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Absolute orbit determination using line-of-sight vector measurements between formation flying spacecraft

  • Yangwei Ou
  • Hongbo Zhang
  • Bin Li
Original Article
  • 79 Downloads

Abstract

The purpose of this paper is to show that absolute orbit determination can be achieved based on spacecraft formation. The relative position vectors expressed in the inertial frame are used as measurements. In this scheme, the optical camera is applied to measure the relative line-of-sight (LOS) angles, i.e., the azimuth and elevation. The LIDAR (Light radio Detecting And Ranging) or radar is used to measure the range and we assume that high-accuracy inertial attitude is available. When more deputies are included in the formation, the formation configuration is optimized from the perspective of the Fisher information theory. Considering the limitation on the field of view (FOV) of cameras, the visibility of spacecraft and the installation of cameras are investigated. In simulations, an extended Kalman filter (EKF) is used to estimate the position and velocity. The results show that the navigation accuracy can be enhanced by using more deputies and the installation of cameras significantly affects the navigation performance.

Keywords

Autonomous orbit determination Formation flying spacecraft Measurement constraint Fisher information theory 

1 Introduction

Spacecraft Formation has been widely used in missions with respect to the earth and other planets. Missions that use spacecraft formation are less expensive, more robust and flexible than those using only one spacecraft (Alfriend et al. 2009). Moreover, some complicated missions can only be fulfilled with formation flying spacecraft (D’Amico et al. 2012). For example, the Magnetospheric Multiscale (MMS) Mission requires four satellites flying in a regular tetrahedron to study the magnetic reconnection and other properties of the earth atmosphere (Roscoe et al. 2011). NASA’s Gravity Recovery and Interior Laboratory (GRAIL) mission used two spacecraft to obtain high-quality gravitational field maps of the moon and determine its interior structure (Bandyopadhyay et al. 2016). Two spacecraft in leader–follower configuration were also applied to measure the gravity field model of the earth (Psiaki 2011). Other two-spacecraft configurations were also investigated in the application of determining the gravity field (Elsaka et al. 2012). Yang used two formation flying spacecraft to realize autonomous navigation in the proximity of an asteroid (Yang et al. 2016).

In the future, spacecraft formation will be equipped with the autonomous navigation system. Researchers have concentrated on the problem of autonomous navigation for many years. For autonomous relative navigation, the inter-spacecraft range and line-of-sight angles are used to determine the relative state (Huxel and Bishop 2009). Wang proposed to use the line-of-sight and range measurements between the chief and deputies as measurements in a novel decentralized relative navigation algorithm for spacecraft formation flying (Wang et al. 2016). Christian (2014) used flash LIDAR and range sensors to realize autonomous rendezvous and docking with a cooperative target. In contrast, absolute navigation requires that absolute information, such as inertial position or velocity, should be included in the measurements (Wang et al. 2014). Markley initially proposed the method to determine the orbits of two spacecraft when they were orbiting around a central body (Markley 1984). In this approach, the inertial attitude information of the chief was assumed to be known (Hill and Born 2007). In addition, Ou clarified the relationship between configuration description parameters and the navigation performance (Ou and Zhang 2018). However, the aforementioned authors assumed that the measurements were available all the time. Actually, the FOV of the camera is limited and the observation is available only when the object is visible. In Huang’s work, field of view limitation of cameras was considered and the relative position and attitude was estimated by a monocular camera (Huang et al. 2017). Similarly, the visibility of the object and the corresponding navigation performance should be investigated in this paper.

Sometimes, more than two spacecraft are used in the formation, and the observability of the system determines the navigation performance (Kang et al. 2009). For the dual-spacecraft formation flying system, special unobservable conditions have been outlined by Markley (1984) and Psiaki (2011). The effect of the chief’s orbital elements on the observability was also clarified by Ou (2016). Maessen (2012) found that both the number of sensors and their installation positions have an effect on the observability. However, these researchers only investigated the system in which only two spacecraft (one deputy) were considered. When more deputies are added, the configuration will be optimally designed to make the system more observable.

In this paper, the chief is orbiting around a central body and it takes the main payload as well as other instruments. Several small spacecraft can be released from the chief and they will operate as moving beacons. As the measurement scheme used by Markley (1984) and Psiaki (2011), we also assume that the chief’s attitude could be determined with high accuracy (e.g., 1 arcsec) using star sensors (Schmidt 2014). Thus, relative line-of-sight vector measured in the body frame could be transformed to the inertial frame. Hence, the relative position vectors expressed in the inertial frame are directly used as measurements in this paper. In view of aforementioned shortcomings, the measurement constraints on the FOV of optical cameras and their installation positions are investigated. In addition, an extended Kalman filter (EKF) that accommodates the measurement missing is used to estimate the state. Moreover, when more deputies are added into the system, the configuration is optimized from the perspective of Fisher information theory (Jauffret 2007).

The rest of the paper is organized as follows. Section 2 gives the dynamical model and measurement model of the navigation scheme. In Sect. 3, the system observability is optimized based on the Fisher information theory. Next, an extended Kalman filter is applied to estimate the position and velocity in Sect. 4. In Sect. 5, the navigation simulations are conducted and finally Sect. 6 concludes this paper.

2 Navigation system design

2.1 Spacecraft dynamics

In this paper, we use Mars as the central body. In the spacecraft formation, the chief takes the main payload and other science instruments. These small cooperative spacecraft act as moving beacons and can be used to conduct other Mars observation missions. The dynamical equations of the formation expressed in the J2000.0 inertial frame of Mars are given as
$$ \left [ \textstyle\begin{array}{c} \dot{\boldsymbol{r}}_{i} \\ \dot{\boldsymbol{v}}_{i} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c} \dot{x}_{i} \\ \dot{y}_{i} \\ \dot{z}_{i} \\ - \mu [ 1 + \frac{3}{2}{J_{2}}( \frac{R_{m}}{r_{i}} ) ^{2} ( 1 - 5\frac{z_{i}^{2}}{r_{i}^{2}}) ] \frac{x_{i}}{r_{i}^{3}} + a_{x} \\ - \mu [ 1 + \frac{3}{2}J_{2} (\frac{R_{m}}{r_{i}} ) ^{2} ( 1 - 5\frac{z_{i}^{2}}{r_{i}^{2}}) ] \frac{y_{i}}{r_{i}^{3}} + a_{y} \\ - \mu [ 1 + \frac{3}{2}J_{2}( \frac{R_{m}}{r_{i}} )^{2} (3 - 5\frac{z_{i}^{2}}{r_{i}^{2}}) ] \frac{z_{i}}{r_{i}^{3}} + a_{z} \end{array}\displaystyle \right ] $$
(1)
where \(\boldsymbol{r}_{i} = [ {{x_{i}},{y_{i}},{z_{i}}} ] ^{{T}}\) represents the position of the spacecraft, \({{\boldsymbol{v}}_{i}} = {[ {{{\dot{x}}_{i}},{{\dot{y}}_{i}},{{\dot{z}}_{i}}}] ^{{T}}}\) denotes the velocity. \(R_{m}= 3396.2~\mbox{km}\) is the equatorial radius. \(\mu = 4.28096\times 10^{4}~\mbox{km}^{3}\,\mbox{s}^{ - 2}\) is the gravitational constant of Mars and \(J_{2} = 1.9555 \times 10^{- 3}\) is the second zonal harmonic perturbation term. \(a_{x}\)\(a_{y}\) and \(a_{z}\) represent other perturbations such as the higher order non-spherical perturbations, the third-body attraction and the atmospheric drag. According to the Goddard Mars Model 2B (GMM2B) (Lemoine et al. 2001), the non-spherical perturbations can be calculated. We find that higher perturbation terms such as \(J_{2,2}\) and \(J_{3}\) are on the order of \(10^{-5}\). Besides, other perturbations are much smaller than this magnitude. For the spacecraft in a 300 km orbit, the perturbation accelerations are approximately on the order of
$$ \begin{aligned} &\frac{\mu }{r^{2}} \biggl( \frac{R_{m}}{r} \biggr) ^{2} J_{2,2} \approx O \bigl(10^{ - 8}\bigr)~\mbox{km}/\mbox{s}^{2}, \\ &\frac{\mu }{r^{2}} \biggl(\frac{R_{m}}{r} \biggr) ^{3}J _{3} \approx O \bigl( 10^{ - 8}\bigr)~\mbox{km}/\mbox{s}^{2}. \end{aligned} $$
(2)

