# A Fast Algorithm for Determining the Optimal Navigation Star for Responsive Launch Vehicles

Authors: Yi Zhao; Hongbo Zhang; Pengfei Li; Guojian Tang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The platform inertial-stellar composite guidance is a composite guidance method supplemented by stellar correction on the basis of inertial navigation, which can effectively improve the accuracy of responsive launch vehicles. In order to solve the problem of rapid determining the optimal navigation star in the system, this paper proposes an algorithm based on the equivalent information compression theory. At first, this paper explains why the single-star scheme can achieve the same accuracy as the dual-star scheme. At the same time, the analytical expression of the optimal navigation star with significant initial error is derived. In addition, the available optimal navigation star determination strategy is also designed according to the arrow-borne navigation star database. The proposed algorithm is evaluated by two representative responsive launch vehicle trajectory simulations. The simulation results demonstrate that the proposed algorithm can determine the optimal navigation star quickly, which greatly shorten the preparation time before the rapid launch of vehicles and improve the composite guidance accuracy.


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## Body

## 1. Introduction

Inertial-stellar composite guidance is a composite guidance method based on inertial guidance supplemented by stellar guidance. It utilizes the inertial space azimuth datum provided by the star to calibrate the error angle between the platform coordinate system and the launch inertial coordinate system and corrects the impact point deviation caused by the platform pointing error [1]. Inertial-stellar composite guidance system corrects the drift error of inertial platform according to the star sensor information, which can not only improve the guidance accuracy and rapid launch ability [2] but also reduce the cost. Moreover, the motion parameters of spacecraft in space can be determined [3–5], and it has strong environmental adaptability.Inertial-stellar guidance is essentially a problem of determining attitude through vector observation. This problem was first proposed by Wahba [6], and various attitude determination algorithms were developed, such as TRIAD [7], QUEST [8, 9], SVD [10], FOAM [11], Euler-q [12], and fast linear attitude estimator method [13–15]. In order to solve the case that there are a large number of outliers, Yang and Carlone formulated the Wahba problem by truncated least squares [16]. Ghadiri et al. [17] proposed a robust multi-objective optimization method to overcome the static attitude determination with bounded uncertainty. These algorithms need at least two vector information to calculate the attitude. However, in some cases, long-term observation of one vector is enough [18]. Reference [19] proposed an attitude determination algorithm based on the minimum squares sum of image point coordinate residuals. The algorithm can still determine the attitude when only one star is observed. Reference [20] derived the attitude analytical solution when only one sensor is used for observation. The analytical solution can be expressed by the combination of two limiting quaternions, and the covariance and singularity analyses were carried out. However, it did not determine the optimal attitude solution. Similarly, according to the number of observation vectors, the inertial-stellar composite guidance can also be divided into single vector observation and double vectors observation, that is, single-star scheme and double-star scheme. For the platform inertial navigation system, the star sensor is usually fixedly installed on the platform. Because the direction of the platform in the inertial space cannot be adjusted after launch, the double-star scheme needs to install two star sensors on the platform, which will greatly complicate the structure. It is found that observing the specific direction star, the single-star scheme can achieve the same accuracy as the double-star scheme [21, 22]. Zhang et al. have proved it theoretically [23]. As it is known, the only practical application is the single-star scheme, such as the American “Trident” submarine long-range ballistic missile. However, the single-star scheme needs to determine the optimal navigation star before the vehicle launch. At present, the optimal navigation star is determined by numerical method [24, 25], which increases the preparation time and limits the wide application.Motivated by the work of Zhang et al., this paper proposes a fast algorithm to determine the optimal navigation star for responsive launch vehicles. Firstly, the relationship equations between the initial error and the impact point deviation and the star sensor measurement are established. Then, our algorithm exploits the equivalent information compression theory [23] to explain why the single-star scheme can achieve the accuracy as the double-star scheme and deduces the optimal navigation star under the condition of significant initial error. The deduced analytical solution can greatly shorten the prelaunch preparation time. On this basis, the local navigation star database is determined according to the deviation angle, and the available optimal navigation star can be determined.The structure of this paper is as follows. Section2 presents the definitions of various coordinate system and the derivations of inertial platform system and star sensor model. Section 3 shows the analytical expression of the optimal navigation star. In Section 4, the available optimal navigation star is determined based on the arrow-borne navigation star database. The simulation results and conclusions are given in Section 5 and Section 6. The contribution of this paper is to provide an analytical solution of optimal navigation star to shorten the prelaunch preparation time and enhance the performance for responsive launch vehicles.


## 2. Inertial Platform System and Star Sensor Modeling

### 2.1. Definitions of Various Coordinate System

#### 2.1.1. Geocentric Inertial Coordinate SystemoE−xIyIzI

The coordinate system originoE is the earth centroid, and the basic plane is the J2000 earth equatorial plane. The oExI axis points from the earth centroid to the J2000 mean equinox in the basic plane. The oEzI axis points to the north pole along the normal of the basic plane. The oEyI axis and the other two axes constitute the right hand system. This coordinate system is abbreviated as the i-system.


#### 2.1.2. Launch Coordinate Systemo−xyz

The system mainly describes the motion of responsive launch vehicle relative to the earth. The launch coordinate system is fixedly connected with the earth, and the origin is taken as the launch pointo. In the system, the ox axis points to the launch aiming direction in the launch horizontal plane, the oy axis points upward perpendicular to the launch point horizontal plane, and the oz axis is perpendicular to the xoy plane. The axes ox, oy, and oz form the right hand coordinate system. This coordinate system is abbreviated as the g-system (Figure 1).Figure 1
Launch coordinate system.


#### 2.1.3. Launch Inertial Coordinate SystemoA−xAyAzA

The launch inertial coordinate system coincides with the launch coordinate system at the launch time. But after launching the vehicle, the origin and the direction of each axis remain stationary in the inertial space. The coordinate system is used to establish the vehicle motion equation in inertial space. This coordinate system is abbreviated as theA-system.


#### 2.1.4. Ideal Inertial Platform Coordinate SystemoP′−xP′yP′zP′

The coordinate system originop′ is located at the platform datum, and the coordinate axis is defined by the platform frame axis or the gyro-sensitive axis. After pre-launch alignment and leveling, each coordinate axis shall be parallel to each coordinate axis of the launch inertial coordinate system. This coordinate system is abbreviated as the p′-system.


#### 2.1.5. Inertial Platform Coordinate SystemoP−xPyPzP

Due to the platform misalignment angle, there is a deviation between the inertial platform coordinate system and the ideal inertial platform coordinate system. This coordinate system is abbreviated as thep-system.


#### 2.1.6. Star Sensor Coordinate Systemos−xsyszs

The coordinate system mainly describes the star sensor measurement. In the system, the coordinate system originos is at the centre of the star sensor imaging device (charge couple device, complementary metal oxide semiconductor, etc.). The osxs axis is consistent with the axis of the optical lens, the osys axis is the vertical to the pixel readout direction, and the oszs axis is the horizontal to the pixel readout direction. The ysoszs plane is consistent with the imaging device plane. The transformation matrix between the star sensor coordinate system and the vehicle body coordinate system is determined by the star sensor installation angle. This coordinate system is abbreviated as the s-system.


### 2.2. Relationship between Impact Point Deviation and Platform Misalignment Angle

The platform misalignment angle represents the inertial reference deviation, that is, the error angle between the inertial platform and the launch inertial coordinate system. It is mainly caused by various initial errors and inertial navigation errors and affects the landing point accuracy. Although the platform misalignment angle is affected by many factors, the initial error accounts for the main part under certain conditions. For the rapid maneuvering launch vehicle, the accuracy of pre-launch orientation and alignment may not be very high, which leads to the significant portion of the initial error in the platform misalignment angle. Therefore, this paper mainly studies the determination of the optimal navigation star under the significant initial error condition.The platform inertial system and the launch inertial system can be coincident with the help of the platform initial alignment. The initial alignment error will be caused due to the equipment inherent error, the external interference influence in the alignment process, and the method error. And the platform alignment error around they-axis will be caused owing to the initial orientation error during launch. Thus, the orientation error can be considered together with the initial alignment error.The initial alignment and orientation errors can be expressed by the three axis misalignment angles between thep′-system and the A-system, which are defined asε0xε0yε0zT. And there are two parts in ε0y: orientation error and aiming error. It is assumed that the adjustment platform adopts the method of yaw first and then pitch; there is
(1)αxαyαz=cosφrcosψrsinφr−cosφrsinψr−sinφrcosψrcosφrsinφrsinψrsinψr0cosψrε0xε0yε0z=CAP′ε0xε0yε0z,whereαxαyαzT are misalignment angles caused by initial alignment and orientation errors, ψr and φr are the rotation angles around the y-axis and z-axis, respectively, and CAP′ is the transformation matrix from the A-system to the p′- system.The inertial guidance accuracy meets the following relationship with the initial alignment error:(2)ΔLΔH=nL1nL2nL3nH1nH2nH3ε0xε0yε0z.In which,nL1, nL2, and nL3 are the partial derivatives of the longitudinal impact point deviation to the initial errors in three directions, respectively. nH1, nH2, and nH3 are the partial derivatives of the lateral impact point deviation to the initial errors in three directions, respectively.It can be obtained by combining Equation (1) and Equation (2).
(3)ΔLΔH=q11q12q13q21q22q23αxαyαz=q1Tq2Tαxαyαz.In which,(4)q11=nL1cosφrcosψr+nL2sinφr−nL3cosφrsinψr,q12=−nL1sinφrcosψr+nL2cosφr+nL3sinφrsinψr,q13=nL1sinψr+nL3cosψr,q21=nH1cosφrcosψr+nH2sinφr−nH3cosφrsinψr,q22=−nH1sinφrcosψr+nH2cosφr+nH3sinφrsinψr,q23=nH1sinψr+nH3cosψr.


