found to be compatible with each other for a wide range of geometric properties. The proposed analytical solutions are demonstrated to be an accurate estimation algorithm with higher efficiency.<\/jats:p>","DOI":"10.3390\/s130506669","type":"journal-article","created":{"date-parts":[[2013,5,17]],"date-time":"2013-05-17T12:10:40Z","timestamp":1368792640000},"page":"6669-6686","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":65,"title":["Fast Estimation of Strains for Cross-Beams Six-Axis Force\/Torque Sensors by Mechanical Modeling"],"prefix":"10.3390","volume":"13","author":[{"given":"Junqing","family":"Ma","sequence":"first","affiliation":[{"name":"Jiangsu Key Lab of Remote Measurement and Control, School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aiguo","family":"Song","sequence":"additional","affiliation":[{"name":"Jiangsu Key Lab of Remote Measurement and Control, School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2013,5,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Siciliano, B., and Khatib, O. (2008). Springer Handbook of Robotics, Springer-Verlag.","DOI":"10.1007\/978-3-540-30301-5"},{"key":"ref_2","unstructured":"Roberts, R., Paul, R., and Hillberry, B. (March, January 25\u2013). The Effect of Wrist Force Sensor Stiffness on the Control of Robot Manipulators. St. Louis, MO, USA."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Siciliano, B., and Villani, L. (1999). Robot Force Control, Springer.","DOI":"10.1007\/978-1-4615-4431-9"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2094","DOI":"10.1109\/TIM.2007.904553","article-title":"Dynamic modeling and compensation of robot six-axis wrist force\/torque sensor","volume":"56","author":"Xu","year":"2007","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0924-4247(93)80153-8","article-title":"Mechanical analysis of a novel 6-degree-of-freedom wrist force sensor","volume":"35","author":"Huang","year":"1993","journal-title":"Sens. Actuat. A Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0924-4247(97)01534-3","article-title":"Shape optimal design and force sensitivity evaluation of six-axis force sensors","volume":"63","author":"Chao","year":"1997","journal-title":"Sens. Actuat. A Phys."},{"key":"ref_7","unstructured":"Vladimir, S. (1987). Robotics: Control, Sensing, Vision, and Intelligence, McGraw-Hill."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1002\/rob.4620060206","article-title":"Six-axis force sensor evaluation and a new type of optimal frame truss design for robotic applications","volume":"6","author":"Bayo","year":"1989","journal-title":"J. Robot. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1016\/j.measurement.2006.11.018","article-title":"A novel self-decoupled four degree-of-freedom wrist force\/torque sensor","volume":"40","author":"Song","year":"2007","journal-title":"Measurement"},{"key":"ref_10","unstructured":"Chen, L., and Song, A. (April, January 11-). A Novel Three Degree-of-Freedom Force Sensor. Zhangjiajie, China."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"299","DOI":"10.4028\/www.scientific.net\/AMM.103.299","article-title":"Development of a novel two-axis force sensor for chinese massage robot","volume":"103","author":"Ma","year":"2012","journal-title":"Appl. Mechan. Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1080\/14786442108636264","article-title":"LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars","volume":"41","author":"Timoshenko","year":"1921","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1080\/14786442208633855","article-title":"On the transverse vibrations of bars of uniform cross-section","volume":"43","author":"Timoshenko","year":"1922","journal-title":"Philos. Mag."},{"key":"ref_14","unstructured":"Timoshenko, S., and Gere, J. (1972). Mechanics of Materials, Brooks\/Cole Engineering Division."},{"key":"ref_15","unstructured":"Vladimir, S. (2006). Mechanics of Structural Elements: Theory and Applications, Springer."},{"key":"ref_16","unstructured":"Pilkey, W.D., and Pilkey, H.O. (1986). Mechanics of Solids, RE Krieger Publishing Company."},{"key":"ref_17","unstructured":"Ugural, A., and Saul, K. (2003). Advanced Strength and Applied Elasticity, Prentice Hall."},{"key":"ref_18","unstructured":"Gere, J., and Goodno, B. (2008). Mechanics of Materials, Cengage Learning."},{"key":"ref_19","unstructured":"Sadd, M.H. (2009). Elasticity: Theory, Applications, and Numerics, Academic Press."},{"key":"ref_20","unstructured":"(2007). Coupled Field Analysis Guide, Help System; Release 14.0; ANSYS Inc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"14537","DOI":"10.3390\/s121114537","article-title":"A robust static decoupling algorithm for 3-axis force sensors based on coupling error model and \u03b5-SVR","volume":"12","author":"Ma","year":"2012","journal-title":"Sensors"},{"key":"ref_22","unstructured":"Voyles, R., Morrow, J., and Khosla, P. (November, January 12-). Shape from Motion Approach to Rapid and Precise Force\/torque Sensor Calibration. San Francisco, CA, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.1007\/s11771-011-0880-6","article-title":"Nonlinear correction of photoelectric displacement sensor based on least square support vector machine","volume":"18","author":"Guo","year":"2011","journal-title":"J. Cent. South Univ. Technol."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/5\/6669\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:46:48Z","timestamp":1760219208000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/5\/6669"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,5,17]]},"references-count":23,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2013,5]]}},"alternative-id":["s130506669"],"URL":"https:\/\/doi.org\/10.3390\/s130506669","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,5,17]]}}}