Augmentation method of XPNAV in Mars orbit based on Phobos and Deimos observations

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ScienceDirect Advances in Space Research 58 (2016) 1864–1878 www.elsevier.com/locate/asr

Augmentation method of XPNAV in Mars orbit based on Phobos and Deimos observations Jiao Rong, Xu Luping, Zhang Hua ⇑, Li Cong Xidian University, School of Aerospace Science and Technology, Xi’an 710126, China Received 18 September 2015; received in revised form 14 July 2016; accepted 15 July 2016 Available online 25 July 2016

Abstract Autonomous navigation for Mars probe spacecraft is required to reduce the operation costs and enhance the navigation performance in the future. X-ray pulsar-based navigation (XPNAV) is a potential candidate to meet this requirement. This paper addresses the use of the Mars’ natural satellites to improve XPNAV for Mars probe spacecraft. Two observation variables of the field angle and natural satellites’ direction vectors of Mars are added into the XPNAV positioning system. The measurement model of field angle and direction vectors is formulated by processing satellite image of Mars obtained from optical camera. This measurement model is integrated into the spacecraft orbit dynamics to build the filter model. In order to estimate position and velocity error of the spacecraft and reduce the impact of the system noise on navigation precision, an adaptive divided difference filter (ADDF) is applied. Numerical simulation results demonstrate that the performance of ADDF is better than Unscented Kalman Filter (UKF) DDF and EKF. In view of the invisibility of Mars’ natural satellites in some cases, a visibility condition analysis is given and the augmented XPNAV in a different visibility condition is numerically simulated. The simulation results show that the navigation precision is evidently improved by using the augmented XPNAV based on the field angle and natural satellites’ direction vectors of Mars in a comparison with the conventional XPNAV. Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: X-ray pulsar; Field angle; Direction vectors; Information fusion; Adaptive divided difference Kalman filtering

1. Introduction Mars exploration is very important part in the deep space exploration. Mars has drawn more space missions than the rest of the Solar System’s planets to date. However, roughly two-thirds of all spacecraft destined for Mars failed before completing their missions and some failed before their observations could begin. The high failure rate of missions attempting to explore Mars is even informally called as the Mars Curse (Curse, 2008). One of the error result in these failures is the orbit error caused ⇑ Corresponding author.

E-mail addresses: [email protected] (J. Rong), [email protected]. edu.cn (X. Luping), [email protected] (H. Zhang), lcongfl[email protected] (L. Cong). http://dx.doi.org/10.1016/j.asr.2016.07.021 0273-1177/Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

by the human operation. For example, in 1999, NASA’s Mars Climate Orbiter, just before orbital insertion, a navigation error sent the satellite into an orbit 100 km lower than its intended 150 km altitude above the planet (Sauser et al., 2009). The probe failure of ‘‘PhobosGrunt” just launched by Russia also demonstrates the importance of the autonomous navigation of Mars probe. Space vehicles operating on close-to-Earth orbits can achieve a complete autonomous navigation solution through the current Global Positioning System (GPS). But, for deep space missions as Mars probe, the GPS is not available due to the signal is too weak to reach the Mars orbit. An alternative is the Deep Space Network (DSN) (Bagri et al., 2007), but such systems suffer from low performance in situations where long range navigation is required due to the geometric dilution of precision

J. Rong et al. / Advances in Space Research 58 (2016) 1864–1878

(GDOP). In currently, the core radius is just determined with a precision of ±200 km (Asmar et al., 2014; Cao et al., 2010), but the XPNAV is expected to achieve less than 1 km (Sheikh, 2005). Autonomous method using celestial sources are proposed to deal with these problem (Fang and Ning, 2009; Rad et al., 2014). The conventional celestial-based system using the star sensor to obtain the geometry relation between the stars and the orbit (Ali et al., 2006; Fang and Ning, 2009; Rad et al., 2014). Generally, this method is just suitable for low orbit, due to that if the precision decrease greatly with the orbit altitude increasing. Recently, the use of the variable celestial sources, such as pulsars, to determine the position draws wide attention in aerospace domain. Pulsar is a natural neutron star, which emits pulse signal periodically. The shape and period of the pulse signal are known to a very high accuracy. These features of pulsars make them good candidates to be employed by autonomous navigation systems (Sheikh, 2005; Graven et al., 2008; Liu et al., 2010a,b; Feng et al., 2013). Comparing with the traditional celestial navigation methods, XPNAV can provide complete navigation solution consisting of time, position, velocity and attitude, and can provide seamless navigation service for the spacecraft from near-Earth orbit to deep space. However, the precision of XPNAV is not high enough by the state-of-the-art technology due to the low Signal Noise Ratio (SNR), phase evolution error and so on (Sheikh, 2005; Ashby and Howe, 2006; Bernhardt et al., 2010). Several methods have been proposed to improve the precision of XPNAV. (Liu et al., 2010a,b; Deng et al., 2012) (Emadzadeh et al., 2009; Huang et al., 2009; Zhang and Xu, 2011) try to refine the phase prediction model and improve the precision of time of arrival (TOA) to improve the performance. However, it is limited by the signal SNR. Another way is to integrated XPNAV and celestial navigation base on star sensor, namely XPNAV/CNS, which is put forward in literature (Liu et al., 2010a,b). Adding inter-satellite baseline information into the XPNAV can also effectively improve the navigation precision in the constellations navigation verified by literatures (Emadzadeh et al., 2009; Chen et al., 2011). One novel XPNAV based on the direction vector from the Earth to the satellite is presented in literature (Qiao et al., 2009) and the simulation results indicate that the valid direction vector information is beneficial to the improvement of XPNAV precision. Motivated by these methods aforementioned, a method taking full advantage of the field angle and direction vectors from the spacecraft to the natural satellites of Mars (Phobos and Deimos) is proposed in this paper. In addition, the adaptive divided difference filter (ADDF) is adopted to integrate the different observations to avoid performance degradation. In XPNAV, the commonly used filters are Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) (Qiao et al., 2009; Liu et al., 2010a,b; Liu et al., 2014). EKF provides insufficient accuracy due to its firstorder approximation of the system equations. Derivation

