Autonomous navigation for a group of satellites with star sensors and inter-satellite links

Acta Astronautica 86 (2013) 10–23

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Autonomous navigation for a group of satellites with star sensors and inter-satellite links Xiong Kai n, Wei Chunling, Liu Liangdong Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing 100190, China

a r t i c l e in f o

abstract

Article history: Received 15 July 2011 Received in revised form 26 October 2012 Accepted 6 December 2012 Available online 29 January 2013

This paper studies the autonomous navigation method for a group of satellites based on relative position measurements, which can be obtained by using inter-satellite links for measuring relative range and navigation star sensors for measuring relative bearing. For the satellites that are far from each other, it may be difficult to obtain relative bearing measurement due to poor visibility. To address this difficulty, this paper proposes a novel scheme, where three satellites, whose relative ranges are rather small such that the relative bearings can be observed, are used as beacons for the navigation of the other satellites that are invisible. The feasibility of the proposed navigation scheme is analyzed by using the Cramer-Rao lower bound (CRLB), with the consideration of the availability of relative bearing measurements. In addition, the multiple model adaptive estimation (MMAE) algorithm is adopted to improve the convergence speed of the estimator in the presence of large initial errors. Simulation results illustrate the high performance of the proposed scheme. & 2012 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Autonomous navigation Constellation Star sensor Visibility CRLB MMAE

1. Introduction Common measurements used to determine the orbit of a satellite are normally generated using ground tracking sensors [29]. The autonomous navigation system estimates the satellite’s position and velocity by using only equipment aboard the satellite. It can improve the survivability of the satellite in the case that communication with the ground is lost. Several different autonomous navigation methods have been proposed using different measurement types, such as optical navigation methods [1–5], magnetometer-based navigation methods [6,7], pulsar based navigation methods [8–12], and navigation methods based on the relative position measurement between a pair of satellites [13–16]. Among the prior mentioned autonomous navigation methods, the navigation method based on the relative

n

Corresponding author. Tel.: þ 86 10 68744801. E-mail address: [email protected] (X. Kai).

position measurement between a pair of satellites has the potential to be highly accurate with current technology. The basic idea of the method is to determine the absolute position of the two satellites by making measurements of the relative position vector from one satellite to the other. The relative position measurement can be obtained by inter-satellite links and star sensors, the former for measuring relative range and the latter for measuring relative bearing. To guarantee the feasibility of absolute navigation, the relative position vector should be referenced to an inertial coordinate system. A time series of the relative position measurements is also required. This navigation method can be easily extended to determine the positions of multiple satellites in a constellation. The inter-satellite range measurement based on a radio ranging signal is a mature technique. The range measurement accuracy is on the order of a few meters. The star sensor is the most accurate attitude sensor for current satellites. It can not only determine the satellite attitude, but also observe the flying satellites due to the satellite reflectivity. In order to distinguish the star sensor

0094-5765/$ - see front matter & 2012 IAA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actaastro.2012.12.001

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

for the relative position measurement and the star sensor for attitude determination, the former is called as navigation star sensor. The navigation star sensors reside on one of the satellites in the constellation with the requirement for autonomous navigation. Call the instrumented satellite ‘‘observer’’ and the other satellites ‘‘observed satellites’’. When the observed satellite and the background stars are imaged together by the navigation star sensor, the bearing of the observed satellite can be determined through image processing with used of stellar ephemerides in the inertial coordinate system [25,28]. The bearing measurement accuracy is on the order of a few arcsec. A major problem with the navigation method based on the relative position measurement is that the observed satellite may be rarely visible to the observer’s navigation star sensor in the case that the satellites in the constellation are far from each other. For the autonomous navigation of the GPS (Global Positioning System) satellite, the apparent magnitude of a GPS satellite at the distance on the order of 20,000 km is estimated to be 11–13, whereas a typical star sensor is capable of observing an object with a magnitude on the order of 7. Thus, it is very difficult for a navigation star sensor to measure the relative bearing between the two GPS satellites. Certainly, the ability of the navigation star sensor to detect the faint target can be enhanced by increasing the lens aperture area of the sensor. However, when the lens aperture area is increased, both the weight of the navigation star sensor and the object density on the Change-Coupled Device (CCD) will be increased. In addition, with high object density, it may be difficult to distinguish a satellite from the background starfield. Once a satellite is acquired, it may be difficult to ensure that the correct satellite is being used. In order to cope with this problem, in this paper, the navigation star sensors and the inter-satellite links are adopted to measure the relative position vectors among the satellites in a low Earth orbit (LEO) constellation. The distances between the satellites in the constellation are controlled to be rather small, such that the navigation star sensor on the observer can detect the observed satellite continuously. In addition, the inter-satellite links between the GPS satellite and the LEO constellation are established. With the relative position measurements among the LEO satellites and the relative range measurements between the GPS satellite and the LEO constellation, the absolute positions of the LEO constellation and the GPS satellite can be estimated simultaneously. This paper makes three contributions. First, a novel scheme is proposed to determine the absolute position of a group of satellites, including the invisible GPS satellite. Second, the performance of the proposed navigation scheme is evaluated based on the Cramer-Rao lower bound (CRLB) [17–20] with the consideration of the visibility. Third, the multiple model adaptive estimation (MMAE) [21–24] is adopted to improve the performance of the navigation system in the presence of large initial error. The remainder of the paper is organized as follows: Section 2 presents the navigation scheme based on the relative position measurement. The visibility analysis method is described in Section 3. Section 4develops a MMAE

