{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T14:05:11Z","timestamp":1766066711704,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,19]],"date-time":"2017-05-19T00:00:00Z","timestamp":1495152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The rise of autonomous systems operating close to humans imposes new challenges in terms of robustness and precision on the estimation and control algorithms. Approaches based on nonlinear optimization, such as moving horizon estimation, have been shown to improve the accuracy of the estimated solution compared to traditional filter techniques. This paper introduces an optimization-based framework for multi-sensor fusion following a moving horizon scheme. The framework is applied to the often occurring estimation problem of motion tracking by fusing measurements of a global navigation satellite system receiver and an inertial measurement unit. The resulting algorithm is used to estimate position, velocity, and orientation of a maneuvering airplane and is evaluated against an accurate reference trajectory. A detailed study of the influence of the horizon length on the quality of the solution is presented and evaluated against filter-like and batch solutions of the problem. The versatile configuration possibilities of the framework are finally used to analyze the estimated solutions at different evaluation times exposing a nearly linear behavior of the sensor fusion problem.<\/jats:p>","DOI":"10.3390\/s17051159","type":"journal-article","created":{"date-parts":[[2017,5,23]],"date-time":"2017-05-23T01:47:33Z","timestamp":1495504053000},"page":"1159","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Optimization-Based Sensor Fusion of GNSS and IMU Using a Moving Horizon Approach"],"prefix":"10.3390","volume":"17","author":[{"given":"Fabian","family":"Girrbach","sequence":"first","affiliation":[{"name":"Xsens Technologies B.V., Enschede 7521 PR, The Netherlands"},{"name":"Department of Microsystems Engineering (IMTEK), University of Freiburg, 79110 Freiburg, Germany"}]},{"given":"Jeroen","family":"Hol","sequence":"additional","affiliation":[{"name":"Xsens Technologies B.V., Enschede 7521 PR, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5514-2503","authenticated-orcid":false,"given":"Giovanni","family":"Bellusci","sequence":"additional","affiliation":[{"name":"Xsens Technologies B.V., Enschede 7521 PR, The Netherlands"}]},{"given":"Moritz","family":"Diehl","sequence":"additional","affiliation":[{"name":"Department of Microsystems Engineering (IMTEK), University of Freiburg, 79110 Freiburg, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,19]]},"reference":[{"key":"ref_1","unstructured":"Verschueren, R., Zanon, M., Quirynen, R., and Diehl, M. (July, January 29). Time-optimal race car driving using an online exact hessian based nonlinear MPC algorithm. Proceedings of the European Control Conference (ECC), Aalborg, Denmark."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gros, S., Zanon, M., Vukov, M., and Diehl, M. (2012, January 23\u201327). Nonlinear MPC and MHE for Mechanical Multi-Body Systems with Application to Fast Tethered Airplanes. Proceedings of the IFAC Conference on Nonlinear Model Predictive Control, Leeuwenhorst, The Netherlands.","DOI":"10.3182\/20120823-5-NL-3013.00061"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chao, Z., Ming, L., Shaolei, Z., and Wenguang, Z. (2011, January 9\u201311). Collision-free UAV formation flight control based on nonlinear MPC. Proceedings of the International Conference on Electronics, Communications and Control (ICECC), Ningbo, China.","DOI":"10.1109\/ICECC.2011.6066578"},{"key":"ref_4","unstructured":"Russo, L.P., and Young, R.E. (1999, January 2\u20134). Moving-Horizon State Estimation Applied to an Industrial Polymerization Process. Proceedings of the American Control Conference, San Diego, CA, USA."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hoffmann, G., Gorinevsky, D., Mah, R., Tomlin, C., and Mitchell, J. (2007, January 20\u201323). Fault Tolerant Relative Navigation Using Inertial and Relative Sensors. Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, SC, USA.","DOI":"10.2514\/6.2007-6789"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Quirynen, R., Vukov, M., and Diehl, M. (2012, January 23\u201327). Auto Generation of Implicit Integrators for Embedded NMPC with Microsecond Sampling Times. Proceedings of the IFAC Nonlinear Model Predictive Control Conference, Noordwijkerhout, The Nederlands.","DOI":"10.3182\/20120823-5-NL-3013.00013"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Frison, G., Kufoalor, D.K.M., Imsland, L., and Jorgensen, J.B. (2014, January 8\u201310). Efficient implementation of solvers for linear model predictive control on embedded devices. Proceedings of the IEEE Conference on Control Applications (CCA), Juan Les Antibes, France.","DOI":"10.1109\/CCA.2014.6981589"},{"key":"ref_8","unstructured":"Hall, D.L., and Llinas, J. (June, January 31). An introduction to multisensor data fusion. Proceedings of the Circuits and Systems, Monterey, CA, USA."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1016\/j.automatica.2004.01.014","article-title":"Multi-sensor optimal information fusion Kalman filter","volume":"40","author":"Sun","year":"2004","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s001900050236","article-title":"Adaptive Kalman Filtering for INS\/GPS","volume":"73","author":"Mohamed","year":"1999","journal-title":"J. Geod."},{"key":"ref_11","unstructured":"Jwo, D.J., and Weng, T.P. (2008, January 6\u201311). An Adaptive Sensor Fusion Method with Applications in Integrated Navigation. the Proceedings of the International Federation of Automatic Control, Seoul, Korea."