Interplanetary Autonomous Navigation using Visible

Interplanetary Autonomous Navigation using Visible Planets Reza Raymond Karimi∗ and Daniele Mortari† Texas A&M University, College Station, Texas 77843-3141

I.

Introduction

Space-based orbit determination problem using line-of-sight measurements appears in two equivalent cases: 1) to estimate the observer’s orbit when observing a known planetary body (navigation problem) and 2) to determine the orbit of an unknown observed body when the observer’s orbit is known (surveillance problem). These two problems are mathematically equivalent. Star trackers provide attitude autonomy to interplanetary missions while the estimation of position still relies on communications to Earth’s ground stations. This paper shows that, in interplanetary missions, autonomy for position estimation can be obtained just by observing visible planets using star trackers. In any space mission and, in particular interplanetary missions, autonomy plays a critical role for the success of the mission, especially if temporary or permanent failure occurs in the communication system. This study first solves the position and velocity (state) estimation problem as a single-point problem using the Pn space-based Initial Orbit Determination (IOD) algorithm8 and then uses this estimation to initiate a filtered estimation of the state using Extended Kalman Filter (EKF). Space observations of line-of-sight directions constitute the measurements of two distinct problems: when estimating the orbit of an unknown observed body (space-based orbit determination problem) and when estimating the observer’s orbit when the observed body is known (navigation problem). There is a perfect duality between these two problems as they share the same mathematics meaning that any method capable of solving one problem can be used to solve the other. However, when the distances between observer and observed become large, as in interplanetary missions, the problem becomes more complicated. In fact, ∗

Texas A&M University Affiliate. E-mail: [email protected] Professor, 746C H.R. Bright Bldg, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, AAS Fellow, AIAA Associate Fellow. IEEE Senior member. E-mail: [email protected] †

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in an interplanetary navigation problem, because of the large distances between the two orbiting bodies and also because of the velocity of the observer, two issues arise which need attention, namely a) the light-time effect, and b) the light aberration effect. The light-time effect is generated by the elapsed (finite) time for the light to reach the observer, while the light aberration effect is an apparent shift in the position of the observed object due to the velocity of the observer. For the latter effect, a (relativistica ) velocity-addition formula is used to compensate for the light aberration. If ignored, these two effects will generate errors in the estimation of the lines-of-sight directions affecting the positioning estimation problem. Therefore, in addition to the measurement noise caused by the star tracker sensor (centroid estimation accuracy), the light-time and the light aberration constitute two additional source of measurements deviations that need to be corrected. This article first summarizes the Pn initial orbit determination algorithm and then, in the subsequent two sections, it describes how to implement the light-time and the light aberration corrections in both Pn IOD and EKF.

II.

Initial Orbit Determination Based on Prescribed Orbits

The geometry of a typical orbit determination problem with Nobs observations can be written as rk = Rk + ρk ρˆk , k = 1, 2, · · · , Nobs (1) where rk and Rk are the k-th spacecraft and tracking site position vectors, respectively. The unknown range ρk is the one to be determined while ρˆk is the measured direction (unitvector) to the spacecraft. The main idea of the Pn technique is to assume a solution to the Keplerian two-body problem and enforce the approximated solution to satisfy the geometry of the problem. The approximated solution can be a polynomial or any other function best describing that part of orbit under determination. The algorithm is first developed with a 3-rd order polynomial to describe the methodology, but the results are generated using a 5-th order polynomial. The trajectory of an orbiting object should satisfy Keplerian two-body problem. Let us now assume an approximated solution in the form of     x(t) = px (t) y(t) = py (t)

  

(2)

z(t) = pz (t)

a

In the restricted relativity theory, any source of light direction appears deflected toward the observer’s velocity direction. In particular, when the observer velocity becomes the speed of light all light directions appear in the velocity direction.

