# A Novel Inertial-Aided Star Pattern Matching Method for Distributed Multiple FoVs

Authors: Rong Wang; Yuxuan Cao; Jianye Liu; Zhi Xiong
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

An aerospace vehicle in high-speed flight involves distributed multiple fields of view (FoVs) of a star sensor, and the maneuvering of the vehicle requires inertial-aided star pattern matching. Accordingly, an inertial-aided star pattern matching method for distributed multiple FoVs is proposed. First, the observation and fusion principles of distributed multiple FoVs are illustrated. Then, an inertial-aided star pattern matching scheme is designed based on the space-time distribution of the line-of sight (LoS) vector and inertial measurement information. Next, an LoS vector estimation method assisted by inertial information and the dynamic establishment of a distributed local catalog are proposed. Finally, in a simulation, the dynamic establishment of the part catalog improves the efficiency of star pattern matching by reducing the catalog scale, and the usability of inertial-aided star pattern matching is analyzed for different gyro and accelerometer precisions.


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## Body

## 1. Introduction

An aerospace vehicle is a kind of advanced reusable vehicle for the future [1]. The multitask mode and maneuvering of an aerospace vehicle have considerable requirements in terms of autonomous navigation. In aviation, the celestial navigation system (CNS) plays an important role and has the advantages of autonomy, highly accurate angle measurements, and nonaccumulating error over time.An analogy can be made between the CNS in the cruise stage of an aerospace vehicle and the small-aperture star sensor that is applied in autonomous identification in a long-range and long-endurance vehicle (LRLEV), where a small-volume sensor and availability during the day and night are ensured [2]. The second-generation CCD star sensor truly realizes full-day autonomous identification without prior information and has the function of “ input starlight and output attitude” [3, 4]. At present, the industrialized series production of star sensors has been formed, and the main products are concentrated in the second-generation CCD star sensors and APS CMOS star sensors [5]. Researchers have studied the optimization and improvement of the triangle matching algorithm. The original Danish University of Technology Liebe proposed a method of storing star distance instead of storing triangle features [6]. A star identification algorithm with star ID shortlisting is researched, in which the star IDs are shortlisted based on worst-case patch mismatch, and stars are identified in the image by an initial match confirmed with a running sequential angular match technique [7]. The star pattern identification algorithm using the vector pattern matching technique on the celestial sphere is also researched, which is formed by the correlation between the original and the reference star patterns on the celestial sphere [8]. The star identification algorithm is researched for the attitude determination of a star sensor in the lost-in-space case, which uses label values to represent each group of stars [9]. A set of lost-in-space star identification algorithms that work effectively for small satellites was proposed [10]. However, a small-FoV star sensor is limited by a small number of visible stars, identifies centrally distributed stars, and requires frame control. In addition, the LRLEV aerospace vehicle has a higher flight speed. In this way, the installation location of the star sensor is constrained further, and multiple FoVs exist in a noncontinuous distribution, which results in visible stars with a poor geometric structure [11]. Therefore, the above issues mean that a single small-aperture star sensor cannot provide continuous and robust navigation information for an aerospace vehicle in flight.Multi-FoV fusion technology can extend the visible area and focus the stars in different FoVs, increasing the number of visible stars and improving the geometric structure of the identified stars [12]. In recent research on a multi-FoV fusion all-sky identification algorithm, Texas A&M University proposed the pyramid algorithm in [13], which identifies a feature based on the star vector inner product of 4 stars, and the SP search algorithm in [14], which searches a spherical polygon in a wide-FoV star sensor. These two algorithms were employed to achieve successful star identification with multi-FoV fusion in an orbit vehicle in [15]. In addition, the applications of mass production multi-FoV star sensors, such as HYDRA [16] and DECOM [17], provided a reference for aerospace vehicle aviation.For an aerospace vehicle with a large turning angle and angular rate, the positions and attitudes from the single CNS have a large error so that only a star tracking algorithm, such as an interframe estimation algorithm, that does not depend on the external navigation information will be more likely to lose lock [18]. Later, the all-sky identification algorithm rapidly recovers the output because of the calculation cost. However, both the CNS and an inertial navigation system (INS) are autonomous navigation systems, and a navigation system integrating these systems can exhibit complementary advantages suitable for the autonomous navigation of an aerospace vehicle. In recent research on a star identification algorithm assisted by an INS, a recursive star identification algorithm used a gyroscope to aid the attitude estimation in [19]; an INS-assisted algorithm estimated the LoS vector with the positions and attitudes from the INS in [20]; and a fast star identification algorithm provided preliminarily analysis of the influence of inertial measurement information. On the basis of these three results, an INS-assisted star identification algorithm can address the problem of a large maneuver in flight through theoretical analysis and an efficient verification of the inertial measurement error in star identification.The perspective inertial-aided approaches for star tracker mainly focused on single FoV. The star sensor with multiple FoVs shows advantage than a single FoV, while stars within the same FoV and across different FoVs are both adopted, so the appropriate inertial-aided approaches need to be investigated. With multiple small distributed FoVs on the aerospace vehicle and the method of star identification assisted by inertial measurement information mentioned above, an inertial-aided star pattern matching method for distributed multiple FoVs is proposed. First, star image fusion under distributed multiple FoVs and an invisible area is analyzed, marking the beginning of the imaging principle of multiple FoVs. Then, an inertial-aided star pattern matching process is proposed based on the spatiotemporal relationship between the LoS vector and the inertial measurement information. Next, multi-LoS vector estimation and the dynamic establishment of a star part catalog are presented, and the inertial-aided star pattern matching method is researched. Finally, the influence of inertial measurement units with different accuracies on aided matching is analyzed, and the results show that aided matching has a higher efficiency due to the scale reduction of the part catalog. The main contribution of this paper lies in two aspects: Firstly, an inertial-aided dynamic predicting method for multiple FoVs’ observed scopes is proposed, in which the biases caused by the errors of inertial navigation system are taken in consideration, respectively. Secondly, a star pattern matching based on predicted local star catalog is proposed, whose magnitude is significantly reduced and the real-time performance is improved than the conventional star pattern matching method based on all-sky star catalog.


