New observation strategy for X‐ray pulsar navigation using a single detector
2016; Institution of Engineering and Technology; Volume: 10; Issue: 6 Linguagem: Inglês
10.1049/iet-rsn.2015.0480
ISSN1751-8792
AutoresHao Liang, Yafeng Zhan, Hailiang Yin,
Tópico(s)Geophysics and Gravity Measurements
ResumoIET Radar, Sonar & NavigationVolume 10, Issue 6 p. 1107-1111 Research ArticleFree Access New observation strategy for X-ray pulsar navigation using a single detector Hao Liang, Hao Liang School of Aerospace Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorYafeng Zhan, Corresponding Author Yafeng Zhan zhanyf@tsinghua.edu.cn School of Aerospace Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorHailiang Yin, Hailiang Yin Department of Electronic Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author Hao Liang, Hao Liang School of Aerospace Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorYafeng Zhan, Corresponding Author Yafeng Zhan zhanyf@tsinghua.edu.cn School of Aerospace Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorHailiang Yin, Hailiang Yin Department of Electronic Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author First published: 01 July 2016 https://doi.org/10.1049/iet-rsn.2015.0480Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract X-ray pulsar navigation using a single detector will be applied in the spacecraft because of limited payloads. For the limited position precision problem of traditional methods which observe pulsars in turn, this study proposed a new observation strategy. On the basis of the posterior covariance matrix in the Bayesian filter structure, the proposed observation strategy gives priority to the pulsar which minimises the relative degree of divergence. Simulation results verify the validity of the observation strategy. 1 Introduction Pulsars are high-speed rotating neutron stars which could generate pulse signals periodically and uniquely in multi-electromagnetic bands [1]. Pulsars have been regarded as nature's 'most stable clocks' [2, 3], and could provide ideal signals for autonomous navigation system of spacecraft. Since Downs [4] first proposed autonomous navigation using pulsars in 1974, a complete theoretical system for pulsar navigation has been built gradually [1, 5-7]. For the practical consideration, it is a best choice to use pulse signals in the X-ray band for spacecraft navigation because of the limited size for detectors onboard. At present, X-ray pulsar navigation (XPNAV) system has been developed from theoretical research to engineering verification. The neutron-star interior composition explorer/station explorer for X-ray timing and navigation technology mission [8, 9] from National Aeronautic and Space Administration (NASA) and the Lightweight Asymmetry and Magnetism Probe (LAMP) from Tsinghua University (THU) are planning to demonstrate XPNAV on-orbit. However, in the initial stage of engineering verification, we may not be able to carry enough X-ray detectors onboard because of various practical restrictions such as payload size and budgets. Thus, it is valuable to study methods for XPNAV using a single detector. The methods for XPNAV using a single detector could be divided into two types: one is to detect a fixed pulsar, which is obviously unobservable [10]; the other is to detect pulsars in time division, which could provide high navigation accuracy [10, 11]. For the method to detect pulsars in time division, most literatures follow a fixed observation order and observe pulsars in turn [9-11]. That is simple but the performance is not good. On the basis of the state error information in the posterior covariance matrix of Bayesian filter, this paper proposes an observation strategy for minimising the relative degree of divergence at each updating time. Comparing with that using the fixed order, we could get a higher navigation accuracy using the minimum relative degree of divergence (MRDD) observation strategy. The structure of this paper is as follows. Section 2 introduces the XPNAV system model. Section 3 describes and analyses the MRDD observation strategy proposed in this paper for XPNAV using a single detector. Section 4 compares the MRDD observation strategy with conventional methods by computer simulations. Section 5 draws the conclusions. 2 XPNAV system model 2.1 State equations The earth-orbiting spacecraft's dynamic characteristic is mainly affected by earth-gravity and planetary perturbations. The dynamic equation could be expressed as [12] (1) where is the spacecraft's position vector, μ is the geocentric gravitational constant, U is the sum of perturbations' potential functions including the Earth's non-spherical shape, lunisolar gravitational perturbation, solar radiation pressure, atmospheric drag etc. and grad(·) is the gradient function. Assuming is the spacecraft's velocity's vector, is the state vector, f(·) is the state transition function and is the state noise which follows the Gaussian distribution with mean zero and covariance . The state function could be expressed as (2) When only considering the earth-gravity, we could build the two-body state equations as (3) where , 03×3, 13×3 and represent the 3 × 3 diagonal matrix whose diagonal elements are 0, 1 and − μ /r3, respectively. 