Hence, the acceleration uncertainties in Eq. (1) are set as gaussian white noises with variance \(10^{ - 16}~\mbox{km}^{2}/\mbox{s}^{4}\).

2.2 Measurement model and measurement constraints

In this navigation system, optical devices are used to measure the relative line-of-sight, i.e., the azimuth and elevation between the chief and deputy. The range information can be obtained by LIDAR or radars. We assume that the deputy is equipped with a beacon which can be detected by the vision-based devices. Therefore, the occultation of the sun by Mars is not considered. As is shown in Fig. 1, we assume that the measurement frame coincides with the body frame and the body frame is aligned to the local-vertical/local-horizontal (LVLH) frame.
Fig. 1

Coordinates and formation flying spacecraft. (a): Mars J2000.0 inertial frame and the local-vertical/local-horizontal (LVLH) frame. (b): The body frame

The measurement information is the inertial line-of-sight vector which is calculated as
$$ \boldsymbol{r}_{2} - \boldsymbol{r}_{1} = {\boldsymbol{C}}_{b}^{I}\left [ \textstyle\begin{array}{c} \rho \cos \phi \cos \theta \\ \rho \cos \phi \sin \theta \\ \rho \sin \phi \end{array}\displaystyle \right ] $$
(3)
where \({\boldsymbol{C}}_{b}^{I}\) is the matrix that transforms the relative measurement information from the body frame to the inertial frame. \({\boldsymbol{r}}_{1}\) and \({\boldsymbol{r}}_{2}\) denote the position vectors of the chief and deputy, respectively. In this paper, we assume that high-accuracy attitude information can be obtained using star sensors, and the attitude determination error is not considered. \(\theta \) is the azimuth and \(\phi \) is the elevation. \(\rho \) is the range measurement. This measurement model shows that the inertial line-of-sight vector includes the information from three parts: the inertial attitude, the range and two relative position angles.
Actually, the FOV of optical cameras is limited. For example, the FOV for high-accuracy star senors and trackers ranges from \(10^{\circ }\) to \(20^{\circ }\) (Schmidt 2014). As in Fig. 2, \(D\) represents the size of FOV and the relative line-of-sight measurement is available only when the deputy is visible. The constraints can be written as
$$ \left \{ \textstyle\begin{array}{l} \phi = \displaystyle\arcsin \biggl(\frac{z}{\rho } \biggr) \in \biggl( - \frac{D}{2},\frac{D}{2} \biggr) , \\ \theta = \displaystyle\arctan \biggl(\frac{y}{x} \biggr) \in \biggl( - \frac{D}{2},\frac{D}{2} \biggr) , \end{array}\displaystyle \right . $$
(4)
where \([ {x,y,z} ] ^{T}\) is the relative position vector.
Fig. 2

The FOV of the optical camera

2.3 Formation configuration design

Relative orbital elements are applied here to design the spacecraft formation (D’Amico and Montenbruck 2009). The keplerian elements are defined as the semi-major axis \(a\), eccentricity \(e\), inclination \(i\), right ascension of ascending node \(\varOmega \), argument of perigee \(\omega \), and mean argument of latitude \(u\). As in Eq. (5), the five parameters \(\{p,s,\alpha ,\theta ,l\}\) are used to design the configuration in the LVLH frame. In this method, the chief is assumed to fly in near-circular orbit. Based on the orbital elements of the chief, the relative orbital elements can be computed with these parameters (Ou et al. 2016):
$$ \left \{ \textstyle\begin{array}{l} x = - p\cos ( {u - \theta _{0} } ) , \\ y = 2p\sin ( {u - \theta _{0} } ) + l, \\ z = s\sin ( {u - \varphi } ) , \end{array}\displaystyle \right . $$
(5)
where \(p=a\delta e\) determines the semi-minor axis of the in-plane ellipse; \(s=a\delta i\) represents the cross-track magnitude in the relative frame; \(\theta _{0}\) is the initial phase angle of the in-plane motion and \(\varphi \) is the initial phase angle of the cross-track motion; \(l\) represents the along-track offset which is set as zero in this paper. These parameters are depicted in Fig. 3.
Fig. 3

Parameters in relative motion frame

3 Configuration optimization method

Researchers have proved that the observability is mainly determined by the configuration. In this paper, we consider the situation when two or more deputies are flying around the chief in the formation. In order to enhance the observability, the formation configuration must be optimized.