## 2. Modeling of Novel Star Sensor with Distributed Multiple FoVs

### 2.1. Framework of Novel Star Sensor with Distributed Multiple FoVs

The flight of an aerospace vehicle includes the aerospace vehicle taking off, landing, and cruising. In the cruise stage, an analogy can be made between an LRLEV with a small-aperture star sensor. The star sensor for an aerospace vehicle should involve distributed multiple FoVs, such that the FoVs do not overlap. A diagram is shown in Figure1.Figure 1
Diagram of a distributed multi-FoV star sensor system.With the core of a multi-FoV star sensor, a CNS can address the problem of aviation usability, but it faces some new challenges. In the existing studies of star identification algorithms with multiple FoVs, the methods of star image fusion and star catalog generation involve continuous multiple FoVs with a small installation angle, and the studies do not present further application analysis of distributed multiple FoVs. In particular, the catalog scale of the multiple FoVs would be larger than that of a single FoV because of the star selection with large angular distance with respect to the setting angles and FoVs. The catalog scale cannot satisfy the efficiency required by aerospace vehicles.Therefore, the design of a CNS should include a full consideration of the influence of distributed multiple FoVs on the star image and the influence of inertial assistant information on star pattern matching. The identification scheme, a star image observation model, and an image fusion method with distributed multiple FoVs, LoS vector estimation, and the dynamic establishment of a part catalog are researched below.


### 2.2. Model of a Single FoV in Star Sensor

A star sensor is a sensor that observes and identifies navigation stars. The optical structure of a single-FoV star sensor can be simplified into two parts: a lens and an image plane, which are shown in Figure2.Figure 2
Imaging of a single-FoV star sensor.Define the coordinate system about the star sensor as follows:(1)
The optic head coordinate system (abbreviated as thes system) takes the image plane center O as the coordinate origin, the LoS vector as the Z axis direction, and the directions of the two vertical sides of the image plane as other axis directions. In terms of a single-FoV star sensor, there is only one s system, which can be called the star sensor coordinate system(2)
The image plane coordinate system (p system) takes the image plane center O as the coordinate origin and the directions of the two vertical sides of the image plane as other axis directionsIn Figure2, the distance OA is denoted by the focal length from the optic center A to the image plane, and the line OZ→ is the optic axis. The size of the star sensor FoV can be regarded as the range observed in the celestial sphere, which is generally described by an angle.According to the definitions of the coordinate systems above, a star imaging model for a single optical head can be built. Here, a single star is used to describe the imaging process. It is assumed that the coordinate of a starq in the s system is xq,yq,zqs. Through a transmission projection, the coordinate of the star q in the p system is xq,yqp given in equation (1):
(1)xqp=−Fxqszqs,yqp=−Fyqszqs,where F is the focal length of the optic head.


### 2.3. New Model for Star Image Fusion between Distributed Multiple FoVs

In the cruise stage of an aerospace vehicle with high speed, the setting of the angle of the multi-FoV star sensor would be constrained due to the strong aerooptical effect. In most cases, multiple optic heads would be installed with large angles to overcome the poor geometric configuration of stars. A double-FoV star sensor is chosen as an example in this section, and the imaging diagram is shown in Figure3, where the area observed by the main optic head is the area Am, and the area Aa is observed by the auxiliary optic head. The invisible area is not observed by the double optical head. That is, the two FoVs do not overlap.Figure 3
Star imaging diagram for distributed FoVs.For multiple FoVs, one optic head corresponds to thes system. The main optic head coordinate system (left) is denoted as the s1 system, and the auxiliary optic head coordinate system (right) is denoted as the s2 system. There are two p systems: the main image plane coordinate system (p1 system) and the auxiliary image plane coordinate system (p2 system). Analogously to the imaging principle in Section 3.1, two stars are used to describe the imaging process for the distributed double-FoV system. It is assumed that there are two stars q1 and q2 observed by the main and auxiliary optic heads, respectively. Their coordinates in the s1 and s2 systems are xq1,yq1,zq1s1 and xq2,yq2,zq2s2, respectively. Through a transmission projection, their coordinates in the imaging planes are
(2)xq1p1=−F1xq1s1zq1s1,yq1p1=−F1yq1s1zq1s1,xq2p2=−F2xq2s2zq2s2,yq2p2=−F2yq2s2zq2s2,where F1 and F2 are the focal lengths of the main and auxiliary optic heads, respectively.The coordinates and magnitudes of invisible stars can be obtained by star image preprocessing. For a small multi-FoV star sensor, star image fusion is the process of transferring the stars in the auxiliary FoV to the main or extended FoV. From Figure4, the two FoVs do not overlap, and there could be a large invisible area. Thus, they cannot be mosaicked. To address the problem for distributed multiple FoVs with a large setting angle, a new star image fusion method is proposed. First, the coordinates of the stars in the s system are rebuilt according to the focal lengths. Second, the stars in the s2 system are transformed into the s1 system, and third, they are projected into the extended plane of the main image plane.Figure 4
Inertial-aided star pattern matching scheme for distributed FoVs.The star image fusion process is shown in Figure4. Because the stars are far from the Earth, their light can be regarded as parallel light. The fused star image contains two parts: one part involves the star points that are observed by the main optic head and projected, and the other part involves the star points that are observed by the auxiliary optic head and transformed. The double-FoV star sensor is used to illustrate the fusion process. In Section 3.2, the coordinates of the star point in the auxiliary image plane are obtained and regarded as xq2,yq2p2 by equation (2), which can be rebuilt into the s2 system by the focal length and can be regarded as xq2,yq2,F2s2.When a star sensor with multiple FoVs is strapped to an aerospace vehicle, the installation matrices describe the relationship between the optical heads and the vehicle body. If the matrices of the main and auxiliary optic heads are regarded asCs1b and Cs2b, the matrix Cs2s1, which describes the transformation of the two s systems, is given by equation (3):
(3)Cs2s1=Cs1bTCs2b.The coordinates of the star pointq2xq2′,yq2′,zq′s1 in the s1 system after the transformation can be obtained by equation (4):
(4)xq2′yq2′zq2′s1=Cs2s1xq2yq2F2s2.Through the projection, the star pointq2 in the p1 system is shown in equation (5):
(5)xq2p1=−F1xq2′zq2′,yq2p1=−F1yq2′zq2′.All the stars in the auxiliary FoV can be transformed by the fusion method above, and the method is not limited to image fusion with double FoVs.


## 2.1. Framework of Novel Star Sensor with Distributed Multiple FoVs

The flight of an aerospace vehicle includes the aerospace vehicle taking off, landing, and cruising. In the cruise stage, an analogy can be made between an LRLEV with a small-aperture star sensor. The star sensor for an aerospace vehicle should involve distributed multiple FoVs, such that the FoVs do not overlap. A diagram is shown in Figure1.Figure 1
Diagram of a distributed multi-FoV star sensor system.With the core of a multi-FoV star sensor, a CNS can address the problem of aviation usability, but it faces some new challenges. In the existing studies of star identification algorithms with multiple FoVs, the methods of star image fusion and star catalog generation involve continuous multiple FoVs with a small installation angle, and the studies do not present further application analysis of distributed multiple FoVs. In particular, the catalog scale of the multiple FoVs would be larger than that of a single FoV because of the star selection with large angular distance with respect to the setting angles and FoVs. The catalog scale cannot satisfy the efficiency required by aerospace vehicles.Therefore, the design of a CNS should include a full consideration of the influence of distributed multiple FoVs on the star image and the influence of inertial assistant information on star pattern matching. The identification scheme, a star image observation model, and an image fusion method with distributed multiple FoVs, LoS vector estimation, and the dynamic establishment of a part catalog are researched below.