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and by considering a 3-rd order polynomial, we have  2 3    px (t) = a1 + a2 t + a3 t + a4 t py (t) = b1 + b2 t + b3 t2 + b4 t3

  

(3)

pz (t) = c1 + c2 t + c3 t2 + c4 t3

The approximated solution p(t) should also be able to satisfy the geometry Eq. (1) so we have  ∼    px (tk ) = Rxk + ρxk ρˆxk (4) py (tk ) ∼ k = 1, 2, · · · , Nobs = Ryk + ρyk ρˆyk    pz (tk ) ∼ = Rzk + ρzk ρˆzk where Rxk indicates the first component of Rk (x) (and similar in y and z), ρˆxk indicates the first component of ρˆk (x) (and similar in y and z), and where tk is the k-th measured time. By taking the left hand side of Eq. (4) to the right, we can define the residuals as     φxk = −px (tk ) + (Rxk + ρxk ρˆxk ) φyk = −py (tk ) + (Ryk + ρyk ρˆyk )

  

k = 1, 2, · · · , Nobs

(5)

φzk = −pz (tk ) + (Rzk + ρzk ρˆzk )

and     Φx =        Φy =           Φz =

N

obs 1X φ2 2 k=1 xk Nobs 1X φ2yk 2 k=1 Nobs 1X φ2 2 k=1 zk

(6)

To have the best fit, the residuals Φx , Φy , and Φz should be minimum with respect to the polynomial coefficients ai , bi , and ci , i = 1, 2, 3, 4 (3-rd order polynomial here)  ∂Φx   =0   ∂ai   ∂Φ y =0  ∂bi   ∂Φz    =0 ∂ci

i = 1, 2, 3, 4

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(7)

So by applying the least square scheme, the coefficients can be determined in terms of the unknown ranges as  PNobs PNobs PNobs −1  {a} = T [ (R + ρ ρ ˆ ) (R + ρ ρ ˆ )t ... ˆxk )t3k ]T  xk xk xk xk xk xk k k=1 k=1 k=1 (Rxk + ρxk ρ  P P P Nobs Nobs obs {b} = T −1 [ N ˆyk ) ˆyk )tk ... ˆyk )t3k ]T k=1 (Ryk + ρyk ρ k=1 (Ryk + ρyk ρ k=1 (Ryk + ρyk ρ   P P P  Nobs Nobs obs {c} = T −1 [ N ˆzk ) ˆzk )tk ... ˆzk )t3k ]T k=1 (Rzk + ρzk ρ k=1 (Rzk + ρzk ρ k=1 (Rzk + ρzk ρ (8) where {a} = {a1 a2 a3 a4 }T , {b} = {b1 b2 b3 b4 }T , {c} = {c1 c2 c3 c4 }T , and T is the 3 × 3 matrix  PNobs PNobs 3  N k=1 tk ... k=1 tk   P obs  Nobs t PNobs t2 ... PNobs t4   k k=1 k  k=1 k (9) T =  k=1    ... ... ... ...   PNobs 6 PNobs 3 PNobs 4 k=1 tk ... k=1 tk k=1 tk Up to now, the polynomial coefficients have been eliminated and we have them in terms of the unknown ρi . The next step towards completing the algorithm is to enforce the assumed solution to satisfy the Keplerian two-body problem. For Nobs observations, we have  µpx (tk )   p¨x (tk ) ∼ =−   2  [px (tk ) + py (tk )2 + pz (tk )2 ]3/2   µpy (tk ) p¨y (tk ) ∼ , =− 2  [px (tk ) + py (tk )2 + pz (tk )2 ]3/2    µpz (tk )   =−  p¨z (tk ) ∼ 2 [px (tk ) + py (tk )2 + pz (tk )2 ]3/2

k = 1, 2, · · · , Nobs

(10)

where µ is the gravitational constant and px , py , and pz are the polynomials approximating x, y, and z coordinates of the observed object, respectively. Equation (10) can be written in the residual form as  µpx (tk )   ψ = p ¨ (t ) +  xk x k   [px (tk )2 + py (tk )2 + pz (tk )2 ]3/2   µpy (tk ) ψyk = p¨y (tk ) + , k = 1, 2, · · · , Nobs (11) 2  [px (tk ) + py (tk )2 + pz (tk )2 ]3/2    µpz (tk )    ψzk = p¨z (tk ) + 2 [px (tk ) + py (tk )2 + pz (tk )2 ]3/2 We can define the the total residual vector (containing all observations) as Ψ = [Ψ1 , Ψ2 , ..., ΨNobs ]T , where Ψk = [ψxk , ψyk , ψzk ] is defined in Eq. (11). The residual Ψ is a 3Nobs × 1 vector. Let us write Ψ in terms of the Taylor’s series expansion up to the 2-nd order, so we have ∂Ψ 1 ∂ 2Ψ ]j + 4ρTj [ 2 ]j 4ρj Ψj+1 ∼ = Ψj + 4ρTj [ ∂ρ 2 ∂ρ 4 of 14