### 2.3. Acquisition of Star Sensor Measurement

The elevation and azimuth angle of the optimal navigation star in theA-system are defined as es and σs, respectively. Thus, the stellar direction unit vector in the A-system can be expressed as
(5)SA=cosescosσssinescosessinσsT.In thes-system, the osxs axis is the optical axis. The angle between the optical axis and the stellar vector is very small, and its directional cosine is approximately 1. The osys and oszs are the output axes. The stellar vector representation in the star sensor coordinate system is shown in Figure 2. It is assumed that the star sensor outputs are ξ and η; the stellar vector can be expressed as
(6)SS=1−ξ−ηT.Figure 2
Representation of the stellar vector in the star sensor coordinate system.The ideal star sensor output should beSs′=100T; then, there is the following equation according to the coordinate transformation relationship:
(7)SS′=CPSCAP′SA,whereCPS is the transformation matrix from the p-system to the s-system and CAP′ is the transformation matrix from the A-system to the p′-system. According to the stellar vector representation in the A-system and the s-system, the following equation can be obtained by the transformation matrix between different coordinate systems.
(8)SS=CPSCP′PCAP′SA,whereCP′P is the transformation matrix from the p′-system to the p-system, which can be expressed as
(9)CP′P=1−αzαyαz1−αx−αyαx1.The stellar vector representation in thep-system is defined as SP, and there is
(10)SP=CSPSS.According to Equation (7), the stellar vector representation in the p′-system can be obtained as follows:
(11)SP′=CSPSS′=CAP′SA.Then, the following equation can be obtained from Equations (10) and (11).
(12)ΔSP=SP−SP′=CSPSS−SS′.AndΔSp can also be represented as
(13)ΔSP=SP′⋅a=CSPSS′⋅a.It can be obtained from Equations (12) and (13).
(14)Ss−Ss′=CPSCSPSS′⋅a.Therefore, the star sensor measurement equation can be expressed as(15)ξη=−sinψ00cosψ0sinφ0cosψ0−cosφ0sinφ0sinψ0αxαyαz=h1Th2Tαxαyαz,whereφ0 and ψ0 are the star sensor installation angles.


## 2.1. Definitions of Various Coordinate System

### 2.1.1. Geocentric Inertial Coordinate SystemoE−xIyIzI

The coordinate system originoE is the earth centroid, and the basic plane is the J2000 earth equatorial plane. The oExI axis points from the earth centroid to the J2000 mean equinox in the basic plane. The oEzI axis points to the north pole along the normal of the basic plane. The oEyI axis and the other two axes constitute the right hand system. This coordinate system is abbreviated as the i-system.


### 2.1.2. Launch Coordinate Systemo−xyz

The system mainly describes the motion of responsive launch vehicle relative to the earth. The launch coordinate system is fixedly connected with the earth, and the origin is taken as the launch pointo. In the system, the ox axis points to the launch aiming direction in the launch horizontal plane, the oy axis points upward perpendicular to the launch point horizontal plane, and the oz axis is perpendicular to the xoy plane. The axes ox, oy, and oz form the right hand coordinate system. This coordinate system is abbreviated as the g-system (Figure 1).Figure 1
Launch coordinate system.


### 2.1.3. Launch Inertial Coordinate SystemoA−xAyAzA

The launch inertial coordinate system coincides with the launch coordinate system at the launch time. But after launching the vehicle, the origin and the direction of each axis remain stationary in the inertial space. The coordinate system is used to establish the vehicle motion equation in inertial space. This coordinate system is abbreviated as theA-system.


### 2.1.4. Ideal Inertial Platform Coordinate SystemoP′−xP′yP′zP′

The coordinate system originop′ is located at the platform datum, and the coordinate axis is defined by the platform frame axis or the gyro-sensitive axis. After pre-launch alignment and leveling, each coordinate axis shall be parallel to each coordinate axis of the launch inertial coordinate system. This coordinate system is abbreviated as the p′-system.


### 2.1.5. Inertial Platform Coordinate SystemoP−xPyPzP

Due to the platform misalignment angle, there is a deviation between the inertial platform coordinate system and the ideal inertial platform coordinate system. This coordinate system is abbreviated as thep-system.


### 2.1.6. Star Sensor Coordinate Systemos−xsyszs

The coordinate system mainly describes the star sensor measurement. In the system, the coordinate system originos is at the centre of the star sensor imaging device (charge couple device, complementary metal oxide semiconductor, etc.). The osxs axis is consistent with the axis of the optical lens, the osys axis is the vertical to the pixel readout direction, and the oszs axis is the horizontal to the pixel readout direction. The ysoszs plane is consistent with the imaging device plane. The transformation matrix between the star sensor coordinate system and the vehicle body coordinate system is determined by the star sensor installation angle. This coordinate system is abbreviated as the s-system.


## 2.1.1. Geocentric Inertial Coordinate SystemoE−xIyIzI

The coordinate system originoE is the earth centroid, and the basic plane is the J2000 earth equatorial plane. The oExI axis points from the earth centroid to the J2000 mean equinox in the basic plane. The oEzI axis points to the north pole along the normal of the basic plane. The oEyI axis and the other two axes constitute the right hand system. This coordinate system is abbreviated as the i-system.


## 2.1.2. Launch Coordinate Systemo−xyz

The system mainly describes the motion of responsive launch vehicle relative to the earth. The launch coordinate system is fixedly connected with the earth, and the origin is taken as the launch pointo. In the system, the ox axis points to the launch aiming direction in the launch horizontal plane, the oy axis points upward perpendicular to the launch point horizontal plane, and the oz axis is perpendicular to the xoy plane. The axes ox, oy, and oz form the right hand coordinate system. This coordinate system is abbreviated as the g-system (Figure 1).Figure 1
Launch coordinate system.


## 2.1.3. Launch Inertial Coordinate SystemoA−xAyAzA

The launch inertial coordinate system coincides with the launch coordinate system at the launch time. But after launching the vehicle, the origin and the direction of each axis remain stationary in the inertial space. The coordinate system is used to establish the vehicle motion equation in inertial space. This coordinate system is abbreviated as theA-system.


## 2.1.4. Ideal Inertial Platform Coordinate SystemoP′−xP′yP′zP′

The coordinate system originop′ is located at the platform datum, and the coordinate axis is defined by the platform frame axis or the gyro-sensitive axis. After pre-launch alignment and leveling, each coordinate axis shall be parallel to each coordinate axis of the launch inertial coordinate system. This coordinate system is abbreviated as the p′-system.


## 2.1.5. Inertial Platform Coordinate SystemoP−xPyPzP

Due to the platform misalignment angle, there is a deviation between the inertial platform coordinate system and the ideal inertial platform coordinate system. This coordinate system is abbreviated as thep-system.


## 2.1.6. Star Sensor Coordinate Systemos−xsyszs

The coordinate system mainly describes the star sensor measurement. In the system, the coordinate system originos is at the centre of the star sensor imaging device (charge couple device, complementary metal oxide semiconductor, etc.). The osxs axis is consistent with the axis of the optical lens, the osys axis is the vertical to the pixel readout direction, and the oszs axis is the horizontal to the pixel readout direction. The ysoszs plane is consistent with the imaging device plane. The transformation matrix between the star sensor coordinate system and the vehicle body coordinate system is determined by the star sensor installation angle. This coordinate system is abbreviated as the s-system.


