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To address the need for scalability and facilitate manufacturing, this study proposes a novel infinite expansion magnetically levitated planar motor (MLPM) based on PCB stator coils. Different from existing magnetic levitation systems that use PCB coils, the design presented in this paper utilizes smaller coil units, with each coil being independent of one another. The coils are structured in a spiral pattern on a 16-layer PCB, comprising 15 layers of coils, while the last layer is dedicated to wiring and other circuits. Magnetic field modeling is conducted for both the stator coil and the 2D Halbach array structure employed in the system. A simple table lookup method is employed to accurately account for the prevalent end effects observed during system motion. Additionally, the decoupling effect of magnetic force and torque is evaluated by solving for the current vector at different points along a specific trajectory. To verify the accuracy of the proposed system\u2019s modeling, a prototype is developed and tested. Experimental results demonstrate that compared to traditional harmonic model methods, the proposed approach improves the calculation accuracy of magnetic force by 50.31% and torque by 70.65%. This study presents a new MLPM system with vast potential applications in precision manufacturing and robotics. The innovative design and improved performance characteristics make it a promising technology for enhancing the capabilities of worktables in precision machining fields.<\/jats:p>","DOI":"10.3390\/s23218735","type":"journal-article","created":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T07:22:15Z","timestamp":1698304935000},"page":"8735","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Force and Torque Model of Magnetically Levitated System with 2D Halbach Array and Printed Circuit Board Coils"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-6162-6633","authenticated-orcid":false,"given":"Menglong","family":"Zou","sequence":"first","affiliation":[{"name":"School of Electrical Information, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8971-3573","authenticated-orcid":false,"given":"Mingxing","family":"Song","sequence":"additional","affiliation":[{"name":"School of Electrical Information, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9508-2867","authenticated-orcid":false,"given":"Shun","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Electrical Information, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4604-6445","authenticated-orcid":false,"given":"Xianze","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Electrical Information, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5718-0853","authenticated-orcid":false,"given":"Fengqiu","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Electrical Information, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Jeong, J.-H., Ryu, J., and Gweon, D.-G. (2017, January 22\u201325). Feedforward reference compensation using bilinear interpolation for long range motion of six degrees-of-freedom magnetic levitation planar motor. Proceedings of the 2017 8th International Conference on Mechanical and Aerospace Engineering (ICMAE), Prague, Czech Republic.","DOI":"10.1109\/ICMAE.2017.8038620"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1109\/TIE.2018.2838110","article-title":"Convergence Rate Oriented Iterative Feedback Tuning With Application to an Ultraprecision Wafer Stage","volume":"66","author":"Li","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5709","DOI":"10.1109\/TIE.2016.2589220","article-title":"Modeling, Analysis, and Advanced Control in Motion Control Systems\u2014Part I","volume":"63","author":"Li","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1109\/28.491486","article-title":"Design and Analysis Framework for Linear Permanent-magnet Machines","volume":"32","author":"Trumper","year":"1996","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Pieters, R., Tung, H.-W., Sargent, S.C.D.F., and Nelson, B.J. (2015, January 26\u201330). RodBot: A rolling microrobot for micromanipulation. Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA.","DOI":"10.1109\/ICRA.2015.7139764"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1109\/TMECH.2011.2121090","article-title":"Bilateral Macro\u2013Micro Teleoperation Using Magnetic Levitation","volume":"16","author":"Mehrtash","year":"2011","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1016\/j.mechatronics.2012.08.002","article-title":"6-DoF miniature maglev positioning stage for application in haptic micro-manipulation","volume":"22","author":"Estevez","year":"2012","journal-title":"Mechatronics"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yu, J., Li, D., Zhu, C., Ouyang, Q., Miao, C., and Yu, H. (2023). A Magnetic Levitation System for Range\/Sensitivity-Tunable Measurement of Density. Sensors, 23.","DOI":"10.3390\/s23083955"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Almobaied, M., Al-Nahhal, H.S., Arrieta, O., and Vilanova, R. (2023). Design a Robust Proportional-Derivative Gain-Scheduling Control for a Magnetic Levitation System. Mathematics, 11.","DOI":"10.3390\/math11194040"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kumar, P., Ansari, M., Toyserkani, E., and Khamesee, M.B. (2023). Experimental Implementation of a Magnetic Levitation System for Laser-Directed Energy Deposition via Powder Feeding Additive Manufacturing Applications. Actuators, 12.","DOI":"10.3390\/act12060244"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kim, S.