，，

College of Mechanical Engineering，Nanjing University of Science and Technology，Nanjing 210094，P.R.China

Abstract: Aiming at the problem of relative navigation for non-cooperative rendezvous of spacecraft，this paper proposes a new angles-only navigation architecture using non-linear dynamics method.This method does not solve the problem of poor observability of angles-only navigation through orbital or attitude maneuvering，but improves the observability of angles-only navigation through capturing the non-linearity of the system in the evolution of relative motion.First，three relative dynamics models and their corresponding line-of-sight（LoS）measurement equations are introduced，including the rectilinear state relative dynamics model，the curvilinear state relative dynamics model，and the relative orbital elements（ROE）state relative dynamics model.Then，an observability analysis theory based on the Gramian matrix is introduced to determine which relative dynamics model could maximize the observability of angles-only navigation.Next，an adaptive extended Kalman filtering scheme is proposed to solve the problem that the angles-only navigation filter using the non-linear dynamics method is sensitive to measurement noises.Finally，the performances of the proposed angles-only navigation architecture are tested by means of numerical simulations，which demonstrates that the angles-only navigation filtering scheme without orbital or attitude maneuvering is completely feasible through improving the modeling of the relative dynamics and LoS measurement equations.

Key words：angles-only navigation；non-linear dynamics；observability analysis；non-cooperative rendezvous；adaptive Kalman filter

0 Introduction

Angles-only navigation is not a new concept，and it was first used in the 19th century when the scientists observed celestial bodies through tele?scopes and calculated their orbits.So far，angles-on?ly navigation has been widely used and studied in sailing［1］，deep space exploration［2］，spacecraft orbit determination［3］，and formation flying［4-5］.

The concept of angles-only navigation is quite simple.The LoS vectors between the chaser space?craft and the space target can be measured by a mon?ocular camera over a period of time to determine the relative motion states between them［6］.The active sensors such as light detection and ranging（Li?DAR）and microwave radars are commonly used to measure relative motion states between two space?craft.However，they cannot be applied to micro-sat?ellite platforms due to their high power consumption and large mass.The passive sensors such as optical and infrared cameras have many advantages［7-9］be?cause they are employable at various inter-satellite separation ranges with small effect on the design of the mass/power of the chaser spacecraft［10］.More?over，most spacecraft are equipped with onboard cameras，if the direction is appropriate，these on?board cameras could be used to capture the space targets within the field of view and perform anglesonly navigation operations［11］.This so-called anglesonly navigation provides a passive，robust，and high-dynamic-range capability.Accordingly，anglesonly navigation represents a critical enabling technol?ogy for a variety of advanced distributed space sys?tem missions，including autonomous rendezvous and docking，space situational awareness，advanced distributed aperture science，and on-orbit servicing of non-cooperative spacecraft［12］.In addition，anglesonly navigation has great application prospects in space-based anti-missile monitoring and active de?bris removal［13-14］.

Due to the lack of depth information of the monocular camera，angles-only navigation suffers from a relative range observability problem during near-range rendezvous［15-17］.Many scholars have studied the methods to improve the observability of angles-only navigation，including the orbital maneu?ver method［18］，the camera bias method［19］，the nonlinear dynamics method［20］，and the multi-satellite multi-sensor method［21］.The camera bias method is only suitable for close ranges；the orbital maneuver method will increase the fuel cost of the satellite platform；and the multi-satellite multi-sensor meth?od has the disadvantage of complicated hardware configuration.

In view of the problems of the above-men?tioned methods，this paper intends to improve the observability of the angles-only navigation system through capturing the non-linearity of the system in the evolution of the relative motion，and an anglesonly navigation architecture using non-linear dynam?ics method is proposed.To solve the problems of angles-only navigation，it is necessary to establish a relative dynamics model and a corresponding LoS measurement equation.First，this paper introduces three spacecraft relative dynamics models，including the rectilinear state relative dynamics model，the curvilinear state relative dynamics model，and the ROE state relative dynamics model.Then，the LoS measurement equation corresponding to each rela?tive dynamics model is established.Second，an ob?servability analysis theory based on the Gramian ma?trix is introduced to determine which relative dynam?ics model can maximize the observability of anglesonly navigation.Third，an adaptive extended Kal?man filtering scheme is proposed to solve the prob?lem that the angles-only navigation filter using nonlinear dynamics method is sensitive to the measure?ment noises.Finally，the performances of the an?gles-only navigation architecture are tested in a highfidelity simulation environment，which verifies the effectiveness of the proposed angles-only navigation architecture.