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of a Jacobian matrix is another drawback to the application of EKF. UKF, as one of the sigma point Kalman filters, avoids a derivative calculation and has a higher theoretic precision than EKF (Wan and Van Der Merwe, 2000). Central Difference Filter (CDF), as another sigma point filter, is modified by employing Stirling’s interpolation formula and named as Divided Difference Filter (DDF) (Schei, 1997). The performance of second-order DDF based on a second order approximation is demonstrably better than second-order EKF and as good as that of UKF (Sˇimandl and Dunı´k, 2009). However, in the XPNAV, the knowledge of process noise statistics is always unavailable and improper assumption of process noise leads to poor estimation performance of the filter. Thus, an adaptive DDF (Dey et al., 2014), called ADDF, is selected and applied to the XPNAV here. It can estimate parameters and states of a nonlinear system with unknown process noise statics by adapting covariance of process noise. In this paper, a significant navigation precision improvement of augmented XPNAV with ADDF will be realized. The rest of the paper is organized as follows. Basic principle of XPNAV Positioning System will be presented in Section 2. Then, the augmented XPNAV model of spacecraft for Mars exploration is described in Section 3. The numerical simulations and the analysis are given in Section 4. Finally, conclusions are drawn in Section 5. 2. Basic principle of XPNAV Positioning System 2.1. Basic observation model In X-ray pulsar navigation system, an inertial coordinate system BCRS (Barycentric of the solar system celestial reference system) with solar system barycenter (SSB) as the origin is adopted in this paper. The BCRS is established in the time framework of TCB (Hellings, 1986a,b) Barycentric Coordinate Time), a time scale in the framework of relativity. The implementation of X-ray Pulsar navigation is realized by comparing the measured pulse TOA at spacecraft with the predicted pulse TOA at the SSB origin. The pulse phase prediction model at the SSB is given by (Sheikh, 2005) uðtÞ ¼ uðt0 Þ þ f ðt  t0 Þ þ

n þ1 n1 X f ðt  t0 Þ n¼2

n!

ð1Þ

where uðt0 Þ is the phase of the pulsar at time t0 at SSB, t0 is the reference epoch, f is the pulse frequency. In general, default value of the t0 is set to zero. In order to get the basic observation in XPNAV, namely the time delay from the spacecraft to SSB along the direction of pulsar radiation, the measured pulse TOA at the spacecraft should be compared with the predicted pulse TOA at the SSB origin. Therefore, the measured pulse TOA must be transferred from spacecraft to SSB origin. In the time transformation model, geometric and

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relativistic effect should be taken into consideration. The simplified time transformation model from spacecraft to SSB in the time scale TCB is (Hellings, 1986a,b; Backer and Hellings, 1986; Sheikh et al., 2007; Kai et al., 2009) t0  tSC ¼

n  RSC 1 2 2 ½ðn  RSC Þ  jjRSC jj þ 2cD c þ 2ðn  bÞðn  RSC Þ  2ðb  RSC Þ   2ls n  RSC þ n  b þ jjRSC jj þ jjbjj þ 3 ln  n  b þ jjbjj c

ð2Þ

where ls is the gravitation of sun and D is the distance between pulsar and SSB. In the same way, a TOA transformed from Mars to SSB in the time scale TCB is (Hellings, 1986a,b; Backer and Hellings, 1986; Sheikh et al., 2007) t0  tM ¼

n  RM 1 2 2 ½ðn  RM Þ  jjRM jj þ 2cD c þ 2ðn  bÞðn  RM Þ  2ðb  RM Þ   2ls n  RM þ n  b þ jjRM jj þ jjbjj þ 3 ln  n  b þ jjbjj c

ð3Þ

Subtracting (4) from (Radosavljevic et al., 2015) yields tM  tSC ¼ nðRSC  RM Þ=c þ Rtel

ð4Þ

where Rtel ¼

1 2 2 f½ðn  RSC Þ  jjRSC jj þ 2ðn  bÞðn  RSC Þ 2cD 2  2ðb  RSC Þ  ½ðn  RM Þ  r2 þ 2ðn  bÞðn  RM Þ   2ls n  RSC þ n  b þ jjRSC jj þ jjbjj  2ðb  RM Þg þ 3 ln n  R þ n  b þ jjR jj þ jjbjj  c M

M

Noticing that RSC ¼ RM þ RSC=M , then the observation for X-ray navigation is obtained Y ¼ tM  tSC  Rtel ¼ n  RSC=M =c

ð5Þ

2.2. Orbital dynamics model The spacecraft state dynamics equation is established within the J2000.0 Mars Centered Inertial (MCI) coordinate system similar to the Earth Centered Inertial (ECI). In MCI coordinate system, the dynamic state of spacecraft can be identified as T

X ¼ ½rv ¼ ½xyz vx vy vz  T

T

ð6Þ T

where r ¼ ½xyz denotes position vector and v ¼ ½vx vy vz  the speed vector in the MCI coordinate system with the Mars as the reference frame origin. Usually the dynamic model of the nonlinear system can be expressed as X_ ¼ f ðX ðtÞ; tÞ þ wðtÞ