To tackle this problem, we turn to the estimation theory. When estimating a vector of parameters \(\boldsymbol{\theta } = [ \theta _{1},\theta _{2}, \ldots ,\theta _{p} ] ^{{T}}\), the covariance matrix of any unbiased estimator \(\hat {\boldsymbol{\theta }}\) satisfies
$$\begin{aligned} &{ \boldsymbol{P}_{\hat {\boldsymbol{\theta }}} \ge \boldsymbol{I}^{ - 1} ( \boldsymbol{\theta } ) ,} \end{aligned}$$
(6)
$$\begin{aligned} &{ \mathrm{var} ( \hat {\boldsymbol{\theta }}_{i} ) \geq {{ \bigl[ {{{ \boldsymbol{I}}^{ - 1}} ( {\boldsymbol{\theta }} ) } \bigr] }_{\mathit{ii}}} ,} \end{aligned}$$
(7)
where \(\boldsymbol{I} ( {\boldsymbol{\theta }} ) \) refers to the Fisher information matrix (FIM) (Bar-Shalom et al. 2002). The FIM can be used to evaluate the observability (Crassidis and Junkins 2011), and its inverse is defined as the Cramér–Rao lower bound (CRLB).
$$ { \bigl[ {{\boldsymbol{I}} ( {\boldsymbol{\theta }} ) } \bigr] _{ij}} = - E \biggl[ { \frac{{{\partial ^{2}}\ln p ( {{\boldsymbol{x}};{\boldsymbol{\theta }}} ) }}{ {\partial {{\boldsymbol{\theta }}_{i}}{{\boldsymbol{\theta }}_{j}}}}} \biggr] . $$
(8)
Consider the discrete-time nonlinear dynamical system
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} {{\boldsymbol{x}}_{k + 1}} = {{\boldsymbol{f}}_{k}} ( {{{\boldsymbol{x}}_{k}},{{\boldsymbol{w}}_{k}}} ) , \quad k = 0,1,2, \ldots , \\ {{\boldsymbol{y}}_{k}} = {{\boldsymbol{h}}_{k}} ( {{{\boldsymbol{x}}_{k}},{{\boldsymbol{v}}_{k}}} ), \quad k = 0,1,2, \ldots . \end{array}\displaystyle \right . \end{aligned}$$
(9)
The recursive calculation of the FIM is concluded as
$$\begin{aligned} &{ {{\boldsymbol{F}}_{k + 1 | k }} = {\boldsymbol{K}}_{k + 1}^{k + 1} - {\boldsymbol{K}} _{k + 1}^{k,k + 1}{ \bigl( {{\boldsymbol{K}}_{k + 1}^{k} + {{\boldsymbol{F}}_{k|k}}} \bigr) ^{ - 1}} {\boldsymbol{K}}_{k + 1}^{k + 1,k} ,} \end{aligned}$$
(10)
$$\begin{aligned} &{ {\boldsymbol{F}}_{k + 1}^{k + 1} = {{\boldsymbol{F}}_{k + 1 \vert k }} + {\boldsymbol{L}} _{k + 1}^{k + 1} ,} \end{aligned}$$
(11)
where the variables above are defined as
$$\begin{aligned} &{ {\boldsymbol{K}}_{k + 1}^{k} = - E \biggl[ { \frac{{{\partial ^{2}}\ln p ( {{{\boldsymbol{x}}_{k + 1}} | {{{\boldsymbol{x}}_{k}}} } ) }}{{\partial {{\boldsymbol{x}}_{k}}\partial {\boldsymbol{x}}_{k}^{T}}}} \biggr] ,} \end{aligned}$$
(12)
$$\begin{aligned} &{ {\boldsymbol{K}}_{k + 1}^{k,k + 1} = - E \biggl[ {\frac{{{\partial ^{2}}\ln p ( {{{\boldsymbol{x}}_{k + 1}} | {{{\boldsymbol{x}}_{k}}} } ) }}{ {\partial {{\boldsymbol{x}}_{k}}\partial {\boldsymbol{x}}_{k + 1}^{T}}}} \biggr] ,} \end{aligned}$$
(13)
$$\begin{aligned} &{ {\boldsymbol{K}}_{k + 1}^{k + 1} = - E \biggl[ {\frac{{{\partial ^{2}}\ln p ( {{{\boldsymbol{x}}_{k + 1}} | {{{\boldsymbol{x}}_{k}}} } ) }}{ {\partial {{\boldsymbol{x}}_{k + 1}}\partial {\boldsymbol{x}}_{k + 1}^{T}}}} \biggr] ,} \end{aligned}$$
(14)
$$\begin{aligned} &{ {\boldsymbol{L}}_{k + 1}^{k + 1} = - E \biggl[ {\frac{{{\partial ^{2}}\ln p ( {{{\boldsymbol{z}}_{k + 1}} | {{{\boldsymbol{x}}_{k + 1}}} } ) }}{ {\partial {{\boldsymbol{x}}_{k}}\partial {\boldsymbol{x}}_{k + 1}^{T}}}} \biggr] ,} \end{aligned}$$
(15)
in which \(p({{\boldsymbol{x}}_{k + 1}} | {{{\boldsymbol{x}}_{k}}} )\) and \(p({{\boldsymbol{z}}_{k+1}} | {{{\boldsymbol{x}}_{k+1}}} )\) denote the probability density function for the dynamical equation and measurement equation, respectively.
In Eq. (10), the FIM of the step \(k\) is propagated by the dynamical equation to obtain \({{\boldsymbol{F}}_{k + 1 | k }}\). As in Eq. (11), the information accumulated from measurements will be appended to the updated FIM. Specifically, consider the linear discrete-time system with Gaussian white noise
$$\begin{aligned} \textstyle\begin{array}{l} {{\boldsymbol{x}}_{k + 1}} = {{\boldsymbol{\varPhi }}_{k}}{{\boldsymbol{x}}_{k}} + {{\boldsymbol{w}}_{k},} \\ {{\boldsymbol{z}}_{k}} = {{\boldsymbol{H}}_{k}}{{\boldsymbol{x}}_{k}} + {{\boldsymbol{v}}_{k},} \end{array}\displaystyle \end{aligned}$$
(16)
where \(\boldsymbol{\varPhi }_{k}\) is the state transition matrix and \({\boldsymbol{H}} _{k}\) is the measurement matrix. The probability density functions are rewritten as
$$\begin{aligned} \begin{aligned} &{ {p ( {{{\boldsymbol{w}}_{k}}} ) = N ( {{\boldsymbol{0}},{{\boldsymbol{Q}}_{ \boldsymbol{k}}}} ) },} \\ &{ {p ( {{{\boldsymbol{v}}_{k}}} ) = N ( {{\boldsymbol{0}},{{\boldsymbol{R}}_{ \boldsymbol{k}}}} ) },} \\ &{ {p ( {{{\boldsymbol{x}}_{k + 1}} | {{{\boldsymbol{x}}_{k}}} } ) = N ( {{{\boldsymbol{x}}_{k + 1}}:{{\boldsymbol{\varPhi }}_{k}}{{\boldsymbol{x}}_{k}},{{\boldsymbol{Q}} _{k}}} ) },} \\ &{ {p ( {{{\boldsymbol{z}}_{k}}|{{\boldsymbol{x}}_{k}}} ) = N ( {{{\boldsymbol{z}} _{k}}:{{\boldsymbol{H}}_{k}}{{\boldsymbol{x}}_{k}},{{\boldsymbol{R}}_{k}}} ) }.} \end{aligned} \end{aligned}$$
(17)
These recursive information propagation equations are similar to the Kalman filter
$$\begin{aligned} &{ {{\boldsymbol{F}}_{k + 1 | k }} = { \bigl( {{{\boldsymbol{\varPhi }}_{k}} {\boldsymbol{F}} _{k | k }^{ - 1}{\boldsymbol{\varPhi }}_{k}^{T} + {{\boldsymbol{Q}}_{k}}} \bigr) ^{ - 1}} ,} \end{aligned}$$
(18)
$$\begin{aligned} &{ {{\boldsymbol{F}}_{k + 1 | {k + 1} }} = {{\boldsymbol{F}}_{k + 1 | k }} + {\boldsymbol{H}}_{k + 1}^{T}{ \boldsymbol{R}}_{k + 1}^{ - 1}{{\boldsymbol{H}}_{k + 1}}.} \end{aligned}$$
(19)
Since the trace of \({\boldsymbol{P}}\) satisfies
$$ {\mathrm{{tr}}} ( {\boldsymbol{P}} ) \ge {\mathrm{{tr}}} \bigl( {{{\boldsymbol{F}}^{ - 1}}} \bigr) = \sum\limits _{i = 1}^{n} {\frac{1}{{{\lambda _{i}}}}} > \frac{n}{ {\sum \nolimits _{i = 1}^{n} {{\lambda _{i}}} }} = \frac{n}{{\mathrm{{tr}} ( {\boldsymbol{F}} ) }} $$
(20)
where \(\lambda _{i}\) is the eigenvalue of \(\boldsymbol{F}\). This relationship reveals that the trace of \(\boldsymbol{F}\) can be used as an index to evaluate the navigation performance. Thus, the summation of all eigenvalues is proposed to assess the observability.
$$ \boldsymbol{J} = \sum\limits _{i=1}^{n} {\lambda_{i}} . $$
(21)

Note that the relative observability among states is not investigated in this paper and it can be obtained by comparing the eigenvalues corresponding to the states. Therefore, this index can only be applied to assess the overall observability.