11

algorithm for the considered autonomous navigation system. Simulations are presented in Section 5, and conclusions are drawn in Section 6. 2. Navigation scheme This section presents the autonomous navigation scheme for a group of satellites with navigation star sensors and inter-satellite links. The concept of the navigation scheme is outlined, followed by the dynamics model and the measurement model of the navigation system. Then an extended Kalman filter (EKF) based on the system model is developed for state estimation. 2.1. Navigation concept The concept of the autonomous navigation method based on the relative position measurement between a pair of satellites is proposed in [13,14]. The relative position measurement is the full 3-dimensional position vector of the observed satellite measured relative to the observer and referenced to the inertial coordinate system. The navigation system measures the relative position vector by measuring the range between the two satellites along with the inertial bearing of the observed satellite as viewed from the observer. The relative range measurement can be made with inter-satellite links, and the relative bearing measurement can be made with navigation star sensors. Except for a few special cases where the two satellites have identical altitudes, the absolute positions of both satellites are observable from a time sequence of the relative position measurements. High position accuracy is achievable if a sufficiently accurate measurement system and a sufficiently accurate dynamic model are available. This concept can be expanded to determine the absolute positions of a group of satellites. The goal of the present paper is to investigate whether the autonomous navigation system based on the relative position measurement can be used to determine the absolute position of a group of satellites in the case that some of the satellites are invisible. The navigation scheme is depicted in Fig. 1. In the figure, satellite 0, satellite 1 and satellite 2 are three satellites in a LEO constellation. Satellite 0 is the observer. Satellite 1 and satellite 2 are the observed satellites. Two navigation star sensors are mounted on satellite 0 to observe satellite 1 and satellite 2 respectively. The distances between the observer and the observed satellites are controlled to be in the scope of 150–300 km, so as to maintain the visibility of the observed satellite. To keep the observed satellite in the navigation star sensor’s field of view (FOV), the observer’s navigation star sensor should be mounted on a gimbaled platform. Dr ð10Þ and Dr ð20Þ are the relative position vectors k k among the LEO satellites, which are obtained by using the navigation star sensors together with the inter-satellite links. 9Dr ðgps0Þ 9, 9Dr ðgps1Þ 9 and 9Dr ðgps2Þ 9 are the relative range k k k measurements between the GPS satellite and the LEO constellation measured by using inter-satellite links. The GPS satellite is invisible to the observer’s navigation star sensor due to the great distance. According to the concept of the absolute navigation based on the relative measurement, the absolute positions

12

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

where 2 xðiÞ ¼4 k

r ðiÞ k vðiÞ k

3 r ðiÞ x,k 6 ðiÞ 7 6 7 r ðiÞ ¼ 6 r y,k 7 k 4 5 r ðiÞ z,k

5

2 ¼ xðgpsÞ k

2

3

r ðgpsÞ 4 k vðgpsÞ k

2

3

2

3 vðiÞ x,k 6 ðiÞ 7 6 7 vðiÞ ¼ 6 vy,k 7i ¼ 0,1,2 k 4 5 vðiÞ z,k r ðgpsÞ x,k

3

6 ðgpsÞ 7 6 7 r ðgpsÞ ¼ 6 r y,k 7 k 4 5 r ðgpsÞ z,k

5

2

vðgpsÞ x,k

ð2Þ

3

6 ðgpsÞ 7 6 7 ¼ 6 vy,k 7 vðgpsÞ k 4 5 vðgpsÞ z,k

ð3Þ

in the prior equations, rðiÞ and vðiÞ are the 3-dimensional k k position and velocity vectors of the LEO satellites. r ðgpsÞ k and vðgpsÞ are the 3-dimensional position and velocity k vectors of the GPS satellite. The sub-label k denotes the discrete time. Note that all the position and velocity vectors are defined with respect to the Earth-centered inertial coordinate system. The dynamic model for the EKF is to model the time evolution of the state vector. A simplified Earth’s gravity model is adopted as the dynamic model. Here the dynamics model are limited to the two-body motion equations augmented by J2 perturbations xk ¼ f ðxk1 Þ þwk where 2

Fig. 1. Concept of the autonomous navigation scheme.

of satellite 0, satellite 1 and satellite 2 can be determined from the measurements Dr ð10Þ and Dr ð20Þ . Furthermore, k k according to the concept of the global positioning system, if the position of the LEO constellation is determined, satellite 0, satellite 1 and satellite 2 can be seen as beacons, and the position of the GPS satellite can be determined from the measurements 9Dr ðgps0Þ 9, 9Dr ðgps1Þ 9 and 9Dr ðgps2Þ 9. k k k Generally, the dynamic model of the GPS satellite for the design of the EKF is more accurate than those of the LEO satellites. Thus, it is expected that the position accuracy of the LEO constellation is improved when the GPS satellite is incorporated into the autonomous navigation system. In the proposed navigation scheme, on the one hand, the LEO constellation helps determining the absolute position of the GPS satellite; on the other hand, the GPS satellite helps improving the position accuracy of the LEO constellation. 2.2. System model The well-known EKF is adopted to estimate the positions of the LEO constellation and the GPS satellite simultaneously. The EKF design requires the definition of the state vector, the establishment of the dynamic and measurement models, the determination of the initial estimates, and the tuning of the noise covariance matrices. Here the dynamic and measurement models for the design of the filter are described. For the considered autonomous navigation system, the state vector is chosen as  T  T  T  T T xk ¼ xkð0Þ xð1Þ xð2Þ xðgpsÞ ð1Þ k k k

ð4Þ 

f xð0Þ k1

3

6  7 6 7 6 f xð1Þ 7 k1 7 6 6 7T   f ðxk1 Þ ¼ xk1 þ 6 7 6 f xð2Þ 7 k1 6  7 4 5 ðgpsÞ f xk1