## 2.2. Model of a Single FoV in Star Sensor

A star sensor is a sensor that observes and identifies navigation stars. The optical structure of a single-FoV star sensor can be simplified into two parts: a lens and an image plane, which are shown in Figure2.Figure 2
Imaging of a single-FoV star sensor.Define the coordinate system about the star sensor as follows:(1)
The optic head coordinate system (abbreviated as thes system) takes the image plane center O as the coordinate origin, the LoS vector as the Z axis direction, and the directions of the two vertical sides of the image plane as other axis directions. In terms of a single-FoV star sensor, there is only one s system, which can be called the star sensor coordinate system(2)
The image plane coordinate system (p system) takes the image plane center O as the coordinate origin and the directions of the two vertical sides of the image plane as other axis directionsIn Figure2, the distance OA is denoted by the focal length from the optic center A to the image plane, and the line OZ→ is the optic axis. The size of the star sensor FoV can be regarded as the range observed in the celestial sphere, which is generally described by an angle.According to the definitions of the coordinate systems above, a star imaging model for a single optical head can be built. Here, a single star is used to describe the imaging process. It is assumed that the coordinate of a starq in the s system is xq,yq,zqs. Through a transmission projection, the coordinate of the star q in the p system is xq,yqp given in equation (1):
(1)xqp=−Fxqszqs,yqp=−Fyqszqs,where F is the focal length of the optic head.


## 2.3. New Model for Star Image Fusion between Distributed Multiple FoVs

In the cruise stage of an aerospace vehicle with high speed, the setting of the angle of the multi-FoV star sensor would be constrained due to the strong aerooptical effect. In most cases, multiple optic heads would be installed with large angles to overcome the poor geometric configuration of stars. A double-FoV star sensor is chosen as an example in this section, and the imaging diagram is shown in Figure3, where the area observed by the main optic head is the area Am, and the area Aa is observed by the auxiliary optic head. The invisible area is not observed by the double optical head. That is, the two FoVs do not overlap.Figure 3
Star imaging diagram for distributed FoVs.For multiple FoVs, one optic head corresponds to thes system. The main optic head coordinate system (left) is denoted as the s1 system, and the auxiliary optic head coordinate system (right) is denoted as the s2 system. There are two p systems: the main image plane coordinate system (p1 system) and the auxiliary image plane coordinate system (p2 system). Analogously to the imaging principle in Section 3.1, two stars are used to describe the imaging process for the distributed double-FoV system. It is assumed that there are two stars q1 and q2 observed by the main and auxiliary optic heads, respectively. Their coordinates in the s1 and s2 systems are xq1,yq1,zq1s1 and xq2,yq2,zq2s2, respectively. Through a transmission projection, their coordinates in the imaging planes are
(2)xq1p1=−F1xq1s1zq1s1,yq1p1=−F1yq1s1zq1s1,xq2p2=−F2xq2s2zq2s2,yq2p2=−F2yq2s2zq2s2,where F1 and F2 are the focal lengths of the main and auxiliary optic heads, respectively.The coordinates and magnitudes of invisible stars can be obtained by star image preprocessing. For a small multi-FoV star sensor, star image fusion is the process of transferring the stars in the auxiliary FoV to the main or extended FoV. From Figure4, the two FoVs do not overlap, and there could be a large invisible area. Thus, they cannot be mosaicked. To address the problem for distributed multiple FoVs with a large setting angle, a new star image fusion method is proposed. First, the coordinates of the stars in the s system are rebuilt according to the focal lengths. Second, the stars in the s2 system are transformed into the s1 system, and third, they are projected into the extended plane of the main image plane.Figure 4
Inertial-aided star pattern matching scheme for distributed FoVs.The star image fusion process is shown in Figure4. Because the stars are far from the Earth, their light can be regarded as parallel light. The fused star image contains two parts: one part involves the star points that are observed by the main optic head and projected, and the other part involves the star points that are observed by the auxiliary optic head and transformed. The double-FoV star sensor is used to illustrate the fusion process. In Section 3.2, the coordinates of the star point in the auxiliary image plane are obtained and regarded as xq2,yq2p2 by equation (2), which can be rebuilt into the s2 system by the focal length and can be regarded as xq2,yq2,F2s2.When a star sensor with multiple FoVs is strapped to an aerospace vehicle, the installation matrices describe the relationship between the optical heads and the vehicle body. If the matrices of the main and auxiliary optic heads are regarded asCs1b and Cs2b, the matrix Cs2s1, which describes the transformation of the two s systems, is given by equation (3):
(3)Cs2s1=Cs1bTCs2b.The coordinates of the star pointq2xq2′,yq2′,zq′s1 in the s1 system after the transformation can be obtained by equation (4):
(4)xq2′yq2′zq2′s1=Cs2s1xq2yq2F2s2.Through the projection, the star pointq2 in the p1 system is shown in equation (5):
(5)xq2p1=−F1xq2′zq2′,yq2p1=−F1yq2′zq2′.All the stars in the auxiliary FoV can be transformed by the fusion method above, and the method is not limited to image fusion with double FoVs.


## 3. Novel Inertial-Aided Star Pattern Matching Method for Distributed Multiple FoVs

### 3.1. Scheme Design

A star catalog contains the possible stars that can be observed by a star sensor. The catalog is preset before star pattern matching and is not changed in the all-sky identification. Assistance identification by external information is a method that utilizes external information with errors to estimate the LoS of the star sensor in real time and constantly maintains a part catalog with a small scale before star pattern matching. The whole scheme is shown in Figure4.Figure4 shows the whole process of the method. It is shown that inertial information helps to estimate the LoS vectors of distributed multiple FoVs, and the errors are calculated as the error radius before the catalogue radius, which helps to select stars and constantly maintain the part catalog. The bias of the calculated LoS vector, caused by the errors in the inertial measurements, could be estimated by the error states of the inertial navigation system. Since the proposed star pattern matching approach is designed for the scenario that stars tracking was just failed and the star sensor could not assist the inertial navigation system during a short episode, the error states of the inertial navigation system could be predicted by time update of its error model in the inertial/celestial fusion filter approximately. Once the stars are retracked after star pattern matching, the measurement update of the fusion filter could be carried out to correct the inertial navigation system again. Thus, the bias information during the star pattern matching episode could be obtained.