## 2.2. Relationship between Impact Point Deviation and Platform Misalignment Angle

The platform misalignment angle represents the inertial reference deviation, that is, the error angle between the inertial platform and the launch inertial coordinate system. It is mainly caused by various initial errors and inertial navigation errors and affects the landing point accuracy. Although the platform misalignment angle is affected by many factors, the initial error accounts for the main part under certain conditions. For the rapid maneuvering launch vehicle, the accuracy of pre-launch orientation and alignment may not be very high, which leads to the significant portion of the initial error in the platform misalignment angle. Therefore, this paper mainly studies the determination of the optimal navigation star under the significant initial error condition.The platform inertial system and the launch inertial system can be coincident with the help of the platform initial alignment. The initial alignment error will be caused due to the equipment inherent error, the external interference influence in the alignment process, and the method error. And the platform alignment error around they-axis will be caused owing to the initial orientation error during launch. Thus, the orientation error can be considered together with the initial alignment error.The initial alignment and orientation errors can be expressed by the three axis misalignment angles between thep′-system and the A-system, which are defined asε0xε0yε0zT. And there are two parts in ε0y: orientation error and aiming error. It is assumed that the adjustment platform adopts the method of yaw first and then pitch; there is
(1)αxαyαz=cosφrcosψrsinφr−cosφrsinψr−sinφrcosψrcosφrsinφrsinψrsinψr0cosψrε0xε0yε0z=CAP′ε0xε0yε0z,whereαxαyαzT are misalignment angles caused by initial alignment and orientation errors, ψr and φr are the rotation angles around the y-axis and z-axis, respectively, and CAP′ is the transformation matrix from the A-system to the p′- system.The inertial guidance accuracy meets the following relationship with the initial alignment error:(2)ΔLΔH=nL1nL2nL3nH1nH2nH3ε0xε0yε0z.In which,nL1, nL2, and nL3 are the partial derivatives of the longitudinal impact point deviation to the initial errors in three directions, respectively. nH1, nH2, and nH3 are the partial derivatives of the lateral impact point deviation to the initial errors in three directions, respectively.It can be obtained by combining Equation (1) and Equation (2).
(3)ΔLΔH=q11q12q13q21q22q23αxαyαz=q1Tq2Tαxαyαz.In which,(4)q11=nL1cosφrcosψr+nL2sinφr−nL3cosφrsinψr,q12=−nL1sinφrcosψr+nL2cosφr+nL3sinφrsinψr,q13=nL1sinψr+nL3cosψr,q21=nH1cosφrcosψr+nH2sinφr−nH3cosφrsinψr,q22=−nH1sinφrcosψr+nH2cosφr+nH3sinφrsinψr,q23=nH1sinψr+nH3cosψr.


## 2.3. Acquisition of Star Sensor Measurement

The elevation and azimuth angle of the optimal navigation star in theA-system are defined as es and σs, respectively. Thus, the stellar direction unit vector in the A-system can be expressed as
(5)SA=cosescosσssinescosessinσsT.In thes-system, the osxs axis is the optical axis. The angle between the optical axis and the stellar vector is very small, and its directional cosine is approximately 1. The osys and oszs are the output axes. The stellar vector representation in the star sensor coordinate system is shown in Figure 2. It is assumed that the star sensor outputs are ξ and η; the stellar vector can be expressed as
(6)SS=1−ξ−ηT.Figure 2
Representation of the stellar vector in the star sensor coordinate system.The ideal star sensor output should beSs′=100T; then, there is the following equation according to the coordinate transformation relationship:
(7)SS′=CPSCAP′SA,whereCPS is the transformation matrix from the p-system to the s-system and CAP′ is the transformation matrix from the A-system to the p′-system. According to the stellar vector representation in the A-system and the s-system, the following equation can be obtained by the transformation matrix between different coordinate systems.
(8)SS=CPSCP′PCAP′SA,whereCP′P is the transformation matrix from the p′-system to the p-system, which can be expressed as
(9)CP′P=1−αzαyαz1−αx−αyαx1.The stellar vector representation in thep-system is defined as SP, and there is
(10)SP=CSPSS.According to Equation (7), the stellar vector representation in the p′-system can be obtained as follows:
(11)SP′=CSPSS′=CAP′SA.Then, the following equation can be obtained from Equations (10) and (11).
(12)ΔSP=SP−SP′=CSPSS−SS′.AndΔSp can also be represented as
(13)ΔSP=SP′⋅a=CSPSS′⋅a.It can be obtained from Equations (12) and (13).
(14)Ss−Ss′=CPSCSPSS′⋅a.Therefore, the star sensor measurement equation can be expressed as(15)ξη=−sinψ00cosψ0sinφ0cosψ0−cosφ0sinφ0sinψ0αxαyαz=h1Th2Tαxαyαz,whereφ0 and ψ0 are the star sensor installation angles.


## 3. Theoretical Optimal Navigation Star Determination Method

For the platform inertial-stellar composite guidance scheme, the single-star scheme for measuring a special navigation star can achieve the same accuracy as the double-star scheme for measuring two stars. This special star is called the optimal navigation star. In terms of the difficulty and cost of realization, the single-star scheme is definitely better than the double-star scheme. Therefore, the single-star scheme is adopted in the practical engineering, which requires the determination of the optimal navigation star.In this section, the equivalent information compression theory is utilized to explain why the single-star scheme can achieve the same accuracy as the double-star scheme firstly. Then, the optimal navigation star is further determined based on the principle. Since it is not combined with the navigation star in the star library, it is also called the theoretical optimal navigation star.


### 3.1. Equivalent Information Compression Theory

The impact point deviation and platform misalignment angle can be expressed in the matrix form(16)p=q⋅a,wherep=ΔLΔHT; q=q1q2T; and a=αxαyαzT. It can be seen from Equation (16) that the rank of q is 2, so there is information compression in the mapping from a to p. It is worth noting that a cannot be uniquely determined by p, which indicates that Equation (16) has numerous solutions. Although there are countless sets of solutions in Equation (16), there is a special solution a0, which belongs to the subspace qs=spanq1q2 formed by each row of vectors. Therefore, a0 can be expressed as
(17)a0=α10q1+α20q2=qT⋅X.Substitute Equation (17) into Equation (16), and there is
(18)p=qqTX.From Equation (17) and Equation (18), we can get
(19)a0=qTqqT−1p.It can be seen from the above equation thata0 and p correspond to each other one by one. If the inner product of two column vectors is defined as a⋅b=aT⋅b, then Equation (16) can be expressed as
(20)p=q1⋅aq2⋅aT,whereqi⋅a reflects the projection of a in the qi direction. q1 and q2 are linearly independent; therefore, p=q⋅a reflects the projection as of a on space qs, and the projection information as⊥ of a on the orthogonal complement qs⊥ of qs is lost. Since qs is a complete subspace on Hilbert space Rn, there is
(21)Rn=qs+qs⊥.According to the projection theorem, we can get(22)a=as+as⊥.It can be seen from the relationship between the impact point deviation and the platform misalignment angle thatq is not full rank. So only the information as in the subspace can be obtained through the impact point deviation, and the information as⊥ in the orthogonal complement cannot be obtained.The star sensor measurement equation can also be expressed in the matrix form(23)Z=h⋅a.Assuming that another set of bases ofqs is h1h2, it can be seen from the above analysis that Z also reflects all the information projected by a on qs, which can be expressed as
(24)a0=hThhT−1Z.Substitute Equation (24) into Equation (16), and we can get
(25)p=q⋅hThhT−1Z.Therefore, from the perspective of information compression,q and h are equal compression maps; that is, the impact point deviation p can be uniquely determined by the single-star observation Z.The impact point deviation is only affected by the projectionas of the misalignment angle a on the subspace Qs=spanq1q2. Therefore, according to Equation (25), it is only necessary to select h1 and h2, so that h and q are equal information compression. Then, the observation information Z contains all the useful information. The schematic diagram of determining the optimal navigation star is shown in Figure 3.Figure 3
Schematic diagram of determining the optimal navigation star.In the figure, all the information of misalignment anglea is composed of as and as⊥, but only as affects the impact point deviation. If h and q are equal information compression maps, the single-star scheme can measure all the information of as. Although the double-star scheme can measure all the information of a, only the as part is used in the correction, and as⊥ belongs to the useless information, so it has the same accuracy as the single-star scheme. In more popular terms, there are only two indicators ΔL and ΔH describing the impact point deviation, which are the reflection of part of the misalignment angle a. When observing a single star, two measurements ξ and η can be obtained, which are also the reflection of part of the misalignment angle a. By selecting the optimal navigation star, ξ and η can include all the information of misalignment angle a contained in ΔL and ΔH. Therefore, the single-star scheme can achieve the same accuracy as the dual-star scheme.


### 3.2. Determining the Optimal Navigation Star

According to the equivalent information compression theory, the optimal navigation star should satisfy(26)h1×h2=q1×q2q1×q2orh1×h2=−q1×q2q1×q2.For the left side of the above equation, it can be obtained according to the Equation (15).
(27)h1×h2=cosφ0cosψ0sinφ0cosφ0sinψ0T.It is assumed that the star sensor is installed on thexoy plane of the platform, and the platform is adjusted to aim at the navigation star by first yaw and then pitch. By substitutingψ0=0∘ into Equation (27), we can get
(28)h1×h2=cosφ0sinφ00T.Define(29)Qc=Qc1Qc2Qc3=q1×q2,where,(30)Qc1=nH1nL3−nH3nL1sinφr+nH3nL2−nH2nL3cosφrcosψr+nH1nL2−nH2nL1cosφrsinψr,Qc2=nH1nL3−nH3nL1cosφr−nH3nL2−nH2nL3sinφrcosψr−nH1nL2−nH2nL1sinφrsinψr,Qc3=−nH1nL2−nH2nL1cosψr+nH3nL2−nH2nL3sinψr.According to Equations (26), (27), and (28), we can get
(31)ψr=tan−1nH1nL2−nH2nL1nH3nL2−nH2nL3,φr=−tan−1nH1nL3−nH3nL1nH2nL3−nH3nL2cosψr+nH2nL1−nH1nL2sinψr−φ0.The optimal navigation star and the rotation angle satisfy the following relationship:(32)σs=−ψr,es=φr+φ0.Therefore, the orientation of the optimal navigation star can be expressed as(33)σs=−tan−1nH1nL2−nH2nL1nH3nL2−nH2nL3,es=−tan−1nH1nL3−nH3nL1nH2nL3−nH3nL2cosψr+nH2nL1−nH1nL2sinψr.According to Equation (26), there is another solution for the orientation of the optimal navigation star.
(34)σ′s=σs−π,e′s=−es.