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1619","DOI":"10.1016\/S0005-1098(01)00115-7","article-title":"Constrained Linear State Estimation\u2014A Moving Horizon Approach","volume":"37","author":"Rao","year":"2001","journal-title":"Automatica"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Girrbach, F., Hol, J.D., Bellusci, G., and Diehl, M. (2017). Towards Robust Sensor Fusion for State Estimation in Airborne Applications Using GNSS and IMU. IFAC-PapersOnLine, in press.","DOI":"10.1016\/j.ifacol.2017.08.1963"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1002\/j.2161-4296.1978.tb01335.x","article-title":"Integration of GPS with Inertial Navigation Systems","volume":"25","author":"Cox","year":"1978","journal-title":"Navigation"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"750","DOI":"10.1109\/TAES.2006.1642588","article-title":"Sigma-point Kalman filtering for integrated GPS and inertial navigation","volume":"42","author":"Crassidis","year":"2006","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1109\/TMECH.2002.1011250","article-title":"An outdoor navigation system using GPS and inertial platform","volume":"7","author":"Panzieri","year":"2002","journal-title":"IEEE\/ASME Trans. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1109\/TAES.2002.1008998","article-title":"Direct Kalman filtering approach for GPS\/INS integration","volume":"38","author":"Honghui","year":"2002","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1177\/0278364910388963","article-title":"Visual-inertial navigation, mapping and localization: A scalable real-time causal approach","volume":"30","author":"Jones","year":"2011","journal-title":"Int. J. Robot. Res."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Gros, S., and Diehl, M. (2012, January 10\u201313). Attitude estimation based on inertial and position measurements. Proceedings of the 51st IEEE Conference on Decision and Control (CDC), Maui, HI, USA.","DOI":"10.1109\/CDC.2012.6426420"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2451","DOI":"10.1021\/ie034308l","article-title":"Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation","volume":"44","author":"Haseltine","year":"2005","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_22","first-page":"519","article-title":"Moving Horizon Estimation for Integrated Navigation Filtering","volume":"48","year":"2015","journal-title":"IFAC-PapersOnLine"},{"key":"ref_23","unstructured":"Gul, H.U., and Kai, D.Y. (2016, January 10\u201313). An Optimal Moving Horizon Estimation for Aerial Vehicular Navigation Application. Proceedings of the IOP Conference Series: Materials Science and Engineering, R\u00e9union Island, France."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"734","DOI":"10.2514\/1.58805","article-title":"Spacecraft Attitude Estimation and Sensor Calibration Using Moving Horizon Estimation","volume":"36","author":"Vandersteen","year":"2013","journal-title":"J. Guid. Control Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1109\/TAES.2003.1261132","article-title":"Survey of maneuvering target tracking. Part I: Dynamic models","volume":"39","author":"Jilkov","year":"2003","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Titterton, D., and Weston, J.L. (2004). Strapdown Inertial Navigation Technology, AIAA.","DOI":"10.1049\/PBRA017E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1109\/TIM.2007.908635","article-title":"Analysis and Modeling of Inertial Sensors Using Allan Variance","volume":"57","author":"Hou","year":"2008","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"19","DOI":"10.2514\/2.4228","article-title":"Strapdown Inertial Navigation Integration Algorithm Design Part 1: Attitude Algorithms","volume":"21","author":"Savage","year":"1998","journal-title":"J. Guid. Control Dyn."},{"key":"ref_29","unstructured":"Kaplan, E.D., and Hegarty, C.J. (2006). Understanding GPS : Principles and Applications, Artech House."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Von Stryk, O. (1993). Numerical Solution of Optimal Control Problems by Direct Collocation. Optimal Control, Birkh\u00e4user Basel.","DOI":"10.1007\/978-3-0348-7539-4_10"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bock, H.G., Diehl, M.M., Leineweber, D.B., and Schl\u00f6der, J.P. (2000). A Direct Multiple Shooting Method for Real-Time Optimization of Nonlinear DAE Processes. Nonlinear Model Predictive Control, Birkh\u00e4user Basel.","DOI":"10.1007\/978-3-0348-8407-5_14"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1090\/S0025-5718-1964-0165693-1","article-title":"Integration Processes Based on Radau Quadrature Formulas","volume":"18","author":"Butcher","year":"1964","journal-title":"Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1016\/j.robot.2014.05.004","article-title":"An Optimization Based Moving Horizon Estimation with Application to Localization of Autonomous Underwater Vehicles","volume":"62","author":"Wang","year":"2014","journal-title":"Robot. Auton. Syst."},{"key":"ref_34","unstructured":"(2017, May 12). Xsens Technologies B.V. Xsens MTi-G-710 Motion Tracker. Available online: https:\/\/www.xsens.com\/products\/mti-g-700\/."},{"key":"ref_35","unstructured":"iMAR Navigation GmbH (2017, May 12). iIMU-FSAS: IMU with Odometer Interface and Integrated Power Regulation. Available online: http:\/\/www.imar-navigation.de\/downloads\/IMU_FSAS.pdf."},{"key":"ref_36","unstructured":"Vydhyanathan, A., Bellusci, G., Luinge, H., and Slycke, P. (2017, May 12). The Next Generation Xsens Motion Trackers for Industrial Applications. Available online: https:\/\/www.xsens.com\/download\/pdf\/documentation\/mti\/mti_white_paper.pdf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1714","DOI":"10.1137\/S0363012902400713","article-title":"A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control","volume":"43","author":"Diehl","year":"2005","journal-title":"SIAM J. 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