(12)

which yields to the iterative solution ρj+1 = ρj −

Jj−1

  1 −1 T −1 ψj + (Jj ψj ) Hj (Jj ψm ) 2

(13)

or up to the 1-st order ρj+1 = ρj − (JjT W Jj )−1 JjT W ψj

(14)

where index j is the number of iteration, Jj , Hj , and W are Jacobian (first derivative ˆ Hessian (second of the residual vector Ψ with respect to the unknown range vector ρ), ˆ and weight matrices, derivative of the residual vector Ψ with respect to the unknown range ρ), respectively. The matrices J and H have dimensions 3Nobs × Nobs and 3Nobs × Nobs × Nobs , respectively. Since the algorithm is based on prescribed orbits, so it is called Pn , P referring to prescribed and n as the number of observations, Nobs . For more details and results regarding this method, see8 and.9

III.

Light-time Correction

Light-time correction is a displacement in the apparent position of a celestial object from its true position (or geometric position) caused by the object’s motion during the time it takes its light to reach an observer, see1 and2 for more on light-time correction. The effect of finite light speed starts playing a role in the accuracy of the solution ρk , k = 1, 2, ..., Nobs as we are dealing with more distant objects than earth-orbiting satellites, specifically an interplanetary mission. For example, it takes light several minutes to reach a typical spacecraft traveling in space (for instance between Mars and Jupiter) that is observing Earth for navigation. Assume that the observations are made at times tk , k = 1, 2, · · · , Nobs with the respective measured directions, ρˆk , k = 1, 2, · · · , Nobs . Since the spacecraft does not have any knowledge of how far it is from the Sun and correspondingly from the Earth, so the known position vectors of the planet, Rk , and the measured directions, ρˆk , k = 1, 2, · · · , Nobs , will be considered the ones at times tk , k = 1, 2, · · · , Nobs while these belong to the time tk − δtk where δtk is the time that takes light to reach the spacecraft. To fix this problem, a modification in the algorithm needs to be made. The first step would be determining the δt as ρk (15) δtk = c where c is the speed of light. The next step would be updating the planet position vector as Rupdated = R(tk − δtk )

(16)

The position vector can be updated using Eq. (15) and Eq. (16). This process is repeated 5 of 14

until the desired accuracy is achieved.

IV.

Light Aberration Correction

The term “aberration” has historically been used to refer to a number of related phenomena concerning the propagation of light in moving bodies.3 Relativistic aberration is described by Einstein’s special theory of relativity, and in other relativistic models such as Newtonian emission theory. It results in aberration of light when the relative motion of observer and light source changes the position of the light source in the field of view of the observer. The effect is independent of the distance between observer and light source. The aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects. At the instant of any observation of an object, the apparent position of the object is displaced from its true position by an amount which depends upon the transverse component of the velocity of the observer, with respect to the vector of the incoming beam of light (i.e., the line actually taken by the light on its path to the observer). Light aberration is independent of the distance of a celestial object from the observer, and depends only on the observer’s instantaneous transverse velocity with respect to the incoming light beam, at the moment of observation. The light beam from a distant object cannot itself have any transverse velocity component, or it could not (by definition) be seen by the observer, since it would miss the observer. Thus, any transverse velocity of the emitting source plays no part in aberration. Another way to state this is that the emitting object may have a transverse velocity with respect to the observer, but any light beam emitted from it which reaches the observer, cannot, for it must have been previously emitted in such a direction for which its transverse component has been “corrected”. Such a beam must come “straight” to the observer along a line which connects the observer with the position of the object when it emitted the light. In our problem, the light aberration is also an issue which needs to be corrected. Assume that the angle between the spacecraft velocity and the observed direction, ρˆobs , is θobs and the aberration angle (between the true and observed directions) is ε, so the corrected (true) planet observed direction could be determined as ρˆtrue =

