{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:51:15Z","timestamp":1760241075284,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2019,11,21]],"date-time":"2019-11-21T00:00:00Z","timestamp":1574294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61603191","61903241"],"award-info":[{"award-number":["61603191","61903241"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper addresses the difficult problem of measuring the attitude of a high-spinning projectile and presents a novel method for estimating the pitch and yaw angles of the projectile in flight. The method is based on analysis of the external moment of the rotating projectile during flight and theoretical derivations obtained from the dynamics\u2019 equations. First, the principle of zero-crossing method is introduced, which explains the process of geomagnetic azimuth and roll measurements by the non-orthogonal geomagnetic sensor combination. Then, the dynamics constraint equations between the Euler angles and flight-path angle, trajectory deflection angle of the projectile are derived using the dynamics equations of the projectile rotating around the centroid, and analysis of the flight characteristics of the projectile in stable flight. Next, the spatial orientation relationship between pitch, yaw angles and magnetic azimuth is established based on the physical principle of geomagnetic azimuth. Finally, the pitch and yaw angles are estimated using the unscented Kalman filter (UKF), with the dynamics constraint equations serving as the driving equations. In the UKF prediction stage, the Runge-Kutta method is used to discretize the state equation that improves the prediction accuracy. Simulation results show that the proposed method can be used to accurately calculate the pitch and yaw angles, and results of experimental data processing also verify the feasibility of the proposed method for real-world applications.<\/jats:p>","DOI":"10.3390\/s19235096","type":"journal-article","created":{"date-parts":[[2019,11,22]],"date-time":"2019-11-22T02:49:27Z","timestamp":1574390967000},"page":"5096","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Novel Method for Estimating Pitch and Yaw of Rotating Projectiles Based on Dynamic Constraints"],"prefix":"10.3390","volume":"19","author":[{"given":"Liangliang","family":"An","sequence":"first","affiliation":[{"name":"School of Energy and Power Engineering, Nanjing University of Science &amp; Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liangming","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Energy and Power Engineering, Nanjing University of Science &amp; Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6552-8207","authenticated-orcid":false,"given":"Ning","family":"Liu","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of High Dynamic Navigation Technology, Beijing Information Science &amp; Technological University, Beijing 100101, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Energy and Power Engineering, Nanjing University of Science &amp; Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yang","family":"Zhong","sequence":"additional","affiliation":[{"name":"School of Energy and Power Engineering, Nanjing University of Science &amp; Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,21]]},"reference":[{"key":"ref_1","first-page":"68","article-title":"Determining inertial orientation of a spinning body with body-fixed sensors","volume":"4025","author":"Hepner","year":"2000","journal-title":"Proc. SPIE Int. Soc. Opt. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.actaastro.2012.02.001","article-title":"The attitude determination system of the RAX satellite","volume":"75","author":"Springmann","year":"2012","journal-title":"Acta Astronaut."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"417","DOI":"10.2514\/3.56190","article-title":"Kalman Filtering for Spacecraft Attitude Estimation","volume":"5","author":"Lefferts","year":"1982","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.2514\/2.4834","article-title":"Kalman Filtering for Spacecraft System Alignment Calibration","volume":"24","author":"Pittelkau","year":"2001","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.2514\/1.G000336","article-title":"Three-Axis Attitude Estimation Using Rate-Integrating Gyroscopes","volume":"39","author":"Crassidis","year":"2016","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Park, C.G., Kim, K., and Kang, W.Y. (2006, January 21\u201324). UKF Based In-Flight Alignment Using Low Cost IMU. Proceedings of the Aiaa Guidance, Navigation, & Control Conference & Exhibit, Keystone, CO, USA.","DOI":"10.2514\/6.2006-6353"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Rhudy, M., Gross, J., Gu, Y., and Napolitano, M.R. (2012, January 13\u201316). Fusion of GPS and Redundant IMU Data for Attitude Estimation. Proceedings of the Aiaa Guidance Navigation & Control Conference, Minneapolls, MN, USA.","DOI":"10.2514\/6.2012-5030"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ettl, J., Schmidt, A.P., Turner, J., and Kim, D.G. (2016, January 16\u201320). Using data fusion of DMARS-R-IMU and GPS data for improving attitude determination accuracy. Proceedings of the International Conference on Space Operations, Daejeon, Korea.","DOI":"10.2514\/6.2016-2637"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Changey, S., Beauvois, D., and Fleck, V. (2006, January 14\u201316). A Mixed Extended-Unscented Filter for attitude estimation with magnetometer sensor. Proceedings of the American Control Conference, Minneapolis, MN, USA.","DOI":"10.1109\/ACC.2006.1657158"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.actaastro.2011.03.010","article-title":"Integrated Attitude Determination and Control System via Magnetic Measurements and Actuation","volume":"69","author":"Abdelrahman","year":"2011","journal-title":"Acta Astronaut."