-K. (2019). Nonlinear Position Stabilizing Control with Active Damping Injection Technique for Magnetic Levitation Systems. Electronics, 8.","DOI":"10.3390\/electronics8020221"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Castellanos Molina, L.M., Galluzzi, R., Bonfitto, A., Tonoli, A., and Amati, N. (2018). Magnetic Levitation Control Based on Flux Density and Current Measurement. Appl. Sci., 8.","DOI":"10.3390\/app8122545"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Jung, D.-H., and Lim, J.S. (2023). A Study on the Control Method of 6-DOF Magnetic Levitation System Using Non-Contact Position Sensors. Sensors, 23.","DOI":"10.3390\/s23020905"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"He, Z., Wen, T., Liu, X., and Suo, Y. (2022). Loss Estimation and Thermal Analysis of a Magnetic Levitation Reaction Flywheel with PMB and AMB for Satellite Application. Energies, 15.","DOI":"10.3390\/en15041584"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/S0141-6359(98)00009-9","article-title":"High-precision magnetic levitation stage for photolithography","volume":"22","author":"Kim","year":"1998","journal-title":"Precis. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1109\/TMECH.2014.2344692","article-title":"Autotuning Controller for Motion Control System Based on Intelligent Neural Network and Relay Feedback Approach","volume":"20","author":"Nguyen","year":"2015","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1320","DOI":"10.1109\/TMECH.2014.2342752","article-title":"A New Flexure-Based Nanomanipulator With Nanometer-Scale Resolution and Millimeter-Scale Workspace","volume":"20","author":"Tang","year":"2015","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_18","first-page":"329","article-title":"Integrated Servo-Mechanical Design of a Fine Stage for a Coarse\/Fine Dual-Stage Positioning System","volume":"21","author":"Zhu","year":"2016","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3522","DOI":"10.1109\/TIE.2016.2522389","article-title":"Multiobjective Optimization of a Magnetically Levitated Planar Motor With Multilayer Windings","volume":"63","author":"Guo","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7621","DOI":"10.1109\/TIE.2016.2592866","article-title":"Parallel Computation of Wrench Model for Commutated Magnetically Levitated Planar Actuator","volume":"63","author":"Xu","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TMAG.2019.2940423","article-title":"Semi-Analytical Solution of Magnetic Force and Torque for a Novel Magnetically Levitated Actuator in Rotary Table","volume":"55","author":"Lu","year":"2019","journal-title":"IEEE Trans. Magn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TMAG.2015.2461611","article-title":"Force Ripple Attenuation of 6-DOF Direct Drive Permanent Magnet Planar Levitating Synchronous Motors","volume":"51","author":"Usman","year":"2015","journal-title":"IEEE Trans. Magn."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wang, L., Huang, S., Cao, G., and Huang, Z. (2021, January 1\u20133). A New Four-Phase Current Distribution Strategy for the Novel 6-DOF Maglev Motor. Proceedings of the 2021 13th International Symposium on Linear Drives for Industry Applications (LDIA), Wuhan, China.","DOI":"10.1109\/LDIA49489.2021.9505904"},{"key":"ref_24","unstructured":"Gloess, R. (2016, January 23\u201328). MAG-6D A New Mechatronics Design of A Levitation Stage with Nanometer Resoultion. Proceedings of the 31st ASPE Annual Meeting, Portland, OR, USA."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1109\/TMAG.2006.886051","article-title":"Modeling of Magnetically Levitated Planar Actuators With Moving Magnets","volume":"43","author":"Jansen","year":"2007","journal-title":"IEEE Trans. Magn."},{"key":"ref_26","first-page":"1","article-title":"Real-Time Data-Driven Force and Torque Modeling on a 2-D Halbach Array by a Symmetric Coil Considering End Effect","volume":"58","author":"Xu","year":"2022","journal-title":"IEEE Trans. Magn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8545","DOI":"10.1109\/TIE.2019.2949519","article-title":"Motion Control of a Magnetic Levitation Actuator Based on a Wrench Model Considering Yaw Angle","volume":"67","author":"Xu","year":"2020","journal-title":"IEEE Trans. Ind. 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