{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T13:11:21Z","timestamp":1769001081535,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T00:00:00Z","timestamp":1717372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Xiamen Municipal Bureau of Science and Technology under the Natural Science Foundation of Xiamen, China","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Fujian Provincial Department of Science and Technology under the Natural Science Foundation of Fujian Province, Chin","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Fujian Provincial Department of Education under the Youth Foundation of Fujian Educational Committee, Fujian, China","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Distinguished Scholars, Xiamen University of Technology of China","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Xiamen University of Technology under the Startup Fund for Graduate Innovation, Xiamen University of Technology of China","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["3502Z20227072"],"award-info":[{"award-number":["3502Z20227072"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["2022J05286"],"award-info":[{"award-number":["2022J05286"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["2023J011431"],"award-info":[{"award-number":["2023J011431"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["JAT210343"],"award-info":[{"award-number":["JAT210343"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["YKJ22019R"],"award-info":[{"award-number":["YKJ22019R"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["YKJCX2023135"],"award-info":[{"award-number":["YKJCX2023135"]}]},{"name":"Key Education and Research Projects of Fujian Province for Young and Middle aged Teachers","award":["JZ230048"],"award-info":[{"award-number":["JZ230048"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A two-axis stabilizing gimbal is a device that ensures a sensor is working properly on a moving platform. When classical mechanics (Newton\u2013Euler and Lagrange) is employed to model a two-axis stable gimbal, its limitations can complicate the modeling process. To address this issue, a method for establishing a dynamic model for a two-axis stabilizing platform based on the Kane method is proposed in this paper. The Kane method offers the advantage of a simple model structure and computational efficiency. Initially, utilizing a generalized coordinate system, expressions of the generalized velocities, deflection velocities and angular velocities are derived. Subsequently, the generalized active forces and inertial forces acting on the two-axis stabilized gimbal are analyzed. Finally, by combining force and velocity with the Kane equation, the dynamic model of the two-axis stable platform is obtained, demonstrating the validity of the Kane method for establishing the two-axis stable platform model. To ensure the pointing accuracy stability of the two-axis stabilizing platform, a Novel Particle Swarm Optimization Proportion Integration Differentiation (NPSO-PID) controller is designed using the PSO algorithm. It is then simulated in MATLAB\/Simulink and compared with a classical PID controller. Simulation results demonstrate that NPSO-PID exhibits superior object tracking performance compared to classical PID controllers and better optimization of control parameters compared to traditional PSO-PID controllers.<\/jats:p>","DOI":"10.3390\/s24113615","type":"journal-article","created":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T12:19:46Z","timestamp":1717417186000},"page":"3615","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Modeling and Control of a Two-Axis Stabilized Gimbal Based on Kane Method"],"prefix":"10.3390","volume":"24","author":[{"given":"Qixuan","family":"Huang","sequence":"first","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiaxing","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiang","family":"Chen","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Satellite Engineering, Shanghai 201109, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Youxin","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuhao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Runjing","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhisheng","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Key Laboratory of Frontier Electric Power Equipment and Intelligent Control, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Flores, J., Gonzalez-Hernandez, I., Salazar, S., Lozano, R., and Reyes, C. (2024). Improving Optical Flow Sensor Using a Gimbal for Quadrotor Navigation in GPS-Denied Environment. Sensors, 24.","DOI":"10.3390\/s24072183"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Jayaweera, H.M.P.C., and Hanoun, S. (2021). UAV Path Planning for Reconnaissance and Look-Ahead Coverage Support for Mobile Ground Vehicles. Sensors, 21.","DOI":"10.3390\/s21134595"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"357","DOI":"10.5772\/56963","article-title":"Research on the cross-coupling of a two axes gimbal system with dynamic unbalance","volume":"10","author":"Abdo","year":"2013","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wang, Y., Sui, X., Zhang, T., Nie, T., and Chen, C. (2023). Design and Experimental Study on the Torque Balancing Mechanism of a Satellite-Borne Two-Axis Rotary Table. Machines, 11.","DOI":"10.3390\/machines11080830"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Niazi, S., Toloei, A., and Ghasemi, R. (2021). Neuro-Predictive Controller for Stabilization of Gimbal Mechanism with Cross-Coupling. Mechatron. Syst. Control., 49.","DOI":"10.2316\/J.2021.201-0198"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.isatra.2022.03.010","article-title":"Adaptive disturbance observer-based dual-loop integral-type fast terminal sliding mode control for micro spacecraft and its gimbal tracking device","volume":"130","author":"Zhang","year":"2022","journal-title":"ISA Trans."