ð7Þ

where wðtÞ is the process noise. On the premise of ignoring the process noise, time derivative of the state vector is obtained X_ ¼ ½_rv_ T ¼ ½vaT

ð8Þ

where a is the sum of a variety of accelerations. When a spacecraft is moving around Mars, the force condition of which is complex and it is extremely difficult to take account of all the forces. Thus only the forces having a significant impact on the movement of the spacecraft is discussed. In this paper, the two-body acceleration aM , the non-spherical Mars gravitational potential aJ 2 and the third body gravitational effects from the Sun aS and the Earth aE are considered to generate the total acceleration of the spacecraft atotal ¼ aM þ aJ 2 þ aS þ aE þ aother

ð9Þ

where aother represents the high-order terms that are considered as the noise of the system model. Each of these expressions of accelerations can be obtained by consulting literature (Qiao et al., 2009). The state model of the spacecraft navigation system can be expressed as 8 dx ¼ vx þ w1x > dt > > > > dy > ¼ vy þ w1y > dt > > > > dz > < dt ¼ vz þ w1z X_ ¼ dv 2 2 x > ¼  lrm3x ½1  32 J 2 ðrrm Þ ð7:5 rz2  1:5Þ þ hax þ w2x > dt > > > > > > dvy ¼  lm3y ½1  3 J 2 ðrm Þ2 ð7:5 z22  1:5Þ þ hay þ w2y > 2 dt r r r > > > : dvz lm z rm 2 3 z2 ¼  r3 ½1  2 J 2 ð r Þ ð7:5 r2  4:5Þ þ haz þ w2z dt ð10Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y 2 þ z2 , rm is the Mars radius and lm is the gravitational parameter of the Mars, Dax Day Daz are the components of perturbation effects from sun and Earth, wðtÞ is the process noise with covariance Q. Thus the motion state of spacecraft in any time can be calculated by integrating the Eq. (8) with the known initial condition X ðt0 Þ. Z t f ðX ðsÞ; sÞds ð11Þ X ðtÞ ¼ X ðt0 Þ þ t0

3. Augmented XPNAV positioning model 3.1. Measurement model of TOA The TOA of XPNAV can be obtained through Eq. (5) which is rewritten as Y 1 ¼ Dt ¼ n  RSC=M =c þ v1 ¼ g1 ½X; t þ v1

ð12Þ

where g1 ½X; t ¼ 1c ½nx ny n z 0 0 0, v1 is the measurement noise of the pulsar with mean zero and variance r2v . The standard deviation rv can be defined as (Sheikh, 2005) rv ¼

1 W 2 SNR

ð13Þ

where W is the pulse width. SNR, Signal-to-Noise Ratio of the pulsar, is defined as

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F X Apf Dtobs SNR ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½BX þ F X ð1  pf ÞðADtobs W =P Þ þ F X Apf Dtobs

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Eq. (15) with Eq. (16), the observation of direction vector can be presented as ð14Þ

Y 2 ¼ ½Da npx npy npz ndx ndy ndz  þ v2 ¼ g2 ½X; t þ v2

ð18Þ

where F X is the observed X-ray photon flux, BX is the X-ray background radiation flux, A is the area of the X-ray detector, pf is the ratio of pulse radiation flux to average pulse radiation flux in a cycle, Dtobs denotes the observation time, and P is the pulse period.

where Da is the field angle of Mars seen from the spacecraft npx , npy , npz are the components of np , and ndx ndy ndz the components of nd . v2 is the observation noise of direction vector which is a zero mean Gaussian function. From Y 1 and Y 2 , the integrated observation of augmented X-ray pulsar navigation method can be written as

3.2. Measurement model of the direction vector

Y ¼ ½Dt Da npx npy npz ndx ndy ndz  þ V

T

As supplemental information, the field angle and direction vectors from spacecraft to satellites of Mars is utilized to augment the XPNAV positioning system, which can be realized by processing the geometrical information in the star image. The position relationships among the spacecraft, Mars and the moons of Mars are shown in Fig. 1. As shown in Fig. 1, oM is the center of mass of Mars. RM is the radius of Mars. Da is the field angel of Mars. Rp and Rd are the vectors of the two moons of Mars with respect to Mars origin. RSC is the vector from Mars origin to spacecraft. RSC=p and RSC=d are the position vectors from spacecraft to the moons of Mars. Through the geometrical relationship shown in Fig. 1, the field angle and direction vector of moons of Mars with respect to spacecraft are obtained. RM jjRSC jj

ð15Þ

np ¼

RSC=p Rp  RSC ¼ jjRSC=p jj jjRp  RSC jj

ð16Þ

nd ¼

RSC=d Rd  RSC ¼ jjRSC=d jj jjRd  RSC jj

ð17Þ

Da ¼ 2arc sin

where jj  jj denotes the norm of vector. Rp and Rd describing the position vectors of the natural satellites of Mars can be accurately obtained by the ephemeris. np denotes the direction vector from spacecraft to Phobos and nd the direction vector from spacecraft to Deimos. Combing