ð5Þ

3 vðiÞ x,k 7 6 7 6 vðiÞ 7 6 y,k 7 6 7 6 ðiÞ 7 6 vz,k 7 6 7 6 mrðiÞ 6 ðiÞ 2 ðiÞ ðiÞ 2 7 ðiÞ x,k 3 fðxk Þ ¼ 6  ðiÞ 3 f1 þ 2 J2 ðRe =9r k 9Þ ½15ðr z,k =9r k 9Þ g 7i ¼ 0,1,2 7 6 9rk 9 7 6 ðiÞ 7 6 6  mry,k3 f1þ 3 J ðRe =9rðiÞ Þ2 ½15ððiÞ =9r ðiÞ 9Þ2 g 7 ðiÞ 2 7 6 2 k z,k k 9r 9 7 6 k 7 6 mrðiÞ 5 4 ðiÞ 2 ðiÞ ðiÞ 2 z,k 3  ðiÞ 3 f1þ 2 J 2 ðRe =9rk Þ ½35ðz,k =9r k 9Þ g 2

9r k 9

ð6Þ 9¼ 9r ðiÞ k



r ðiÞ x,k

2

 2  2 0:5 þ r ðiÞ þ r ðiÞ y,k z,k

ð7Þ

in the prior equations, m is the Earth’s gravitational constant, Re is the Earth’s radius, J2 is the second zonal coefficient that is used to describe the non-spherical Earth’s gravity effects, T is the sample interval of the filter. The formulation of the orbital dynamic equation of   the GPS satellite f xðgpsÞ is the same as that of the LEO k   ðiÞ satellite f xk . It is omitted here for the sake of simplicity. The process noise wk is assumed to be a Gaussian white noise with zero-mean and covariance   ð8Þ E wk wTk ¼ Q k

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

where Qk is a known positive definite matrix. wk is introduced here to describe the un-modeled terms in the actual satellite dynamic. Note that the un-modeled terms in the dynamic equation of the GPS satellite are relatively small due to the GPS satellite’s high altitude. Although it is somewhat arbitrary to treat the unmodeled terms as Gaussian white noise for the design of the EKF, practically, this slight inaccuracy has little impact on the filtering performance. In general, to facilitate the implementation, the Kalman filter model is a simplified version of the truth model. Similar orbital dynamics model are adopted in [5,10,11]. The model is believed to be sufficiently accurate for the considered navigation system in the case that no orbital maneuvers are performed. The relative position measurement among the LEO satellites and the relative range measurement between the LEO constellation and the GPS satellite are used together for navigation. The measurement model for these measurements is shown as follows: yk ¼ hðxk Þ þvk where 2

Drð10Þ k

ð9Þ 3

6 7 6 Dr ð20Þ 7 6 7 k 6 7 6 9Dr ðgps0Þ 9 7 yk ¼ 6 7 k 6 7 6 9Dr ðgps1Þ 9 7 6 7 k 4 5 ðgps2Þ 9 9Dr k

2

r ð1Þ r kð0Þ k

3

6 7 6 r ð2Þ r ð0Þ 7 6 k 7 k 6 7 6 9r ðgpsÞ r ð0Þ 9 7 hðxk Þ ¼ 6 k 7, k 6 7 6 9r ðgpsÞ r ð1Þ 9 7 6 k 7 k 4 5 ðgpsÞ ð2Þ 9r k r k 9

ð10Þ

in the prior equations, Dr kðj0Þ (j ¼1, 2) is the measured relative position vector between satellite j and satellite 0 referenced to the inertial coordinate system, 9Dr ðgpsiÞ 9 k (i¼0, 1, 2) is the relative range measurement between the GPS satellite and satellite i. The measurement noise vk is assumed to be a zero-mean white noise with covariance   E vk r Tk ¼ Rk ð11Þ where Rk is a known positive definite matrix. The relative position vector is obtained by using the navigation star sensor together with the inter-satellite link. The navigation star sensor is used to measure the unit vector Dr kðj0Þ =9Dr kðj0Þ 9 that represents the relative bearing between the observer and the observed satellite in the inertial coordinate system, and the inter-satellite link is used to measure the range 9Dr ðj0Þ 9. Combining the k ðj0Þ measurements Dr kðj0Þ =9Dr ðj0Þ 9 and 9 D r 9, we obtain the k k relative position vector Drðj0Þ . The relative bearing meak surement error and the relative range measurement error are denoted by vfk and vrk respectively. We assume that   E vfk vTfk ¼ Rfk   E vrk vTrk ¼ Rrk where Rfk and Rrk are known positive definite matrices. The relative position measurement error vrk is formulated as    vrk ¼ Drðj0Þ =9Dr kðj0Þ 9 þ vfk 9Dr kðj0Þ 9 þ vrk Dr kðj0Þ k ¼

Drkðj0Þ 9Dr kðj0Þ 9

vrk þ9Dr kðj0Þ 9vfk þ vfk vrk

13

Considering that the minor term vfkvrk is negligible, the covariance of vrk is "   Drðj0Þ ðj0Þ k 9vfk Rrk ¼ E vrk vTrk  E ðj0Þ vrk þ 9Dr k 9 Dr 9 k