### 3.2. Prediction of the Optical Axis Direction Based on Inertial Measurements

Star identification assisted by an INS for a single-FoV star sensor was studied in [21]. The LoS vector estimation method can be applied to a multi-FoV star sensor. It is assumed that the star sensor has NB distributed FoVs sorted from 1 to NB. Obviously, each installation matrix Csnb is certain. In theory, the nth LoS vector rni (1≤n≤NB) is estimated by equation (6):
(6)rni=CeiCneCbnCsbrsn.Actually, the real LoS vector will be affected by the errors in the inertial information as shown in equation (7):
(7)r~ni=CeiC~neC~bnCsbrsn.


### 3.3. Prediction of Multiple FoVs’ Observed Scopes

The matricesC~ne and C~bn contain the errors. When the errors are small, they are shown in equations (8) and (9) [22]:
(8)C~bn=CbnI−ϕ×,(9)C~ne=I+ΔP×Cne,where ϕ is the platform error angle vector defined in the navigation system (ENU), ΔP=−ΔLΔλcosLΔλsinLT is the error vector from the position, and δL is the latitude error and δλ is the longitude error. The inertial errors are derived from the INS error equations.Inserting equations (8) and (9) into equation (7) and comparing the simplified result with equation (6), the LoS estimation bias ψn of the nth FoV can be obtained as follows:
(10)ψn=C~bnϕ−ΔP.According to equation (9), ψn is a rotation Euler vector, giving the rotation between the real LoS vector (without errors) and the estimated LoS vector (with errors) of the nth FoV. It can also be defined as a unique bias angle ηn for ψn. If the estimated LoS vector r~ni is obtained, there is another vector r0i that is on the cone generatrix whose cone line is the estimated LoS r~ni and cone angle is ηn. Thus, the vector r0i for the estimated LoS vector r~ni by equation (7) is given by equation (11):
(11)r0i=I+ψn×r~ni.Then, the LoS bias angleηn can be calculated by equation (12).
(12)ηn=dr0ir~ni=arccosr0i⋅r~nir0i⋅r~ni.The relationship between the estimated and real LoS is shown in Figure5. The point O is the optic center, and OC¯ is the real LoS vector. The arc ACB⌢ is the real visible range of the star sensor. The errors from the inertial information are small, so the LoS bias angle is also small. Therefore, OC1¯ is the estimated LoS, and∠C1OC is the LoS bias angle. The visible range becomes the arc A1C1B1⌢. The inertial errors concern the error radius A2A1¯ so that the catalog radius A2D¯ contains the FoV radius A1D¯ and the error radius. In this way, the range of the catalog is the arc A2C1B2⌢.Figure 5
Inertial error analysis for the FoV.In fact, the celestial sphere area observed by the star sensor is very small and can be regarded as a plane. If the LoS bias angle is small, the celestial sphere area with error is still approximately a plane. Therefore, the LoS bias angle can only affect the range of the catalog.If the LoS bias angle is written asηn, then the error radius can be given by equation (13):
(13)ΔRn=sinηn.In other words, from equations (3)–(13), each real LoS vector r~ni and error radius ΔRn can be calculated.With the assistance of LoS vector estimation and error radius certainty, the stars in the part catalog are approximately equal to those in the star image. For Section4.2, the error radius for the nth FoV is written as ΔRn, and the catalog radius R~n is given by equation (14):
(14)R~n=Rn+ΔRn.The part catalog is established with the estimated LoS vector and the catalog radius. The method for generating the part catalog has two parts: one part involves star selection, and the other part involves the pattern structure.One star is used to illustrate the star selection method. If the estimated LoS vector is written asr~ni and the catalog radius is R~n for the nth FoV, the condition for star q in the nth FoV is shown in equation (15):
(15)dr~niq≤R~,where dr~niq indicates the argument between the star q and the estimated LoS vector r~ni. Equation (15) is used to select stars in an all-sky catalog and choose the stars in each FoV.


### 3.4. Star Pattern Matching Based on Predicted Local Star Catalog

The pattern of the catalog depends on the star identification algorithm. A triangle matching algorithm is chosen, and the pattern is based on the angular distance between the stars. The catalog structure of packet storage for a single FoV in [18] is employed, and the section below analyzes the extended application of distributed FoVs.However, unlike traditional star sensor with single-FoV, the angular distances between the stars show different characters for star sensor with distributed multiple FoVs.In Figure6, there is not only angular distance between the stars within the same FoV but also angular distance between the stars across different FoVs. Thus, two situations are taken in consideration according to the angular distances between stars during the star pattern matching:
(1)
Angular distance within a single FoV: the star sets are obtained by the star selection. Thenth set is written as starn and the argument set in each starn is dpqn. For a star sensor with multiple distributed small FoVs, the FoV size is smaller than the installation angle for each optical head, and generally, each FoV has the same size. Therefore, the angular distance of each FoV together form the whole argument set dpqn (1≤n≤NB)(2)
Angular distance across a pair of FoVs: each star setstarn is selected, and then the set with a number larger than n is selected. The angular distance across each two FoVs is calculated. The calculation process is as follows: If a star set starn with number n is chosen, then the following sets with a number from n+1 to NB are chosen to calculate the angular distance. If there is a following set starmn+1≤m≤NB, then the argument is written as dpnqm. Accordingly, the argument set is written as dpnqmnm. According to the calculation, the argument sets dpnqmnm for each two FoVs and the whole set dpnqmnm can be obtainedFigure 6
Two situations of the angular distances for distributed multiple FoVs.The two argument sets have different values. One needs to choose the appropriate argument value as the grouping interval as in reference [18]. The two sets are divided into intervals, and the argument patterns are all based on packet storage. These patterns make up the part catalog. The whole catalog is shown in Figure 7.Figure 7
Dynamic establishment of the local star catalog.With the combination of the inertial-aided star identification scheme (Figure1) and the part catalog, the inertial-aided star pattern matching process for distributed multiple FoVs is as follows:Step 1. use the inertial information to estimate the LoS vector r~ni of each FoVStep 2. calculate the error radius ΔRn from the inertial errors and further obtain the catalog radius R~n of each FoVStep 3. utilize the LoS vector r~ni from Step 1 and two radii (ΔRn and R~n) from Step 2 to select stars for each FoV from the all-sky catalog (external data) and then generate the part catalog in real timeStep 4. use the fused image (external data) and the part catalog from Step 3 to match the star pattern


## 3.1. Scheme Design

A star catalog contains the possible stars that can be observed by a star sensor. The catalog is preset before star pattern matching and is not changed in the all-sky identification. Assistance identification by external information is a method that utilizes external information with errors to estimate the LoS of the star sensor in real time and constantly maintains a part catalog with a small scale before star pattern matching. The whole scheme is shown in Figure4.Figure4 shows the whole process of the method. It is shown that inertial information helps to estimate the LoS vectors of distributed multiple FoVs, and the errors are calculated as the error radius before the catalogue radius, which helps to select stars and constantly maintain the part catalog. The bias of the calculated LoS vector, caused by the errors in the inertial measurements, could be estimated by the error states of the inertial navigation system. Since the proposed star pattern matching approach is designed for the scenario that stars tracking was just failed and the star sensor could not assist the inertial navigation system during a short episode, the error states of the inertial navigation system could be predicted by time update of its error model in the inertial/celestial fusion filter approximately. Once the stars are retracked after star pattern matching, the measurement update of the fusion filter could be carried out to correct the inertial navigation system again. Thus, the bias information during the star pattern matching episode could be obtained.