## 3.1. Equivalent Information Compression Theory

The impact point deviation and platform misalignment angle can be expressed in the matrix form(16)p=q⋅a,wherep=ΔLΔHT; q=q1q2T; and a=αxαyαzT. It can be seen from Equation (16) that the rank of q is 2, so there is information compression in the mapping from a to p. It is worth noting that a cannot be uniquely determined by p, which indicates that Equation (16) has numerous solutions. Although there are countless sets of solutions in Equation (16), there is a special solution a0, which belongs to the subspace qs=spanq1q2 formed by each row of vectors. Therefore, a0 can be expressed as
(17)a0=α10q1+α20q2=qT⋅X.Substitute Equation (17) into Equation (16), and there is
(18)p=qqTX.From Equation (17) and Equation (18), we can get
(19)a0=qTqqT−1p.It can be seen from the above equation thata0 and p correspond to each other one by one. If the inner product of two column vectors is defined as a⋅b=aT⋅b, then Equation (16) can be expressed as
(20)p=q1⋅aq2⋅aT,whereqi⋅a reflects the projection of a in the qi direction. q1 and q2 are linearly independent; therefore, p=q⋅a reflects the projection as of a on space qs, and the projection information as⊥ of a on the orthogonal complement qs⊥ of qs is lost. Since qs is a complete subspace on Hilbert space Rn, there is
(21)Rn=qs+qs⊥.According to the projection theorem, we can get(22)a=as+as⊥.It can be seen from the relationship between the impact point deviation and the platform misalignment angle thatq is not full rank. So only the information as in the subspace can be obtained through the impact point deviation, and the information as⊥ in the orthogonal complement cannot be obtained.The star sensor measurement equation can also be expressed in the matrix form(23)Z=h⋅a.Assuming that another set of bases ofqs is h1h2, it can be seen from the above analysis that Z also reflects all the information projected by a on qs, which can be expressed as
(24)a0=hThhT−1Z.Substitute Equation (24) into Equation (16), and we can get
(25)p=q⋅hThhT−1Z.Therefore, from the perspective of information compression,q and h are equal compression maps; that is, the impact point deviation p can be uniquely determined by the single-star observation Z.The impact point deviation is only affected by the projectionas of the misalignment angle a on the subspace Qs=spanq1q2. Therefore, according to Equation (25), it is only necessary to select h1 and h2, so that h and q are equal information compression. Then, the observation information Z contains all the useful information. The schematic diagram of determining the optimal navigation star is shown in Figure 3.Figure 3
Schematic diagram of determining the optimal navigation star.In the figure, all the information of misalignment anglea is composed of as and as⊥, but only as affects the impact point deviation. If h and q are equal information compression maps, the single-star scheme can measure all the information of as. Although the double-star scheme can measure all the information of a, only the as part is used in the correction, and as⊥ belongs to the useless information, so it has the same accuracy as the single-star scheme. In more popular terms, there are only two indicators ΔL and ΔH describing the impact point deviation, which are the reflection of part of the misalignment angle a. When observing a single star, two measurements ξ and η can be obtained, which are also the reflection of part of the misalignment angle a. By selecting the optimal navigation star, ξ and η can include all the information of misalignment angle a contained in ΔL and ΔH. Therefore, the single-star scheme can achieve the same accuracy as the dual-star scheme.


## 3.2. Determining the Optimal Navigation Star

According to the equivalent information compression theory, the optimal navigation star should satisfy(26)h1×h2=q1×q2q1×q2orh1×h2=−q1×q2q1×q2.For the left side of the above equation, it can be obtained according to the Equation (15).
(27)h1×h2=cosφ0cosψ0sinφ0cosφ0sinψ0T.It is assumed that the star sensor is installed on thexoy plane of the platform, and the platform is adjusted to aim at the navigation star by first yaw and then pitch. By substitutingψ0=0∘ into Equation (27), we can get
(28)h1×h2=cosφ0sinφ00T.Define(29)Qc=Qc1Qc2Qc3=q1×q2,where,(30)Qc1=nH1nL3−nH3nL1sinφr+nH3nL2−nH2nL3cosφrcosψr+nH1nL2−nH2nL1cosφrsinψr,Qc2=nH1nL3−nH3nL1cosφr−nH3nL2−nH2nL3sinφrcosψr−nH1nL2−nH2nL1sinφrsinψr,Qc3=−nH1nL2−nH2nL1cosψr+nH3nL2−nH2nL3sinψr.According to Equations (26), (27), and (28), we can get
(31)ψr=tan−1nH1nL2−nH2nL1nH3nL2−nH2nL3,φr=−tan−1nH1nL3−nH3nL1nH2nL3−nH3nL2cosψr+nH2nL1−nH1nL2sinψr−φ0.The optimal navigation star and the rotation angle satisfy the following relationship:(32)σs=−ψr,es=φr+φ0.Therefore, the orientation of the optimal navigation star can be expressed as(33)σs=−tan−1nH1nL2−nH2nL1nH3nL2−nH2nL3,es=−tan−1nH1nL3−nH3nL1nH2nL3−nH3nL2cosψr+nH2nL1−nH1nL2sinψr.According to Equation (26), there is another solution for the orientation of the optimal navigation star.
(34)σ′s=σs−π,e′s=−es.


## 4. Determining the Available Optimal Navigation Star Based on the Star Database

### 4.1. Angle Analysis of the Deviation from the Optimal Navigation Star

For the single-star platform inertial-stellar composite guidance scheme, only observing the optimal navigation star can achieve the same accuracy as the double star guidance scheme. However, in star database, there are not necessarily stars in the optimal navigation star direction. And only one real star can be selected as the navigation star in the star library according to a certain principle. This star is called the available optimal navigation star. In this section, the angle that the available navigation star deviates from the optimal navigation star is analyzed.In thei-system, several groups of optimal navigation stars are randomly generated, in which the elevation angles are evenly distributed within −90°,90°, and azimuth angles are evenly distributed within −180°,180°. Each combination of elevation angles and azimuth angles eNi,σNi represents a group of possible optimal navigation star, and its direction vector in the i-system is
(35)VNi=coseNicosσNicoseNisinσNisineNiT.For any star above 5.5 mag in the star database, its elevation angle and azimuth angle areeSj,σSj; then, the direction vector in the i-system can be expressed as
(36)VSj=coseSjcosσSjcoseSjsinσSjsineSjT.The angle between the optimal navigation star and the available navigation star can be calculated according to the following equation:(37)αij=arccosVNi⋅VSj.By traversingj, the minimum angular distance between the optimal navigation star and the available navigation star can be obtained.100000 samples are sampled, and the results are shown in Figures4 and 5.Figure 4
Statistical histogram of the angles that the available navigation stars deviate from the optimal navigation star.Figure 5
Probability density histogram of the angles that the available navigation stars deviate from the optimal navigation star.Figures4 and 5, respectively, show the statistical histogram and probability density histogram of the angles that the available navigation stars deviate from the optimal navigation star. Here, each straight bar represents 0.1°, and the sum of all the sampling times is 100000. In Figure 4, the angular deviations are between 0 and 6° mostly, which mainly concentrated in 1°~3° and relatively few more than 5° or less than 1°. Compared with Figure 4, the shapes of the statistical histogram and probability density histogram are basically the same. Figure 5 also shows the probability density function diagram of the corresponding normal distribution. However, it is obvious that the distribution is not quite consistent with the normal distribution.Tables1 and 2 provide the corresponding numerical statistical results. It can be seen from Table 1 that the maximum deviation angle is 7.4221° and the mean deviation angle is 2.0949°. The more detailed statistical analysis results of the available navigation star deviation from the optimal navigation star are illustrated in Table 2. The table counts the single probability and cumulative probability of the deviation angle. It can be observed that 2°>α≧1° is the most, accounting for 33.558% and the deviation angle greater than 7° accounts for only 0.014%.Table 1
The basic analysis results of the available navigation star deviation from the optimal navigation star.
EventMaximumMinimumMeanSquare3σ rangeValue (deg)7.42210.00412.09491.1139[-1.2467, 5.4365]Table 2
statistical analysis results of the available navigation star deviation from the optimal navigation star.
Deviation angleNumberProbability (%)Cumulative numberCumulative probability (%)1°>α≧0°1544915.4491544915.4492°>α≧1°3355833.5584900749.0073°>α≧2°2939229.3927839978.3994°>α≧3°1484214.8429324193.2415°>α≧4°52165.2169845798.4576°>α≧5°12891.2899974699.7467°>α≧6°2400.2409998699.986α≧7°140.014100000100.000Therefore, if the upper limit of star-sensitive measurement magnitude is 5.5 mag, the available navigation star can be found within the angular distance range within 7° from the optimal navigation star.In the above analysis, constraints such as occlusion of the sun, moon, and earth have not been taken into account, and the deviation angle will be much larger after consideration.