Drkðj0Þ

vrk þ 9Dr kðj0Þ 9vfk 9Dr kðj0Þ 9 

¼

Drkðj0Þ Drkðj0Þ 2

9Drðj0Þ 9 k

!T 3 5

T 2

Rrk þ9Dr kðj0Þ 9 Rfk

Obviously, the measurement error covariance of the relative position is related to the relative bearing and the relative range of the satellites. The navigation star sensor is adopted to image both the observed satellite and the background stars, and the relative bearing between the two satellites is extracted directly from the image with the use of the stellar ephemerides in the inertial coordinate system. The advantage of this strategy is that it does not depend on the attitude of the observer, or on the misalignments of the navigation star sensors. Thus, it can reduce the possible error sources. In contrast, if the attitude determination star sensor is used to provide the observer’s inertial attitude knowledge, and the navigation star sensor is used to measure the bearing of the observed satellite relative to the observer’s body-fixed coordinate system, then some sort of misalignment would be present in the resultant relative bearing measurement. It should be mentioned that, the availability of the relative bearing measurement depends on the visibility of the observed satellite. If the observed satellite is not visible to the observer, the observed satellite cannot be imaged by the navigation star sensor, and the relative bearing measurement is not achievable. In this case, only the relative range measurement is used for navigation, and the corresponding measurement model is degenerated as 2 3 2 3 9Drð10Þ 9r ð1Þ r kð0Þ 9 9 k 6 7 6 k 7 6 9Drð20Þ 9 7 6 9r ð2Þ r ð0Þ 9 7 6 7 6 k 7 k k 6 7 6 7 6 9Dr ðgps0Þ 9 7 6 9r ðgpsÞ r ð0Þ 9 7 yk ¼ 6 ð12Þ 7 hðxk Þ ¼ 6 k 7 k k 6 7 6 7 6 9Dr ðgps1Þ 9 7 6 9r ðgpsÞ r ð1Þ 9 7 6 7 6 7 k k 4 5 4 k 5 9 r ð2Þ 9 9Dr ðgps2Þ 9r ðgpsÞ k k k The visibility of the observed satellite is affected by the positions of Earth, Sun, the observed satellite and the observer. The apparent magnitude of the observed satellite varies with time, and sometimes it may be rather large. As the detection ability of the navigation star sensor is limited, the relative bearing measurement between the LEO satellites is not always available. In fact, the measurement model may switch from (10) to (12) frequently. The visibility analysis method for the observed satellite is given in Section 3. The ability of the autonomous navigation system to cope with the switched measurements is tested in Section 5.

14

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

2.3. EKF algorithm In the proposed autonomous navigation scheme, all the measurements are sent to one satellite through the intersatellite links and processed in a large filter. The EKF is implemented to incorporate the measurements in (10), and estimate the state vector in (1). The EKF algorithm is based on the linearization of the nonlinear time propagation and measurement update equations around each estimate. The system model formulated by (4) and (9) are used to define the time propagation and measurement update equations. Here we briefly review the necessary computations. Initial conditions are assumed as Gaussian random variables with the mean x^ 0 and the corresponding error covariance P0. The time propagation from k  1 to k uses a linearization of (4) about the value xk1 ¼ x^ k1 . It performs the following computations:   x^ k9k1 ¼ f x^ k1 P k9k1 ¼ F k P k1 F Tk þ Q k ð13Þ where x^ k is the state estimate vector, x^ k9k1 is the prediction of the state vector, Pk is the state estimate error

 covariance, F k ¼ @f ðxk1 Þ=@xk1 x ¼ x^ is the linearized k1 k1 dynamic matrix. When the measurement is available, the measurement update is carried out using the following calculations: h  i x^ k ¼ x^ k9k1 þK k yk h x^ k9k1 ð14Þ  1 K k ¼ P k9k1 H Tk H k P k9k1 H Tk þRk

ð15Þ

H k ¼ ðIK k H k ÞP k9k1 ðIK k H k ÞT þ K k Rk K Tk

ð16Þ

where Kk is the Kalsman filter gain, H k ¼ @hðxk Þ= @H k xk ¼ x^ k9k1 is the measurement sensitivity matrix. The outputs x^ k and Pk complete the filter cycle for one sample interval and prepare it for recursive operation over the next sample interval. The steady state estimate of the EKF can be reached by iterating (13)–(16). 3. Visibility analysis method An important problem remained to be solved is the calculation of the availability of the relative bearing measurements. The relative bearing measurement is not achievable unless the observed satellites and the background stars are visible to the observer’s navigation star sensor. Three criteria are applied to determine whether an observed satellite is visible. To be visible, the observed satellite must (1) be of sufficient apparent magnitude to be detected by the navigation star sensor, (2) be fully illuminated by Sun and (3) for the navigation star sensor, Earth and Sun cannot be in the FOV. 3.1. Apparent magnitude condition The absolute magnitude of non-star objects (such as planets, comets, asteroids, and reflecting objects) is defined as the apparent magnitude that the object would have if it were at 1 AU distance from both Sun and Earth and at a phase angle x ¼ 03 . As shown in Fig. 2, the observed satellite’s phase angle x is the angle between

Fig. 2. Geometry of phase angle (the sun-satellite-observer angle).

the position vector from the observed satellite to the observer Dr(0j) and the position vector from the observed satellite to the Sun’s center Dr(Sunj). In this section, the time label k is omitted for simplicity. In particular, the absolute magnitude M of an observed spherical object (e.g., a planet or a satellite) can be approximated by the following expression:  r pffiffiffi M ¼ mSun 5log10 d a ð17Þ d0 where mSun ¼  26.73 is the Sun’s apparent magnitude, d0 ¼ 1 AU¼1.4959787  108 km, rd is the radius of the observed satellite, and a is the body albedo (reflectivity). Generally, rd can be approximated by the extent of the satellite. The apparent magnitude m of the observed satellite can be calculated from the absolute magnitude M by [25] " # 2 2 9DrðSunjÞ 9 9Drð0jÞ 9 m ¼ M þ 2:5log10 j ¼ 1,2 ð18Þ 4 d0 pðxÞ where 9Dr(Sunj)9 is the range between Sun and the observed satellite, 9Dr(0j)9 is the range between the observer and the observed satellite, p(x) is the Phase Integral (integration of reflected light) that for a spherical body can be well approximated by pðxÞ ¼