## 3.2. Prediction of the Optical Axis Direction Based on Inertial Measurements

Star identification assisted by an INS for a single-FoV star sensor was studied in [21]. The LoS vector estimation method can be applied to a multi-FoV star sensor. It is assumed that the star sensor has NB distributed FoVs sorted from 1 to NB. Obviously, each installation matrix Csnb is certain. In theory, the nth LoS vector rni (1≤n≤NB) is estimated by equation (6):
(6)rni=CeiCneCbnCsbrsn.Actually, the real LoS vector will be affected by the errors in the inertial information as shown in equation (7):
(7)r~ni=CeiC~neC~bnCsbrsn.


## 3.3. Prediction of Multiple FoVs’ Observed Scopes

The matricesC~ne and C~bn contain the errors. When the errors are small, they are shown in equations (8) and (9) [22]:
(8)C~bn=CbnI−ϕ×,(9)C~ne=I+ΔP×Cne,where ϕ is the platform error angle vector defined in the navigation system (ENU), ΔP=−ΔLΔλcosLΔλsinLT is the error vector from the position, and δL is the latitude error and δλ is the longitude error. The inertial errors are derived from the INS error equations.Inserting equations (8) and (9) into equation (7) and comparing the simplified result with equation (6), the LoS estimation bias ψn of the nth FoV can be obtained as follows:
(10)ψn=C~bnϕ−ΔP.According to equation (9), ψn is a rotation Euler vector, giving the rotation between the real LoS vector (without errors) and the estimated LoS vector (with errors) of the nth FoV. It can also be defined as a unique bias angle ηn for ψn. If the estimated LoS vector r~ni is obtained, there is another vector r0i that is on the cone generatrix whose cone line is the estimated LoS r~ni and cone angle is ηn. Thus, the vector r0i for the estimated LoS vector r~ni by equation (7) is given by equation (11):
(11)r0i=I+ψn×r~ni.Then, the LoS bias angleηn can be calculated by equation (12).
(12)ηn=dr0ir~ni=arccosr0i⋅r~nir0i⋅r~ni.The relationship between the estimated and real LoS is shown in Figure5. The point O is the optic center, and OC¯ is the real LoS vector. The arc ACB⌢ is the real visible range of the star sensor. The errors from the inertial information are small, so the LoS bias angle is also small. Therefore, OC1¯ is the estimated LoS, and∠C1OC is the LoS bias angle. The visible range becomes the arc A1C1B1⌢. The inertial errors concern the error radius A2A1¯ so that the catalog radius A2D¯ contains the FoV radius A1D¯ and the error radius. In this way, the range of the catalog is the arc A2C1B2⌢.Figure 5
Inertial error analysis for the FoV.In fact, the celestial sphere area observed by the star sensor is very small and can be regarded as a plane. If the LoS bias angle is small, the celestial sphere area with error is still approximately a plane. Therefore, the LoS bias angle can only affect the range of the catalog.If the LoS bias angle is written asηn, then the error radius can be given by equation (13):
(13)ΔRn=sinηn.In other words, from equations (3)–(13), each real LoS vector r~ni and error radius ΔRn can be calculated.With the assistance of LoS vector estimation and error radius certainty, the stars in the part catalog are approximately equal to those in the star image. For Section4.2, the error radius for the nth FoV is written as ΔRn, and the catalog radius R~n is given by equation (14):
(14)R~n=Rn+ΔRn.The part catalog is established with the estimated LoS vector and the catalog radius. The method for generating the part catalog has two parts: one part involves star selection, and the other part involves the pattern structure.One star is used to illustrate the star selection method. If the estimated LoS vector is written asr~ni and the catalog radius is R~n for the nth FoV, the condition for star q in the nth FoV is shown in equation (15):
(15)dr~niq≤R~,where dr~niq indicates the argument between the star q and the estimated LoS vector r~ni. Equation (15) is used to select stars in an all-sky catalog and choose the stars in each FoV.


## 3.4. Star Pattern Matching Based on Predicted Local Star Catalog

The pattern of the catalog depends on the star identification algorithm. A triangle matching algorithm is chosen, and the pattern is based on the angular distance between the stars. The catalog structure of packet storage for a single FoV in [18] is employed, and the section below analyzes the extended application of distributed FoVs.However, unlike traditional star sensor with single-FoV, the angular distances between the stars show different characters for star sensor with distributed multiple FoVs.In Figure6, there is not only angular distance between the stars within the same FoV but also angular distance between the stars across different FoVs. Thus, two situations are taken in consideration according to the angular distances between stars during the star pattern matching:
(1)
Angular distance within a single FoV: the star sets are obtained by the star selection. Thenth set is written as starn and the argument set in each starn is dpqn. For a star sensor with multiple distributed small FoVs, the FoV size is smaller than the installation angle for each optical head, and generally, each FoV has the same size. Therefore, the angular distance of each FoV together form the whole argument set dpqn (1≤n≤NB)(2)
Angular distance across a pair of FoVs: each star setstarn is selected, and then the set with a number larger than n is selected. The angular distance across each two FoVs is calculated. The calculation process is as follows: If a star set starn with number n is chosen, then the following sets with a number from n+1 to NB are chosen to calculate the angular distance. If there is a following set starmn+1≤m≤NB, then the argument is written as dpnqm. Accordingly, the argument set is written as dpnqmnm. According to the calculation, the argument sets dpnqmnm for each two FoVs and the whole set dpnqmnm can be obtainedFigure 6
Two situations of the angular distances for distributed multiple FoVs.The two argument sets have different values. One needs to choose the appropriate argument value as the grouping interval as in reference [18]. The two sets are divided into intervals, and the argument patterns are all based on packet storage. These patterns make up the part catalog. The whole catalog is shown in Figure 7.Figure 7
Dynamic establishment of the local star catalog.With the combination of the inertial-aided star identification scheme (Figure1) and the part catalog, the inertial-aided star pattern matching process for distributed multiple FoVs is as follows:Step 1. use the inertial information to estimate the LoS vector r~ni of each FoVStep 2. calculate the error radius ΔRn from the inertial errors and further obtain the catalog radius R~n of each FoVStep 3. utilize the LoS vector r~ni from Step 1 and two radii (ΔRn and R~n) from Step 2 to select stars for each FoV from the all-sky catalog (external data) and then generate the part catalog in real timeStep 4. use the fused image (external data) and the part catalog from Step 3 to match the star pattern