2.2 Measurement equations The principle of XPNAV is shown in Fig. 1 [13]. Fig. 1Open in figure viewerPowerPoint Principle of XPNAV As the distance between pulsars and the earth are over 1 kpc (1 kpc ; 3.086e19 m), we could consider the direction vectors of pulsars to the earth and the solar system barycenter (SSB) are the same. Measuring the time when an X-ray pulsar signal arrives at the spacecraft in the geocentric coordinate time (TCG) system, transferring to the barycentric coordinate time (TCB) system and subtracting from the time when the same X-ray pulsar signal arrives at SSB, we could get the time difference of arrival (TDOA) between the spacecraft and SSB. could be achieved by XPNAV database. Multiplying the TDOA by the light velocity, we could get the spacecraft's position vector relative to SSB in the pulsar's directional projection. When observing no less than three pulsars, the spacecraft's exact position could be determined. On the basis of the above principle of XPNAV, we could get the equations (4) where is the ith pulsar's arrive time at SSB in TCG, denotes the ith pulsar's arrive time at the spacecraft in TCB, is the time transfer error of the relativity theory, Δti is the ith TDOA between the spacecraft and SSB, is the ith pulsar's unit direction vector, is the spacecraft's position vector, c is the light velocity, δtc is the clock bias. could be calculated by [1] (5) where D0 is the distance between the pulsar to SSB, b is the SSB's position vector and μs is the heliocentric geocentric gravitational constant. Assuming , , , is the measurement noise which follows the Gaussian distribution with mean zero and covariance . The measurement equations in matrix could be expressed as (6) 3 Observation strategy In consideration of computational accuracy, complexity and reliability, the navigation filtering algorithm used in XPNAV is generally based on the Bayesian filter structure [1, 14, 15]. Taking (2) and (6) into the Bayesian filter, the navigation filter of XPNAV could be expressed as [16] Time update (7) (8) Measurement update (9) (10) (11) (12) (13) (14) where is the estimation of at the discrete time k, is the state covariance matrix of Bayesian filter, represents the Gaussian density function with mean and covariance , . Assuming is the error of estimation, is a unit direction matrix, , and then to 's directional projection could be defined as [17] (15) The variance of ɛk|k is (16) Assuming is the unit direction vector when obtains the maximal value . When the random variable is in 's directional projection, the variance of the random variable ɛk|k could reach the maximum. Therefore, reflects the highest possible of 's divergence direction, and reflects the relatively divergence possibility. The more is, the more possible the Bayesian filter could be divergence. Then, we could define that the relative degree of divergence is . Using Lagrange multiplier method, and assuming (17) then (18) (19) Thus, λ is equal to the eigenvalue of , is equal to the eigenvector of . Then (20) (21) λmax is the maximal eigenvalue of , which is the relative degree of divergence that we are seeking for. To minimise the relative degree of divergence is an obvious way to reduce the estimation error and divergence possibility of the state. For XPNAV using a single detector and observing pulsars in time division, we may take each pulsars' measurement functions into (9)–(13) and predict the maximal eigenvalue of . In order to minimise the relative degree of divergence, we propose to observe the pulsar whose maximal eigenvalue of is minimum at each updating time. The MRDD observation strategy is as follows: (a) By using the state and the state covariance matrix at the discrete time k − 1, do the time updating and calculate the prior covariance matrix from (7) and (8). (b) Build the measurement function assemble from (6), where represents the measurement function of observing the ith pulsar, and m is the total number of the pulsars. (c) Take each AZ into (9)–(13) and calculate the posterior covariance matrix assemble , where represents the posterior covariance matrix of the ith pulsar. (d) Select the pulsar whose maximal eigenvalue of is minimum to observe, update the observed pulsars' TDOA and get the measurement value . (e) Update the system by using to get the state at the discrete time k from (14), and then back to step a). 4 Computer simulations 4.1 Simulation conditions The SEPNet01 is a low Earth orbit mobile communication satellite launched by THU. On the basis of XPNAV system, we simulate the SEPNet01 navigation by observing three pulsars which is minimum needed for XPNAV. The parameters of X-ray pulsar sources are shown in Table 1 [1], and the orbital elements of SEPNet01 are shown in Table 2. The σΔt in Table 1 is based on 500 s observation time and 1 m2 detector. Table 1. Parameters of X-ray pulsar sources ID NAME RA (J2000), h:min:s DE (J2000), dd:mm:ss Flux, photons/cm2/s Period, s σΔt, ns 1 B0531 + 21 05:34:31.973 +22:00:52.06 1.54e00 0.0334 3.63e2 2 B1821−24 18:24:32.01 −24:52:10.74 1.93e-4 0.0031 1.08e3 3 B1939 + 21 19:39:38.6 +21:34:59.14 4.99e-5 0.0016 1.15e3 Table 2. Orbital elements