ð19Þ

where Y ¼ ½Y 1 Y 2 T , V ¼ ½v1 v2 T is the measurement noise with the covariance. 3.3. Visibility analysis of the Mars’ natural satellites Utilizing the field angle of Mars and the line-of-sight direction of Mars’ natural satellite as an additional observation for XPNAV positioning system, the visibility of Mars’ satellite can’t be ignored (Wang and Xia, 2015; Lan et al., 2008). A geometric illustration of the visibility of Mars’ moons is shown in Fig. 2, where S is the center of mass of spacecraft and RM is the radius of Mars, P is a tangency point on the edge of the Mars image. OD1 and OD2 are the position vectors of Mars’ satellites which can be accurately obtained by ephemeris. The visibility condition of Phobos is the same as Deimos, therefore only the Deimos’ visibility is discussed here. The positions of Mars’ natural satellites in two different cases are shown in Fig. 2. In case one, when RM b1 6 a or b1 > a and jjOD1 jj > cosðb ; Mars’ satellite 1 aÞ   is visible to spacecraft, where a ¼ arcos jjRRM and jj SC   OD1 b1 ¼ arcos jjRRSCSCjjjjOD : In case two, when b2 P a and 1 jj RM jjOD2 jj < cosðb , the Mars’ satellite is invisible to space2 aÞ   OD2 craft, where b2 ¼ arcos jjRRSCSCjjjjOD . In other word, the vis2 jj

ibility of Phobos and Deimos can be determined by comparing the viewing angle (2a) of Mars with the angle

PSR

ZM

S

R SC/p

Δα

Rp

oM

D2

R SC

R SC/d

RM

R

YM

o 2

Rd D2

XM Fig. 1. Geometrical relationships among Mars, spacecraft and the moons of Mars.

Spacecraft

R SC

M

Deimos in case 2

1

P

D1 Deimos in case 1

Fig. 2. The visibility of the Mars’ natural satellites.

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between the position vectors of Mars’ natural satellites and spacecraft. When the latter falls within the scope of the former, the natural satellite is visible and vice versa. It can be seen that the visibility of the Mars’ satellite is determined by the radius of Mars and the orbit height of spacecraft and Mars’ natural satellites (Phobos and Deimos). According to the above geometric relationship, a higher orbit of spacecraft has a large probability to see two Mars’ natural satellites. The analysis above is testified using the orbital parameters of Mars Pathfinder, Test Satellite, Phobos, Deimos and Satellite Tool Kit (STK) software. Mars Pathfinder is a Mars probe spacecraft launched in 1997, while the Test Satellite is a test case whose parameters is shown in Table 1. The parameters of Mars Pathfinder, Phobos, Deimos are shown in Table 1. The visibility of Phobos and Deimos are illustrated in Figs. 3 and 4. In the visibility simulation of Phobos and Deimos, all the parameters needed in the procedure are given in the Section 4.1. In the STK simulation Figs. 3 and 4, horizontal axis denotes the simulation from the time October 1, 2009 0:00:00(UTCG) to October 4, 2009 0:00:00(UTCG). In the Fig. 4, full line denotes the Phobos or Deimos are visible to Mars Pathfinder or Test satellite and the blank space indicates the Phobos or Deimos are invisible to Mars Pathfinder or Test satellite indicated by arrow above. The visibility time percent of Phobos and Deimos to Mars Pathfinder and Test Satellite are shown in Table 2 respectively. From Table 2, it can be seen that spacecraft moving in a higher orbit has a longer visibility time of Phobos and

Deimos than that in a low one. Table 2 also indicates that, for the high orbit, both Phobos and Deimos are visible in most of the time (Test Satellite orbit). As the spacecraft running in a low orbit (Mars Pathfinder orbit), the invisibility time of both the natural satellites is only 5%. From the two different orbit situations, it can be concluded that the position vector information of the natural satellites is available in the proposed method. 3.4. Filter model ADDF is adopted as iteration filter. ADDF (Subrahmanya and Shin, 2009) has the advantage of UKF with a second order Taylor’s series approximation. In addition, it can also estimate process noise covariance matrix and automatically adjust the noise covariance matrix depending on the level of the process noise to achieve a better state estimation. The details of ADDF is presented in the Appendix, and the numerical simulation is presented in Section 4. 4. Numerical simulation experiments To verify the performance of the proposed method, we select two kinds of orbit. One is the orbit of American Mars Pathfinder and the other is a hypothetical orbit whose semi-major axis is much long than Mars Pathfinder. Firstly, to explain the advantage of the ADDF, it is compared with the UKF、DDF and EKF in the same condition. Then the augmented XPNAV method with the

Table 1 Initial orbital elements of the Mars’ natural satellites and spacecraft. Orbit

Semimajor axis

Eccentricity

Inclination

RAAN

Perigee

Mean anomaly

Phobos Deimos Mars Pathfinder Test Satellite

9378.9757 km 23458.5796 km 8500 km 26000 km

0.0152 0.0003 0.2364 0.0020

1.0708° 1.4769° 23.4550° 45°

57.1742° 320.3742° 0.2580° 0°

76.6225° 7.8752° 71.3470° 0°

293.2612° 49.4993° 85.1520° 0°

Fig. 3. The visibility of the Phobos and Deimos to Mars Pathfinder.

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Fig. 4. The visibility of Phobos and Deimos to Test Satellite.