## 4. Simulation Results and Analysis

### 4.1. Analysis of Star Observability Performance

In this section, we choose the double-FoV star sensor for a simulation experiment. The main parameters are shown in Table1.Table 1
Parameters of the star sensor.
ParameterValueFocal length (mm)54.95Limited magnitude (M)6FoV size (°)4°×4°The installation angle is 60°. The all-sky catalog is based on the Smithsonian Astrophysical Observatory (SAO) with a 6.0 limited magnitude and double and variable star disposing. The simulation experiment is designed with ten thousand Monte Carlo star observations, and the numbers of visible stars in the main, auxiliary, and fused FoVs are recorded. The mean value results are shown in Figure8.Figure 8
Mean value of visible stars for different FoVs.When the number of visible stars is below 3, star pattern matching does not work due to the triangle matching algorithm chosen in this paper. Thus, the number of instances that the number of visible stars is below 3 is counted for the different limited magnitudes. In particular, this number is also recorded when the number of visible stars is below 3 in both single FoVs but above 3 in the fused FoV (called remedy times). The results are all shown in Figure9.Figure 9
Numbers of instances of a number of visible stars below 3.According to Figures8 and 9, the stars in the fused FoV contain the stars in the main and auxiliary FoVs from the corresponding relationship. When the limited magnitude is small (<5), there are a large number of results with a number of stars below 3 because the all-sky catalog has few stars for a small limited magnitude. At the same time, the remedy times are also small. As the magnitude increases, the number of visible stars in each FoV obviously increases. When the magnitude is 5, the remedy times reach a maximum, which corresponds to the number of stars above 3 in the fused FoV. When the magnitude is 6, each FoV basically satisfies the requirement for triangle matching. When the magnitude increases above 6, each FoV shows an obvious increase in the number of visible stars. However, on the contrary, the number of instances of a number of visible stars below 3 is reduced.


### 4.2. Analysis of the Observed Scope Prediction Performance

According to the inertial-aided star identification principle in Section4, assistance information helps to estimate the LoS vector, but the errors affect the estimation accuracy and the number of stars that are the same in both the star image and catalog. Obviously, the more identical stars there are, the more successful the matching results will be. In addition, the influence of inertial errors on the LoS vector estimation is independent of the multi-FoV star sensor when there are no errors in the installation. Therefore, only a single FoV is analyzed in this section.An error model of the inertial sensor is established. For a gyroscope, the errors are mainly due to the gyro drift errorε, which consists of a random constant εb, random errors of a first-order Markov process εr, and white noise wg. For an accelerometer, its drift ∇a is only due to a first-order Markov process. In addition, the error model is the same for the three axes in both the gyroscope and the accelerometer.The value of the gyroscope bias is based on the value of SIGI in [1]. To compare the different influences on accuracy, three conditions (denoted by cond. hereafter) from high to low accuracy are given for the gyro in this section and shown in Table 2.Table 2
Three gyroscope bias conditions.
Gyroscope bias conditionGyroscope bias valueCond. 10.004°/hCond. 21°/hCond. 35°/hAt the same time, the accelerometer bias is fixed at100μg. The correlation times of the Markov process for the gyroscope and accelerometer are 3600 s and 1800 s, respectively.The following variables are defined in this section to describe the degree of influence. The LoS vector bias angle isη0. If a star image has NO visible stars, having the same stars means that the stars exist in both the star image and the part catalog, whose number is NRP. Thus, the rate of identical stars COR is given by equation (16):
(16)COR=NRPNO×100%.However, the COR will be affected by the stars in the image. The COR concerns stars that do not have a strict uniform distribution in the all-sky catalog. Therefore, the number of stars affecting matchingNai is given by equation (17):
(17)Nai=NO−NRP.According to the simulation conditions, the experiment simulates an aerospace vehicle cruise. The influences of different gyroscope biases on the LoS bias angle, the COR and the number of stars affecting matching are shown in Figures10–12, respectively.Figure 10
LoS vector bias angle for different gyroscope biases.Figure 11
COR for different gyroscope biases.Figure 12
Number of stars affecting matching for different gyroscope biases.From Figure10, the LoS vector bias angle is minimum in cond. 1 (0.004°/h), and all the values are below 0.15° because the maximum gyroscope accuracy means that the assistance information does not diverge and hardly affects the accuracy of the estimated LoS. The results of cond. 2 (1°/h) are in the middle, and the maximum result of cond. 4 is below 3. The results of cond. 3 (5°/h) are the worst because the inertial positions and attitudes experience larger error accumulations, which is not small for this condition. The maximum is 16.30°, which is close to the FoV diameter (122°≈16.97°) and theoretically has a serious influence on the COR.The situation of identical stars is shown in Figures11 and 12. The results of cond. 3 (5°/h) are affected more strongly than the result of the other two conditions. The COR of cond. 3 exhibits a larger decrease in the fluctuation and is below 10% after 4300 s, which corresponds to the obvious increase in the number of stars affecting matching. The LoS bias angle has a limited influence on the COR of cond. 2, which is stable at approximately 80% at approximately 4000 s, corresponding to 5 stars affecting matching. The COR of cond. 1 (0.004°/h) is always above 80%, and the number of stars affecting matching is below 5.Consider the 4060 s epoch as an example. The LoS vector is RA of 182.909° and Dec of 20.000°. The number of visible stars is 26. At this time, the results of the 5 conditions are shown in Table3. The identical stars are shown in Figure 13.Table 3
Statistics in the 4060 s epoch for 5 conditions.
Accuracy conditionEstimated LoSLoS bias angle (°)CORPart catalog starsIdentical starsRA (°)Dec (°)0.004°/h182.88720.0700.0739100%26260.05°/h182.88719.7490.2516100%26260.1°/h182.98020.7550.757796.15%26251°/h181.39717.9882.467269.23%27185°/h189.83531.85713.384538.46%2712Figure 13
The identical stars in the 4060 s epoch.
(a)
Identical stars in cond. 4 (1°/h)(b)
Identical stars in cond. 5 (5°/h)From Table3, in this epoch, the CORs of cond. 3 (5°/h) and cond. 2 (1°/h) are poor. The number of identical stars of cond.3 (5°/h) is very limited. Compared with the other 2 conditions, the number of identical stars is obviously reduced.In addition, defining the matching success rate is the percent rate of the epochs with a correct identification in all the epochs. The success rates of the above three conditions are 98.80%, 88.58%, and 41.67%, respectively. The distribution of these success rates is similar to that of the LoS vector, bias angle, and the COR: the first condition gives the beset result, the second condition is in the middle, and the third condition gives the worst result.


