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

An aerospace vehicle in high-speed ﬂ ight involves distributed multiple ﬁ elds 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 e ﬃ ciency of star pattern matching by reducing the catalog scale, and the usability of inertial-aided star pattern matching is analyzed for di ﬀ erent gyro and accelerometer precisions.


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 longrange and long-endurance vehicle (LRLEV), where a smallvolume 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 trian-gle 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 INSassisted 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.  Figure 1.

Modeling of Novel
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   International Journal of Aerospace Engineering 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.

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 Figure 2.
Define the coordinate system about the star sensor as follows: (1) The optic head coordinate system (abbreviated as the s 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 directions In Figure 2, the distance jOAj 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 star q in the s system is ðx q , y q , z q Þ s . Through a transmission projection, the coordinate of the star q in the p system is ðx q , y q Þ p given in equation (1): where F is the focal length of the optic head.

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 Figure 3, 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. For multiple FoVs, one optic head corresponds to the s system. The main optic head coordinate system (left) is denoted as the s 1 system, and the auxiliary optic head coordinate system (right) is denoted as the s 2 system. There are two p systems: the main image plane coordinate system (p 1 system) and the auxiliary image plane coordinate system (p 2 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 q 1 and q 2 observed by the main and auxiliary optic heads, respectively. Their coordinates in the s 1 and s 2 systems are ðx q 1 , y q 1 , z q 1 Þ s 1 and ðx q 2 , y q 2 , z q 2 Þ s2 , respectively. Through a transmission projection, their coordinates in the imaging planes are  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 Figure 4, 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 s 2 system are transformed into the s 1 system, and third, they are projected into the extended plane of the main image plane.
The star image fusion process is shown in Figure 4. 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 ðx q 2 , y q 2 Þ p 2 by equation (2), which can be rebuilt into the s 2 system by the focal length and can be regarded as ðx q 2 , y q 2 , F 2 Þ s2 . 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 as C b s 1 and C b s 2 , the matrix C s 1 s 2 , which describes the transformation of the two s systems, is given by equation (3): The coordinates of the star point in the s 1 system after the transformation can be obtained by equation (4): Through the projection, the star point q 2 in the p 1 system is shown in equation (5): 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. Main bore Auxiliary bore Star vector (parallel light)

Auxiliary optic head
Main optic head  International Journal of Aerospace Engineering

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 allsky 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 Figure 4. Figure 4 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.

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 N B distributed FoVs sorted from 1 to N B . Obviously, each installation matrix C b s n is certain. In theory, the nth LoS vector r i n (1 ≤ n ≤ N B ) is estimated by equation (6): Actually, the real LoS vector will be affected by the errors in the inertial information as shown in equation (7):  (8) and (9) [22]: where ϕ is the platform error angle vector defined in the navigation system (ENU), ΔP = ½−ΔL Δλ cos L Δλ sin L T 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 ψ n of the nth FoV can be obtained as follows: According to equation (9), ψ n 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 η n for ψ n . If the estimated LoS vectorr i n is obtained, there is another vector r i 0 that is on the cone generatrix whose cone line is the estimated LoSr i n and cone angle is η n . Thus, the vector r i 0 for the estimated LoS vectorr i n by equation (7) is given by equation (11): Then, the LoS bias angle η n can be calculated by equation (12).   Figure 4: Inertial-aided star pattern matching scheme for distributed FoVs.

International Journal of Aerospace Engineering
The relationship between the estimated and real LoS is shown in Figure 5. 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, OC 1 is the estimated LoS, and∠C 1 OC is the LoS bias angle. The visible range becomes the arc The inertial errors concern the error radius A 2 A 1 so that the catalog radius A 2 D contains the FoV radius A 1 D and the error radius. In this way, the range of the catalog is the arc 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 as η n , then the error radius can be given by equation (13): In other words, from equations (3)-(13), each real LoS vectorr i n and error radius ΔR n 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 Section 4.2, the error radius for the nth FoV is written as ΔR n , and the catalog radiusR n is given by equation (14): 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 i n and the catalog radius isR n for the nth FoV, the condition for star q in the nth FoV is shown in equation (15): where dri n q indicates the argument between the star q and the estimated LoS vectorr i n . Equation (15) is used to select stars in an all-sky catalog and choose the stars in each FoV.

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 Figure 6, 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. The nth set is written as fstarg n and the argument set in each fstarg n is fd pq g n . 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 ffd pq g n g (1 ≤ n ≤ N B ) (2) Angular distance across a pair of FoVs: each star set fstarg n 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 fstarg n with number n is chosen, then the following sets with a number from n + 1 to N B are chosen to calculate the angular distance. If there is a following set fstarg m ðn + 1 ≤ m ≤ N B Þ, then the argument is written as d ðpÞ n ðqÞ m . Accordingly, the argument set is written as fd ðpÞ n ðqÞ m g nm . According to the calculation, the argument sets fd ðpÞ n ðqÞ m g nm for each two FoVs and the whole set ffd ðpÞ n ðqÞ m g nm g can be obtained 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.
With the combination of the inertial-aided star identification scheme (Figure 1) and the part catalog, the inertial-

Real LoS
Optic center

LoS bias angle
Es tim at ed Lo S Er ro r rad ius Fo V ra di us Figure 5: Inertial error analysis for the FoV. 6 International Journal of Aerospace Engineering aided star pattern matching process for distributed multiple FoVs is as follows: Step 1. use the inertial information to estimate the LoS vectorr i n of each FoV Step 2. calculate the error radius ΔR n from the inertial errors and further obtain the catalog radiusR n of each FoV Step 3. utilize the LoS vectorr i n from Step 1 and two radii (ΔR n andR 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 time Step 4. use the fused image (external data) and the part catalog from Step 3 to match the star pattern

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 Table 1.
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 thou-sand 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 Figure 8.
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 Figure 9.
According to Figures 8 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   International Journal of Aerospace Engineering 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.

Analysis of the Observed Scope Prediction Performance.
According to the inertial-aided star identification principle in Section 4, 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 addi-tion, 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 error ε, which consists of a random constant ε b , random errors of a first-order Markov process ε r , and white noise w g . 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.
At the same time, the accelerometer bias is fixed at 100μg. 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 is η 0 . If a star image has N O visible stars, having the same stars means that the stars exist in both the star image and the part catalog, whose number is N RP . Thus, the rate of identical stars COR is given by equation (16): 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 matching N ai is given by equation (17): 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 Figures 10-12, respectively.
From Figure 10   16:97°) and theoretically has a serious influence on the COR.
The situation of identical stars is shown in Figures 11 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 Table 3. The identical stars are shown in Figure 13.
From Table 3 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.

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 Table 1. The installation angle is 30°. A trace of the aerospace vehicle in cruise is simulated in Figure 14.
The 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 Figure 18.
From Figure 18, 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.  Figure 19.     10 International Journal of Aerospace Engineering 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 Figure 20.
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 Figure 20. 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.

Conclusions
An inertial-aided star pattern matching method for distributed multiple FoVs is proposed, because the existing method