Now we investigate the situation in which \(N\) independent identically distributed (IID) observations \({\boldsymbol{x}} [ n ] \) exist, and they own identical probability density function (pdf) \({p ( {{\boldsymbol{x}} [ n ] ;{\boldsymbol{\theta }}})}\). The FIM becomes
$$\begin{aligned} \boldsymbol{I} ( {\boldsymbol{\theta }} ) =& - E \biggl[ \frac{{{\partial ^{2}}\ln p ( {{\boldsymbol{x}};{\boldsymbol{\theta }}} ) }}{{\partial {\boldsymbol{\theta }} \partial {{\boldsymbol{\theta }}^{T}}}} \biggr] \\ =& - \sum\limits _{n = 0}^{N - 1} {E \biggl[ {\frac{{{\partial ^{2}}\ln p ( {{\boldsymbol{x}} [ n ] ;{\boldsymbol{\theta }}} ) }}{{\partial {\boldsymbol{\theta }}\partial {{\boldsymbol{\theta }}^{T}}}}} \biggr] } \\ =& Ni ( {\boldsymbol{\theta }} ) , \end{aligned}$$
(22)
which is \(N\) times of the single observation. Moreover, the CRLB for \(N\) IID observations is \(1/N\) times that for one observation (Steven 1993). Therefore, in order to achieve the best observability, the observations between the chief and deputies should be independent and identically distributed, i.e., they should own the same mean and covariance in the sense of Gaussian distribution. Firstly, we consider the case when there are two deputies. The observations with respect to two deputies are denoted as \({\boldsymbol{h}}_{1}\) and \({\boldsymbol{h}}_{2}\) as below
$$ \begin{aligned} {{\boldsymbol{h}}_{1}} ( k ) = {\boldsymbol{C}}_{b}^{I}\left [ { \textstyle\begin{array}{c} {{\rho _{1}} ( k ) \cos {\phi _{1}} ( k ) \cos {\theta _{1}} ( k ) } \\ {{\rho _{1}} ( k ) \cos {\phi _{1}} ( k ) \sin {\theta _{1}} ( k ) } \\ {{\rho _{1}} ( k ) \sin {\phi _{1}} ( k ) } \end{array}\displaystyle } \right ] + {\boldsymbol{v}} ( k ) , \\ {{\boldsymbol{h}}_{2}} ( k ) = {\boldsymbol{C}}_{b}^{I}\left [ { \textstyle\begin{array}{c} {{\rho _{2}} ( k ) \cos {\phi _{2}} ( k ) \cos {\theta _{2}} ( k ) } \\ {{\rho _{2}} ( k ) \cos {\phi _{2}} ( k ) \sin {\theta _{2}} ( k ) } \\ {{\rho _{2}} ( k ) \sin {\phi _{2}} ( k ) } \end{array}\displaystyle } \right ] + {\boldsymbol{v}} ( k ) . \end{aligned} $$
(23)
Note that \({\boldsymbol{C}}_{b}^{I}\) is the attitude transformation matrix for the chief. Given the IID condition, the covariances for two observations should be identical when the deputies are flying around the chief. According to Eq. (3), the measurement error is
$$\begin{aligned} &{ \Delta ( {{{\boldsymbol{r}}_{2}} - {{\boldsymbol{r}}_{1}}} )} \\ &{ \quad = {\boldsymbol{C}}_{b}^{I}\left [ { \textstyle\begin{array}{c} { ( {\rho + \Delta \rho } ) \cos ( {\phi + \Delta \phi } ) \cos ( {\theta + \Delta \theta } ) } \\ { ( {\rho + \Delta \rho } ) \cos ( {\phi + \Delta \phi } ) \sin ( {\theta + \Delta \theta } ) } \\ { ( {\rho + \Delta \rho } ) \sin ( {\phi + \Delta \phi } ) } \end{array}\displaystyle } \right ]} \\ &{ \qquad {}- {\boldsymbol{C}}_{b}^{I}\left [ { \textstyle\begin{array}{*{20}{l}} {\rho \cos \phi \cos \theta } \\ {\rho \cos \phi \sin \theta } \\ {\rho \sin \phi } \end{array}\displaystyle } \right ]} \end{aligned}$$
(24)
where \(\Delta \rho , \Delta \phi , \Delta \theta \) are the measurement errors and they satisfy
$$ \textstyle\begin{array}{l} \Delta \rho \sim N \bigl( {0,\sigma _{\rho }^{2}} \bigr) , \\ \Delta \phi \sim N \bigl( {0,\sigma _{\phi }^{2}} \bigr) , \\ \Delta \theta \sim N \bigl( {0,\sigma _{\theta }^{2}} \bigr) . \end{array} $$
(25)
Since the angles’ errors are small, the following approximations exist for any small angle \(\varepsilon \):
$$ \left \{ \textstyle\begin{array}{l} \cos ( {\varepsilon + \Delta \varepsilon } ) = \cos \varepsilon - \sin \varepsilon \Delta \varepsilon , \\ \sin ( {\varepsilon + \Delta \varepsilon } ) = \sin \varepsilon + \cos \varepsilon \Delta \varepsilon . \end{array}\displaystyle \right . $$
(26)
Expanding the error in Eq. (24), we find that the expectation of measurement error is zero and the covariance matrix \({{{\boldsymbol{R}}_{k}}}\) is computed as
$$\begin{aligned} {\boldsymbol{R_{k}}} =& E \bigl[ {\Delta ( {{{\boldsymbol{r}}_{2}} - {{\boldsymbol{r}}_{1}}} ) \Delta {{ ( {{{\boldsymbol{r}}_{2}} - {{\boldsymbol{r}}_{1}}} ) }^{{T}}}} \bigr] \\ =& {\boldsymbol{C}}_{b}^{I}\left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c} {\mathrm{{var}}} ( x ) &0&0 \\ 0&\mathrm{{var}} ( y ) &0 \\ 0&0&\mathrm{{var}} ( z ) \end{array}\displaystyle } \right ] {{\boldsymbol{C}}_{b}^{I}}^{{T}} \end{aligned}$$
(27)
where the elements in the matrix are calculated as
$$\begin{aligned} {\mathrm{{var}}} ( x ) =& E \bigl[ {{\rho ^{2}}{{\sin }^{2}}\phi {{\cos }^{2}}\theta {{ ( {\Delta \phi } ) }^{2}}} \bigr] \\ &{} + E \bigl[ {{\rho ^{2}}{{\cos }^{2}}\phi {{\sin }^{2}}\theta {{ ( {\Delta \theta } ) }^{2}}} \bigr] \\ &{}+ E \bigl[ {{{\cos }^{2}}\phi {{\cos }^{2}} \theta {{ ( {\Delta \rho } ) }^{2}}} \bigr] \\ =& {\rho ^{2}}{\sin ^{2}}\phi {\cos ^{2}}\theta \sigma _{\phi }^{2} + {\rho ^{2}}{\cos ^{2}}\phi {\sin ^{2}}\theta \sigma _{\theta }^{2} \\ &{} + {\cos ^{2}}\phi {\cos ^{2}}\theta \sigma _{\rho }^{2}, \end{aligned}$$
(28)
$$\begin{aligned} {\mathrm{{var}}} ( y ) =& E \bigl[ {{\rho ^{2}}{{\sin }^{2}}\phi {{\sin }^{2}}\theta {{ ( {\Delta \phi } ) }^{2}}} \bigr] \\ &{} + E \bigl[ {{\rho ^{2}}{{\cos }^{2}}\phi {{\cos }^{2}}\theta {{ ( {\Delta \theta } ) }^{2}}} \bigr] \\ &{} + E \bigl[ {{{\cos }^{2}}\phi {{\sin }^{2}}\theta {{ ( {\Delta \rho } ) }^{2}}} \bigr] \\ =& {\rho ^{2}}{\sin ^{2}}\phi {\sin ^{2}}\theta \sigma _{\phi }^{2} + {\rho ^{2}}{\cos ^{2}}\phi {\cos ^{2}}\theta \sigma _{\theta }^{2} \\ &{} + {\cos ^{2}}\phi {\sin ^{2}}\theta \sigma _{\rho }^{2}, \end{aligned}$$
(29)
$$\begin{aligned} {\mathrm{{var}}} ( z ) =& E \bigl[ {{\rho ^{2}}{{\cos }^{2}}\phi \Delta {\phi ^{2}} + \Delta {\rho ^{2}}{{\sin }^{2}}\phi } \bigr] \\ =& {\rho ^{2}}{\cos ^{2}}\phi \sigma _{\phi }^{2} + {\sin ^{2}}\phi \sigma _{\rho }^{2}. \end{aligned}$$
(30)
It requires that the covariances for two Gaussian errors should be the same. Hence, the variances along each direction should be identical as well. We have
$$ \left \{ \textstyle\begin{array}{l} {\rho _{1}} ( k ) = {\rho _{2}} ( k ) , \\ {\sin ^{2}}{\phi _{1}} ( k ) = {\sin ^{2}}{\phi _{2}} ( k ) , \\ {\sin ^{2}}{\theta _{1}} ( k ) = {\sin ^{2}}{\theta _{2}} ( k ) . \end{array}\displaystyle \right . $$
(31)
Consider the ranges of angles
$$ \textstyle\begin{array}{l} {\phi _{1}} ( k ) ,{\phi _{2}} ( k ) \in \bigl[ { - 90^{\circ },90^{\circ }} \bigr] , \\ {\theta _{1}} ( k ) ,{\theta _{2}} ( k ) \in \bigl[ { - 180^{\circ },180^{\circ }} \bigr] . \end{array} $$
(32)
We obtain the solutions
$$ \left \{ { \textstyle\begin{array}{*{20}{l}} {{\rho _{1}} ( k ) = {\rho _{2}} ( k ) } \\ {{\phi _{1}} ( k ) = \pm {\phi _{2}} ( k ) } \\ {{\theta _{1}} ( k ) = \pm {\theta _{2}} ( k ) , {\theta _{2}} ( k ) + 180^{\circ } } \end{array}\displaystyle } \right . $$
(33)
For example, we consider the in-plane configuration shown in Fig. 4. According to the solutions obtained, the out-of-plane phase angle is zero and the difference between the in-plane phase angle \(\theta _{1}\) and \(\theta _{2}\) should be \(180^{\circ }\). The optimal configuration occurs when one deputy always lags behind another for half of a period. When more than two deputies are added into the configuration, the system will suffer from great computational burden even though the observability will be enhanced. As a result, we only consider the case when there are two deputies.
Fig. 4