ρˆobs sin θtrue − vˆ sin ε sin θobs

(17)

where vˆ is the spacecraft velocity unit vector and ϑtrue is the angle between the velocity and true observation direction which is θtrue = θobs + ε. The aberration angle ε, at any time, can be obtained as p (c/v) 1 − (ρˆTobs vˆ)2 (v/c) sin θobs tan ε = = (18) T 1 − (c/v)(ρˆobs vˆ) 1 − (v/c) cos θobs 6 of 14

where c and v are the speed of light and spacecraft velocity magnitude respectively. Note: The combination of light aberration and light-time correction is called planetary aberration. The relativistic aberration is basically the light aberration including the special relativity. According to Einstein’s special relativity theory, light directions are affected by the aberration toward inertial velocity vector. The apparent (observed) and true light directions can be related as cos θtrue + (v/c) (19) cos θobs = 1 + (v/c) cos θtrue The apparent angle between the observed planet direction and spacecraft velocity vector is known, and the true direction can be obtained using Eq. (19). This ends the last modification we need to apply to the orbit determination (spacecraft navigation) algorithm. See4 and5 for more on relativistic aberration. Implementing the Corrections The two corrections (light-time and light abberation effect) are implemented in IOD algorithm simultaneously. A brief description is presented here: the inputs to the angle-only IOD routine are the known position vectors of the observed visible planet Rk , measured line-ofsights ρˆk , and guessed values of the unknown ranges ρk where k = 1, 2, ..., Nobs . Once the IOD algorithm solves for the ranges, the estimated ranges are used to compute the estimated position and velocity vectors of the spacecraft, rk and vk respectively. The estimated range ρk then is used to compute the estimated time-delay δt (the time that takes light to get to the spacecraft from the planet) using Eq. (15) and consequently an update of the position of the visible planet is made through Eq. (16). At the same time, the estimated velocity of the spacecraft is used to estimate the angle between the velocity vector and true (updated) line-of-sight vector θtrue using Eq. (19).The true (updated) line-of-sight vector then can be determined using Eq. (17). Once the updated position vector of the planet Rupdated and updated line of-sight vector ρˆupdated are computed, the spacecraft position vector can be obtained using Eq. (1). The flowchart of the corrections implementation is illustrated in Fig.(1). Note that the IOD method used for the purpose is Pn which was descried earlier in the section.

V.

Extended Kalman Filter Implementation

Up to now, we have developed some modifications through which an initial orbit determination technique can be applied to an interplanetary orbit determination scenario. To have an autonomous navigation system, we need to perform a sequential state estimation so we can have the real-time position and velocity of the spacecraft. To this end, the Ex-

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Figure 1. Light-time and Light Aberration Effect Correction in the IOD Algorithm

tended Kalman Filter (EKF) is employed as the estimation problem under study involves nonlinear model. Considering a continuous-time truth nonlinear model and discrete-time measurement, we have   X(t) ˙ = f (X(t), u(t), t) + G(t)w(t)  y˜i = h(Xi ) + νi

(20)

where x is the system state vector, y˜ is the measurement, h(x) are measurement functions, f is the system, u is the input control, and G is the model error. And also, ν and w are zero-mean Gaussian white-noise processes meaning that the errors are not correlated forward or backward in time. Their covariances are given by   0, i 6= j E{νi νjT } =  Ri , i = j

(21)

E{w(t)wT (τ )} = Q(t)δ(t − τ )

(22)

and

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and since ν and w are uncorrelated E{νk wT (tk )} = 0

(23)

A summary of the continuous-discrete extended Kalman filter can be found in.6 Now let us apply extended Kalman filter to our problem. The system under study is wellknown and the truth model is available. For the spacecraft traveling in the solar system under the Sun gravitational field, the motion is governed by Keplerian two-body equations. Note that no perturbations (solar pressure and gravity from other celestial bodies) are considered in this formulation, so we have µ (24) r¨ = − 3 r r where µ is the Sun gravitational constant and r is the spacecraft position vector. Defining the state vector X = [x y z x˙ y˙ z] ˙ T , the system dynamic would be