### 4.2. Determining the Available Optimal Navigation Star Based on the Arrow-Borne Navigation Star Database

In practical application, the navigation star must be selected in the arrow-borne navigation star database. According to the above analysis, within the range of 7° from the theoretical optimal navigation star, the probability of finding the navigation star is 100%. Therefore, a method determining the available optimal navigation star based on the local navigation star database is proposed to improve the efficiency of star selection.


#### 4.2.1. Determining the Local Navigation Star Database

Strong light sources should be avoided when determining the local navigation star database (this paper takes avoiding the sun as an example). The right ascension and declination of the sun obtained from the ephemeris table are defined asαsun and δsun, and the unit sun direction vector in the i-system can be expressed as
(38)iI=cosδsuncosαsuncosδsunsinαsunsinαsun,whereiI is the unit sun direction vector in the i-system.Then, the unit sun direction vector in theA-system can be further obtained:
(39)isun=CIA⋅iI,whereisun is the unit sun direction vector in the A-system and CIA is the transformation matrix from the i-system to the A-system.Therefore, the angular distanceθs between the theoretical optimal navigation star and the sun can be calculated as
(40)θs=arccosSI⋅isun.Ifθs is less than the sum of the solar avoidance angle αsun and deviation angle Δα, the deviation angle can be recalculated according to the following equation:
(41)Δβ=−αsunαsun+Δαθs+αsun+Δα,whereΔβ is the recalculated deviation angle.The navigation star orientation in the star database is defined ase0σ0, and its unit vector in the A-system can be expressed as
(42)i0=cose0cosσ0sine0cose0sinσ0.Then, the angular distanceθI between the navigation star and the theoretical optical navigation star and the angular distance θ0 between the navigation star and the sun can be calculated, respectively.
(43)θI=arccosi0⋅SI,θ0=arccosi0⋅isun.Therefore, the value ofθI and the deviation angle can be compared, so as θ0 and αsun.
(44)θI<Δα,θs≥αsun+Δα,θI<Δβ,θs<αsun+Δα,θ0>αsun.If the above equation is valid, it means that the navigation star is within the deviation angle range of the optimal navigator star and outside the sun avoidance angle range. And the navigation star can be put into the local navigation star database. After calculating all the navigation stars in the star database, the local star database for determining the available optimal navigation star can be obtained.


#### 4.2.2. Determining the Available Optical Navigation Star

Considering that the navigation star with the smallest angular distance from the theoretical optimal navigation star is not necessarily the available optimal navigation star, this paper utilizes the combination of minimum angular distance and minimum accuracy change to determine the available optimal navigation star. Firstly, the angular distance between the stars in the local navigation star database and the optimal navigation star is calculated, and the one with the smallest angular distance is the first available navigation star. Secondly, estimate the accuracy variation of any star in the local navigation star database, and the smallest is the second available navigation star. The calculation method for estimating the accuracy change caused by navigation star deviation is as follows.The gradient can be calculated from the partial derivative of the composite guidance accuracy at the optimal navigation star varying with the navigation star orientation.(45)d∇=∂CEP∂esi+∂CEP∂σsj,whereCEP is the circular error probable and ∂CEP/∂es and ∂CEP/∂σs are the partial derivative of composite guidance CEP to elevation and azimuth angle at the optimal navigation star. The direction perpendicular to the gradient is the direction with the slowest change in the composite guidance accuracy.
(46)d∇⊥=−∂CEP∂σsi+∂CEP∂esj.Therefore, for any star in the local navigation star database, the accuracy change ∆CEP can be estimated according to(47)ΔCEP=∂CEP∂σsΔes2+∂CEP∂esΔσs2,whereΔCEP is the estimated value of the accuracy change between the navigation star and the optimal navigation star. Δes and Δσs are the difference of elevation angle and azimuth angle between the star in the local navigation star database and the optimal navigation star. When the star is smallest, it is selected as the second available navigation star. For the first and the second available navigation star, the one with smaller CEP is the available optimal navigation star.


## 4.1. Angle Analysis of the Deviation from the Optimal Navigation Star

For the single-star platform inertial-stellar composite guidance scheme, only observing the optimal navigation star can achieve the same accuracy as the double star guidance scheme. However, in star database, there are not necessarily stars in the optimal navigation star direction. And only one real star can be selected as the navigation star in the star library according to a certain principle. This star is called the available optimal navigation star. In this section, the angle that the available navigation star deviates from the optimal navigation star is analyzed.In thei-system, several groups of optimal navigation stars are randomly generated, in which the elevation angles are evenly distributed within −90°,90°, and azimuth angles are evenly distributed within −180°,180°. Each combination of elevation angles and azimuth angles eNi,σNi represents a group of possible optimal navigation star, and its direction vector in the i-system is
(35)VNi=coseNicosσNicoseNisinσNisineNiT.For any star above 5.5 mag in the star database, its elevation angle and azimuth angle areeSj,σSj; then, the direction vector in the i-system can be expressed as
(36)VSj=coseSjcosσSjcoseSjsinσSjsineSjT.The angle between the optimal navigation star and the available navigation star can be calculated according to the following equation:(37)αij=arccosVNi⋅VSj.By traversingj, the minimum angular distance between the optimal navigation star and the available navigation star can be obtained.100000 samples are sampled, and the results are shown in Figures4 and 5.Figure 4
Statistical histogram of the angles that the available navigation stars deviate from the optimal navigation star.Figure 5
Probability density histogram of the angles that the available navigation stars deviate from the optimal navigation star.Figures4 and 5, respectively, show the statistical histogram and probability density histogram of the angles that the available navigation stars deviate from the optimal navigation star. Here, each straight bar represents 0.1°, and the sum of all the sampling times is 100000. In Figure 4, the angular deviations are between 0 and 6° mostly, which mainly concentrated in 1°~3° and relatively few more than 5° or less than 1°. Compared with Figure 4, the shapes of the statistical histogram and probability density histogram are basically the same. Figure 5 also shows the probability density function diagram of the corresponding normal distribution. However, it is obvious that the distribution is not quite consistent with the normal distribution.Tables1 and 2 provide the corresponding numerical statistical results. It can be seen from Table 1 that the maximum deviation angle is 7.4221° and the mean deviation angle is 2.0949°. The more detailed statistical analysis results of the available navigation star deviation from the optimal navigation star are illustrated in Table 2. The table counts the single probability and cumulative probability of the deviation angle. It can be observed that 2°>α≧1° is the most, accounting for 33.558% and the deviation angle greater than 7° accounts for only 0.014%.Table 1
The basic analysis results of the available navigation star deviation from the optimal navigation star.
EventMaximumMinimumMeanSquare3σ rangeValue (deg)7.42210.00412.09491.1139[-1.2467, 5.4365]Table 2
statistical analysis results of the available navigation star deviation from the optimal navigation star.
Deviation angleNumberProbability (%)Cumulative numberCumulative probability (%)1°>α≧0°1544915.4491544915.4492°>α≧1°3355833.5584900749.0073°>α≧2°2939229.3927839978.3994°>α≧3°1484214.8429324193.2415°>α≧4°52165.2169845798.4576°>α≧5°12891.2899974699.7467°>α≧6°2400.2409998699.986α≧7°140.014100000100.000Therefore, if the upper limit of star-sensitive measurement magnitude is 5.5 mag, the available navigation star can be found within the angular distance range within 7° from the optimal navigation star.In the above analysis, constraints such as occlusion of the sun, moon, and earth have not been taken into account, and the deviation angle will be much larger after consideration.


## 4.2. Determining the Available Optimal Navigation Star Based on the Arrow-Borne Navigation Star Database

In practical application, the navigation star must be selected in the arrow-borne navigation star database. According to the above analysis, within the range of 7° from the theoretical optimal navigation star, the probability of finding the navigation star is 100%. Therefore, a method determining the available optimal navigation star based on the local navigation star database is proposed to improve the efficiency of star selection.


