{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:57:02Z","timestamp":1760237822693,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,6,27]],"date-time":"2020-06-27T00:00:00Z","timestamp":1593216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Nie guigen","award":["41074023"],"award-info":[{"award-number":["41074023"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, X-ray pulsar-based navigation (XPNAV) as a significant navigation method has been widely used in deep space exploration. However, the accuracy of XPNAV is limited to the existence of the pulsar direction error. To improve the performance of XPNAV, we have proposed a novel algorithm named \u201cthe modified augmented state extended Kalman filter\u201d (MASEKF). The algorithm considers the high-order terms of direction error and then adds a more precise direction error into state equation and measurement equation. In the simulation, by comparing the performance of MASEKF, EKF, and ASEKF at the same time, it is found that MASEKF has better performance in the accuracy and stability, and the results also demonstrate that MASEKF algorithm has faster convergence speed. This paper provides a strong reference for other improvements of algorithms towards direction error. The purpose of this study is to establish MASEKF and add the direction error into the measurement equation and the state equation, so as to realize the coordination and symmetry of the algorithm.<\/jats:p>","DOI":"10.3390\/sym12071059","type":"journal-article","created":{"date-parts":[[2020,6,30]],"date-time":"2020-06-30T14:24:24Z","timestamp":1593527064000},"page":"1059","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An Improved Augmented Algorithm for Direction Error in XPNAV"],"prefix":"10.3390","volume":"12","author":[{"given":"Xiaobin","family":"Ren","sequence":"first","affiliation":[{"name":"GNSS Research Center, Wuhan University; Wuhan 430019, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1271-7968","authenticated-orcid":false,"given":"Guigen","family":"Nie","sequence":"additional","affiliation":[{"name":"GNSS Research Center, Wuhan University; Wuhan 430019, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lianyan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5542","DOI":"10.1016\/j.ijleo.2016.03.043","article-title":"A robust compressed sensing based method for X-ray Pulsar Profile construction","volume":"127","author":"Shen","year":"2016","journal-title":"Optik"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.actaastro.2016.08.007","article-title":"Performance enhancement of X-ray pulsar navigation using autonomous optical sensor","volume":"128","author":"Xiong","year":"2016","journal-title":"Acta Astronaut."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.2514\/1.G000789","article-title":"Validation of Pulsar Phase Tracking for Spacecraft Navigation","volume":"38","author":"Anderson","year":"2015","journal-title":"J. Guid. Control Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.ijleo.2018.08.116","article-title":"A novel X-ray pulsar integrated navigation method for ballistic aircraft","volume":"175","author":"Xu","year":"2018","journal-title":"Optik"},{"key":"ref_5","first-page":"4369","article-title":"Application of STEKF in X-ray Pulsar Based Autonomous Navigation","volume":"29","author":"Du","year":"2012","journal-title":"Int. Workshop Inf. Electron. Eng."},{"key":"ref_6","unstructured":"Wan, E.A., van der Merwe, R., and IEEE (2000, January 1\u20134). The unscented Kalman Filter for nonlinear estimation. Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium 2000, Lake Louise, AB, Canada."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jprocont.2007.11.004","article-title":"Applying the unscented Kalman filter for nonlinear state estimation","volume":"18","author":"Kandepu","year":"2008","journal-title":"J. Process Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1016\/j.asr.2010.08.019","article-title":"X-ray pulsar navigation method for spacecraft with pulsar direction error","volume":"46","author":"Liu","year":"2010","journal-title":"Adv. Space Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5640309","DOI":"10.1155\/2017\/5640309","article-title":"State Estimation for Sampled-Data Descriptor Nonlinear System: A Strong Tracking Unscented Kalman Filter Approach","volume":"2017","author":"Liang","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1017\/S0373463318000632","article-title":"Orbit Determination Using Pulsar Timing Data and Orientation Vector","volume":"72","author":"Zhang","year":"2019","journal-title":"J. Navig."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ast.2011.04.010","article-title":"Pulsar navigation for interplanetary missions using CV model and ASUKF","volume":"22","author":"Liu","year":"2012","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_12","first-page":"17","article-title":"Formation of a Satellite Navigation System Using X-Ray Pulsars","volume":"131","author":"Zhang","year":"2019","journal-title":"Publ. Astron. Soc. Pac."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1016\/j.asr.2018.08.026","article-title":"An improved augmented X-ray pulsar navigation algorithm based on the norm of pulsar direction error","volume":"62","author":"Xu","year":"2018","journal-title":"Adv. Space Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1068","DOI":"10.1109\/7.165368","article-title":"Observability Analysis of Piece-Wise Constant Systems.2. Application to Inertial Navigation In-Flight Alignment","volume":"28","author":"Goshenmeskin","year":"1992","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/j.asr.2012.10.009","article-title":"Autonomous navigation of Mars probe using X-ray pulsars: Modeling and results","volume":"51","author":"Wei","year":"2013","journal-title":"Adv. Space Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1639","DOI":"10.1016\/j.asr.2019.07.038","article-title":"Designing observation scheme in X-ray pulsar-based navigation with probability ellipsoid","volume":"64","author":"Lin","year":"2019","journal-title":"Adv. Space Res."