 2

ðpxÞcosðxÞ þsinðxÞ 3p

ð19Þ

The Phase Integral shown in (19) is also a reasonable first approximation for the satellite of unknown shape. From Fig. 2, it is straightforward that the observed satellite’s phase angle is calculated as " # Drð0jÞ DrðSunjÞ 1 x ¼ cos ð20Þ 9Dr ð0jÞ 99Dr ðSunjÞ 9 The higher the apparent magnitude, the dimmer an observed satellite will appear. Given the observed satellite’s apparent magnitude m and an apparent magnitude bound m0 limited by the navigation star sensor’s detection ability, the condition to ensure that the observed satellite to be of sufficient apparent magnitude to be detected by the navigation star sensor is m r m0

ð21Þ

3.2. Illuminating condition A necessary condition to ensure the visibility of the observed satellite is that the observed satellite is fully

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

15

Fig. 4. Geometry for positions of Earth, observed satellite and observer.

"

y0 ¼ cos1

Fig. 3. Geometry of shadow cast by Earth on satellite orbit.

illuminated by the Sun. Earth-orbiting satellites may be hidden in Earth’s shadow for some portion of the orbit. Fig. 3 provides a diagram of a satellite in Earth orbit, as well as the shadow cast by Earth from Sun. To determine whether the observed satellite is located inside the Earth’s shadow, we should determine the size of the shadow cast by the body and whether the satellite’s path intersects this shadow. The angle c between the observed satellite’s position vector r (j) and the pointing vector toward the Sun’s center r (Sunj) is determined from " # r ðjÞ r ðSunjÞ c ¼ cos1 ðjÞ ðSunjÞ ð22Þ 9r 9 9r 9 The observed satellite will be illuminated in the case that it is not within the Earth’s shadow. From Fig. 3, given the Sun-Earth-satellite angle c, the illuminating condition is written as

c o903 þ b

ð23Þ

where 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 9r ðjÞ 9 R2e 5 b ¼ sin1 4 2 9r ðjÞ 9

ð24Þ

Re is the Earth’s radius. 3.3. No glare or occultation condition If the sunlight portion of Earth fells within the FOV, then the navigation star sensor could not be used because of too much glare. Alternatively, if the FOV is occulted by the dark side of Earth, then the observed background stars cannot provide an inertial reference. It is assumed that the navigation star sensor is always slewed so that the observed satellite lies in the center of its FOV. As the FOV of the navigation star sensor is rather small (o11), the angle of the FOV is neglected in the analysis. The positions of Earth, the observed satellite and the observer are shown graphically in Fig. 4. In the figure, r (j) and r (0) are the position vectors of the observed satellite and the observer respectively. From the figure, to determine whether the FOV is occulted by Earth, the follow condition is used:

y 4 y0 where

y ¼ cos1

ð25Þ "

#  r ð0Þ r ð1Þ r ð0Þ 9rð0Þ r ð1Þ 99r ð0Þ 9

ð26Þ

# Re 9r ð0Þ 9

ð27Þ

y is the Earth-observer-satellite angle, and y0 is the corresponding angle bound. Similar methods can be used to determine whether the Sun or Moon are in the FOV of the navigation star sensor. 4. Multiple model adaptive estimation The performance of the EKF may be degraded in the case of large initial error. Motivated by the encouraging results in [21,22], the MMAE is adopted to cope with this problem. The MMAE, as a well-known adaptive estimation method, has received a great deal of attention in recent years due to its unique power to handle model uncertainties. In the MMAE, it is assumed that the model uncertainty can be adequately approximated by a set of models;a bank of filters, each based on a particular model, run in parallel to obtain the model-conditional estimates; and the overall state estimate is a certain combination of these modelconditional estimates. Various multiple model estimators can be found in the literature, and numerous publications have appeared reporting the successful application of these approaches in a variety of estimation problems. However, to the authors’ knowledge, the MMAE has not been used in the satellite autonomous navigation system based on the relative position measurement. The mean of the initial estimate is assumed to be x^ 0 . To deal with the uncertainty in the initial estimate, the mean of the initial estimate in the MMAE is assumed to be contained in the follow model set: x^ 0 2 fx^ 0 ð1Þ, x^ 0 ð2Þ,    , x^ 0 ðLÞg

ð28Þ

where x^ 0 ðtÞ (t ¼ 1,2,    ,L) is the element of the parameter set, t is the model index, LAN is the total number of the elements in the model set. The error covariance of x^ 0 ðtÞ is assumed to be P 0 ðtÞ ¼ P 0

ð29Þ

The initial weight of the estimate x^ 0 ðtÞ is set to be equally

m0 ðtÞ ¼

1 L

ð30Þ

For the system with large initial error, the implementation procedure of the MMAE algorithm is formulated as follows: Step 1: Model individual filtering At time k, each filter predicts and updates its state and covariance individually under the assumption that the t-th model matches the true model. For the t-th filter,