### 4.2.1. Determining the Local Navigation Star Database

Strong light sources should be avoided when determining the local navigation star database (this paper takes avoiding the sun as an example). The right ascension and declination of the sun obtained from the ephemeris table are defined asαsun and δsun, and the unit sun direction vector in the i-system can be expressed as
(38)iI=cosδsuncosαsuncosδsunsinαsunsinαsun,whereiI is the unit sun direction vector in the i-system.Then, the unit sun direction vector in theA-system can be further obtained:
(39)isun=CIA⋅iI,whereisun is the unit sun direction vector in the A-system and CIA is the transformation matrix from the i-system to the A-system.Therefore, the angular distanceθs between the theoretical optimal navigation star and the sun can be calculated as
(40)θs=arccosSI⋅isun.Ifθs is less than the sum of the solar avoidance angle αsun and deviation angle Δα, the deviation angle can be recalculated according to the following equation:
(41)Δβ=−αsunαsun+Δαθs+αsun+Δα,whereΔβ is the recalculated deviation angle.The navigation star orientation in the star database is defined ase0σ0, and its unit vector in the A-system can be expressed as
(42)i0=cose0cosσ0sine0cose0sinσ0.Then, the angular distanceθI between the navigation star and the theoretical optical navigation star and the angular distance θ0 between the navigation star and the sun can be calculated, respectively.
(43)θI=arccosi0⋅SI,θ0=arccosi0⋅isun.Therefore, the value ofθI and the deviation angle can be compared, so as θ0 and αsun.
(44)θI<Δα,θs≥αsun+Δα,θI<Δβ,θs<αsun+Δα,θ0>αsun.If the above equation is valid, it means that the navigation star is within the deviation angle range of the optimal navigator star and outside the sun avoidance angle range. And the navigation star can be put into the local navigation star database. After calculating all the navigation stars in the star database, the local star database for determining the available optimal navigation star can be obtained.


### 4.2.2. Determining the Available Optical Navigation Star

Considering that the navigation star with the smallest angular distance from the theoretical optimal navigation star is not necessarily the available optimal navigation star, this paper utilizes the combination of minimum angular distance and minimum accuracy change to determine the available optimal navigation star. Firstly, the angular distance between the stars in the local navigation star database and the optimal navigation star is calculated, and the one with the smallest angular distance is the first available navigation star. Secondly, estimate the accuracy variation of any star in the local navigation star database, and the smallest is the second available navigation star. The calculation method for estimating the accuracy change caused by navigation star deviation is as follows.The gradient can be calculated from the partial derivative of the composite guidance accuracy at the optimal navigation star varying with the navigation star orientation.(45)d∇=∂CEP∂esi+∂CEP∂σsj,whereCEP is the circular error probable and ∂CEP/∂es and ∂CEP/∂σs are the partial derivative of composite guidance CEP to elevation and azimuth angle at the optimal navigation star. The direction perpendicular to the gradient is the direction with the slowest change in the composite guidance accuracy.
(46)d∇⊥=−∂CEP∂σsi+∂CEP∂esj.Therefore, for any star in the local navigation star database, the accuracy change ∆CEP can be estimated according to(47)ΔCEP=∂CEP∂σsΔes2+∂CEP∂esΔσs2,whereΔCEP is the estimated value of the accuracy change between the navigation star and the optimal navigation star. Δes and Δσs are the difference of elevation angle and azimuth angle between the star in the local navigation star database and the optimal navigation star. When the star is smallest, it is selected as the second available navigation star. For the first and the second available navigation star, the one with smaller CEP is the available optimal navigation star.


## 4.2.1. Determining the Local Navigation Star Database

Strong light sources should be avoided when determining the local navigation star database (this paper takes avoiding the sun as an example). The right ascension and declination of the sun obtained from the ephemeris table are defined asαsun and δsun, and the unit sun direction vector in the i-system can be expressed as
(38)iI=cosδsuncosαsuncosδsunsinαsunsinαsun,whereiI is the unit sun direction vector in the i-system.Then, the unit sun direction vector in theA-system can be further obtained:
(39)isun=CIA⋅iI,whereisun is the unit sun direction vector in the A-system and CIA is the transformation matrix from the i-system to the A-system.Therefore, the angular distanceθs between the theoretical optimal navigation star and the sun can be calculated as
(40)θs=arccosSI⋅isun.Ifθs is less than the sum of the solar avoidance angle αsun and deviation angle Δα, the deviation angle can be recalculated according to the following equation:
(41)Δβ=−αsunαsun+Δαθs+αsun+Δα,whereΔβ is the recalculated deviation angle.The navigation star orientation in the star database is defined ase0σ0, and its unit vector in the A-system can be expressed as
(42)i0=cose0cosσ0sine0cose0sinσ0.Then, the angular distanceθI between the navigation star and the theoretical optical navigation star and the angular distance θ0 between the navigation star and the sun can be calculated, respectively.
(43)θI=arccosi0⋅SI,θ0=arccosi0⋅isun.Therefore, the value ofθI and the deviation angle can be compared, so as θ0 and αsun.
(44)θI<Δα,θs≥αsun+Δα,θI<Δβ,θs<αsun+Δα,θ0>αsun.If the above equation is valid, it means that the navigation star is within the deviation angle range of the optimal navigator star and outside the sun avoidance angle range. And the navigation star can be put into the local navigation star database. After calculating all the navigation stars in the star database, the local star database for determining the available optimal navigation star can be obtained.


## 4.2.2. Determining the Available Optical Navigation Star

Considering that the navigation star with the smallest angular distance from the theoretical optimal navigation star is not necessarily the available optimal navigation star, this paper utilizes the combination of minimum angular distance and minimum accuracy change to determine the available optimal navigation star. Firstly, the angular distance between the stars in the local navigation star database and the optimal navigation star is calculated, and the one with the smallest angular distance is the first available navigation star. Secondly, estimate the accuracy variation of any star in the local navigation star database, and the smallest is the second available navigation star. The calculation method for estimating the accuracy change caused by navigation star deviation is as follows.The gradient can be calculated from the partial derivative of the composite guidance accuracy at the optimal navigation star varying with the navigation star orientation.(45)d∇=∂CEP∂esi+∂CEP∂σsj,whereCEP is the circular error probable and ∂CEP/∂es and ∂CEP/∂σs are the partial derivative of composite guidance CEP to elevation and azimuth angle at the optimal navigation star. The direction perpendicular to the gradient is the direction with the slowest change in the composite guidance accuracy.
(46)d∇⊥=−∂CEP∂σsi+∂CEP∂esj.Therefore, for any star in the local navigation star database, the accuracy change ∆CEP can be estimated according to(47)ΔCEP=∂CEP∂σsΔes2+∂CEP∂esΔσs2,whereΔCEP is the estimated value of the accuracy change between the navigation star and the optimal navigation star. Δes and Δσs are the difference of elevation angle and azimuth angle between the star in the local navigation star database and the optimal navigation star. When the star is smallest, it is selected as the second available navigation star. For the first and the second available navigation star, the one with smaller CEP is the available optimal navigation star.


## 5. Simulation Results

This section mainly include two parts: (1) determining the theoretical optimal navigation star and (2) determining the available optimal navigation star based on the star database. The simulations are primarily aimed at verifying the effectiveness of the proposed method.In the simulation, two representative responsive launch vehicle trajectories are adopted. The launch time is 00 : 00 : 00, 1 January 2019 (UTC). The first whole flight time is 1300 s, and the second whole flight time is 2300 s. The initial position is (0°N, 0°E). The star sensor works beyond the atmosphere. And the star sensor installation angle isφ0,ψ0=20°,0°. The simulation parameters for the initial alignment error and star sensor error are listed in Table 3. Two trajectories can better verify the effectiveness of the proposed method.Table 3
The value of each error in the simulation.
Error typesError symbolsValue (3σ)UnitsInitial orientation (alignment) errorε0x100″ε0y300ε0z100Star sensor measurement errorεξ, εη0″Star sensor installation errorΔφ0, Δψ00