The in-plane configuration with two deputies

4 The extended Kalman filter

Since the dynamical equation is nonlinear, an extended Kalman filter (EKF) is applied here to estimate the position and velocity. When there is one deputy in the system, the Jacobian matrix is calculated as
$$\begin{aligned} {\boldsymbol{A}} =& \frac{{\partial ( {{{{\dot{\boldsymbol{r}}}}_{1}},{{{\dot{\boldsymbol{v}}}} _{1}},{{{\dot{\boldsymbol{r}}}}_{2}},{{{\dot{\boldsymbol{v}}}}_{2}}} ) }}{{\partial ( {{{\boldsymbol{r}}_{1}},{{\boldsymbol{v}}_{1}},{{\boldsymbol{r}}_{2}},{{\boldsymbol{v}}_{2}}} ) }} \\ =&\left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{I}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{ \boldsymbol{G}}_{1}} ( t ) }&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}} _{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{I}}_{3 \times 3}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{G}}_{2}} ( t ) }&{{{\boldsymbol{0}}_{3 \times 3}}} \end{array}\displaystyle } \right ] \end{aligned}$$
(34)
where \({\boldsymbol{G}}_{1}\) and \({\boldsymbol{G}}_{2}\) are the gravity gradient tensors for the chief and the deputy. The elements of this two matrices are calculated in Appendix 1. The state transition matrix in EKF is computed using the method proposed by Markley (1986).
$$ {\boldsymbol{\varPhi }} = \left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{{\boldsymbol{\varPhi }}_{1rr}}}&{{{\boldsymbol{\varPhi }}_{1rv}}}&{{{\boldsymbol{0}}_{3 \times 3}}}& {{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{ \boldsymbol{\varPhi }}_{1vr}}}&{{{\boldsymbol{\varPhi }}_{1vv}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{\varPhi }} _{2rr}}}&{{{\boldsymbol{\varPhi }}_{2rv}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{\varPhi }} _{2vr}}}&{{{\boldsymbol{\varPhi }}_{2vv}}} \end{array}\displaystyle } \right ] . $$
(35)
The elements in the state transition matrix for the chief and deputies are
$$ \left\{ \textstyle\begin{array}{l} {{\boldsymbol{\varPhi }}_{\mathit{irr}}} = {{\boldsymbol{I}}_{3 \times 3}} + ( {2{{\boldsymbol{G}} _{i0}} + {{\boldsymbol{G}}_{i}}} ) {{{T^{2}}}/6}, \\ {{\boldsymbol{\varPhi }}_{\mathit{irv}}} = {{\boldsymbol{I}}_{3 \times 3}}\cdot T + ( {{{\boldsymbol{G}} _{i0}} + {{ \boldsymbol{G}}_{i}}} ) {{{T^{3}}} /12,} \\ {{\boldsymbol{\varPhi }}_{\mathit{ivr}}} = ( {{{\boldsymbol{G}}_{i0}} + {{\boldsymbol{G}}_{i}}} ) {T /2,} \\ {{\boldsymbol{\varPhi }}_{\mathit{ivv}}} = {{\boldsymbol{I}}_{3 \times 3}} + ( {{{\boldsymbol{G}} _{i0}} + 2{{\boldsymbol{G}}_{i}}} ) {{{T^{2}}} /6,} \end{array}\displaystyle \right . \quad i = 1,2, $$
(36)
where \({{{\boldsymbol{G}}_{i0}}}\) and \({{{\boldsymbol{G}}_{i}}}\) equal to the gravity gradient tensor matrix evaluated at the beginning and end of the propagation interval. \(T\) is the state update period. The measurement sensitivity matrix is computed as
$$\begin{aligned} {\boldsymbol{H}} =& \frac{{\partial ( {{{\boldsymbol{r}}_{2}} - {{\boldsymbol{r}}_{1}}} ) }}{ {\partial ( {{{\boldsymbol{r}}_{1}},{{\boldsymbol{v}}_{1}},{{\boldsymbol{r}}_{2}},{{\boldsymbol{v}} _{2}}} ) }} \\ =& \left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {-{{\boldsymbol{I}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{I}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \end{array}\displaystyle } \right ] \end{aligned}$$
(37)
where \({ [ {{{\boldsymbol{r}}_{1}},{{\boldsymbol{v}}_{1}}} ] ^{T}}\) is the state of the chief and \({ [ {{{\boldsymbol{r}}_{2}},{{\boldsymbol{v}}_{2}}} ] ^{T}}\) is the state of the deputy. Based on these preparations, the extended Kalman filter algorithm is given as follows.
Consider the discrete-time system in Eq. (9), we have
$$ \textstyle\begin{array}{l} E \bigl[ {{{\boldsymbol{w}}_{k}}{\boldsymbol{w}}_{k}^{T}} \bigr] = {{\boldsymbol{Q}}_{k},} \\ E \bigl[ {{{\boldsymbol{v}}_{k}}{\boldsymbol{v}}_{k}^{T}} \bigr] = {{\boldsymbol{R}}_{k},} \\ {{\boldsymbol{x}}_{0}} \sim N ( {0,{{\boldsymbol{P}}_{0}}} ) . \end{array} $$
(38)
Note that the acceleration uncertainties in Eq. (1) satisfy