1 Relative Dynamics Models

The relative dynamics model describes how the relative motion states between the chaser spacecraft and the space target evolve over time.In order to meet the requirements of on-board implementation，the relative dynamics model should be simplified as much as possible to save the resources of the onboard computer.The simplified form of the relative dynamics model established in this paper can be ex?pressed as

wherex（t0）is the relative motion state between the chaser spacecraft and the space target at the initial timet0；andΦ（tf，t0）the state transition matrix（STM）.

Next，three types of relative dynamics models will be introduced，including the rectilinear state rel?ative dynamics model，the curvilinear state relative dynamics model，and the ROE state relative dynam?ics model.

1.1 Rectilinear state relative dynamics model

For the modeling of the rectilinear state relative dynamics，it is necessary to establish a Cartesian frame with the centroid of the chaser spacecraft as the origin.In this Cartesian frame，thex-axis is aligned with the radial direction（R）of the chaser spacecraft；they-axis is aligned with the tangential direction（T）of the chaser spacecraft；and thez-ax?is is aligned with the angular momentum direction（N）of the chaser spacecraft.The rectilinear state vectorxrecincludes the rectilinear relative position vectorrrecand the rectilinear relative velocity vec?torvrec

wherex，y，andzrepresent the rectilinear relative position components in theR，T，andNdirections，respectively；andvx，vy，andvzthe rectilinear rela?tive velocity components in theR，T，andNdirec?tions，respectively.

For the unperturbed near-circular orbits，the Hill-Clohessy-Wiltshire（HCW）equation can be used to establish the rectilinear state relative dynam?ics model，and its STM can be given as

where

wherenis the mean motion of the chaser space?craft；andτ=tf-t0the time interval.

1.2 Curvilinear state relative dynamics model

In addition to the rectilinear state mentioned above，the curvilinear state can also be used to de?scribe the relative motion states between the chaser spacecraft and the space target，and it can better capture the orbital curvature of the relative motion.The curvilinear state vectorxcurincludes the curvilin?ear relative position vectorrcurand the curvilinear rel?ative velocity vectorvcur，where the curvilinear rela?tive position vectorrcurincludes the orbital radius dif?ferencer，the angular in-plane separationθ，and the angular out-of-plane separationψ，as shown in Fig.1.For a near-circular orbit，in order to ensure the consistency of dimensions，it is necessary to multiply the angular in-plane separationθand the an?gular out-of-plane separationψby the orbital semimajor axisa，namely

Fig.1 shows the relationship between the curvi?linear state vectorxcurand the absolute position vec?torρof the chaser spacecraft and the space target.Since the first-order curvilinear state and rectilinear state relative dynamics models are the same in the near-circular orbits［22］，the STM in Eqs.（3—7）can also be used to propagate the curvilinear relative mo?tion states.

Fig.1 Definition of curvilinear state

1.3 ROE state relative dynamics model

In addition，a set of mean ROE can be used to describe the relative motion states between the chas?er spacecraft and the space target.The parameter?ized form of the mean ROE used in this paper is giv?en as［23］

wherea，e，i，Ω，ω，andMare the classic Kepler orbital elements；δarepresents the relative semi-ma?jor axis，δλthe relative mean longitude，u=M+ωthe mean argument of latitude，δe=（δex，δey）Tthe relative eccentricity vector，andδi=（δix，δiy）Tthe relative inclination vector.

The set of ROE is particularly suitable for the problems of angles-only navigation because the weakly observable inter-satellite distance is almost equal to the componentaδλ.The relative dynamics model established based on ROE is effective for cir?cular orbits（i.e.，ec=0），but it is still singular for equatorial orbits（i.e.，ic=0）［24］.For orbits with ar?bitrary eccentricity，the STM defined by the mean ROE includingJ2perturbations，the atmospheric drag，and the solar radiation pressure，and the third body gravity is obtained.In the case of onlyJ2per?turbations，the STM over the time interval［t0，tf］is expressed as［25］

For a near-circular orbit（i.e.，ec≈0），the STM given by Eq.（7）can be simplified as［26］

The argument of latitudeuof the chaser space?craft could also be used as the independent variable instead of time，and theτin Eq.（11）can also be ex?pressed as［24］

where Δuis the change in the argument of the lati?tude of the chaser spacecraft over the time interval［t0，tf］.

