Autonomous relative navigation is one of the keytechnologies for the accompanying satellite, the space RVD(Rendezvous and Docking) mission, the capture andmaintenance of the on orbit satellites, and deep spaceexplore. Autonomous operations reduce the dependence ofspace missions to human interaction and communication with Earth, if not make it totally independent, especially fora cluster of multiple spacecraft. Formation flight of multiplespacecraft in deep space is another area that has attracted agreat deal of interest in recent years. To accomplish theformation, accurate estimation of relative position betweenthe space vehicles is necessary. Spacecrafts operating onclose-to-Earth orbits can obtain a complete autonomousnavigation solution through the current Global NavigationSatellite System (GNSS) (B. W. Parkinson and J. J. Spiker,Jr., 1996). But, for deep space missions or situations whereGNSS is not available, an alternative navigation approach isneeded. Employing Earth-based navigation systems, such asthe Deep Space Network (DSN), is a possibility. However,such systems suffer from low performance in situationswhere long range navigation is required. Furthermore, theyare highly based on communicating with Earth to analyzetheir data. Because of these problems, the need for higheraccuracy, and also continuingly increasing cost ofspacecrafts operations, spacecraft navigation is evolvingfrom Earth-based solutions towards more autonomousmethods (D. Folta, C. Gramling, A. Long, D. Leung, andS. Belur, 1999 and R. Gounley, R. White, and E. Gai,1984). An autonomous navigation system internallycomputes its own navigation and guidance information byusing onboard sensors. A possibility is to use the signalsemitted from X-ray celestial sources. One of the mostreliable X-ray sources is pulsars. Relative navigation ofspacecrafts may be accomplished by observing X-raysources and indirectly determining the spacecrafts' relativeposition.For X-ray pulsars navigation, which has been employed andstudied in both contexts of absolute navigation (Sheikh, S. I.,et al. 2006, Sheikh, S. I. and Pines, D. J. 2006, and Sala,J., et al, 2004), and relative navigation (Sheikh, S. I.,Golshan, A. R., and Pines, D. J. 2007, Emadzadeh, A. A.,Speyer, J. L., and Hadaegh, F. Y. 2007, Emadzadeh, A.A., Speyer, J. L., and Golshan, A. R. 2009, andEmadzadeh, A. A., Golshan, A. R., and Speyer, J. L2009), is shown that a key task is estimation of the pulsephase (Emadzadeh, A. A., Speyer, J. L., and Golshan, A.R. 2009, Emadzadeh, A. A., Golshan, A. R., and Speyer,J. L 2009, Golshan, A. R. and Sheikh, S. I. 2007, Hanson,J., et al. 2008, and Emadzadeh, A. A. and Speyer, J. L.2010). In (Hanson, J., et al. 2008) utilizing the epochfolding procedure is proposed for estimation of the pulsephase, and the proposed estimator is analyzed assuming thatthe procedure noise is Gaussian. In (madzadeh, A. A. andSpeyer, J. L. 2010) the epoch folding procedure ismathematically formulated, and it is shown how it results inretrieving the pulsar intensity. The procedure noise isanalyzed, and its statistical properties such as its mean,variance, and autocorrelation function are presented. In thiswork, the pulse delay estimation problem is introduced, andit is explained how employing epoch folding, the relativeposition between two space vehicles, can be estimated.Based on epoch folding, two different pulse delay estimatorsare proposed, and their performance is compared against theCramér-Rao lower bound (CRLB). It is also investigatedhow imprecise absolute velocity data can affect the positionestimation accuracy.Then based on above research, the algorithm of relativenavigation for formation flying spacecrafts using X-raypulsars was investigated. And a novel relative navigationalgorithm fo r multiple-satellite formation using X-raypulsars measurements is proposed. The problem of relativenavigation between formation flights utilizing X-ray pulsarsmeasurements is investigated. The time difference of signalarrival (TDOA) is estimated by signal's cross-correlatedprocessing, which is further used as measurement to achievethe relative navigation. A Constrained Adaptive KalmanFilter is employed to estimate the relative positions andvelocities between the formation flights. Numericalsimulations are performed to assess the proposed navigationalgorithm. Furthermore, errors of the navigation areanalyzed in order to improve the accuracy of this method.So in a word, this paper presents a method of determining arelative navigation solution between two formation flightsby using celestial X-ray pulsar sources. The rest of the paperis organized as follows. In Section II, the principle of therelative navigation based on X-ray pulsar is described, andmathematical models describing the X-ray pulsar signals aredeveloped. Furthermore, the epoch folding procedure ispresented and analyzed. Section IV presents the CRLB forestimation of the pulse phase, and how by employing theepoch folding approach and fitting the photon count data to the known pulsar rate function the pulse phase can beestimated is exp lains. It also compares the proposedestimator’s performance against the CRLB. In Section VI,using the measured photon TOAs, a ML estimator isdeveloped, and its statistical properties are studied. SectionVII discusses the relative dynamic equations between twoformation flights, as well as the statement function used inlater simulation processes. Section VIII introduces thealgorithm and procedures for using constrained adaptiveKalman Filter for non-linear systems to estimate thestatements and filter the errors. Section IX providesprecondition of simulation, and results of three-dimensionrelative navigation between two formation flights are given.Finally, some concluding remarks are given in Section X.And some proofs, derivations, and clarifying remarks arepresented in the Appendix.
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