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X. Kai et al. / Acta Astronautica 86 (2013) 10–23

the state estimate x^ k ðtÞ and the corresponding covariance Pk(t) are calculated by

 x^ k9k1 ðtÞ ¼ f x^ k1 ðtÞ ð31Þ P k9k1 ðtÞ ¼ F k ðtÞP k1 ðtÞF Tk ðtÞ þ Q k

ð32Þ

h i x^ k ðtÞ ¼ x^ k9k1 ðtÞ þ K k ðtÞfyk h x^ k9k1 ðtÞ g

ð33Þ

h i1 K k ðtÞ ¼ P k9k1 ðtÞH Tk ðtÞ H k ðtÞP k9k1 ðtÞH Tk ðtÞ þRk

ð34Þ

P k ðtÞ ¼ ½IK k ðtÞH k ðtÞP k9k1 ðtÞ½IK k ðtÞH k ðtÞT þK k ðtÞRk K Tk ðtÞ ð35Þ



 where F k ðtÞ ¼ @f ðxk1 Þ=@xk1 x ¼ x^ ðtÞ , H k ðtÞ ¼ @h k1 k1 ðxk Þ=@xk xk ¼ x^ k9k1 ðtÞ . Note that the state estimates of the individual filters are calculated from different initial estimates. Step 2: Weight calculation The weights of the individual filters are calculated from the residuals and the corresponding covariance matrices. With the Gaussian assumption, the likelihood for the t-th filter is   1 1 ~ Lk ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  y~ Tk ðtÞS1 ð t Þ y ð t Þ ð36Þ k k 2 92pSk ðtÞ9 where

h i y~ k ðtÞ ¼ yk h x^ k9k1 ðtÞ

ð37Þ

S k ðtÞ ¼ H k ðtÞP k9k1 ðtÞH Tk ðtÞ þRk

ð38Þ

The weight for the state estimate x^ k ðtÞ is calculated as

mk1 ðtÞLk ðtÞ

mk ðtÞ ¼

L P

t¼1

ð39Þ

mk1 ðtÞLk ðtÞ

Step 3: Combination The combined state estimate x^ k and its covariance Pk are calculated from the weight state estimate x^ k ðtÞ(t ¼ 1,2,    ,L) and the covariance matrices Pk(t) (t ¼ 1,2,    ,L) as x^ k ¼

L X

mk ðtÞx^ k ðtÞ

ð40Þ

t¼1

Pk ¼

L X







mk ðtÞfP k ðtÞ þ x^ k x^ k ðtÞ x^ k x^ k ðtÞ T g

ð41Þ

t¼1

By implementing Step 1–Step 3 recursively, we obtain the state estimate x^ k from the measurement yk for k¼1,2,yK(KAN).

The MMAE algorithm has a limitation. Generally, a large number of models are required to describe the model uncertainty effectively. However, the use of more models may increase the computational burden considerably. The heavy computational burden limits the practical application of the algorithm. In order to circumvent this difficulty, in this paper, only the initial position and velocity uncertainties of the observer (satellite 0) are described by a model set. Accordingly, a limited number of parallel filters are implemented to obtain the state estimate. The performance of the MMAE with a small quantity of models is evaluated in the next section. 5. Simulation This section presents the feasibility analysis results for the proposed autonomous navigation scheme by using the CRLB. The validity of the analysis is confirmed by the navigation results. In addition, the EKF and the MMAE are used separately to estimate the positions of a group of satellites in the presence of large initial error. 5.1. Simulation conditions The numerical simulation is implemented under the following conditions. The initial orbit elements of the three satellites in the LEO constellation and the GPS satellite are presented in Table 1. The high-fidelity orbit data of the satellites are produced by using a high accurate numerical orbit propagator. The geopotential perturbation, the luni-solar gravity perturbation, the solar radiation pressure perturbation and the atmospheric drag perturbation are taken into consideration for the orbit propagator. Note that this is for the truth model but not for the dynamic model in the filter. The relative measurement data is generated from the orbit data according to the method in [15]. The accuracies of the relative bearing measurement from the navigation star sensors and the relative range measurement from the inter-satellite links are set as 5 arcsec and 3 m respectively, based on the characteristics of the real instruments. The calculation method of the relative position measurement error covariance is shown in Section 2. It is assumed that the navigation star sensor is capable of observing stars of magnitude 6. The availability of the relative bearing measurement is determined by using the method shown in Section 3. In the case that the relative bearing measurement is not available, only the relative range measurement is used for navigation. The simulations are run for 5 orbits. The relative position sensors are assumed to sample 1 time per second.

Table 1 Initial orbit elements of the satellites. Satellite

Semi-major axis (km)

Eccentricity

Inclination (deg.)

Right ascending node (deg.)

Argument of perigee (deg.)

Mean anomaly (deg.)

0 1 2 GPS

7471.004 7471.004 7471.004 26563

0 0 0 0.0012

63.41 63.41 63.41 52.8

0 0 0 213.0

1.077 0 0 334.1

0  0.074 1.038 147.0

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

17

The simulated orbit data and measurement data are used to evaluate the performance of the proposed navigation scheme. First, the orbit data is used for the feasibility analysis. Then the navigation algorithm operates on the measurement data and produces the state estimate. The comparison between the state estimate and the orbit data yields the estimation error. 5.2. Feasibility analysis The feasibility of the proposed navigation scheme is analyzed with the consideration of the availability of the relative bearing measurement. Fig. 5 plots the visibility of the observed satellites due to the apparent magnitude condition during 5 orbits. The solid line plots the apparent magnitudes of satellite 1 and satellite 2. The dashed line plots the apparent magnitude bound m0 limited by the navigation star sensor’s detection ability. This plot shows that satellite 2 is bright enough for the entire orbit of the observer, and satellite 1 is bright enough except for a small portion of the observer’s orbit. Fig. 6 plots the visibility of the observed satellite due to the illuminating condition. The solid line plots the angle c between the observed satellite’s position vector r(j) and the pointing vector toward the Sun’s center Dr(Sunj). The dashed line plots the angle bound 901þ b defined by the Earth shadow. From the figure, for only a small portion of the observer’s orbit, satellite 1 and satellite 2 are not visible due to Earth’s shadow. Fig. 7 plots the visibility of the observed satellite due to the no glare or occultation condition. The solid line plots the Earth-observer-satellite angle y, and the dashed line plots the corresponding angle bound y0. This plot shows that for the observation of satellite 1 and satellite 2, the observer’s navigation star sensor does not suffer from the glare or occultation of Earth. From the prior analysis, according to the visibility conditions shown in (21), (23) and (25), the observed satellites in the LEO constellation are visible for the observer’s navigation star sensor most of the time. In this

Fig. 5. Apparent magnitudes of the observed satellites and the corresponding bounds.