},{"key":"ref_17","first-page":"105","article-title":"The use of x-ray pulsars for spacecraft navigation","volume":"119","author":"Sheikh","year":"2005","journal-title":"Adv. Astronaut. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2514\/1.13331","article-title":"Spacecraft navigation using x-ray pulsars","volume":"29","author":"Sheikh","year":"2006","journal-title":"J. Guid. Control Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2310","DOI":"10.2514\/1.G003677","article-title":"Characterization of X-Ray Pulsar Navigation for Tracking Closed Earth Orbits","volume":"42","author":"Yu","year":"2019","journal-title":"J. Guid. Control Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3847\/1538-4365\/ab3718","article-title":"In-orbit Demonstration of X-Ray Pulsar Navigation with the Insight-HXMT Satellite","volume":"244","author":"Zheng","year":"2019","journal-title":"Astrophys. J. Suppl. Ser."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yu, W.H., and Inst, N. (2016, January 12\u201316). Application of X-Ray Pulsar Navigation: A Characterization of the Earth Orbit Trade Space. Proceedings of the 29th International Technical Meeting of the Satellite Division of the Institute of Navigation, Portland, OR, USA.","DOI":"10.33012\/2016.14750"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s10569-010-9320-4","article-title":"Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009","volume":"109","author":"Archinal","year":"2011","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_23","first-page":"123","article-title":"XANV\/GNSS-Based Integrated Navigation Using a Robust Filtering Algorithm","volume":"48","author":"Zhou","year":"2016","journal-title":"J. Aeronaut. Astronaut. Aviat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1109\/JSEN.2018.2879355","article-title":"A Novel Celestial Aided Time-Differenced Pulsar Navigation Method Against Ephemeris Error of Jupiter for Jupiter Exploration","volume":"19","author":"Gui","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_25","first-page":"367","article-title":"A new HST measurement of the Crab pulsar proper motion","volume":"344","author":"Caraveo","year":"1999","journal-title":"Astron. Astrophys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1049\/iet-rsn.2017.0571","article-title":"X-ray pulsar navigation using multiple detectors based on a new observation strategy","volume":"12","author":"Guo","year":"2018","journal-title":"IET Radar Sonar Navig."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10569-007-9072-y","article-title":"Report of the IAU\/IAG Working Group on cartographic coordinates and rotational elements: 2006","volume":"98","author":"Seidelmann","year":"2007","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.ast.2018.07.019","article-title":"Autonomous orbit and attitude determination for Earth satellites using images of regular-shaped ground objects","volume":"80","author":"Li","year":"2018","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Scott, R., Ploen, D.P.S., Fred, Y., Hadaegh, A., and Acikmese, B. (2004). Dynamics of Earth Orbiting Formations. Collect. Tech. Pap. Aiaa Guid. Navig. Control Conf.","DOI":"10.2514\/6.2004-5134"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.asr.2010.07.007","article-title":"Numerical modeling of a Global Navigation Satellite System in a general relativistic framework","volume":"47","author":"Delva","year":"2011","journal-title":"Adv. Space Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1016\/j.asr.2016.02.001","article-title":"X-ray pulsars\/Doppler integrated navigation for Mars final approach","volume":"57","author":"Cui","year":"2016","journal-title":"Adv. Space Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.ejcon.2018.12.010","article-title":"Long baseline navigation with explicit pseudo-range clock offset and propagation speed estimation","volume":"49","author":"Batista","year":"2019","journal-title":"Eur. J. Control"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Pines, J., Ray, P.S., and Wood, K.S. (2006). Spacecraft Navigation Using X-Ray Pulsars. J. Guid. Control Dyn., 29.","DOI":"10.2514\/1.13331"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1864","DOI":"10.1016\/j.asr.2016.07.021","article-title":"Augmentation method of XPNAV in Mars orbit based on Phobos and Deimos observations","volume":"58","author":"Rong","year":"2016","journal-title":"Adv. Space Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2394","DOI":"10.1016\/j.asr.2013.02.007","article-title":"X-ray pulsar-based navigation system with the errors in the planetary ephemerides for Earth-orbiting satellite","volume":"51","author":"Wang","year":"2013","journal-title":"Adv. Space Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1059\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:43:34Z","timestamp":1760175814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1059"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,27]]},"references-count":35,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071059"],"URL":"https:\/\/doi.org\/10.3390\/sym12071059","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,6,27]]}}}