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.actaastro.2016.11.045","article-title":"Advanced Numerical Study of the Three-Axis Magnetic Attitude Control and Determination with Uncertainties","volume":"132","author":"Ivanov","year":"2017","journal-title":"Acta Astronaut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"244","DOI":"10.2514\/2.5065","article-title":"Spacecraft Attitude Rate Estimation from Geomagnetic Field Measurements","volume":"26","author":"Psiaki","year":"2003","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_13","unstructured":"Christin, S.H. (2009, January 5\u20138). Satellite Attitude Determination Using Magnetometer Data Only. Proceedings of the 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, FL, USA."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.2514\/1.G002591","article-title":"Magnetometer-Based Attitude Determination for Deployed Spin-Stabilized Spacecraft","volume":"40","author":"Roberts","year":"2017","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_15","first-page":"276","article-title":"Real-Time Attitude-Independent Magnetometer Bias Estimation for Spinning Spacecraft","volume":"41","year":"2017","journal-title":"J. Guid. Control Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2184","DOI":"10.2514\/1.G001804","article-title":"Spacecraft Attitude Stabilization using Magnetorquers with Separation between Measurement and Actuation","volume":"39","author":"Celani","year":"2015","journal-title":"J. Guid. Control Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Robinson, J.B., and Richie, D.J. (2016). Stabilization and Attitude Determination Methods for FalconSAT-3. J. Spacecr. Rocket., 507\u2013519.","DOI":"10.2514\/1.A33362"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1002\/j.2161-4296.2011.tb02581.x","article-title":"Effective Use of Magnetometer Feedback for Smart Projectile Applications","volume":"58","author":"Rogers","year":"2011","journal-title":"Navigation"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sedlak, J.E. (2006, January 21\u201324). Iterative magnetometer calibration. Proceedings of the AIAA\/AAS Astrodynamics Specialist Conference, Keystone, CO, USA.","DOI":"10.2514\/6.2006-6386"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"S130","DOI":"10.1002\/tee.22345","article-title":"Magnetometer calibration improvement using wavelet and genetic algorithm","volume":"11","author":"Hu","year":"2016","journal-title":"IEEJ Trans. Electr. Electron. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"115","DOI":"10.2514\/1.6278","article-title":"Real-Time Attitude-Independent Three-Axis Magnetometer Calibration","volume":"28","author":"John","year":"2005","journal-title":"J. Guid. Control Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Wilson, M.J. (2005, January 10\u201313). Onboard Attitude Determination for Gun-Launched Projectiles. Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA.","DOI":"10.2514\/6.2005-1217"},{"key":"ref_23","unstructured":"Wilson, M.J. (2007). Projectile Navigation and the Application to Magnetometers, University of Delaware. Dissertations & Theses\u2014Gradworks."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"123","DOI":"10.3182\/20090610-3-IT-4004.00026","article-title":"Real time Estimation of Supersonic Projectile Roll Angle using Magnetometers: In-lab Experimental Validation","volume":"42","author":"Changey","year":"2009","journal-title":"IFAC Proc. Vol."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Changey, S., Pecheur, E., Bernard, L., and Sommer, E. (2012, January 23\u201326). Real time estimation of projectile roll angle using magnetometers: In-flight experimental validation. Proceedings of the Position Location & Navigation Symposium, Myrtle Beach, SC, USA.","DOI":"10.1109\/PLANS.2012.6236904"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"339","DOI":"10.2514\/1.G001104","article-title":"Efficient Attitude Estimation for a Spin-Stabilized Projectile","volume":"39","author":"Maley","year":"2015","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1002\/navi.5","article-title":"A Low-Cost Orientation Estimator for Smart Projectiles Using Magnetometers and Thermopiles","volume":"59","author":"Rogers","year":"2012","journal-title":"Navigation"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"152","DOI":"10.2514\/3.22911","article-title":"Recursive attitude determination from vector observations Euler angle estimation","volume":"10","author":"Idan","year":"1987","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12","DOI":"10.2514\/1.22452","article-title":"Survey of Nonlinear Attitude Estimation Methods","volume":"30","author":"Crassidis","year":"2007","journal-title":"J. Guid. Control Dyn."},{"key":"ref_30","unstructured":"Theil, S., Appel, P., and Schleicher, A. (2003, January 11\u201314). Low Cost, Good Accuracy\u2014Attitude Determination using Magnetometer and Simple Sun Sensor. Proceedings of the 17th Annual AIAA\/USU Conference on Small Satellites, Logan, UT, USA."