Table 2 Time visibility percent of Phobos and Deimos. Orbit

Time visibility of Phobos

Time visibility of Deimos

Time visibility of any of Phobos and Deimos

Mars Pathfinder Test Satellite

68.3147% 93.1640%

83.7090% 100.0000%

94.8233% 100.000%

Table 3 Parameters of the pulsars. Pulsar number

Right ascension (J2000.0) (rad)

Declination (J2000.0) (rad)

Periods (s)

Photon flux (ph/cm2/s)

B0531+21 B1937+21 B182124

1.4597 5.1472 4.8194

0.3493 0.3767 0.4341

3.34e2 1.56e3 3.05e3

6.41e2 4.99e5 1.93e4

observation information of Mars’ satellites is numerically simulated. Since the natural satellites of Mars are not visible within a certain period of time, three different situations are considered in the simulation. In the first case, none of the Mars’ satellites are visible, in other words, only three different pulsars are utilized by the navigation system and the three pulsars are observed at the same time. In the second case, only one satellite of the Mars available. In the third case, both of the satellites of Mars are visible in all the orbit. In this paper, all the numerical simulations are based on 100 times Monte Carlo experiments. Three pulsars which are considered as good candidates for XPNAV are use in the paper. All the parameter of the orbits and pulsars are given in Section 4.1. 4.1. Initial conditions The parameters of pulsars, Phobos, Deimos, and orbits of spacecraft are provide here. Three pulsars (B0531+21, B1937+21, B182124) are selected as the navigation sources whose parameters are shown in Table 3. These pulsars are used for XPNAV by several literatures, and are regarded as good candidates for the XPNAV (Sheikh, 2005; Bernhardt et al., 2010; Becker et al., 2013). The area

of the X-ray detector is 1 m2 which is closed to the RXTE X-ray detector and the background noise is about 4.45e1 ph=cm2 =s according the data in (Sheikh, 2005; Sheikh and Pines, 2005). The instantaneous orbit parameters of natural satellites of Mars can be obtained from the JPL ephemeris DE405 which can be found in (Ephemerides, 2016 [Accessed: 22Mar-2016]). The instantaneous parameters of Phobos and Deimos at the initial time October 1, 2009 0:00:00 UT obtained from the JPL ephemeris DE405 are shown in Table 1. Two orbits are adopted for simulation. (1) One is the orbit of the American Mars Pathfinder launched on 4 December 1996 at 06:58:1000 UTCG. (2) The other (Test Satellite) is the hypothetical orbit of satellite whose parameters are shown in Table 1. In the numerical simulation, the two orbits are produced by STK software. The measurement time step for the pulsar observations is set up to 300 s as it was used in (Sheikh, 2005; Sheikh and Pines, 2006) and It can be deduced by Eqs. (14) and (15) if the position variance need to be less than 500 m. The filtering step runs immediately after the pulsar observation step. The total time of the simulation is 150000 s by filtering 500 times. The simulation parameters of filter for augmented XPNAV are described as follows.

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Initial state error of object satellite is

4.2. Simulation and analysis

dX ð0Þ ¼ ½2km 2km 2km 5m=s 5m=s 5m=s 

T

ð20Þ

Initial state covariance matrix is 2

2

2

2

2

2

P ¼ diagðð2kmÞ ð2kmÞ ð2kmÞ ð5m=sÞ ð5m=sÞ ð5m=sÞ Þ ð21Þ Initial noise covariance matrix is Q ¼ diagðð20mÞ2 ; ð20mÞ2 ; ð20mÞ2 ; ð0:1m=sÞ2 ;ð0:1m=sÞ2 ; ð0:1m=sÞ2 Þ ð22Þ

The accuracy of unit vector pointing to the center of Mars’ natural satellites from spacecraft is assumed to be 0:001rad and the timing observation noises of the three pulsars are calculated as, 0.109 km, 0.325 km and 0.344 km. The observation noises are calculated by Eq. (13) and Eq. (14) where the observation time Dtobs ¼ 300s, the pulse period parameters are list in the Table 3 and others parameters are set with reference to (Sheikh, 2005). Thus the covariance matrix of measurement noise including three pulsars field angle, and two direction vectors can be expressed as 2

R ¼ diag

2

ð0:109kmÞ2 ;ð0:325kmÞ2 ;ð0:344kmÞ2 ð104 radÞ ;ð104 radÞ ;  2

2

2

!

2

ð104 radÞ ;ð104 radÞ ;ð104 radÞ ;ð104 radÞ

ð23Þ

Mars Pathfinder orbit is selected to compare the performance of four different filters (EKF,DDF, UKF, and ADDF) in three different cases. In the first case, none of the Mars’ satellites are visible, the accuracy of position and speed of spacecraft with four different filters are compared. In the second case, only one Mars’ satellite is visible and without loss of generality, the Deimos is selected for the simulation in this paper. In the last case, both the Deimos and Phobos are visible to Mars Pathfinder. In this paper, the mean and mean square error of Position and velocity statistically calculated based on 100 times Monte Carlo experiments are applied. Assuming that both the Deimos and Phobos are visible to Mars Pathfinder, the comparison of differences among EKF,DDF,UKF and ADDF are shown in Fig. 5. Then, the statistical comparison results in all the three different cases are shown in Table 4 for clarity. In Fig. 5, the abscissa corresponds to iteration times of filtering and ordinate denotes the position or velocity estimation error. From the simulation results of three different cases, it can be seen that the performance of DDF is comparable to UKF, while the latter is slightly better and EKF has the worst performance. Of these three filters, ADDF filter has the best navigation precision by adjusting the Q matrix based on the statistical estimation of process noise.

Fig. 5. Navigation precision of using three pulsars, field angle and two direction vectors on Mars Pathfinder orbit.