Fig. 6. Sun–Earth-satellite angle and the corresponding bounds.

Fig. 7. Earth-observer-satellite angle and the corresponding bounds.

case, the relative position measurement is frequently available for the navigation system. Further analysis shows that the GPS satellite is seldom visible for the navigation star sensors on the LEO satellite (or other GPS satellite). The apparent magnitude of a GPS satellite for another GPS satellite is shown in Fig. 8. From Fig. 8, the observed GPS satellite is not visible for the considered navigation star sensor. Thus, it is reasonable to adopt the autonomous navigation scheme presented in Section 2. It should be pointed out that, although visibility duration for the observed satellite can be determined along the observer’s orbit, additional visibility limitations such as satellite component obstruction or detector gimbaled axis limitations may reduce these durations and would require further analysis for a specific implementation. Now we are on the point of answering the question whether the autonomous navigation system based on the intermittent relative position measurement can be used to determine the absolute positions of the satellites. The CRLB is calculated to analyze the feasibility of the proposed scheme. With a certain system model, the CRLB can provide a theoretical lower bound on the mean square

18

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

Fig. 8. Apparent magnitude for a pair of GPS satellites. Fig. 9. CRLB values for the position estimate of satellite 2.

error (MSE) of a state estimate. It can be used to evaluate the best achievable performance before running the filtering algorithm [19]. For clarity, a brief description for the calculation of the CRLB is collected in Appendix A. The CRLB is calculated to analyze the feasibility of the proposed navigation scheme based on the navigation star sensors and the inter-satellite links. The square root of the CRLB values for the position vector estimates of satellite 2 and GPS satellite are plotted versus time in Figs. 9 and 10. As the results for the three LEO satellites are similar, only the result for satellite 2 in the LEO constellation is shown here. According to the results, the calculated CRLB values converge gradually during the simulation. It indicates that the position vectors of both the LEO constellation and the GPS satellite can be obtained from a properly designed estimator. 5.3. Navigation results First, the EKF presented in Section 2 is adopted to estimate the positions of the satellites in the case that the initial position error is relatively small. For the design of the EKF, the process noise covariance matrix Qk is designed according to the magnitude of the un-modeled terms in the dynamic equation 2 3 Q 0,k 6Q 7 6 1,k 7 7 Qk ¼ 6 6 Q 2,k 7 4 5 Q gps,k " Q 0,k ¼ Q 1,k ¼ Q 2,k ¼ Q gps,k ¼

s2r I 33 033

033

#

2 v I 33

s

where sr ¼2  10  5 m, sv ¼1  10  4 m/s. The measurement noise covariance matrix Rk is designed according to the precision of the bearing and range measurements 2 2 3 spos I 33 6 2 7 s I 7 Rk ¼ 6 4 pos 33 5 s2rng I 33

Fig. 10. CRLB values for the position estimate of GPS satellite.

where spos ¼5 m, srng ¼3 m. The initial state estimates x^ 0 for both the LEO satellites and the GPS satellite are generated randomly. The initial position errors are on the order of 1000 m and the initial velocity errors are on the order of 0.1 m/s. Unlike the LEO satellite, the dynamic model of the GPS satellite is known with a high degree of accuracy. The effect of the GPS satellite to improve the navigation performance of the LEO constellation is illustrated in Figs. 11 and 12. When the positions of the LEO constellation and the GPS satellite are estimated individually, the satellite 2’s position and velocity error curves (solid line) and the 3s bounds (dashed line) computed from the filter’s error covariance matrix are shown in Fig. 11. When the positions of the LEO constellation and the GPS satellite are estimated together, the error curves and the 3s bounds are shown in Fig. 12. The numerical performances for the EKF are presented in Table 2. From Table 2, when the GPS satellite is incorporated into the navigation system, the position accuracy of the

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

19

Fig. 11. Estimation error curve of the EKF for satellite 2 in the absence of the GPS satellite. (a) Position and (b) Velocity.

Fig. 12. Estimation error curve of the EKF for satellite 2 in the presence of the GPS satellite. (a) Position and (b) Velocity.

Table 2 Steady-state position error for satellite 2 and GPS satellite. Initial position error

GPS satellite

Algorithm

Parallel filter number

RMS error (m) Satellite 2

Small Small Large Large Large

Absence Presence Presence Presence Presence

EKF EKF EKF MMAE MMAE

1 1 1 64 729

LEO satellite is slightly improved due to the high-accuracy of the GPS satellite dynamic model. A satisfactory result is obtained by using the proposed navigation scheme. It is

GPS satellite

rx

ry

rz

rx

ry

rz

24.0 12.6 1.4  107 0.5  103 17.6

42.3 27.7 0.6  107 1.2  103 29.1

29.6 17.5 1.5  107 0.9  103 19.3

– 26.3 2.1  107 0.7  103 63.1

– 24.0 1.2  107 1.5  103 67.0

– 22.2 1.6  107 1.6  103 40.6

apparent that that a better navigation strategy is to estimate the position of the LEO constellation together with a GPS satellite.

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X. Kai et al. / Acta Astronautica 86 (2013) 10–23

Fig. 13. Estimation error curve of the EKF for GPS satellite in the case of small initial error. (a) Position and (b) Velocity.

Fig. 14. Estimation error curve of the EKF for GPS satellite in the case of large initial error. (a) Position and (b) Velocity.

