{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T19:28:08Z","timestamp":1769455688746,"version":"3.49.0"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["201908320035"],"award-info":[{"award-number":["201908320035"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51405237"],"award-info":[{"award-number":["51405237"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The solution of the dynamic equations of the six-axis accelerometer is a prerequisite for sensor calibration, structural optimization, and practical application. However, the forward dynamic equations (FDEs) and inverse dynamic equations (IDEs) of this type of system have not been completely solved due to the strongly nonlinear coupling relationship between the inputs and outputs. This article presents a comprehensive study of the FDEs and IDEs of the six-axis accelerometer based on a parallel mechanism. Firstly, two sets of dynamic equations of the sensor are constructed based on the Newton\u2013Euler method in the configuration space. Secondly, based on the analytical solution of the sensor branch chain length, the coordination equation between the output signals of the branch chain is constructed. The FDEs of the sensor are established by combining the coordination equations and two sets of dynamic equations. Furthermore, by introducing generalized momentum and Hamiltonian function and using Legendre transformation, the vibration differential equations (VDEs) of the sensor are derived. The VDEs and Newton\u2013Euler equations constitute the IDEs of the system. Finally, the explicit recursive algorithm for solving the quaternion in the equation is given in the phase space. Then the IDEs are solved by substituting the quaternion into the dynamic equations in the configuration space. The predicted numerical results of the established FDEs and IDEs are verified by comparing with virtual and actual experimental data. The actual experiment shows that the relative errors of the FDEs and the IDEs constructed in this article are 2.21% and 7.65%, respectively. This research provides a new strategy for further improving the practicability of the six-axis accelerometer.<\/jats:p>","DOI":"10.3390\/s21010233","type":"journal-article","created":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T22:35:48Z","timestamp":1609540548000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-8663","authenticated-orcid":false,"given":"Linkang","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1205-6653","authenticated-orcid":false,"given":"Jingjing","family":"You","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaolong","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huaxin","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chenggang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongtao","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2114","DOI":"10.1177\/0142331218780224","article-title":"An acceleration-based hybrid learning-adaptive controller for robot manipulators","volume":"41","author":"Unel","year":"2019","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_2","first-page":"1","article-title":"Strapdown Sculling Velocity Algorithms Using Novel Input Combinations","volume":"2018","author":"Huang","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Balfas, M., Ahamed, S.I., Tamma, C., Arif, M., Kattan, A.J., and Chu, W. (2018, January 23\u201327). A Study and Estimation a Lost Person Behavior in Crowded Areas Using Accelerometer Data from Smartphones. Proceedings of the 2018 IEEE 42nd Annual Computer Software and Applications Conference (COMPSAC), Tokyo, Japan.","DOI":"10.1109\/COMPSAC.2018.00149"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.asr.2016.09.001","article-title":"Gyroscope-reduced inertial navigation system for flight vehicle motion estimation","volume":"59","author":"Wang","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1364","DOI":"10.1016\/j.medengphy.2008.09.005","article-title":"Direct measurement of human movement by accelerometry","volume":"30","author":"Godfrey","year":"2008","journal-title":"Med. Eng. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mohammed, Z., Elfadel, I., Abe, M., and Rasras, M. (2018). Monolithic Multi Degree of Freedom (MDoF) Capacitive MEMS Accelerometers. Micromachines, 9.","DOI":"10.3390\/mi9110602"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1108\/SR-05-2018-0133","article-title":"A novel multiple-axis MEMS gyroscope-accelerometer with decoupling frames","volume":"39","author":"Baranov","year":"2019","journal-title":"Sens. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.sna.2014.04.033","article-title":"Design and analysis of gyro-free inertial measurement units with different configurations","volume":"214","author":"Liu","year":"2014","journal-title":"Sens. Actuators Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2520","DOI":"10.1109\/TIM.2005.858129","article-title":"Design of accelerometer-based inertial navigation systems","volume":"54","author":"Tan","year":"2005","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1109\/TIM.2010.2083331","article-title":"A Six-Degree-of-Freedom Acceleration Sensing Method Based on Six Coplanar Single-Axis Accelerometers","volume":"60","author":"Wang","year":"2011","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1016\/j.jbiomech.2005.03.016","article-title":"Lightweight low-profile nine-accelerometer package to obtain head angular accelerations in short-duration impacts","volume":"39","author":"Yoganandan","year":"2006","journal-title":"J. Biomech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.sna.2016.10.005","article-title":"A 6-DOF acceleration sensor with cylindrical configuration","volume":"251","author":"Zou","year":"2016","journal-title":"Sens. Actuators Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.sna.2005.03.060","article-title":"A scheme for improving the performance of a gyroscope-free inertial measurement unit","volume":"121","author":"Park","year":"2005","journal-title":"Sens. Actuators Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1016\/j.proche.2009.07.330","article-title":"Sensor Fusion Algorithm and Calibration for a Gyroscope-free IMU","volume":"1","author":"Schopp","year":"2009","journal-title":"Procedia Chem."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"074001","DOI":"10.1088\/1361-6501\/ab7612","article-title":"An improved self-calibration method with consideration of inner lever-arm effects for a dual-axis rotational inertial navigation system","volume":"31","author":"Song","year":"2020","journal-title":"Meas. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3301","DOI":"10.1109\/JSEN.2014.2387829","article-title":"Principle Research on a Single Mass Six-Degree-of-Freedom Accelerometer With Six Groups of Piezoelectric Sensing Elements","volume":"15","author":"Lv","year":"2015","journal-title":"IEEE Sens. J."},{"key":"ref_17","unstructured":"Meng, M., Wu, Z.C., Yu, Y., Ge, Y., and Ge, Y.J. (2005, January 18\u201322). Design and Characterization of a Six-axis Accelerometer. Proceedings of the 2005 IEEE International Conference on Robotics and Automation (ICRA 2005), Barcelona, Spain."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/j.sna.2006.05.044","article-title":"Development of miniaturized 6-axis accelerometer utilizing piezoresistive sensing elements","volume":"134","author":"Amarasinghe","year":"2007","journal-title":"Sens. Actuators Phys."},{"key":"ref_19","unstructured":"Gaillet, A., and Reboulet, C. (1983, January 17\u201321). An Isostatic Six Component Force and Torque Sensor. Proceedings of the 13th International Symposium on Industrial Robotics, Chicago, IL, USA."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Zhang, C., Zhang, D., Shi, Z., and Zhao, T. (2016). Mathematical Model and Calibration Experiment of a Large Measurement Range Flexible Joints 6-UPUR Six-Axis Force Sensor. Sensors, 16.","DOI":"10.3390\/s16081271"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Niu, Z., Zhao, T., Zhao, Y., Hu, Q., and Ding, S. (2017). Design and Analysis of the Measurement Characteristics of a Bidirectional-Decoupling Over-Constrained Six-Dimensional Parallel-Mechanism Force Sensor. Sensors, 17.","DOI":"10.3390\/s17091985"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mechmachtheory.2019.103697","article-title":"Influence of flexible spherical joints parameters on accuracy of the six-axis force\/torque sensor with three-three orthogonal parallel mechanism","volume":"145","author":"Zhou","year":"2020","journal-title":"Mech. Mach. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sna.2019.111772","article-title":"Multi-axis force sensors: A state-of-the-art review","volume":"304","author":"Templeman","year":"2020","journal-title":"Sens. Actuators Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1108\/SR-06-2018-0162","article-title":"Sensitivity and sensitivity isotropy of an 8\/4\u20134 parallel six-axis force sensor","volume":"40","author":"Li","year":"2020","journal-title":"Sens. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.mechmachtheory.2015.03.018","article-title":"Isotropy analysis of redundant parallel six-axis force sensor","volume":"91","author":"Yao","year":"2015","journal-title":"Mech. Mach. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1016\/j.sna.2006.10.024","article-title":"Single-mass 6-DOF isotropic accelerometer with segmented PSD sensors","volume":"135","author":"Chapsky","year":"2007","journal-title":"Sens. Actuators Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1016\/j.sna.2010.08.021","article-title":"Design, analysis and fabrication of a multidimensional acceleration sensor based on fully decoupled compliant parallel mechanism","volume":"163","author":"Gao","year":"2010","journal-title":"Sens. Actuators Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Sun, Z., Liu, J., Yu, C., and Zheng, Y. (2016). A Small Range Six-Axis Accelerometer Designed with High Sensitivity DCB Elastic Element. Sensors, 16.","DOI":"10.3390\/s16091552"},{"key":"ref_29","first-page":"36","article-title":"Structural research on a six-axis accelerometer for picking robot wrist","volume":"20","author":"Yu","year":"2011","journal-title":"Int. Agric. Eng. J."},{"key":"ref_30","first-page":"67","article-title":"Influences analysis of configurations on the performance of parallel type six-axis accelerometers","volume":"37","author":"Xia","year":"2013","journal-title":"Trans. Famena"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"12637","DOI":"10.1109\/JSEN.2020.3001250","article-title":"Decoupling Algorithm and Maximum Operation Frequency of a Novel Parallel Type Six-Axis Accelerometer","volume":"20","author":"You","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5485","DOI":"10.1109\/JSEN.2018.2834727","article-title":"Multi-Axial Force\/Torque Sensor Calibration Method Based on Deep-Learning","volume":"18","author":"Oh","year":"2018","journal-title":"IEEE Sens. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.3788\/OPE.20152309.2522","article-title":"Error and fault-tolerant processing of parallel type six-axis accelerometer","volume":"23","author":"You","year":"2015","journal-title":"Opt. Precis. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Yang, E.C.-Y. (2016). Design and Sensitivity Analysis Simulation of a Novel 3D Force Sensor Based on a Parallel Mechanism. Sensors, 16.","DOI":"10.3390\/s16122147"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7129","DOI":"10.1109\/JSEN.2020.2966013","article-title":"Multiparameter Modeling of Piezoelectric Six-Degree-of-Freedom Accelerometer About Sensitivity Characteristics","volume":"20","author":"Zhang","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.mechmachtheory.2017.05.009","article-title":"Forward kinematic analysis of the 3-RPRS parallel manipulator","volume":"116","author":"Nag","year":"2017","journal-title":"Mech. Mach. Theory"},{"key":"ref_37","first-page":"1216","article-title":"Research on the decoupling algorithm of pre-stressed parallel six-axis accelerometer","volume":"38","author":"You","year":"2017","journal-title":"Chin. J. Sci. Instrum."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.mechmachtheory.2017.05.002","article-title":"Forward dynamic analysis of parallel robots using modified decoupled natural orthogonal complement method","volume":"115","author":"Raoofian","year":"2017","journal-title":"Mech. Mach. Theory"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.rcim.2013.10.009","article-title":"A novel approach for forward dynamic analysis of 3-PRS parallel manipulator with consideration of friction effect","volume":"30","author":"Yuan","year":"2014","journal-title":"Robot. Comput. Integr. Manuf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.ymssp.2018.03.015","article-title":"Research on static decoupling algorithm for piezoelectric six axis force\/torque sensor based on LSSVR fusion algorithm","volume":"110","author":"Li","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_41","unstructured":"Wan, Z., Cui, F., Zhang, Y.K., Liu, W., Chen, W.Y., and Zhang, W.P. (2011, January 16\u201318). Design of Proof Mass and System-Level Simulation of a Micromachined Electrostatically Suspended Accelerometer. Proceedings of the 2011 International Conference on Advanced Design and Manufacturing Engineering, Guangzhou, China."},{"key":"ref_42","first-page":"134","article-title":"Parameter identification and perturbation algorithm of parallel type six-axis accelerometer","volume":"38","author":"You","year":"2019","journal-title":"J. Vib. Shock"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"29627","DOI":"10.1109\/ACCESS.2020.2971711","article-title":"Ocean Wave Buoy Based on Parallel Six-Dimensional Accelerometer","volume":"8","author":"Yunping","year":"2020","journal-title":"IEEE Access"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/1\/233\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:06:09Z","timestamp":1760159169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/1\/233"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,1]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["s21010233"],"URL":"https:\/\/doi.org\/10.3390\/s21010233","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,1]]}}}