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Table 4 Position and velocity estimation error on Mars Pathfinder orbit. In the first case, none of the Mars’ satellites are visible, in other words, only three different pulsars are utilized by the navigation system. In the second case, only one satellite of the Mars available. In the third case, both of the satellites of Mars are visible in all the orbit. Conditions

Filters

Position estimation error (km)

Speed estimation error (km/s)

Mean

RMSE

Mean

RMSE

Case one

EKF DDF UKF ADDF

0.5108 0.2571 0.2554 0.1167

0.3170 0.2245 0.1780 0.0918

1.94e4 7.16e5 6.79e5 2.93e5

9.09e5 8.11e5 5.48e5 2.90e5

Case two

EKF DDF UKF ADDF

0.4599 0.2326 0.2124 0.0655

0.2944 0.1340 0.1328 0.0437

1.69e4 6.11e5 5.65e5 1.57e5

8.08e5 4.73e5 3.99e5 1.31e5

Case three

EKF DDF UKF ADDF

0.3781 0.2167 0.1969 0.0660

0.2492 0.1368 0.1209 0.0359

1.35e4 6.04e5 5.47e5 1.51e5

7.59e5 4.67e5 3.73e5 1.04e5

Position estimation error (km)

3.5 3

0.4

2.5

0.2

ADDFX1 ADDFX2 ADDFX3

0 280 300 320 340 360

2 1.5 1 0.5 0 0

50

100

150

200 250 300 Iteration times of filter

-3

Velocity estimation error (km/s)

1

x 10

15 10 5 0

0.8 0.6

x 10

350

400

450

500

-5

ADDFV1 ADDFV2 ADDFV3

280 300 320 340 360

0.4 0.2 0

50

100

150

200 250 300 Iteration times of filter

350

400

450

500

Fig. 6. Navigation precision of using ADDF in three different cases on Mars Pathfinder orbit.

A Careful observation of the filtering error curves of ADDF will find that ADDF’s performance in the previous 50 times of filtering is not the best. The reason is that a sliding window with size N (N = 50) is adopted in this paper and the adaptation does not start until the length of residual sequences is equal to epoch length N. Therefore, the filtering statistical data, namely the mean and root mean squared error (RMSE) value of position and velocity error

are abstracted from the fiftieth time iterative filtering to the end of filtering duration for EFK, DDF and UKF filters while for the ADDF from the seventieth iterative filtering at which time, the adaptive filtering iterative times is only twenty times. The numerical filtering results in three different cases are shown in Table 4. From the aforementioned visibility analysis of the Mars’ natural satellites, it can be seen that in some times the

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Fig. 7. Navigation precision of using ADDF and DDF with different initial Q on Mars Pathfinder orbit.

J. Rong et al. / Advances in Space Research 58 (2016) 1864–1878

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Table 5 Position and velocity estimation error with different initial Q on Mars Pathfinder orbit. The term ‘‘Expanding 100 times or 10,000 times” denotes that the noise covariance matrix is initialized with 100 times or 10,000 times greater than the value of Eq. (22). Filters

DDF DDF(Expanding 100 times) DDF(Expanding 10,000 times) ADDF ADDF(Expanding 100 times) ADDF(Expanding 10,000 times)

Position estimation error (km)

Speed estimation error (km/s)

Mean

RMSE

Mean

RMSE

0.2167 0.4420 0.8046 0.0660 0.0994 0.1234

0.1368 0.2607 0.5132 0.0359 0.1741 0.1731

6.04e5 1.83e4 9.50e4 1.51e5 1.11e5 1.65e5

4.67e5 8.59e05 4.84e4 1.04e5 1.79e5 2.75e5

Fig. 8. Navigation precision of using three pulsars and two direction vectors on Test Satellite orbit.

Phobos and Deimos are not visible to the spacecraft, thus the impact of the visibility of the natural satellites of Mars on the performance of the proposed method is studied. The same filter will be used in three different cases. The statistical results are also shown in the Table 4 and a simulation diagram displaying the navigation precision of ADDF is shown in Fig. 6. It should be noted that in the simulation waveforms, the name suffixes of filter X1, X2, X3 indicate that the filter is working on the first, second and third case respectively. From the Fig. 6, it can be seen that the position and velocity estimation error in the case one is larger than the other cases with the same ADDF filter. In the case two and case three, the performance of the ADDF filter has a similar performance and it is obvious that the position and velocity estimation error in the case three is small than

that in case two. Having a look through of the three cases, one can see that the performance of ADDF is improved with the increasing effective observation information, which clarify the effectiveness of the improved method. From the Table 4, it can be seen that the position and velocity estimation error with the EKF is largest among these four filters. The data shown in Table 4 indicate that in case one, by using ADDF the precision of position and velocity is improved by 20% and 30% respectively compared with UKF and DDF, while in the second and last situation, the improvement percent reaches up to over 40%. Table 4 also shows that the accuracy of position and speed by using the direction vectors of Mars’ natural satellites based XPNAV is improved by 20% over the traditional XPNAV with the DDF or UKF filter. As shown in Fig. 6 and Table 4, the position and velocity accuracy with

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Position estimation error (km)

3.5

0.6

3

0.4

2.5

0.2

ADDFX1 ADDFX2 ADDFX3

0

2

180 200 220 240 260

1.5 1 0.5 0 0

50

100

150

200 250 300 Iteration times of filter

-4

Velocity estimation error (km/s)

5

x 10

350

400

450

500

-5

8 6 4 2 0

4 3

x 10

ADDFV1 ADDFV2 ADDFV3

180 200 220 240 260

2 1 0

50

100

150

200 250 300 Iteration times of filter

350

400

450

500

Fig. 9. Navigation precision of using ADDF in three different cases on Test Satellite orbit.