of SEPNet01 satellite Semi-major axis, km Inclination, deg RAAN, deg Eccentricity, deg Average of perigee, deg True anomaly, deg 7163.87 98.40 310.15 0.0041 96.84 202.30 In the simulation, we calculate the accuracy of XPNAV with three methods: (i) observe three pulsars at the same time; (ii) observe three pulsars in turn; (iii) observe three pulsars with the MRDD observation strategy. The orbit data of SEPNet01 is generated by two-body models. As a result of the high computational accuracy and robustness of the cubature Kalman filter (CKF) algorithm [18], we adopt CKF as the Gaussian Bayesian filter for navigation. The updating time Ts is set as 100 and 600 s, respectively. The single simulation time is 5 days, and 100 times Monte Carlo simulations are performed for each method. The state transition equation of the spacecraft is based on the two-body model, whose state equations are shown in (3). The fourth-order Runge–Kutta method is adopted for computing the state equations [19]. The measurement equation is described as (6), where is determined from σΔt in Table 1, diag(·) means the diagonal matrix. 4.2 Simulation results 100 s updating time represents the relatively short observing time interval. In the condition of 100 s updating time, simulation results of the algorithms are shown in Figs. 2 and 3. Fig. 2Open in figure viewerPowerPoint Position error of XPNAV (100 s updating time) Fig. 3Open in figure viewerPowerPoint Velocity error of XPNAV (100 s updating time) The average position errors and the velocity errors of three methods are shown in Table 3. Table 3. Comparison of three methods' position errors and velocity errors with 100 s updating time Methods Average position error, km Accuracy decline ratio, % Average velocity error, m/s Accuracy decline ratio, % observe three pulsars at the same time 0.308 — 0.318 — observe three pulsars in turn 0.445 44.48 0.445 40.00 observe three pulsars with the MRDD observation strategy 0.362 17.53 0.363 14.15 From Table 3, the method of observing three pulsars at the same time has the highest accuracy, observing with the MRDD observation strategy is in the second place and much better than observing pulsars in turn. Therefore, in the condition of short observing time interval, observing with the MRDD observation strategy using a single detector is more accurate than observing pulsars in turn. During two orbit periods, each pulsar's observation time distribution of the MRDD observation strategy with 100 s updating time is shown in Fig. 4. Fig. 4Open in figure viewerPowerPoint Pulsars' observation time distribution (100 s updating time) As we can obtain from Fig. 4, the pulsar 'B1821–24' is never observed. Two reasons could explain this phenomenon. First, from Table 1, 'B1821–24's direction vector is almost parallel to 'B0531 + 21's, and it is more probable to choose the one with small measurement error. Second, because the orbit model is precise, the divergence speed of the navigation filter is slow, and observing 'B1821–24' is not yet acquired. 600 s updating time represents the relatively long observing time interval. In the condition of 600 s updating time, simulation results of the algorithms are shown in Figs. 5 and 6. Fig. 5Open in figure viewerPowerPoint Position error of XPNAV (600 s updating time) Fig. 6Open in figure viewerPowerPoint Velocity error of XPNAV (600 s updating time) The average position errors and the velocity errors of three methods are shown in Table 4. Table 4. Comparison of three methods' position errors with 600 s updating time Methods Average position error, km Accuracy decline ratio, % Average velocity error, m/s Accuracy decline ratio, % observing at the same time 0.203 — 0.199 — observing in turn 0.305 50.25 0.291 46.23 observing with the MRDD observation strategy 0.241 18.72 0.233 17.09 From Table 4, in the condition of 600 s updating time, observing with the MRDD observation strategy's performance is still better than observing pulsars in turn. Besides that, the accuracy decline ratio of position errors and velocity errors are almost the same. Therefore, in the condition of both short and long observing time interval, observing with the MRDD observation strategy using a single detector is more accurate than observing pulsars in turn. Each pulsar's observation time distribution of the MRDD observation strategy with 600 s updating time is similar to the one shown in Fig. 4 with 100 s updating time. 5 Conclusion The single detector navigation is an important research aspect in XPANV. For the low precision problem of observing pulsars in turn, based on the study of the posterior covariance matrix in the Bayesian filter structure, this paper proposes an MRDD observation strategy. Simulation results show that, compared with traditional observing method, observing with the MRDD observation strategy could achieve a higher accuracy. 6 Acknowledgments This work is supported by National Natural Science Foundation of China (61271265, 61132002), and Tsinghua University Independent Scientific Research Project (2013089244). 7 References 1Sheikh, S.I.: ' The use of variable celestial X-ray sources for spacecraft navigation'. 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