2 LoS Measurement Equations

The LoS vectors of the space target relative to the chaser spacecraft are modeled as a function of state variables，and the general form of the non-lin?ear measurement modelhis recorded as

where the non-linear measurement modelhdepends on the state variablex.

2.1 Rectilinear state LoS measurement equa?tion

Before defining the LoS measurement vectors，the camera frame“cam”needs to be defined.With?out loss of generality，it is assumed that the camera fixed boresight is aligned with the anti-flight direc?tion.Under this assumption，the relationship be?tween the relative position vectors in the camera frame“cam”and the Cartesian frame“car”are giv?en as

The LoS vector could be represented by a set of azimuthαand elevationε，which can be ex?pressed as a function of the relative position vectorrcamin the camera frame，as shown in Fig.2.Thus，the rectilinear state LoS measurement equation can be expressed as

Fig.2 LoS measurement geometry

2.2 Curvilinear state LoS measurement equa?tion

The LoS vector defined by the curvilinear statexcuris calculated through mapping the curvilinear statexcurto the rectilinear statexrecand applying Eqs.（14—15）.The mapping relationship between the rectilinear state and the curvilinear state is given as

2.3 ROE state LoS measurement equation

In order to define the LoS vector by the ROE state，it is necessary to map the ROE state to the rectilinear state.Four different mapping relation?ships are considered in this paper：（1）The ROE state is linearly mapped to the rectilinear state，which is recorded as the measurement modelh1；（2）the ROE state is linearly mapped to the curvilinear state，and then the curvilinear state is mapped to the rectilinear state，which is recorded as the measure?ment modelh2；（3）the ROE state is non-linearly mapped to the rectilinear state，which is recorded as the measurement modelh3；（4）the ROE state is non-linearly mapped to the rectilinear state with the transformation from the mean orbital elements to the osculating orbital elements［27］，which is recorded as the measurement modelh4.For the measurement modelh3，the ROE state and the absolute orbital el?ements of the chaser spacecraft are used to calculate the absolute orbital elements of the space target.Then，the absolute position vectors of the chaser spacecraft and the space target are calculated in the geocentric inertial frame according to their absolute orbital elements.Finally，the relative position vec?tors in the Cartesian frame are obtained according to the absolute position vectors of the chaser spacecraft and the space target in the geocentric inertial frame.The linear map between the ROE state and the recti?linear or curvilinear state is formulated by treating the ROE as the integration constant of the HCW equation［28］.If necessary，Eq.（16）can be used to map the curvilinear state to the relative position vec?tor in the Cartesian frame.The mapping relationship between the ROE state and the translational state is given as

3 Observability Analysis of Angles?Only Navigation Based on Gramian Matrix

An observability analysis theory based on the Gramian matrix is introduced to determine which relative dynamics model could maximize the ob?servability of angles-only navigation.Assume that the relative motion state at timet0isx（t0），andkLoS measurements are performed between the timet1andtk，we can obtain the measurement vec?torZas

In order to prove that the angles-only naviga?tion system has local weak observability，it can be proved that the partial derivative matrixχof the LoS measurements relative to the disturbance state at the reference time is full rank，andχcan be ex?pressed as

where

In addition，the observability of angles-only navigation can also be judged according to the condi?tion number of the Gramian matrix，which is ex?pressed as cond（χTχ）.The larger the cond（χTχ），the worse the observability of angles-only naviga?tion.

3.1 Rectilinear state measurement sensitivity matrix Hrec

In order to calculate the measurement sensitivi?ty matrixHrecof the LoS measurements relative to the rectilinear state，the measurement sensitivity matrix of the LoS measurements relative to the rela?tive position vector in the camera frame needs to be calculated.According to the Ref.［29］，it is given as

Using Eq.（14）to map the partial derivative ob?tained from Eq.（21）to the Cartesian frame，we ob?tain the rectilinear state measurement sensitivity ma?trixHrecas