Table 6 Position and velocity estimation error on Test Satellite orbit. Conditions

Filters

Position estimation error (km)

Speed estimation error (km/s)

Mean

RMSE

Mean

RMSE

Case one

EKF DDF UKF ADDF

0.5763 0.3691 0.3650 0.1744

0.3681 0.2449 0.2273 0.2042

2.11e4 4.51e5 4.48e5 1.44e5

9.64e5 2.56e5 2.26e5 2.36e5

Case two

EKF DDF UKF ADDF

0.4978 0.3423 0.3485 0.1446

0.3105 0.2019 0.1806 0.1537

1.81e4 4.38e5 4.78e5 9.78e06

8.66e5 1.94e5 2.37e5 1.61e5

Case three

EKF DDF UKF ADDF

0.4038 0.2846 0.2827 0.1066

0.2451 0.1554 0.1754 0.1165

1.37e4 4.29e5 4.27e5 7.83e6

7.4e5 1.91e5 1.84e5 1.25e5

the improved method can reach up to 50% compared to the traditional XPNAV by employing ADDF. Another advantage of the ADDF is the ability to adapt the process noise. In general, we assume that the process noise in the system model is constant. However, in fact the process noise is variable with the mismatch degree between the established model and actual model. Thus a further simulation is carried out to investigate the adaption of ADDF to the process noise, where the noise covariance

matrix Q has been initialized with 100 times or 10,000 times greater than the value of Eq. (22). That is to say, the new expanded initial noise covariance matrix is Q100 ¼ diagðð200mÞ2 ;ð200mÞ2 ;ð200mÞ2 ;ð1m=sÞ2 ;ð1m=sÞ2 ;ð1m=sÞ2 Þ 2

2

2

2

Q10000 ¼ diagðð2000mÞ ; ð2000mÞ ; ð2000mÞ ; ð10m=sÞ ; 2

2

ð10m=sÞ ; ð10m=sÞ Þ

J. Rong et al. / Advances in Space Research 58 (2016) 1864–1878

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Fig. 10. Navigation precision of using ADDF and DDF with different initial Q on Test Satellite orbit.

The simulation waveforms and statistical data of the ADDF with different initial noise covariance matrix in the third situation are shown in Fig. 7 and Table 5.

In Fig. 7, the performance of ADDF filter with the initial noise covariance matrix be expanded by 1 time, 100 times and 10,000 times are compared and the results show

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Table 7 Position and velocity estimation error with different initial Q on Test Satellite orbit. Filters

DDF DDF(Expanding 100 times) DDF(Expanding 10,000 times) ADDF ADDF(Expanding 100 times) ADDF(Expanding 10,000 times)

Position estimation error (km)

Speed estimation error (km/s)

Mean

RMSE

Mean

RMSE

0.2846 0.5331 0.8430 0.1066 0.1257 0.1357

0.1554 0.3221 0.5568 0.1165 0.1428 0.1584

4.29e5 2.10e4 9.87e4 7.83e6 1.10e5 1.95e5

1.91e5 1.10e4 5.09e4 1.25e5 1.76e5 3.96e5

that the ADDF filter converges to as Table value even in a situation where the initial noise covariance is far away from the actual value. Table 5 further shows that performance of ADDF (adjusting the Q matrix)is outstanding in a situation where the priori knowledge of actual process noise covariance is absent. The character of ADDF filter will be extremely valuable in a situation where the process noise is hard to calculate in XPNAV. In addition to the Mars Pathfinder orbit, a high orbit (a hypothetical satellite whose orbit confined to the parameters shown in Table 3) is also taken into account to verify the performance of new filter and the augmented XPNAV positioning system proposed in this paper. Under the same simulation condition as that of Mars Pathfinder, the simulation waveforms and statistical data are obtained as shown in Figs. 8 and 9, and Table 6. Verification test results of the adaption of ADDF is shown in Fig. 10 and Table 7. From the statistical data and filtering figures above, it can be seen that in the case of adopting Test Satellite orbit the performance of ADDF is also superior to DDF and UKF. The accuracy of position and velocity is improved by more than 45%. Employing the direction vectors of Mars’ moon system, the accuracy of position and speed of the augmented XPNAV is improved by 20–50% over the traditional XPNAV positioning system with different filters. Comparing the simulation results in Table 6 and Table 7 with that in Table 4 and Table 5, one can see that the performance of spacecraft moving in the lower orbit (Mars Pathfinder orbit) is slight better than that in a higher orbit. Since the angle changing rate between the direction vector from Phobos to Mars barycenter and direction vector from spacecraft to Mars barycenter in the lower orbit is higher than that in the high orbit, the geometry configuration in the lower orbit is better than that in the high orbit. In a word, the simulation results in the higher and lower orbits both verify the effectiveness of the proposed augmented XNAV method in this paper. 5. Conclusion In this paper, we addressed the use of the Mars’ natural satellites to improve XPNAV for Mars probe spacecraft. Two observation variables of the field angle and natural

satellites’ direction vectors of Mars are added into the XPNAV positioning system. Taking into account that Phobos or Deimos may be eclipsed by the Mars under Some circumstances, we carried on the visibility analysis of the Mars’ natural satellites and the visibility simulation experiments in two different situations. Simulation results indicate that the visibility time percent of any of Mars’ natural satellites is more than 94% with the spacecraft orbit height vary from 8500 km to 26000 km. In order to verify the validity of augmented XPNAV positioning system, a numerical simulation is carried out in three cases. In the first case, none of the Mars’ satellites are visible. In the second case, only one satellite of the Mars available and in the last case, both of the satellites of Mars are visible. Numerical simulation results show that the positioning precision is the highest in the last case where the direction vectors of Phobos and Deimos were fully utilized and the position precision and velocity precision reaches to 0.066 km and 1.51e5 km/s with the ADDF filter on Mars Pathfinder orbit. In view of effects of system noise on the augmented XPNAV positioning system, an ADDF filter was adopted to accurately estimate position and velocity error of the spacecraft. A set of comparison experiments of ADDF with UKF and DDF in three cases were carried out and the results show that the performance of ADDF is improved by 40% compared with UKF and DDF. A further study of ADDF performance by expanding the initial noise covariance matrix is made. The results show that even the noise covariance matrix of ADDF is Expanded by 10,000 times the position precision can reach to 0.1234 km and the velocity precision to 1.65e5 km/s which demonstrate that ADDF has a better system noise adaptive capacity. All of the simulations indicate that augmented XPNAV positioning system with ADDF filter proposed in this paper shows a significant improvement over the traditional autonomous navigation systems. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 61573059, 61401340), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JM6035), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130203120004), the

