{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T01:03:34Z","timestamp":1776733414038,"version":"3.51.2"},"reference-count":40,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2021,9,26]],"date-time":"2021-09-26T00:00:00Z","timestamp":1632614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04111\/2020"],"award-info":[{"award-number":["UIDB\/04111\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The need to estimate the orientation between frames of reference is crucial in spacecraft navigation. Robust algorithms for this type of problem have been built by following algebraic approaches, but data-driven solutions are becoming more appealing due to their stochastic nature. Hence, an approach based on convolutional neural networks in order to deal with measurement uncertainty in static attitude determination problems is proposed in this paper. PointNet models were trained with different datasets containing different numbers of observation vectors that were used to build attitude profile matrices, which were the inputs of the system. The uncertainty of measurements in the test scenarios was taken into consideration when choosing the best model. The proposed model, which used convolutional neural networks, proved to be less sensitive to higher noise than traditional algorithms, such as singular value decomposition (SVD), the q-method, the quaternion estimator (QUEST), and the second estimator of the optimal quaternion (ESOQ2).<\/jats:p>","DOI":"10.3390\/s21196419","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T22:16:38Z","timestamp":1632780998000},"page":"6419","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Static Attitude Determination Using Convolutional Neural Networks"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2095-7056","authenticated-orcid":false,"given":"Guilherme Henrique","family":"dos Santos","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Federal University of Santa Catarina, Florian\u00f3polis 88040-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6806-9122","authenticated-orcid":false,"given":"Laio Oriel","family":"Seman","sequence":"additional","affiliation":[{"name":"Graduate Program in Applied Computer Science, University of Vale do Itaja\u00ed, Itaja\u00ed 88302-901, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2191-6064","authenticated-orcid":false,"given":"Eduardo Augusto","family":"Bezerra","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Federal University of Santa Catarina, Florian\u00f3polis 88040-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0446-9271","authenticated-orcid":false,"given":"Valderi Reis Quietinho","family":"Leithardt","sequence":"additional","affiliation":[{"name":"VALORIZA, Research Center for Endogenous Resources Valorization, Instituto Polit\u00e9cnico de Portalegre, 7300-555 Portalegre, Portugal"}]},{"given":"Andr\u00e9 Sales","family":"Mendes","sequence":"additional","affiliation":[{"name":"Expert Systems and Applications Lab, Faculty of Science, University of Salamanca, Plaza de los Ca\u00eddos s\/n, 37008 Salamanca, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3723-616X","authenticated-orcid":false,"given":"St\u00e9fano Frizzo","family":"Stefenon","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Applied Science, University of Regina, Regina, SK 3737, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"103775","DOI":"10.1016\/j.robot.2021.103775","article-title":"6D pose estimation with combined deep learning and 3D vision techniques for a fast and accurate object grasping","volume":"141","author":"Le","year":"2021","journal-title":"Robot. Auton. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"74865","DOI":"10.1109\/ACCESS.2018.2882502","article-title":"A Method of Robot Base Frame Calibration by Using Dual Quaternion Algebra","volume":"6","author":"Wang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"29173","DOI":"10.1109\/ACCESS.2018.2833160","article-title":"Design of an Anthropomorphic, Compliant, and Lightweight Dual Arm for Aerial Manipulation","volume":"6","author":"Suarez","year":"2018","journal-title":"IEEE Access"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"106839","DOI":"10.1016\/j.knosys.2021.106839","article-title":"Graph neural network for 6D object pose estimation","volume":"218","author":"Yin","year":"2021","journal-title":"Knowl.-Based Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"153804","DOI":"10.1109\/ACCESS.2020.3018225","article-title":"Efficient Ego-Motion Estimation for Multi-Camera Systems with Decoupled Rotation and Translation","volume":"8","author":"Tian","year":"2020","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"149","DOI":"10.34768\/amcs-2020-0012","article-title":"An algorithm for quaternion-based 3D rotation","volume":"30","author":"Cariow","year":"2020","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1137\/1007077","article-title":"A least squares estimate of satellite attitude","volume":"7","author":"Wahba","year":"1965","journal-title":"SIAM Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"16857","DOI":"10.1007\/s00521-018-03975-z","article-title":"Intelligent active fault-tolerant system for multi-source integrated navigation system based on deep neural network","volume":"32","author":"Guo","year":"2020","journal-title":"Neural Comput. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Tan, T.N., Khenchaf, A., Comblet, F., Franck, P., Champeyroux, J.M., and Reichert, O. (2020). Robust-Extended Kalman Filter and Long Short-Term Memory Combination to Enhance the Quality of Single Point Positioning. Appl. Sci., 10.","DOI":"10.3390\/app10124335"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1109\/LRA.2019.2959507","article-title":"OriNet: Robust 3-D Orientation Estimation With a Single Particular IMU","volume":"5","author":"Esfahani","year":"2019","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Shuster, M.D. (1978, January 7\u20139). Approximate algorithms for fast optimal attitude computation. Proceedings of the Guidance and Control Conference, Palo Alto, CA, USA.","DOI":"10.2514\/6.1978-1249"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"70","DOI":"10.2514\/3.19717","article-title":"Three-axis attitude determination from vector observations","volume":"4","author":"Shuster","year":"1981","journal-title":"J. Guid. Control."},{"key":"ref_13","first-page":"245","article-title":"Attitude determination using vector observations and the singular value decomposition","volume":"36","author":"Markley","year":"1988","journal-title":"J. Astronaut. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"885","DOI":"10.2514\/2.4618","article-title":"Second estimator of the optimal quaternion","volume":"23","author":"Mortari","year":"2000","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_15","unstructured":"Davenport, P.B. (1968). A Vector Approach to the Algebra of Rotations with Applications, NASA. NASA Technical Note D-4696."},{"key":"ref_16","unstructured":"Keat, J. (1977). Analysis of Least-Squares Attitude Determination Routine DOAOP (CSC\/TM-77\/6034), Computer Sciences Corp.. Technical Report CSC\/TM-77\/6034."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"93","DOI":"10.3390\/a2010093","article-title":"A survey on star identification algorithms","volume":"2","author":"Spratling","year":"2009","journal-title":"Algorithms"},{"key":"ref_18","unstructured":"Hashim, H.A. (2020). Attitude determination and estimation using vector observations: Review, challenges and comparative results. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Barnes, C., Lu, J., Yang, J., and Li, H. (2019, January 16\u201320). On the continuity of rotation representations in neural networks. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA.","DOI":"10.1109\/CVPR.2019.00589"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Xiang, Y., Schmidt, T., Narayanan, V., and Fox, D. (2017). Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes. arXiv.","DOI":"10.15607\/RSS.2018.XIV.019"},{"key":"ref_21","unstructured":"Do, T.T., Cai, M., Pham, T., and Reid, I. (2018). Deep-6dpose: Recovering 6d object pose from a single rgb image. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gao, G., Lauri, M., Zhang, J., and Frintrop, S. (2018, January 8\u201314). Occlusion resistant object rotation regression from point cloud segments. Proceedings of the 15th European Conference on Computer Vision (ECCV), Munich, Germany.","DOI":"10.1007\/978-3-030-11009-3_44"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"140310","DOI":"10.1109\/ACCESS.2020.3013291","article-title":"D3PointNet: Dual-Level Defect Detection PointNet for Solder Paste Printer in Surface Mount Technology","volume":"8","author":"Park","year":"2020","journal-title":"IEEE Access"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"37121","DOI":"10.1109\/ACCESS.2019.2905546","article-title":"Using Deep Learning in Semantic Classification for Point Cloud Data","volume":"7","author":"Yao","year":"2019","journal-title":"IEEE Access"},{"key":"ref_25","first-page":"820","article-title":"Pointcnn: Convolution on x-transformed points","volume":"31","author":"Li","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2644","DOI":"10.1109\/TGRS.2019.2953092","article-title":"Separating the Structural Components of Maize for Field Phenotyping Using Terrestrial LiDAR Data and Deep Convolutional Neural Networks","volume":"58","author":"Jin","year":"2020","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3793","DOI":"10.1109\/TII.2020.3016591","article-title":"Learning to Estimate the Body Shape Under Clothing From a Single 3-D Scan","volume":"17","author":"Hu","year":"2021","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"30969","DOI":"10.1109\/ACCESS.2020.2973003","article-title":"Sparse-to-Dense Multi-Encoder Shape Completion of Unstructured Point Cloud","volume":"8","author":"Peng","year":"2020","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"72727","DOI":"10.1109\/ACCESS.2020.2987829","article-title":"RMAF: Relu-Memristor-Like Activation Function for Deep Learning","volume":"8","author":"Yu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"62830","DOI":"10.1109\/ACCESS.2020.2983774","article-title":"Sequence-Dropout Block for Reducing Overfitting Problem in Image Classification","volume":"8","author":"Qian","year":"2020","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3450","DOI":"10.1109\/TSP.2020.2997940","article-title":"Improved Most Likely Heteroscedastic Gaussian Process Regression via Bayesian Residual Moment Estimator","volume":"68","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"94140","DOI":"10.1109\/ACCESS.2020.2995656","article-title":"Long Short-Term Memory With Attention Mechanism for State of Charge Estimation of Lithium-Ion Batteries","volume":"8","author":"Mamo","year":"2020","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2045","DOI":"10.1109\/TAES.2018.2881353","article-title":"Accurate Direct Strapdown Direction Cosine Algorithm","volume":"55","author":"Xu","year":"2019","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1109\/LSP.2019.2929863","article-title":"On Estimating the Norm of a Gaussian Vector Under Additive White Gaussian Noise","volume":"26","author":"Dytso","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5667","DOI":"10.1049\/iet-gtd.2020.0814","article-title":"Hybrid deep learning for power generation forecasting in active solar trackers","volume":"14","author":"Stefenon","year":"2020","journal-title":"IET Gener. Trans. Distrib."},{"key":"ref_36","first-page":"261","article-title":"Attitude Determination from Vector Observations: A Fast Optimal Matrix Algorithm","volume":"41","author":"Markley","year":"1993","journal-title":"J. Astronaut. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2696108","DOI":"10.1155\/2017\/2696108","article-title":"Orientation uncertainty characteristics of some pose measuring systems","volume":"2017","author":"Franaszek","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_38","unstructured":"Curtis, W.D., Janin, A.L., and Zikan, K. (1993, January 18\u201322). A note on averaging rotations. Proceedings of the IEEE Virtual Reality Annual International Symposium, Seattle, WA, USA."},{"key":"ref_39","unstructured":"Paiva, L.P.S., de Melo, F.M.S.R., Menezes Filho, R., Vieira, L.A., and Oliveira, F. (2020). Error analysis for alignment in AHRS: QUEST and SAAM algorithms. An. Soc. Bras. Autom., 2."},{"key":"ref_40","first-page":"8288","article-title":"Rigid 3-D Registration: A Simple Method Free of SVD and Eigendecomposition","volume":"69","author":"Wu","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/19\/6419\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:05:11Z","timestamp":1760166311000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/19\/6419"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,26]]},"references-count":40,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["s21196419"],"URL":"https:\/\/doi.org\/10.3390\/s21196419","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,26]]}}}