3.2 Curvilinear state measurement sensitivity matrix Hcur

The curvilinear state measurement sensitivity matrixHcuris the product of the rectilinear state mea?surement sensitivity matrixHrecevaluated at the rela?tive position vector computed using Eq.（16）and the partial derivative of the rectilinear state with re?spect to the current curvilinear state，namely

where

3.3 ROE state measurement sensitivity matrix HROE

For the case of linearly mapping the ROE state to the rectilinear state，the measurement sensitivity matrixHrecROEis the product of the rectilinear state measurement sensitivity matrixHrecevaluated atΠ（t）xROEand the partial derivative of the rectilinear state with respect to the ROE state，namely

For the case of linearly mapping the ROE state to the curvilinear state，the curvilinear state is then mapped to the rectilinear state.The measurement sensitivity matrixis the product of the curvilin?ear state measurement sensitivity matrixHcurevalu?ated atΠ（t）xROEand the partial derivative of the curvilinear state with respect to the ROE state，namely

For the case of non-linearly mapping the ROE state to the rectilinear state，the absolute orbital ele?ments of the space target is calculated by the ROE state and the absolute orbital elements of the chaser spacecraft to obtain the relative position vector in the Cartesian frame，and the nonlinear mapping rela?tionship is denoted asg（xROE（t），eco（t））.The mea?surement sensitivity matrixis the product of the rectilinear measurement sensitivity matrixHrecevaluated atg（xROE（t），eco（t））and the partial deriv?ative of the rectilinear state with respect to the ROE state，namely

Finally，the measurement sensitivity matrix in Eqs.（25—27）could be approximated as follows

4 Adaptive Extended Kalman Filter

Extended Kalman filter（EKF）［30］is used to perform real-time filtering of the relative motion states between the chaser spacecraft and the space target.EKF uses a linear approach to transform a nonlinear system into an approximate linear sys?tem.The statistics of the process and measure?ment noises are critical to the stability and perfor?mances of the EKF.Thus，proper selection of the statistics of the process and measurement noises can improve the accuracy of dynamic relative state estimation.

This paper introduces the“innovation”covari?ance matching method［31］to adjust the measurement noises online to solve the problem that the anglesonly navigation filter using non-linear dynamics method is sensitive to the measurement noises.The basic idea of the“innovation”covariance matching method is to use the statistical and theoretical values of the sample to infer the measurement noises of the Kalman filter and make adaptive adjustments.The measurement prediction error of the Kalman filter is also called“innovation”or“residual”［32］can be giv?en by

By substituting Eq.（30）into Eq.（29），one can obtain“innovation”as

On the other hand，if a sample ofNinnova?tions is collected in a sliding window，the sample co?varianceof the sample can be calculated as

According to the equivalent relationship of Eq.（32）and Eq.（33），the estimation of the mea?surement noise covariance matrixis calculated as

Then

Finally，the corrected measurement noise cova?riance matrix is obtained as

5 Numerical Simulation

5.1 Problem configuration

The theoretical relative motion states between the chaser spacecraft and the space target are numer?ically propagated using a 20 × 20 gravity field and includingJ2perturbations，the solar radiation pres?sure，the third-body and atmospheric drag perturba?tions.A set of LoS measurements is finally created from the theoretical relative motion states.In this paper，the performances of the proposed angles-on?ly navigation architecture are tested by means of nu?merical simulations.The numerical simulation is mainly carried out in three typical orbital scenarios：ROE1 which represents a possible configuration for the beginning of an approach to an non-cooperative space target，ROE2 which presents a drift of almost 1 km per orbit toward an non-cooperative space tar?get；ROE3 which represents the starting point of a docking phase.The main simulation parameters are shown in Table 1.The relative motion trajectories are shown in Fig.3.

Table 1 Main simulation parameters

Fig.3 Relative trajectories in three typical orbital sce?narios

5.2 Simulation results and analysis

5.2.1 Simulation results of the observability analysis of angles?only navigation

This study compares and analyzes the observ?ability of angles-only navigation under three relative dynamics models，including the rank of the matrixχand the condition number cond（χTχ）of the Gramian matrix.In the process of analyzing the observability of angles-only navigation，the variables include not only various relative dynamics and measurement models，but also the sampling period and total sam?pling time.First，a comparative analysis of the ob?servability of angles-only navigation under the recti?linear and curvilinear state relative dynamics models is carried out，and the results are shown in Tables 2—4.Table 2 shows the results of the observability analysis of angles-only navigation when the sam?pling period is 10 s and the total sampling time is 1 000 s under the rectilinear and curvilinear state rel?ative dynamics models，including the rank of the matrixχand the condition number cond（χTχ）of theGramian matrix；Table 3 shows the condition num?ber cond（χTχ）of the Gramian matrix under the rec?tilinear and curvilinear state relative dynamics mod?els，the total sampling time is 1 000 s，and the dif?ferent sampling periods；Table 4 shows the condi?tion number cond（χTχ）of the Gramian matrix under the rectilinear and curvilinear state relative dynamics models，the sampling period is 10 s，and the differ?ent total sampling time.