### 4.3. Analysis of the Star Pattern Matching Performance

In this section, a simulation experiment is designed for the dynamic flight of an aerospace vehicle.The main parameters of the double-FoV star sensor are shown in Table1. The installation angle is 30°. A trace of the aerospace vehicle in cruise is simulated in Figure 14.Figure 14
Cruise trace of an aerospace vehicle.The number of visible stars and the main and auxiliary LoS vectors per second around the 3000 s trace are recorded, and the results are shown in Figures15–17.Figure 15
Number of visible stars in the fused FoV.Figure 16
Main LoS vector.Figure 17
Auxiliary LoS vector.From Figure15, the number of visible stars in the fused FoV exhibits a fluctuation. The mean number of stars is 5.4 (3.0 in the main FoV and 2.4 in the auxiliary FoV). During the whole simulation, there are 757 epochs with a number of visible stars below 3 in the fused FoV (1433 epochs in the main FoV and 1639 epochs in the auxiliary FoV), which shows that the fused FoV can efficiently extend the number of visible stars.The matching success rate is 80.36% in the fused FoV. After calculating the statistics, the epochs with a number of visible stars below 5 mainly affect the success rate. The results of four epochs that can match are shown in in Figure18.Figure 18
Four kinds of epoch for matching.
(a)
Identified stars only in the main FoV(b)
Identified stars only in the auxiliary FoV(c)
Identified stars across FoVs case1(d)
Identified stars across FoVs case2From Figure18, the star pattern matching algorithm for distributed FoVs can not only identify the stars only in a single FoV (in subgraphs (a) and (b)) but also provide opportunities to identify stars across FoVs (in subgraphs (c) and (d)). The algorithm takes full advantage of stars across multiple FoVs.However, there is not a star image with stars existing in both the main and auxiliary FoVs for the single-FoV star sensor. Thus, a dynamic simulation is designed that chooses an installation angle of 45° and simulates the possible 100 attitudes in one position of an aerospace vehicle. The results of successful matching are shown in Figure19.Figure 19
Number of successful matching for the whole trace.From Figure19, due to the small FoV (4°×4°) and limited number of visible stars, approximately 50% of the matches fail (50 successful matches of 100 in the main FoV and 52 successful matches in the auxiliary FoV), for the whole trace. This means that star identification only in the single small FoV cannot provide robust output. In contrast, the fused FoV has a failure rate of approximately 10% (90 successful matches in the fused FoV). This result proves the efficiency of the method.As an evaluation of real-time performance, compared to conventional approaches, the dynamic local star catalog established by the proposed method and the conventional all-sky star catalog are compared in Figure20.Figure 20
Comparison of dynamic local star catalog and all-sky star catalog.In the proposed method, the optical axis direction is estimated based on inertial measurements, and then the multiple FoVs’ observed scopes are estimated. Beneficial from the estimation of the observed scopes, the local star catalog for star pattern matching could be established dynamically, whose magnitude is significantly reduced, as shown in Figure20. Thus, the real-time performance is improved than the conventional star pattern matching method based on all-sky star catalog.In summary, the simulation above proves that the star identification method for distributed multiple FoVs can best utilize the visible stars in each FoV to increase the match success rate not only in a Monte Carlo simulation but also in a dynamic flight experiment.


## 4.1. Analysis of Star Observability Performance

In this section, we choose the double-FoV star sensor for a simulation experiment. The main parameters are shown in Table1.Table 1
Parameters of the star sensor.
ParameterValueFocal length (mm)54.95Limited magnitude (M)6FoV size (°)4°×4°The installation angle is 60°. The all-sky catalog is based on the Smithsonian Astrophysical Observatory (SAO) with a 6.0 limited magnitude and double and variable star disposing. The simulation experiment is designed with ten thousand Monte Carlo star observations, and the numbers of visible stars in the main, auxiliary, and fused FoVs are recorded. The mean value results are shown in Figure8.Figure 8
Mean value of visible stars for different FoVs.When the number of visible stars is below 3, star pattern matching does not work due to the triangle matching algorithm chosen in this paper. Thus, the number of instances that the number of visible stars is below 3 is counted for the different limited magnitudes. In particular, this number is also recorded when the number of visible stars is below 3 in both single FoVs but above 3 in the fused FoV (called remedy times). The results are all shown in Figure9.Figure 9
Numbers of instances of a number of visible stars below 3.According to Figures8 and 9, the stars in the fused FoV contain the stars in the main and auxiliary FoVs from the corresponding relationship. When the limited magnitude is small (<5), there are a large number of results with a number of stars below 3 because the all-sky catalog has few stars for a small limited magnitude. At the same time, the remedy times are also small. As the magnitude increases, the number of visible stars in each FoV obviously increases. When the magnitude is 5, the remedy times reach a maximum, which corresponds to the number of stars above 3 in the fused FoV. When the magnitude is 6, each FoV basically satisfies the requirement for triangle matching. When the magnitude increases above 6, each FoV shows an obvious increase in the number of visible stars. However, on the contrary, the number of instances of a number of visible stars below 3 is reduced.