$$ \left \{ \textstyle\begin{array}{l} E \bigl[ {{a_{x}} ( t ) {a_{x}}{{ ( \tau ) }}} \bigr] = {q^{2}} ( t ) \delta ( {t - \tau } ) , \\ E \bigl[ {{a_{y}} ( t ) {a_{y}}{{ ( \tau ) }}} \bigr] = {q^{2}} ( t ) \delta ( {t - \tau } ) , \\ E [ {{a_{z}} ( t ) {a_{z}}{{ ( \tau ) }}} ] = {q^{2}} ( t ) \delta ( {t - \tau } ) . \end{array}\displaystyle \right . $$
(39)
The spectral density matrix (Gelb 1974) of the process noise is
$$\begin{aligned} {\boldsymbol{Q}} ( t ) = \mathrm{diag}\left ( {\left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{{\boldsymbol{0}}_{1 \times 3}}}&{{{\boldsymbol{I}}_{1 \times 3}} \cdot {q^{2}}}& {{{\boldsymbol{0}}_{1 \times 3}}}&{{{\boldsymbol{I}}_{1 \times 3}} \cdot {q^{2}}} \end{array}\displaystyle } \right ] } \right ) \end{aligned}$$
(40)
where \({{{\boldsymbol{0}}_{1 \times 3}}}\) denotes \([ {0,0,0} ] \) and \({{\boldsymbol{I}}_{1 \times 3}}\) denotes \([ {1,1,1} ] \). The covariance matrix for the discrete process noise is (Brown and Hwang 1997)
$$ {{\boldsymbol{Q}}_{k}} = \int _{{t_{k}}}^{{t_{k + 1}}} {\boldsymbol{\varPhi }} ( {{t_{k + 1}},\tau } ) {\boldsymbol{Q}} ( \tau ) {\boldsymbol{\varPhi }} { ( {{t_{k + 1}},\tau } ) ^{T}}d\tau , $$
(41)
whose calculation is given in Appendix 2.
The time update of the filter is
$$\begin{aligned} &{ {{{\hat{\boldsymbol{x}}}}_{k|k - 1}} = {{\boldsymbol{f}}_{k|k - 1}} ( {{{{\hat{\boldsymbol{x}}}} _{k - 1}}} ) ,} \end{aligned}$$
(42)
$$\begin{aligned} &{ {{{\boldsymbol{P}}_{k|k - 1}} = {{\boldsymbol{\varPhi }}_{k|k - 1}} {{ \boldsymbol{P}}_{k - 1}} {\boldsymbol{\varPhi }}_{k|k - 1}^{T} + {{ \boldsymbol{Q}}_{k}}}} \end{aligned}$$
(43)
And, the Kalman gain is calculated as
$$ {{{\boldsymbol{K}}_{k}} = {{\boldsymbol{P}}_{k|k - 1}} {\boldsymbol{H}}_{k}^{T}{{ \bigl[ {{{\boldsymbol{H}} _{k}} {{\boldsymbol{P}}_{k|k - 1}} { \boldsymbol{H}}_{k}^{T} + {{\boldsymbol{R}}_{k}}} \bigr] } ^{ - 1}}}. $$
(44)
Consider the measurement constraints on the FOV, we modified the standard EKF to accommodate intermittent observations (Sinopoli et al. 2004). A Bernoulli random variable \(\gamma _{k}\) is defined which means \(\gamma =1\) when the measurement is available and \(\gamma =0\), otherwise;
$$\begin{aligned} &{ {{{{\hat{\boldsymbol{x}}}}_{k}} = {{{\hat{\boldsymbol{x}}}}_{k|k - 1}} + { \gamma _{k}} {{\boldsymbol{K}}_{k}} \bigl( {{{\boldsymbol{y}}_{k}} - {{\boldsymbol{h}}_{k}} ( {{{{\hat{\boldsymbol{x}}}} _{k|k - 1}}} ) } \bigr) } ,} \end{aligned}$$
(45)
$$\begin{aligned} &{ {{{\boldsymbol{P}}_{k}} = ( {{\boldsymbol{I}} - {\gamma _{k}} {{ \boldsymbol{K}}_{k}} {{\boldsymbol{H}} _{k}}} ) {{\boldsymbol{P}}_{k|k - 1}}} .} \end{aligned}$$
(46)
Note that the covariance matrix of the measurement error in Eq. (27) is not static and thus should be computed online. The modified EKF algorithm taking into account the measurement constraints is shown in Fig. 5.
Fig. 5

The flow chart of a modified EKF algorithm

5 Simulations

5.1 Navigation analysis without measurement constraints

In this section, we assume that the relative measurement information is available in the whole period. The configuration presented in Fig. 4 is considered in this section and the orbital elements of three spacecraft are given in Table 1. The configuration design parameters for each cases are presented in Table 2. It can be seen that the in-plane phase angle differences between two deputies from Case 1 to Case 3 are \(180^{ \circ }\), \(30^{\circ }\) and \(120^{\circ }\), respectively.
Table 1

Orbital elements of three formation flying spacecraft

Orbital elements

Chief

Deputy 1

Deputy 2: Case 1

Deputy 2: Case 2

Deputy 2: Case 3

Semi-major axis a (km)

3697

3697

3697

3697

3697

Eccentricity e

0.001

0.0037

0.0017

0.0036

0.0024

Inclination i ()

90.00

90.00

90.00

90.00

90.00

RAAN Ω ()

0.000

0.000

0.000

0.000

0.000

Perigee ω ()

0.000

0.000

180.0

−22.03

−98.56

True anomaly f ()

0.000

0.000

180.0

22.18

98.82

Table 2

Configuration design parameters in three cases

Parameters

Deputy 1

Deputy 2: Case 1

Deputy 2: Case 2

Deputy 2: Case 3

p (km)

10

10

10

10

s (km)

0

0

0

0

α ()

90

90

90

90

\(\theta _{0}\) ()