Then the unfavorable effect of the large initial error is illustrated. In this scenario, the initial position errors are set to be on the order of 10,000 m and the initial velocity errors are set to be on the order of 1 m/s. The navigation results of the EKF for the GPS satellite in the cases of small initial error and those in the case of large initial error are presented in Figs. 13 and 14. From the figures, we can see that the filtering performance is degraded due to the large initial error. This result is consistent with the well recognized notion that the EKF is sensitive to the initial error. In order to cope with the unfavorable effect of the large initial error, the MMAE presented in Section 4 is adopted for navigation. Multiple parallel filters with different initial state estimate x^ 0 ðtÞ are implemented to capture the positions of the satellites. A set of models is used to describe the initial position and velocity of satellite 0. We assume that

the initial position error and the initial velocity error are contained in a fixed set of parameters, i.e.,

drð0Þ x,0 2 f5000m,0,5000mg drð0Þ y,0 2 f5000m,0,5000mg drð0Þ z,0 2 f5000m,0,5000mg dvð0Þ x,0 2 f0:5m,0,0:5mg dvð0Þ y,0 2 f0:5m,0,0:5mg dvð0Þ z,0 2 f0:5m,0,0:5mg h ð0Þ where dr x,0

ð0Þ dr y,0

drð0Þ z,0

iT

h ð0Þ and dvx,0

dvð0Þ y,0

ð0Þ dvz,0

iT

are

the initial position error and the initial velocity error. With this assumption, the number of filters in the MMAE is

X. Kai et al. / Acta Astronautica 86 (2013) 10–23

21

Fig. 15. Estimation error curve of the MMAE for GPS Satellite in the case of large initial error. (a) Position and (b) Velocity.

Fig. 16. Estimation error curve of the MMAE for satellite 2 in the case of large initial error. (a) Position and (b) Velocity.

36 ¼729. If the number of the parameters in the parameter set is reduced, the number of the models in the model set will be reduced. Accordingly, the number of filters will be reduced. For example, if the parameter number is 2, the number of filters is 26 ¼64. The covariance matrices Qk and Rk of the parallel filters in the MMAE are same as those of the EKF. The position and velocity error curves of the MMAE for the GPS satellite are shown in Fig. 15. The error curves of the MMAE for satellite 2 in the LEO constellation are shown in Fig. 16. In comparison with the result in Fig. 14, the estimation error decreases considerably. It indicates that the unfavorable effect of the large initial error is partly eliminated. The corresponding root mean square (RMS) errors are listed in Table 2. A better state estimate is obtained by using the MMAE instead of the EKF due to the ability of the MMAE to deal with the uncertainty in the initial estimate.

The number of the parallel EKFs affects the granularity of the different initial estimates and ultimately the accuracy of the state estimation. To analyze the effect of the parallel filter number to the estimation accuracy, we attempt to adopt different number of the parallel EKFs in the MMAE. Some results are shown in Table 2. It shows that, when more parallel EKFs (or more models) are adopted, the performance of the MMAE is improved. However, the use of more parallel EKFs will increase the computational burden. For the EKF, the computation time is several minutes. For the MMAE, the computation time is a few hours. The selection of the parallel filter number can be seen as a tradeoff between the estimation accuracy and the computational burden. A promising method to reduce the computational burden is to use the variable-structure multiple-model estimation [26,27].

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X. Kai et al. / Acta Astronautica 86 (2013) 10–23

The CRLB of an unbiased estimate of the state vector at time index k is

6. Conclusion This paper proposes an autonomous navigation scheme for a group of satellite based on the relative position measurements among the LEO satellites and the relative range measurements between the LEO constellation and the GPS satellite. The relative bearing and range measurements are made with the navigation star sensor and the inter-satellite link respectively. The main advantage of the proposed scheme is that, the relative bearing between the GPS satellites or that between the GPS satellite and the LEO constellation, which is difficult to observe, is not required in the navigation system. The feasibility analysis shows that the absolute positions of the LEO satellites and the GPS satellite can be determined precisely using the proposed scheme. This result is confirmed through numerical simulations with the consideration of the availability of the measurement. The results show that, it is better to estimate the positions of the LEO constellation together with the GPS satellite with a highaccuracy dynamic model, and the MMAE outperforms the commonly used EKF in the presence of large initial error. This proposed scheme is also appropriate for geostationary satellites autonomous navigation. Certainly, before practical application, further work is required to test the performance of the scheme by using the telemetry data from real satellites.

Acknowledgment This work was supported in part by China Natural Science Foundation (61074103). Appendix A The calculation of the CRLB is briefly described in the appendix. For the system described by xk ¼ f ðxk1 Þ þwk

ðA1Þ

yk ¼ hðxk Þ þ vk

ðA2Þ

where xk and yk are the state and the measurement respectively, wk and vk are zero-mean white noise processes with known covariance matrices     ðA3Þ E wk wTk ¼ Q k E vk vTk ¼ Rk we can obtain a matrix Jk, such that   T E x~ k x~ k ZJ 1 k

ðA4Þ

where x~ k is the estimation error. Jk is called the Fisher information matrix, which can be calculated iteratively by [20]  T  

1  1 J k ¼ Q 1 J k1 k þE H k Rk H k Q k E F k  T i1  T  E F k Q 1 ðA5Þ þ E F k Q 1 k Fk k where F k ¼ @f =@xx ¼ xk1 , Hk ¼ @h=@xx ¼ xk . Note that the matrices F k and H k are different from the Jacobian matrices Fk and Hk in the EKF algorithm in that F k and H k are calculated at the true state.

CRLBðxk ðjÞÞ ¼ J1 k ðj,jÞ

ðA6Þ

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