## 4.2. Analysis of the Observed Scope Prediction Performance

According to the inertial-aided star identification principle in Section4, assistance information helps to estimate the LoS vector, but the errors affect the estimation accuracy and the number of stars that are the same in both the star image and catalog. Obviously, the more identical stars there are, the more successful the matching results will be. In addition, the influence of inertial errors on the LoS vector estimation is independent of the multi-FoV star sensor when there are no errors in the installation. Therefore, only a single FoV is analyzed in this section.An error model of the inertial sensor is established. For a gyroscope, the errors are mainly due to the gyro drift errorε, which consists of a random constant εb, random errors of a first-order Markov process εr, and white noise wg. For an accelerometer, its drift ∇a is only due to a first-order Markov process. In addition, the error model is the same for the three axes in both the gyroscope and the accelerometer.The value of the gyroscope bias is based on the value of SIGI in [1]. To compare the different influences on accuracy, three conditions (denoted by cond. hereafter) from high to low accuracy are given for the gyro in this section and shown in Table 2.Table 2
Three gyroscope bias conditions.
Gyroscope bias conditionGyroscope bias valueCond. 10.004°/hCond. 21°/hCond. 35°/hAt the same time, the accelerometer bias is fixed at100μg. The correlation times of the Markov process for the gyroscope and accelerometer are 3600 s and 1800 s, respectively.The following variables are defined in this section to describe the degree of influence. The LoS vector bias angle isη0. If a star image has NO visible stars, having the same stars means that the stars exist in both the star image and the part catalog, whose number is NRP. Thus, the rate of identical stars COR is given by equation (16):
(16)COR=NRPNO×100%.However, the COR will be affected by the stars in the image. The COR concerns stars that do not have a strict uniform distribution in the all-sky catalog. Therefore, the number of stars affecting matchingNai is given by equation (17):
(17)Nai=NO−NRP.According to the simulation conditions, the experiment simulates an aerospace vehicle cruise. The influences of different gyroscope biases on the LoS bias angle, the COR and the number of stars affecting matching are shown in Figures10–12, respectively.Figure 10
LoS vector bias angle for different gyroscope biases.Figure 11
COR for different gyroscope biases.Figure 12
Number of stars affecting matching for different gyroscope biases.From Figure10, the LoS vector bias angle is minimum in cond. 1 (0.004°/h), and all the values are below 0.15° because the maximum gyroscope accuracy means that the assistance information does not diverge and hardly affects the accuracy of the estimated LoS. The results of cond. 2 (1°/h) are in the middle, and the maximum result of cond. 4 is below 3. The results of cond. 3 (5°/h) are the worst because the inertial positions and attitudes experience larger error accumulations, which is not small for this condition. The maximum is 16.30°, which is close to the FoV diameter (122°≈16.97°) and theoretically has a serious influence on the COR.The situation of identical stars is shown in Figures11 and 12. The results of cond. 3 (5°/h) are affected more strongly than the result of the other two conditions. The COR of cond. 3 exhibits a larger decrease in the fluctuation and is below 10% after 4300 s, which corresponds to the obvious increase in the number of stars affecting matching. The LoS bias angle has a limited influence on the COR of cond. 2, which is stable at approximately 80% at approximately 4000 s, corresponding to 5 stars affecting matching. The COR of cond. 1 (0.004°/h) is always above 80%, and the number of stars affecting matching is below 5.Consider the 4060 s epoch as an example. The LoS vector is RA of 182.909° and Dec of 20.000°. The number of visible stars is 26. At this time, the results of the 5 conditions are shown in Table3. The identical stars are shown in Figure 13.Table 3
Statistics in the 4060 s epoch for 5 conditions.
Accuracy conditionEstimated LoSLoS bias angle (°)CORPart catalog starsIdentical starsRA (°)Dec (°)0.004°/h182.88720.0700.0739100%26260.05°/h182.88719.7490.2516100%26260.1°/h182.98020.7550.757796.15%26251°/h181.39717.9882.467269.23%27185°/h189.83531.85713.384538.46%2712Figure 13
The identical stars in the 4060 s epoch.
(a)
Identical stars in cond. 4 (1°/h)(b)
Identical stars in cond. 5 (5°/h)From Table3, in this epoch, the CORs of cond. 3 (5°/h) and cond. 2 (1°/h) are poor. The number of identical stars of cond.3 (5°/h) is very limited. Compared with the other 2 conditions, the number of identical stars is obviously reduced.In addition, defining the matching success rate is the percent rate of the epochs with a correct identification in all the epochs. The success rates of the above three conditions are 98.80%, 88.58%, and 41.67%, respectively. The distribution of these success rates is similar to that of the LoS vector, bias angle, and the COR: the first condition gives the beset result, the second condition is in the middle, and the third condition gives the worst result.


## 4.3. Analysis of the Star Pattern Matching Performance

In this section, a simulation experiment is designed for the dynamic flight of an aerospace vehicle.The main parameters of the double-FoV star sensor are shown in Table1. The installation angle is 30°. A trace of the aerospace vehicle in cruise is simulated in Figure 14.Figure 14
Cruise trace of an aerospace vehicle.The number of visible stars and the main and auxiliary LoS vectors per second around the 3000 s trace are recorded, and the results are shown in Figures15–17.Figure 15
Number of visible stars in the fused FoV.Figure 16
Main LoS vector.Figure 17
Auxiliary LoS vector.From Figure15, the number of visible stars in the fused FoV exhibits a fluctuation. The mean number of stars is 5.4 (3.0 in the main FoV and 2.4 in the auxiliary FoV). During the whole simulation, there are 757 epochs with a number of visible stars below 3 in the fused FoV (1433 epochs in the main FoV and 1639 epochs in the auxiliary FoV), which shows that the fused FoV can efficiently extend the number of visible stars.The matching success rate is 80.36% in the fused FoV. After calculating the statistics, the epochs with a number of visible stars below 5 mainly affect the success rate. The results of four epochs that can match are shown in in Figure18.Figure 18
Four kinds of epoch for matching.
(a)
Identified stars only in the main FoV(b)
Identified stars only in the auxiliary FoV(c)
Identified stars across FoVs case1(d)
Identified stars across FoVs case2From Figure18, the star pattern matching algorithm for distributed FoVs can not only identify the stars only in a single FoV (in subgraphs (a) and (b)) but also provide opportunities to identify stars across FoVs (in subgraphs (c) and (d)). The algorithm takes full advantage of stars across multiple FoVs.However, there is not a star image with stars existing in both the main and auxiliary FoVs for the single-FoV star sensor. Thus, a dynamic simulation is designed that chooses an installation angle of 45° and simulates the possible 100 attitudes in one position of an aerospace vehicle. The results of successful matching are shown in Figure19.Figure 19
Number of successful matching for the whole trace.From Figure19, due to the small FoV (4°×4°) and limited number of visible stars, approximately 50% of the matches fail (50 successful matches of 100 in the main FoV and 52 successful matches in the auxiliary FoV), for the whole trace. This means that star identification only in the single small FoV cannot provide robust output. In contrast, the fused FoV has a failure rate of approximately 10% (90 successful matches in the fused FoV). This result proves the efficiency of the method.As an evaluation of real-time performance, compared to conventional approaches, the dynamic local star catalog established by the proposed method and the conventional all-sky star catalog are compared in Figure20.Figure 20
Comparison of dynamic local star catalog and all-sky star catalog.In the proposed method, the optical axis direction is estimated based on inertial measurements, and then the multiple FoVs’ observed scopes are estimated. Beneficial from the estimation of the observed scopes, the local star catalog for star pattern matching could be established dynamically, whose magnitude is significantly reduced, as shown in Figure20. Thus, the real-time performance is improved than the conventional star pattern matching method based on all-sky star catalog.In summary, the simulation above proves that the star identification method for distributed multiple FoVs can best utilize the visible stars in each FoV to increase the match success rate not only in a Monte Carlo simulation but also in a dynamic flight experiment.


## 5. Conclusions

An inertial-aided star pattern matching method for distributed multiple FoVs is proposed, because the existing method for multiple FoVs cannot provide a continuous and robust star vector for an aerospace vehicle in cruise. First, the matching scheme is researched in the analysis of a distributed multi-FoV CNS. Then, the observation model for a multi-FoV star sensor is presented. Next, a specific matching method is proposed on the basis of LoS vector estimation and the dynamic establishment of a part catalog. Finally, simulation experiments are designed to prove the efficiency of the matching method and robustness for an aerospace vehicle. The results from the simulation will help realize autonomous star identification for aerospace vehicles.



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*Source: 2676210-2020-08-06.xml*