Table 2 Observability analysis of angles?only navigation when the sampling period is 10 s and the total sampling time is 1 000 s under the rectilinear and curvilinear state relative dynamics models

Table 3 Condition number cond(χTχ) of the Gramian matrix under the rectilinear and curvilinear state relative dynam?ics models with the total sampling time of 1 000 s and different sampling periods

Table 4 Condition number cond(χTχ) of the Gramian matrix under the rectilinear and curvilinear state relative dynam?ics models with the sampling period of 10 s and different total sampling time

It can be seen from Table 2 that the matrixχunder the rectilinear state relative dynamics model is not full rank，so the rectilinear state relative dynam?ics model cannot provide complete observability for angles-only navigation.In addition，comparing and analyzing the condition number cond（χTχ）of the Gramian matrix，it can be concluded that the observ?ability of angles-only navigation in ROE1 and ROE2 is better than that in ROE3.In addition，ma?trixχunder the curvilinear state relative dynamics model is full rank，so the curvilinear state relative dynamics model can provide complete observability for the angles-only navigation system.It can be seen from Table 3 that when other conditions are fixed，the sampling period has a relatively small effect on the observability of angles-only navigation，and the maximum and minimum condition number of the Gramian matrix do not exceed one order of magni?tude，which indicates that the performances of the angles-only navigation system are robust against low sampling rates or short-term data interruptions.It can be seen from Table 4 that when other condi?tions are fixed，the greater the total sampling time，the better the observability of the angles-only navi?gation system.Thus，the observability of angles-on?ly navigation can be improved by increasing the total sampling time.

In addition，the observabilities of angles-only navigation under the ROE state relative dynamics model with four measurement models are compared and analyzed，and the results are shown in Tables 5—6.Table 5 shows that in the results of the ob?servability analysis of angles-only navigation under the simplified and complex STM with the measure?ment modelh1，the sampling period is 10 s，and the total sampling time is 1 000 s；Table 6 shows the re?sults of the observability analysis of angles-only nav?igation under the simplified STM with four measure?ment models.The sampling period is 10 s，and the total sampling time is 1 000 s.

It can be seen from Table 5 that the matrixχunder the measurement modelh1is not full rank，so the angles-only measurement navigation system is not completely observable.In addition，the differ?ence between the condition number of the Gramian matrix under the complex and simplified STM is very small.Thus，in order to save the resources of on-board computer，it is completely reasonable to use a simplified STM for near-circular orbits.It can be seen from Table 6 that the measurement modelsh2，h3，andh4can provide complete observability for the angles-only navigation system，and the condi?tion number of the Gramian matrix under the mea?surement modelh4is smaller than those of the other three scenarios.Thus，the transformation from the mean orbital elements to the osculating orbital ele?ments can also improve the observability of anglesonly navigation.

Table 5 Observability analysis of angles?only navigation under the simplified and complex STM with the measurement model h1 when the sampling period is 10 s,and the total sampling time is 1 000 s

Table 6 Observability analysis of angles?only navigation under the simplified STM with four measurement models when the sampling period is 10 s and the total sampling time is 1 000 s

5.2.2 Simulation results of the angles?only navi?gation filter

This paper also compares and analyzes the per?formances of the angles-only navigation filter under different relative dynamics and measurement models in ROE1，and the results are shown in Figs.4—7.Fig.4 shows the estimation errors of the angles-only navigation filter under the rectilinear state relative dynamics model.Fig.5 shows the estimation errors of the angles-only navigation filter under the curvilin?ear state relative dynamics model.The estimation errors of the angles-only navigation filter under the ROE state relative dynamics model with four mea?surement models are compared and analyzed，and the results are shown in Fig.6.In addition，the esti?mation errors of the angles-only navigation filter un?der the rectilinear state relative dynamics model and the ROE state relative dynamics model with the measurement modelh4are compared and analyzed，and the results are shown in Fig.7.