### 5.1. Determining the Optimal Navigation Star

This section is used to evaluate the effectiveness of the algorithm in Section3. In the simulation, the optimal navigation star is determined by three methods, which are traversal method, simplex evolutionary method, and analytical method proposed in this paper.The traversal method searches in the full dimensional space with−90°≤es≤90° and −180°<σs≤180°, and the step is 1°. Taking the 6000 km trajectory (first trajectory) as an example, when the optimal navigation star is determined by the traversal method, the composite guidance accuracy under different measurement orientations is shown in Figure 6, and the composite guidance CEP contour is shown in Figure 7.Figure 6
Composite guidance CEP variation diagram for 6000 km.Figure 7
Composite guidance CEP contour map for 6000 km.Figure6 shows the composite guidance CEP variation diagram for 6000 km. It can be observed that there are two minimum points in the composite guidance accuracy variation diagram corresponding to the single-star tuning platform; that is, there are two optimal navigation stars. And the two optimal navigation stars azimuth is approximately on the same line as the emission point, that is, e′s=−es and σ′s=σs−π. This is consistent with the analysis conclusion of Equation (33). Besides, the composite guidance accuracy is approximately symmetric with respect to the line according to Figure 7.In the simplex evolutionary method, the initial vertex isX0=es0σs0T=20°0°T. Take the distance between vertices Δd=10° to construct the initial simplex, and the iteration termination condition is taken as ε=0.1m. The change of the simplex optimal vertex in the iteration process is shown in Figure 8, and the convergence error is shown in Figure 9.Figure 8
Simplex optimal vertex iterative change diagram with a range of 6000 km.Figure 9
Simplex convergent error variation diagram with a range of 6000 km.It can be seen from the above figures that when utilizing the simplex evolutionary method to determine the optimal navigation star, the simplex converges quickly in the solution process, and the algorithm has a large search range. At the same time, it can achieve high accuracy by controlling the convergence domain.Table4 represents the required time and the optimal navigation stars determined by the three methods. In the table, CEPINS is the pure inertial guidance accuracy, and CEPCOM is the composite guidance accuracy.Table 4
Optimal navigation star at different range.
RangeMethodes (deg)σs (deg)CEPINS (m)CEPCOM (m)t (s)6000kmTraversing3802628.896.782963.05Simplex37.6011-0.04112628.890.01312.09Analysis37.6011-0.04042628.890.0010.00112000kmTraversing3003198.364.022985.53Simplex30.0226-0.13163198.360.01513.22Analysis30.0264-0.13183198.360.0010.003The simulation results show that under the condition of only considering the initial alignment error, the results obtained by analytical method are consistent with those obtained by traversal method and simplex evolutionary method, which verify the effectiveness of the proposed method. At the same time, the azimuth angle of the navigation star is about 0°, which indicates that the optimal navigation star is near the shooting plane when the star sensor is installed on thexoy plane of the platform.According to the results of the optimal navigation stars and the corresponding composite guidance accuracy, the accuracy of the traversal method is limited because the traversal method is searched with a fixed step, while the analytic method and simplex evolutionary method have no such limitation. Thus, the optimal navigation star azimuth can achieve high accuracy. When comparing the calculation time of the three methods, the results are calculated on PC. By contrast, the traversal calendar takes about 50 minutes, while the analytic method can be completed in a very short time. And the composite guidance accuracy corresponding to the optimal navigation star obtained by the analytical method is 99.99% (from 6.78 m to 0.001 m) and 92.31% (from 0.013 m to 0.001 m) higher than that obtained by the traversal method and the simplex evolutionary method. Moreover, the time-consuming of the traversal method is related to the traversal step size. The smaller the step size, the more time-consuming, but the more accurate the optimal star azimuth is determined. Therefore, under the condition of significant initial error, the method proposed in this paper can be used to help determine the optimal navigation star quickly. Since only the initial orientation error is considered in the simulation, the stellar guidance can correct all the effects of the error, and the corrected accuracy is close to 0 m. Of course, it is impossible to achieve when all error factors are considered.Taking the responsive launch vehicle with a range of 6000 km as an example, the influence of the star sensor installation error and measurement error on the optimal navigation star is analyzed. In the simulation, simplex evolutionary method and analytic method are utilized to determine the optimal navigation star. Table5 represents the optimal navigation stars when considering the star sensor installation error, and Table 6 represents the optimal navigation stars when considering the star sensor measurement error.Table 5
Optimal navigation stars with different star sensor installation error.
Δφ0, Δψ0″,3σMethodes (deg)σs (deg)0Simplex37.6011-0.0411Analysis37.6011-0.040410Simplex37.5998-0.0417Analysis37.6011-0.040420Simplex37.5991-0.0423Analysis37.6011-0.040430Simplex37.5979-0.0440Analysis37.6011-0.0404Table 6
Optimal navigation star with different star sensor measurement error.
εξ, εη″,3σMethodes (deg)σs (deg)0Simplex37.6011-0.0411Analysis37.6011-0.040410Simplex37.6011-0.0404Analysis37.6011-0.040420Simplex37.6012-0.0403Analysis37.6011-0.040430Simplex37.6012-0.0404Analysis37.6011-0.0404The star sensor installation error has a certain impact on the optimal navigation star, but the impact is small. The range of changes in elevation angle and azimuth angle is both within 0.01°. Comparing Tables5 and 6, it can be seen that the star sensor measurement error has less impact on the optimal navigation star. Therefore, the method proposed in this paper can determine the optimal navigation star effectively.


### 5.2. Determining the Optimal Available Navigation Star

Stars are basically evenly distributed in the celestial coordinate system, and the earth shielding range of in the star sensor view field is basically fixed. Due to the physical realization of the inertial platform frame angle, there will be some restrictions on the azimuth and elevation angles. It is assumed that the azimuth angle has a limit of ±45°, and the elevation angle has a limit of ±60°. At the same time, it is assumed that the sun’s avoidance angle is 20°, the moon’s avoidance angle is 10°, the horizon’s additional avoidance angle is 5°, and the large planet’s avoidance angle is 2°. The arrow-borne navigation star database is shown in Figure10, and the generated local navigation star database based on the Section 4.2.1 is shown in Table 7.Figure 10
Arrow-borne navigation star database.Table 7
Local navigation star database.
Numberes (deg)σs (deg)CEPINS (m)CEPCOM (m)135.0459-7.38242628.89139.409236.4093-3.56982628.8967.152340.6013-3.48932628.8983.301432.84180.94062628.8979.633541.21341.42352628.8969.357631.51622.50292628.89108.593738.15456.39152628.89117.695839.92437.91222628.89150.393Figure10 shows the generated arrow-borne navigation star database when the launch time is January 1, 2019. Due to the influence of constraints, the final number of available navigation stars is 292. Compared with Figure 10, it can be seen that there are only 8 alternative navigation stars in the local navigation star database, indicating that most stars in the array-borne navigation star database can be excluded based on the maximum deviation angle from the theoretical optimal navigation star, thus shortening the time to determine the available optimal navigation star. Tables 8 and 9 represent the available optimal navigation stars for the 6000 km and 12000 km launch vehicle.Table 8
The available optimal navigation stars for the 6000 km launch vehicle.
Date2019/1/12019/3/102019/8/20Theoretical optimal navigation stares (deg)37.601137.601137.6011σs (deg)-0.0404-0.0404-0.0404CEPINS (m)2628.892628.892628.89CEPCOM (m)0.0010.0010.001Available optimal navigation star (traversal method)e′s (deg)36.409336.582737.0740σ′s (deg)-3.56981.0344-2.9698CEPINS (m)2628.892628.892628.89CEPCOM (m)67.15226.32054.120Available optimal navigation star (proposed method)e′s (deg)36.409336.582737.0740σ′s (deg)-3.56981.0344-2.9698CEPINS (m)2628.892628.892628.89CEPCOM (m)67.15226.32054.120Table 9
The available optimal navigation stars for the 12000 km launch vehicle.
Date2019/1/12019/3/102019/8/20Theoretical optimal navigation stares (deg)30.026430.026430.0264σs (deg)-0.1318-0.1318-0.1318CEPINS (m)3198.363198.363198.36CEPCOM (m)0.0010.0010.001Available optimal navigation star (traversal method)e′s (deg)30.571832.259325.3675σ′s (deg)0.78250.3548-0.4691CEPINS (m)3198.363198.363198.36CEPCOM (m)26.33233.22464.516Available optimal navigation star (proposed method)e′s (deg)30.571832.259325.3675σ′s (deg)0.78250.3548-0.4691CEPINS (m)3198.363198.363198.36CEPCOM (m)26.33233.22464.516Table8 and 9 show the comparison results of the proposed method and the traversal method to determine the available optimal navigation star. The traversal method in the table refers to traversing all stars in the local navigation star database, and the results obtained can be considered as accurate. The proposed method refers to the available optimal navigation star determined according to Section 4.2.2. The above results are the navigation stars selected from the real local navigation star database after considering various star selection constraints. It can be observed from the tables that the navigation star determined by the proposed method is the same as the traversal method, which proves that this method in this paper is effective. At the same, the angular distance between the theoretical optimal navigation star and the available optimal navigation star is within 5°, and the variation of composite guidance accuracy is less than 70 m, indicating that the available optimal navigation star still has a good correction effect.