0

180

30

120

When there is only one deputy, the initial covariance matrix in the EKF is given as
$$ {{{\boldsymbol{P}}_{0}} = \mathrm{diag} \bigl( { \bigl[ {{{\boldsymbol{I}}_{1 \times 3}} \cdot {{10}^{8}},{{\boldsymbol{I}}_{1 \times 3}} \cdot 1,{{ \boldsymbol{I}}_{1 \times 3}} \cdot {{10}^{8}},{{\boldsymbol{I}}_{1 \times 3}} \cdot 1} \bigr] } \bigr) ,} $$
(47)
which means the standard deviation (STD) of initial position uncertainty along each axis is 10 km and the STD of initial velocity uncertainty along each axis is 1 m/s.
In order to achieve a satisfactory navigation accuracy, we find that the STD of range error must be under 1 m and the STD of angle measurement error should be less than 10 arcsec. Note that all errors mentioned later refer to the STD. When the range error is 1 m and the angle measurement error is 5 arcsec, Fig. 6 and Fig. 7 show the position and velocity estimation errors, respectively.
Fig. 6

The position estimation result with high-accuracy measurements

Fig. 7

The velocity estimation result with high-accuracy measurements

After another deputy is added into the configuration, the position and velocity estimation errors are compared in Figs. 8 and 9. It can be seen that the estimation accuracies for two-deputy scenario are nearly two times of using one deputy. The navigation performance for Case 1 is better than Case 2 and Case 3, which confirms that the optimal in-plane phase angle difference should be \(180^{\circ }\). After adding another deputy, Fig. 10 illustrates that the performance index in Eq. (21) is almost twice as large as before. This result coincides with the result of Fisher information theory as discussed above.
Fig. 8

Comparison of position estimation errors when using two deputies (\(1\sigma \))

Fig. 9

Comparison of velocity estimation errors when using two deputies (\(1\sigma \))

Fig. 10

The comparison of the performance index after adding another deputy

5.2 Navigation analysis with measurement constraints

Due to limited FOV of optical camera, the line-of-sight measurement is only available when the deputy is visible. As in Fig. 4, we assume that there are four cameras mounted on the \(\pm x\) and \(\pm y\) directions, respectively. This in-plane configuration can make full use of each camera by prolonging the time that the object can be observed. We consider the situation when only one deputy exists in the formation. The measurement constraints denoted in the LVLH frame are
$$ \left \{ \textstyle\begin{array}{l} \phi = \arcsin \biggl( \displaystyle\frac{z}{\rho } \biggr) \in \biggl( { - \frac{D}{2},\frac{D}{2}} \biggr) \\ \theta = \arctan \biggl( \displaystyle\frac{y}{x} \biggr) \\ \hphantom{\theta}\in \left\{ \textstyle\begin{array}{l} ( { - \frac{D}{2},\frac{D}{2}} ) ,\; ( { - \frac{D}{2} + \frac{\pi }{2},\frac{D}{2} + \frac{\pi }{2}} ) \\ ( {\pi - \frac{D}{2},\pi } ) , ( { - \pi , - \pi + \frac{D}{2}} ) \\ ( { - \frac{D}{2} - \frac{\pi }{2},\frac{D}{2} - \frac{\pi }{2}} ) \end{array}\displaystyle \right\} \end{array}\displaystyle \right . $$
(48)
When the size of FOV is \(20^{\circ }\), Fig. 11 shows that the deputy is visible when \(\theta \) lies on the solid line. We find that the number of observations in the vicinity of \(y\) axis is five times of the observations near \(x\) axis. In this simulation, only one deputy is considered in the formation. The range error is 1 m and the relative position angle error is 5 arcsec. In Figs. 12 and 13, it can be seen that the navigation error increases after taking into account the measurement constraints. When the observation is missing, only the time update is performed in the filter. Therefore, the error keeps increasing until the measurement is available again. The navigation errors along each axis are illustrated in Figs. 14 and 15, which are apparently larger than those in Figs. 6 and 7.
Fig. 11

The visible region when the FOV is \(20^{\circ }\)

Fig. 12

Position estimation error with measurement constraints (\(1 \sigma \))

Fig. 13

Velocity estimation error with measurement constraints (\(1 \sigma \))

Fig. 14

Position estimation errors along each axis with measurement constraints

Fig. 15

The velocity estimation errors along each axis with measurement constraints

In Figs. 16 and 17, the position and velocity estimation errors of five cases are compared when optical cameras are installed in different locations. Table 3 gives the estimation errors along each axis. It can be seen that the cameras mounted along \(y\) axis are more effective than those mounted along \(x\) axis. The reason of this phenomenon is that the number of observations in the proximity of \(y\) axis is much larger than that near \(x\) axis. This is because the relative velocity near \(y\) axis is much smaller than that in the neighborhood of \(x\) axis. Therefore, more information will be accumulated from observations near \(y\) axis under the same sampling frequency. When cameras are installed along the \({\pm} x\) directions, we find that the navigation performance will be enhanced after increasing the sampling frequency near \(x\) axis. The comparisons of position and velocity estimation errors are illustrated in Figs. 18 and 19.
Fig. 16

The position errors considering the installation of cameras (\(1\sigma \))

Fig. 17

The velocity errors considering the installation of cameras (\(1\sigma \))

Fig. 18

Position estimation errors using different sampling frequency (\(1\sigma \))

Fig. 19

Velocity estimation errors using different sampling frequency (\(1\sigma \))

Table 3

Orbit determination errors for the chief under different camera installations (\(1\sigma \))

Camera installations

x (m)

y (m)

z (m)

\(v_{x}\) (m/s)

\(v_{y}\) (m/s)

\(v_{z}\) (m/s)

x(±), y(±)

5.237

7.348

26.53

0.016

0.029

0.005

x(+)

2080

1826

2102

1.5294

0.712

0.347

y(+)

232.1

959.3

407.9

0.342

0.756

0.157

x(±)

141.4

20.31

556.7

0.416

0.105

0.126

y(±)

7.898

7.491

58.08

0.044

0.031

0.007

There is no doubt that larger FOV can provide more effective measurements. Nevertheless, the measurement accuracy will diminish when the FOV becomes larger. Therefore, the navigation errors of four cases considering various FOVs and measurement accuracies are compared in Figs. 20 and 21. The result reveals that the relative position angle measurement errors dominate the navigation accuracy although the FOV is increasing.
Fig. 20

Position estimation errors with various FOVs (\(1\sigma \))

Fig. 21

Velocity estimation errors with various FOVs (\(1\sigma \))

6 Conclusions

In this paper, autonomous navigation is achieved by using formation flying spacecraft. The inertial line-of-sight vectors between the chief and deputies are used as the measurements. The constraint on the camera is considered and the installation of cameras is investigated. Moreover, when there are two deputies in the formation, optimal configurations are obtained from the perspective of Fisher information theory and an index is proposed to evaluate the system’s observability. The extended Kalman filter is modified to tackle the problem of measurement missing. The navigation simulations reveal that the installation of cameras influences the navigation performance. The results also show that the camera installed along \(y\) axis can receive more measurement information. Thus, the cameras mounted along this axis can significantly enhance the navigation accuracy. After the configuration is optimized, the simulation confirms that the navigation accuracy can be enhanced by using more deputies. Future work may concentrate on how to reduce the computation when more deputies are included in the system.

Notes

Acknowledgement

This work is supported by the Major Program of National Natural Science Foundation of China under Grant Numbers 61690210 and 61690213.