It can be seen from Fig.4 that the angles-only navigation filter under the rectilinear state relative dynamics model cannot converge to the true intersatellite range，and it exhibits continuous oscillation errors during the entire simulation process.It can be seen from Fig.5 that the angles-only navigation filter under the curvilinear state relative dynamics model can slowly converge to the true inter-satellite range，and the errors of the other state elements show the same oscillation as the orbital period.In addition，the oscillation amplitude of the errors of the relative velocity in tangential direction is twice the oscilla?tion amplitude of the errors of the relative velocity in radial direction，and they are out of phase with each other.Thus，the above results show that the geo?metric shape of the relative motion are observable.

Fig.4 Estimation errors of the angles-only navigation filter under the rectilinear state relative dynamics model

Fig.5 Estimation errors of the angles-only navigation filter under the curvilinear state relative dynamics model

It can be seen from Fig.6 that under the ROE state relative dynamics model，the estimation errors of the angles-only navigation filter with the measure?ment modelh1cannot converge to the trueaδλ，while the angles-only navigation filter with the mea?surement modelsh2，h3，andh4can converge to the trueaδλ，and the angles-only navigation filter with the measurement modelsh3andh4can reduce the es?timation errors ofaδλto a few hundred meters.In addition，it can be seen from Fig.6 that the transfor?mation from the mean orbital elements to the oscu?lating orbital elements can also improve the perfor?mances of the angles-only navigation filter，which is also consistent with the conclusions drawn by the observability analysis theory of angles-only naviga?tion based on the Gramian matrix.It can be seen from Fig.7 that the performances of the angles-only navigation filter under the ROE state relative dy?namics model with the measurement modelh4is sig?nificantly better than that under the rectilinear state relative dynamics model.The convergence rate is faster and the navigation accuracy is higher.

Fig.6 Estimation errors of the angles-only navigation filter under the ROE state relative dynamics model with four mea?surement models

Fig.7 Estimation errors of the angles-only navigation filter under the rectilinear state relative dynamics model and the ROE state relative dynamics model with the measurement model h4

The above results further confirm the conclu?sions drawn by the observability analysis theory of angles-only navigation based on the Gramian ma?trix.That is，capturing the non-linearity of the sys?tem in the evolution of relative motion can improve the performances of the angles-only navigation fil?ter.It demonstrates that the angles-only navigation filtering scheme without orbital or attitude maneu?vering is completely feasible.

6 Conclusions

This paper proposes an angles-only navigation architecture using non-linear dynamics method for non-cooperative rendezvous of spacecraft.The an?gles-only navigation architecture improves the ob?servability of angles-only navigation through captur?ing the non-linearity of the system in the evolution of relative motion.Then，the performances of the angles-only navigation architecture are tested in a high-fidelity simulation environment，and the fol?lowing conclusions are drawn：

（1）The rectilinear state relative dynamics model cannot provide complete observability for an?gles-only navigation，while the curvilinear state rela?tive dynamics model can.Thus，the curvilinear state relative dynamics model provides a novel scheme to solve the problem of poor observability of angles-only navigation.

（2）When other conditions are fixed，the sam?pling period has a relatively small effect on the ob?servability of angles-only navigation.However，the condition number of the Gramian matrix can be re?duced through increasing the total sampling time，so as to improve the observability of angles-only navigation.

（3）In the case of the ROE state relative dy?namics model，including the non-linear mapping re?lationship from the ROE state to the rectilinear state and the transformation from the mean orbital ele?ments to the osculating orbital elements in the LoS measurement equations can improve the observabili?ty of angles-only navigation.

Finally，the comparative analysis of the results of the performances of the angles-only navigation fil?ter under different relative dynamics models further confirm the conclusions drawn through the observ?ability analysis theory based on the Gramian matrix.

In a nutshell，this paper demonstrates that the angles-only navigation filtering scheme without or?bital or attitude maneuvering is completely feasible through improving the modeling of the relative dy?namics and LoS measurement equations.Thus，the angles-only navigation architecture using non-linear dynamics method proposed in this paper could pro?vide theoretical guidelines for any missions that use angles-only navigation for non-cooperative rendez?vous of spacecraft.