## 5.1. Determining the Optimal Navigation Star

This section is used to evaluate the effectiveness of the algorithm in Section3. In the simulation, the optimal navigation star is determined by three methods, which are traversal method, simplex evolutionary method, and analytical method proposed in this paper.The traversal method searches in the full dimensional space with−90°≤es≤90° and −180°<σs≤180°, and the step is 1°. Taking the 6000 km trajectory (first trajectory) as an example, when the optimal navigation star is determined by the traversal method, the composite guidance accuracy under different measurement orientations is shown in Figure 6, and the composite guidance CEP contour is shown in Figure 7.Figure 6
Composite guidance CEP variation diagram for 6000 km.Figure 7
Composite guidance CEP contour map for 6000 km.Figure6 shows the composite guidance CEP variation diagram for 6000 km. It can be observed that there are two minimum points in the composite guidance accuracy variation diagram corresponding to the single-star tuning platform; that is, there are two optimal navigation stars. And the two optimal navigation stars azimuth is approximately on the same line as the emission point, that is, e′s=−es and σ′s=σs−π. This is consistent with the analysis conclusion of Equation (33). Besides, the composite guidance accuracy is approximately symmetric with respect to the line according to Figure 7.In the simplex evolutionary method, the initial vertex isX0=es0σs0T=20°0°T. Take the distance between vertices Δd=10° to construct the initial simplex, and the iteration termination condition is taken as ε=0.1m. The change of the simplex optimal vertex in the iteration process is shown in Figure 8, and the convergence error is shown in Figure 9.Figure 8
Simplex optimal vertex iterative change diagram with a range of 6000 km.Figure 9
Simplex convergent error variation diagram with a range of 6000 km.It can be seen from the above figures that when utilizing the simplex evolutionary method to determine the optimal navigation star, the simplex converges quickly in the solution process, and the algorithm has a large search range. At the same time, it can achieve high accuracy by controlling the convergence domain.Table4 represents the required time and the optimal navigation stars determined by the three methods. In the table, CEPINS is the pure inertial guidance accuracy, and CEPCOM is the composite guidance accuracy.Table 4
Optimal navigation star at different range.
RangeMethodes (deg)σs (deg)CEPINS (m)CEPCOM (m)t (s)6000kmTraversing3802628.896.782963.05Simplex37.6011-0.04112628.890.01312.09Analysis37.6011-0.04042628.890.0010.00112000kmTraversing3003198.364.022985.53Simplex30.0226-0.13163198.360.01513.22Analysis30.0264-0.13183198.360.0010.003The simulation results show that under the condition of only considering the initial alignment error, the results obtained by analytical method are consistent with those obtained by traversal method and simplex evolutionary method, which verify the effectiveness of the proposed method. At the same time, the azimuth angle of the navigation star is about 0°, which indicates that the optimal navigation star is near the shooting plane when the star sensor is installed on thexoy plane of the platform.According to the results of the optimal navigation stars and the corresponding composite guidance accuracy, the accuracy of the traversal method is limited because the traversal method is searched with a fixed step, while the analytic method and simplex evolutionary method have no such limitation. Thus, the optimal navigation star azimuth can achieve high accuracy. When comparing the calculation time of the three methods, the results are calculated on PC. By contrast, the traversal calendar takes about 50 minutes, while the analytic method can be completed in a very short time. And the composite guidance accuracy corresponding to the optimal navigation star obtained by the analytical method is 99.99% (from 6.78 m to 0.001 m) and 92.31% (from 0.013 m to 0.001 m) higher than that obtained by the traversal method and the simplex evolutionary method. Moreover, the time-consuming of the traversal method is related to the traversal step size. The smaller the step size, the more time-consuming, but the more accurate the optimal star azimuth is determined. Therefore, under the condition of significant initial error, the method proposed in this paper can be used to help determine the optimal navigation star quickly. Since only the initial orientation error is considered in the simulation, the stellar guidance can correct all the effects of the error, and the corrected accuracy is close to 0 m. Of course, it is impossible to achieve when all error factors are considered.Taking the responsive launch vehicle with a range of 6000 km as an example, the influence of the star sensor installation error and measurement error on the optimal navigation star is analyzed. In the simulation, simplex evolutionary method and analytic method are utilized to determine the optimal navigation star. Table5 represents the optimal navigation stars when considering the star sensor installation error, and Table 6 represents the optimal navigation stars when considering the star sensor measurement error.Table 5
Optimal navigation stars with different star sensor installation error.
Δφ0, Δψ0″,3σMethodes (deg)σs (deg)0Simplex37.6011-0.0411Analysis37.6011-0.040410Simplex37.5998-0.0417Analysis37.6011-0.040420Simplex37.5991-0.0423Analysis37.6011-0.040430Simplex37.5979-0.0440Analysis37.6011-0.0404Table 6
Optimal navigation star with different star sensor measurement error.
εξ, εη″,3σMethodes (deg)σs (deg)0Simplex37.6011-0.0411Analysis37.6011-0.040410Simplex37.6011-0.0404Analysis37.6011-0.040420Simplex37.6012-0.0403Analysis37.6011-0.040430Simplex37.6012-0.0404Analysis37.6011-0.0404The star sensor installation error has a certain impact on the optimal navigation star, but the impact is small. The range of changes in elevation angle and azimuth angle is both within 0.01°. Comparing Tables5 and 6, it can be seen that the star sensor measurement error has less impact on the optimal navigation star. Therefore, the method proposed in this paper can determine the optimal navigation star effectively.


## 5.2. Determining the Optimal Available Navigation Star

Stars are basically evenly distributed in the celestial coordinate system, and the earth shielding range of in the star sensor view field is basically fixed. Due to the physical realization of the inertial platform frame angle, there will be some restrictions on the azimuth and elevation angles. It is assumed that the azimuth angle has a limit of ±45°, and the elevation angle has a limit of ±60°. At the same time, it is assumed that the sun’s avoidance angle is 20°, the moon’s avoidance angle is 10°, the horizon’s additional avoidance angle is 5°, and the large planet’s avoidance angle is 2°. The arrow-borne navigation star database is shown in Figure10, and the generated local navigation star database based on the Section 4.2.1 is shown in Table 7.Figure 10
Arrow-borne navigation star database.Table 7
Local navigation star database.
Numberes (deg)σs (deg)CEPINS (m)CEPCOM (m)135.0459-7.38242628.89139.409236.4093-3.56982628.8967.152340.6013-3.48932628.8983.301432.84180.94062628.8979.633541.21341.42352628.8969.357631.51622.50292628.89108.593738.15456.39152628.89117.695839.92437.91222628.89150.393Figure10 shows the generated arrow-borne navigation star database when the launch time is January 1, 2019. Due to the influence of constraints, the final number of available navigation stars is 292. Compared with Figure 10, it can be seen that there are only 8 alternative navigation stars in the local navigation star database, indicating that most stars in the array-borne navigation star database can be excluded based on the maximum deviation angle from the theoretical optimal navigation star, thus shortening the time to determine the available optimal navigation star. Tables 8 and 9 represent the available optimal navigation stars for the 6000 km and 12000 km launch vehicle.Table 8
The available optimal navigation stars for the 6000 km launch vehicle.
Date2019/1/12019/3/102019/8/20Theoretical optimal navigation stares (deg)37.601137.601137.6011σs (deg)-0.0404-0.0404-0.0404CEPINS (m)2628.892628.892628.89CEPCOM (m)0.0010.0010.001Available optimal navigation star (traversal method)e′s (deg)36.409336.582737.0740σ′s (deg)-3.56981.0344-2.9698CEPINS (m)2628.892628.892628.89CEPCOM (m)67.15226.32054.120Available optimal navigation star (proposed method)e′s (deg)36.409336.582737.0740σ′s (deg)-3.56981.0344-2.9698CEPINS (m)2628.892628.892628.89CEPCOM (m)67.15226.32054.120Table 9
The available optimal navigation stars for the 12000 km launch vehicle.
Date2019/1/12019/3/102019/8/20Theoretical optimal navigation stares (deg)30.026430.026430.0264σs (deg)-0.1318-0.1318-0.1318CEPINS (m)3198.363198.363198.36CEPCOM (m)0.0010.0010.001Available optimal navigation star (traversal method)e′s (deg)30.571832.259325.3675σ′s (deg)0.78250.3548-0.4691CEPINS (m)3198.363198.363198.36CEPCOM (m)26.33233.22464.516Available optimal navigation star (proposed method)e′s (deg)30.571832.259325.3675σ′s (deg)0.78250.3548-0.4691CEPINS (m)3198.363198.363198.36CEPCOM (m)26.33233.22464.516Table8 and 9 show the comparison results of the proposed method and the traversal method to determine the available optimal navigation star. The traversal method in the table refers to traversing all stars in the local navigation star database, and the results obtained can be considered as accurate. The proposed method refers to the available optimal navigation star determined according to Section 4.2.2. The above results are the navigation stars selected from the real local navigation star database after considering various star selection constraints. It can be observed from the tables that the navigation star determined by the proposed method is the same as the traversal method, which proves that this method in this paper is effective. At the same, the angular distance between the theoretical optimal navigation star and the available optimal navigation star is within 5°, and the variation of composite guidance accuracy is less than 70 m, indicating that the available optimal navigation star still has a good correction effect.


## 6. Conclusion

The demand for the application of single-star inertial-stellar guidance system in responsive launch vehicles is to determine the optimal navigation star quickly. However, the current optimal navigation star selection schemes are to determine the star by numerical method, which increase the preparation time before launch. This paper proposes a fast algorithm to determine the star. The key of this algorithm is to deduce the optimal navigation star based on the equivalent information compression theory under the condition of significant initial error. It is obvious that the analytical solution is less time-consuming than the numerical solution. And the analytical solution can achieve the same accuracy as the numerical solution, or even higher.On the basis of determining the optimal navigation star, the available optimal navigation star should be further determined in combination with the arrow-borne navigation star database. There are certain deviations between the optimal navigation star and the navigation stars in the database. Therefore, the deviation angles between them without considering constraints are analyzed firstly. Based on the deviation angle, the navigation stars are selected to the local navigation database. Then, the available optimal navigation star can be determined according to certain criteria. The algorithm proposed in this paper can quickly determine the optimal navigation star and the available optimal navigation star.



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*Source: 1000865-2022-04-19.xml*