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S0019-0578(00)00038-0","article-title":"Composite control of a gyro mirror line-of-sight stabilization platform-design and auto-tuning","volume":"40","author":"Tan","year":"2001","journal-title":"ISA Trans."},{"key":"ref_8","first-page":"1","article-title":"Proxy-based sliding mode stabilization of a two-axis gimbaled platform","volume":"3","author":"Hasturk","year":"2011","journal-title":"Target"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109477","DOI":"10.1016\/j.oceaneng.2021.109477","article-title":"Sliding-mode control of ship-mounted Stewart platforms for wave compensation using ve-locity feedforward","volume":"236","author":"Cai","year":"2021","journal-title":"Ocean. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.actaastro.2022.02.003","article-title":"Design and multi-body dynamic analysis of the Archimede space exploration rov-er","volume":"194","author":"Caruso","year":"2022","journal-title":"Acta Astronaut."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"105770","DOI":"10.1016\/j.ast.2020.105770","article-title":"Dynamic modeling of helicopter-slung load system under the flexible sling hypothesis","volume":"99","author":"Cao","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1109\/TRO.2020.3014519","article-title":"Kinematics and dynamics of flexible robotic manipulators using dual screws","volume":"37","author":"Cibicik","year":"2020","journal-title":"IEEE Trans. Robot."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"93133","DOI":"10.1109\/ACCESS.2020.2994105","article-title":"Design and implementation of novel fractional-order controllers for stabilized platforms","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_14","first-page":"1","article-title":"A novel self-tuning fuzzy logic-based PID controllers for two-axis gimbal stabilization in a missile seeker","volume":"2021","author":"Anitha","year":"2021","journal-title":"Int. J. Aerosp. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"\u015eahin, M. (2023). Stabilization of Two Axis Gimbal System with Self Tuning PID Control. Politek. Derg., 1.","DOI":"10.2339\/politeknik.1210906"},{"key":"ref_16","first-page":"69","article-title":"A geometrical interpretation of Kane\u2019s equations","volume":"436","author":"Lesser","year":"1992","journal-title":"Proc. R. Soc. London. Ser. Math. Phys. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20169","DOI":"10.3390\/s150820169","article-title":"Control parameters optimization based on co-simulation of a mechatronic system for an UA-based two-axis inertially stabilized platform","volume":"15","author":"Zhou","year":"2015","journal-title":"Sensors"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhou, X., Jia, Y., Zhao, Q., and Yu, R. (2016). Experimental validation of a compound control scheme for a two-axis inertially stabilized platform with multi-sensors in an unmanned helicopter-based airborne power line inspection system. Sensors, 16.","DOI":"10.3390\/s16030366"},{"key":"ref_19","unstructured":"Long, B.J. (2015). Research on Control Algorithm of Strapdown Seeker. [Master\u2019s Thesis, Harbin Institute of Technology]."},{"key":"ref_20","unstructured":"Liu, B.J. (2023). Research on Control Algorithm of Two-axis Stabilized Platform, University of Electronic Science and Technology of China."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Hussain, Z., and Azlan, N.Z. (2016, January 22). KANE\u2019s Method for Dynamic Modeling. Proceedings of the 2016 IEEE International Conference on Automatic Control and Intelligent Systems (I2CACIS), Selangor, Malaysia.","DOI":"10.1109\/I2CACIS.2016.7885310"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1049\/iet-epa.2019.0568","article-title":"Identification and control of the motor-drive servo turntable with the switched friction model","volume":"14","author":"Zhang","year":"2020","journal-title":"IET Electr. Power Appl."},{"key":"ref_23","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle Swarm Optimization. Proceedings of the ICNN\u201995-International Conference on Neural Networks, IEEE, Perth, WA, Australia."},{"key":"ref_24","first-page":"206","article-title":"Comparing and optimum seeking of PID tuning methods base on ITAE index","volume":"23","author":"Xu","year":"2003","journal-title":"Proc. CSEE"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1109\/TCST.2005.847331","article-title":"PID control system analysis, design, and technology","volume":"13","author":"Ang","year":"2005","journal-title":"IEEE Trans. Control. Syst. Technol."},{"key":"ref_26","first-page":"5","article-title":"Simulation Research on Neural Network PID Control System of Turntable Based on PSO-BP Combina-tion Algorithm Optimization","volume":"59","author":"Wang","year":"2023","journal-title":"Aviat. Precis. Manuf. Technol."},{"key":"ref_27","unstructured":"Zhang, P. (2011). Modeling and Control of Airborne\/Missile-borne Vision-Guidance Stabilized Platform, National Defense Industry Press."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3615\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:53:08Z","timestamp":1760107988000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3615"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,3]]},"references-count":27,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["s24113615"],"URL":"https:\/\/doi.org\/10.3390\/s24113615","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,3]]}}}