J. Rong et al. / Advances in Space Research 58 (2016) 1864–1878

Fundamental Research Funds for the Central Universities (Grant No. JB161303), AreoSpace T.T.&.C. Innovation Program (Grant No. 201515A). Appendix A Considering Eqs. (7) and (19), the state equation and measurement equation of augmented XPNAV with a discretization can be described as xk ¼ f ðxk1 Þ þ wk

ð24Þ

y k ¼ gðxk Þ þ vk

where xk 2 R is a state vector, y k 2 R is output vector. The process and measurement noise of the systems are assumed to be Gaussian white noise with zero mean and the covariance matrices conforming to 8 T > < EfwðiÞw ðjÞg ¼ QðkÞdi;j ð25Þ EfvðiÞvT ðjÞg ¼ RðkÞdi;j > : T EfwðiÞv ðjÞg ¼ 0 n

A set of Sigma points of state with a length of 2n + 1 is selected and the sampling strategy is 8 v ¼ xk1 > < 0 vi ¼ xk1 þ h b S x;k1 ; i ¼ 1    n ð26Þ > : b x;k1 ; i ¼ 1    n viþn ¼ xk1  h S pffiffiffi where i ¼ 1    n; interval length is h ¼ 3 for Gaussian noise distribution (Nørgaard et al., 2000), b S x;k1 is the ChoS Tx;k1 . The xk1 is S x;k1 b lesky factor of Pb k1 , namely Pb k1 ¼ b obtained as xk1 ¼ ½v0 vi viþn 

ð27Þ

xi;kjk1 ¼ f ðxi;k1 Þ; ;i ¼ 0;    2n

ð28Þ

b x kjk1 ¼

2n X ðmÞ W i xi;kjk1 ; i ¼ 0;    2n

ð29Þ

i¼0 T

ð1Þ ð2Þ ð1Þ ð2Þ Pb kjk1 ¼ ½S ðkÞ S ðkÞ   ½S ðkÞ S ðkÞ  þ Q x x x x xb xb xb xb pffiffiffiffiffiffiffiffiffiffiffi ð1Þ S i; k ¼ W ðc1Þ f ðviþn;k1 Þ; i ¼ 1;    n; x xb pffiffiffiffiffiffiffiffiffiffiffi ð2Þ S i; k ¼ W ðc2Þ ½f ðvi;k1 Þ  2f ðv0;k1 Þ; i ¼ 1;    n; x xb ðmÞ

W 0 ¼ ðh2  nÞ=h2 ; ðmÞ

W 1:2n ¼ 1=2h2 ; W ðc1Þ ¼ 1=4h2 ; W ðc2Þ ¼ ðh2  1Þ=4h4

(3) measurement updating steps Pb kjk1 ¼ Sb x;kjk1  b S Tx;kjk1 8 v ¼b x kjk1 > < 0 vi ¼ b x kjk1 þ h b S x;kjk1 ; i ¼ 1    n > : viþn ¼ b x kjk1  h Sb x;kjk1 ; i ¼ 1    n

ð30Þ

ð31Þ

xkjk1 ¼ ½v0 vi viþn  y i;kjk1 ¼ gðxi;kjk1 Þ; i ¼ 0;    2n b y kjk1 ¼

2n X

ð32Þ

ðmÞ

W i y i;kjk1 ; i ¼ 0;    2n

ð33Þ

i¼0

m

where di;j is Kronecker Delta function. The algorithm procedures for adaptive divided difference filter are as follows (Dey et al., 2014): (1) The initial state estimate is given as b x 0 and Pb 0 is the covariance matrix. (2) Time updating steps

1877

T

ð1Þ ð2Þ ð1Þ ð2Þ y Pb kjk1 ¼ ½S ðkÞ S ðkÞ   ½S ðkÞ S ðkÞ  þ Rk x x x x yb yb yb yb pffiffiffiffiffiffiffiffiffiffiffi ð1Þ S i; k ¼ W ðc1Þ gðb v iþn;k1 Þ; i ¼ 1;    n x yb pffiffiffiffiffiffiffiffiffiffiffi ð2Þ S i; k ¼ W ðc2Þ ½gðb v 0;k1 Þ; i ¼ 1;    n: v i;k1 Þ  2gðb yb x ð1Þ b Pb xy i; k kjk1 ¼ ½ S x;kjk1 ½S yb x

T

ð34Þ

ð35Þ

1 by K k ¼ Pb xy kjk1 ð P kjk1 Þ

ð36Þ

#k ¼ y k  b y kjk1

ð37Þ

b x kjk1 þ K k #k xk ¼ b

ð38Þ

Pb k ¼

ð39Þ

S ky



T ðS ky Þ

þ

K k Rk K Tk

ð1Þi;k ð2Þi;k S x;kjk1  K k S KkS : S ky ¼ ½ b yb yb x x

Then a recursive calculation from the beginning K = 1 can be conducted. (4) Steps for DDF adaption k X b # ðkÞ ¼ 1 C #k ðjÞ#Tk ðjÞ k N j¼kN þ1

ð40Þ

b k ¼ Kk C b # ðkÞK T Q k k

ð41Þ

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