References

  1. Alfriend, K.T., Vadali, S.R., Gurfil, P., How, J., Breger, L.S.: Spacecraft Formation Flying: Dynamics, Control and Navigation (2009). Butterworth-Heinemann Google Scholar
  2. Bandyopadhyay, S., Foust, R., Subramanian, G.P., Chung, S.J., Hadaegh, F.Y.: Review of formation flying and constellation missions using nanosatellites. J. Spacecr. Rockets 53, 567 (2016).  https://doi.org/10.2514/1.A33291 ADSCrossRefGoogle Scholar
  3. Bar-Shalom, Y., Li, X.R., Kirubarajan, T.: Estimation with Applications to Tracking and Navigation. Wiley, New York (2002).  https://doi.org/10.1002/0471221279.ch3 Google Scholar
  4. Brown, R.G., Hwang, P.Y.C.: Introduction to Random Signals and Applied Kalman Filtering. Wiley, New York (1997) zbMATHGoogle Scholar
  5. Christian, J.A., Robinson, S.B., D’Souza, C.N., Ruiz, J.P.: Cooperative relative navigation of spacecraft using flash light detection and ranging sensors. J. Guid. Control Dyn. 37, 452 (2014).  https://doi.org/10.2514/1.61234 ADSCrossRefGoogle Scholar
  6. Crassidis, J.L., Junkins, J.L.: Optimal Estimation of Dynamic Systems, 2nd edn. Chapman & Hall, London (2011) zbMATHGoogle Scholar
  7. D’Amico, S., Montenbruck, O.: Proximity operations of formation-flying spacecraft using an eccentricity/inclination vector separation. J. Guid. Control Dyn. 29, 554 (2009).  https://doi.org/10.2514/1.15114 CrossRefGoogle Scholar
  8. D’Amico, S., Ardaens, J.S., Larsson, R.: Spaceborne autonomous formation-flying experiment on the PRISMA mission. J. Guid. Control Dyn. 35, 834 (2012).  https://doi.org/10.2514/1.55638 ADSCrossRefGoogle Scholar
  9. Elsaka, B., Kusche, J., Ilk, K.H.: Recovery of the Earth’s gravity field from formation-flying satellites: temporal aliasing issues. Adv. Space Res. 50, 1534 (2012).  https://doi.org/10.1016/j.asr.2012.07.016 ADSCrossRefGoogle Scholar
  10. Gelb, A.: Applied Optimal Estimation. MIT Press, Cambridge (1974). Chap. 3 Google Scholar
  11. Hill, K., Born, G.: Autonomous interplanetary orbit determination using satellite-to-satellite tracking. J. Guid. Control Dyn. 30, 679 (2007).  https://doi.org/10.2514/1.24574 ADSCrossRefGoogle Scholar
  12. Huang, P.F., Chen, L., Zhang, B., Meng, Z.J., Liu, Z.X.: Autonomous rendezvous and docking with nonfull field of view for tethered space robot. Int. J. Aerosp. Eng. 2017, 3162349 (2017) Google Scholar
  13. Huxel, P.J., Bishop, R.H.: Navigation algorithms and observability analysis for formation flying missions. J. Guid. Control Dyn. 32, 1218 (2009).  https://doi.org/10.2514/1.41288 ADSCrossRefGoogle Scholar
  14. Jauffret, C.: Observability and Fisher information matrix in nonlinear regression. IEEE Trans. Aerosp. Electron. Syst. 43, 756 (2007).  https://doi.org/10.1109/TAES.2007.4285368 ADSCrossRefGoogle Scholar
  15. Kang, W., Ross, I.M., Pham, K., Gong, Q.: Autonomous observability of networked multisatellite systems. J. Guid. Control Dyn. 32, 869 (2009).  https://doi.org/10.2514/1.38826 ADSCrossRefGoogle Scholar
  16. Lemoine, F.G., Smith, D.E., Rowlands, D.D., Zuber, M.T., Neumann, G.A., Chinn, D.S., Pavlis, D.E.: An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor. J. Geophys. Res. 106, 23359 (2001).  https://doi.org/10.1029/2000JE001426 ADSCrossRefGoogle Scholar
  17. Maessen, D., Gill, E.: Relative state estimation and observability analysis for formation flying satellites. J. Guid. Control Dyn. 35, 321 (2012).  https://doi.org/10.2514/1.55125 ADSCrossRefGoogle Scholar
  18. Markley, F.L.: Autonomous navigation using landmark and intersatellite data. In: AIAA/AAS Astrodynamics Conference, Seatle, WA, Hawaii (1984). AIAA Paper Google Scholar
  19. Markley, F.L.: Approximate Cartesian state transition matrix. J. Astronaut. Sci. 34, 161 (1986) ADSGoogle Scholar
  20. Ou, Y., Zhang, H.: Observability-based Mars autonomous navigation using formation flying spacecraft. J. Navig. 71, 21 (2018).  https://doi.org/10.1017/S0373463317000510 CrossRefGoogle Scholar
  21. Ou, Y., Zhang, H., Xing, J.: Autonomous orbit determination and observability analysis for formation satellites. In: 2016 35th Chinese Control Conference, CCC, Chengdu (2016).  https://doi.org/10.1109/ChiCC.2016.7554179 Google Scholar
  22. Psiaki, M.L.: Absolute orbit and gravity determination using relative position measurements between two satellites. J. Guid. Control Dyn. 34, 1285 (2011).  https://doi.org/10.2514/1.47560 ADSCrossRefGoogle Scholar
  23. Roscoe, C.W., Vadali, S.R., Alfriend, K.T., Desai, U.P.: Optimal formation design for magnetospheric multiscale mission using differential orbital elements. J. Guid. Control Dyn. 34, 1070 (2011).  https://doi.org/10.2514/1.52484 ADSCrossRefGoogle Scholar
  24. Schmidt, U.: ASTRO APS autonomous star sensor first year flight experience and operations on AlphaSat (invited). In: AIAA/AAS Astrodynamics Specialist Conference, vol. 701 (2014). American Institute of Aeronautics and Astronautics Google Scholar
  25. Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M.I., Sastry, S.S.: Kalman filtering with intermittent observations. IEEE Trans. Autom. Control 49, 9 (2004).  https://doi.org/10.1109/TAC.2004.834121 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Steven, M.K.: In: Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall Signal Processing Series. Prentice Hall, New York (1993) Google Scholar
  27. Wang, Y., Zheng, W., Sun, S., Li, L.: X-ray pulsar-based navigation using time-differenced measurement. Aerosp. Sci. Technol. 36, 27 (2014).  https://doi.org/10.1016/j.ast.2014.03.007 CrossRefGoogle Scholar
  28. Wang, X., Qin, W., Bai, Y., Cui, N.: A novel decentralized relative navigation algorithm for spacecraft formation flying. Aerosp. Sci. Technol. 48, 28 (2016).  https://doi.org/10.1016/j.ast.2015.10.014 CrossRefGoogle Scholar
  29. Yang, H.X., Vetrisano, M., Vasile, M., Zhang, W.: Autonomous navigation of spacecraft formation in the proximity of minor bodies. Proc. Inst. Mech. Eng., G J. Aerosp. Eng. 230, 189 (2016).  https://doi.org/10.1177/0954410015590465 CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Aerospace Science and EngineeringNational University of Defense TechnologyChangshaChina

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