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/MAES.2015.150069","article-title":"Attitude estimation and sensor identification utilizing nonlinear filters based on a low-cost MEMS magnetometer and sun sensor","volume":"30","author":"Teshnizi","year":"2015","journal-title":"Aerosp. Electron. Syst. Mag. IEEE"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"533","DOI":"10.2514\/1.32395","article-title":"Attitude Estimation for Sounding Rockets Using Microelectromechanical System Gyros","volume":"31","author":"Bekkeng","year":"2008","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Thompson, A.A. (2002). A Procedure for Calibrating Magnetic Sensors, Army Research Laboratory (ARL).","DOI":"10.21236\/ADA399851"},{"key":"ref_34","first-page":"60","article-title":"MAGSONDE (patent pending): A device for making angular measurements on spinning projectiles using magnetic sensors","volume":"4025","author":"Harkins","year":"2000","journal-title":"Proc. SPIE Int. Soc. Opt. Eng."},{"key":"ref_35","first-page":"106","article-title":"Attitude Measurement on High-spinning Projectile Using Magnetic Sensors and Acceletometers","volume":"25","author":"Li","year":"2008","journal-title":"Trans. Nanjing Univ. Aeronaut. Astronaut."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhu, J., Wu, P., and Bo, Y. (2016). A Novel Attitude Estimation Algorithm Based on the Non-Orthogonal Magnetic Sensors. Sensors, 16.","DOI":"10.3390\/s16050730"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"22509","DOI":"10.1109\/ACCESS.2019.2896612","article-title":"Magnetometer-Based Phase Shifting Ratio Method for High Spinning Projectile\u2019s Attitude Measurement","volume":"7","author":"Zhao","year":"2019","journal-title":"IEEE Access"},{"key":"ref_38","first-page":"65381G-11","article-title":"A strong tracking extended Kalman observer for projectile attitude and position estimation","volume":"6538","author":"Boutayeb","year":"2007","journal-title":"Proc SPIE"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2546","DOI":"10.1109\/TAES.2013.6621835","article-title":"Ballistic Roll Estimation using EKF Frequency Tracking and Adaptive Noise Cancellation","volume":"49","author":"Allik","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst"},{"key":"ref_40","first-page":"802","article-title":"Projectile attitude and position determination using magnetometer sensor only","volume":"13","author":"Changey","year":"2005","journal-title":"Proc. SPIE Int. Soc. Opt. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"929","DOI":"10.2514\/1.G003225","article-title":"Unscented Partial-Update Schmidt\u2013Kalman Filter","volume":"41","author":"Brink","year":"2018","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/9.847726","article-title":"A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators","volume":"45","author":"Julier","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/JPROC.2003.823141","article-title":"Unscented Filtering and Nonlinear Estimation","volume":"93","author":"Julier","year":"2004","journal-title":"Proc. IEEE"},{"key":"ref_44","unstructured":"Lisano, M.E. (2006, January 4\u20138). Nonlinear Consider Covariance Analysis Using a Sigma-Point Filter Formulation. Proceedings of the Guidance and Control 2006: Proceedings of the 29th Annual AAS Rocky Mountain Guidance and Control Conference, Univelt, San Diego, CA, USA."},{"key":"ref_45","unstructured":"Lisano, M.E. (2007, January 24\u201328). Comparing Consider-Covariance Analysis with Sigma-Point Consider Filter and Linear-Theory Consider Filter Formulas. Proceedings of the 20th International Symposium on Space Flight Dynamics, Annapolis, MD, USA."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"117","DOI":"10.2514\/1.G000467","article-title":"Unscented Schmidt\u2013Kalman Filter Algorithm","volume":"38","author":"Stauch","year":"2015","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Zingg, D., and Chisholm, T. (1995, January 19\u201322). Runge-Kutta methods for linear problems. Proceedings of the 12th Computational Fluid Dynamics Conference, San Diego, CA, USA.","DOI":"10.2514\/6.1995-1756"},{"key":"ref_48","unstructured":"Hempel, P.R. (2012, January 13\u201316). Application of implicit Runge-Kutta methods. Proceedings of the AIAA & AAS, Astrodynamics Conference, Minneapolis, MN, USA."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Aristoff, J.M., and Poore, A. (2012, January 13\u201316). Implicit Runge-Kutta Methods for Orbit Propagation. Proceedings of the AIAA & AAS, Astrodynamics Specialist Conference, Minneapolis, MN, USA.","DOI":"10.2514\/6.2012-4880"},{"key":"ref_50","unstructured":"Han, Z. (2008). Exterior Ballistics for Projectiles and Rockets, Beijing Institute of Technology Press."},{"key":"ref_51","unstructured":"Pu, F., and Rui, X. (1980). Exterior Ballistics, National Defense Industry Press."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/23\/5096\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:36:24Z","timestamp":1760189784000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/23\/5096"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,21]]},"references-count":51,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["s19235096"],"URL":"https:\/\/doi.org\/10.3390\/s19235096","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,11,21]]}}}