X˙ = f (X(t)) =

              

x˙ y˙

              

z˙ µ − 2 x  (x + y 2 + z 2 )3/2        µ     − y     2 2 2 3/2   (x + y + z )     µ     z − 2  (x + y 2 + z 2 )3/2

(25)

˜ contains of azimuth φ and elevation ϑ angles The input measurement vector y

y˜ =

  ϑ˜ φ˜

=

  ϑ + ν  ϑ

(26)

φ + ν φ 

where νϑ and νφ are the measurement noise associated with the elevation and azimuth angles respectively. The elevation and azimuth angles can be computed as    x − Rx  −1   ϑ = sin )2 + (y − Ry )2 + (z − Rz )2 ]1/2 [(x − Rx y − Ry    φ = tan−1 x − Rx For matrix H we have

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(27)

∂ϑ ∂h  ∂x H= =  ∂φ ∂X ∂x 

∂ϑ ∂y ∂φ ∂y

 ∂ϑ 0 0 0  ∂z  0 0 0 0

(28)

and components of matrix H                               

∂ϑ ∂x ∂ϑ ∂y ∂ϑ ∂z ∂φ ∂x ∂φ ∂y

(x − Rx )(z − Rz ) [(x − Rx + (y − Ry + (z − Rz )2 ][(x − Rx )2 + (y − Ry )2 ]1/2 (y − Ry )(z − Rz ) =− 2 2 [(x − Rx ) + (y − Ry ) + (z − Rz )2 ][(x − Rx )2 + (y − Ry )2 ]1/2 [(x − Rx )2 + (y − Ry )2 ]1/2 =− (x − Rx )2 + (y − Ry )2 + (z − Rz )2 y − Ry =− (x − Rx )2 + (y − Ry )2 x − Rx = (x − Rx )2 + (y − Ry )2

=−

)2

)2

(29)

Note that in the heart of the extended Kalman filter algorithm, the two corrections of light-time and light aberration are implemented. The procedure is very similar to that of angle-only initial orbit determination explained in the previous sections. Figure (2) illustrates how the light-time and light aberration effect corrections are implemented into EKF. For more details on extended Kalman filter, see.7

Figure 2. Light-time and Light Aberration Effect Correction in EKF

VI.

Results

In this section, the performance of the autonomous interplanetary spacecraft navigation system using visible planets is tested. The planet Earth is considered as the known visible sun-orbiting object. A spacecraft is traveling on a trajectory with an initial position of 10 of 14

more than two hundred million kilometers from the Sun with a distance of approximately seventy million kilometers from Earth (almost the same distance from Earth as Mars). The spacecraft is observing Earth for the navigation purposes. The methods Pn showed excellent performance for the interplanetary space-based initial orbit determination. The results from Pn was used as the input to EKF. For the initial orbit determination part, the number of observations used was Nobs = 8 with time interval ∆t = 3 ∗ 105 s (or almost three and half days). To make the problem more challenging (interesting), the spacecraft orbit was considered coplanar with the sun-earth orbit plane. Also note that the other planets have very small inclination angles. The position and velocity vectors of the spacecraft (r and v) and Earth (R and V ) at the initial time t0 of observation are       −2  

                2   −1.521   0   8 r = −0.5 108 km, v = −30 km/s, R = 10 km, V = 0 −29.29 km/s                         0 0 0 0 The noise involved in the measurements has the standard deviation of 3σ = 1000 . The last (at t8 ) estimated position and velocity were used as the output of the IOD algorithm and input to EKF. The state vector X0 (initial conditions for EKF) is defined as [x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 ]T and for this problem, x0 = −183, 553, 246.154077 km, y0 = −129, 751, 886.849561 km, z0 = 47.8566691563623 km, x˙ 0 = 9.2804459659289 km/s, y˙ 0 = −26.6655452207945 km/s, z˙0 = 0.0 km/s. For EKF part, 150 observations with time interval ∆t = 3 ∗ (103 )s (or 50 min) were considered. For the navigation purpose,the estimated position and velocity of the spacecraft are desired. No process noise was considered since we assumed the model has no error(no perturbations due to gravitational fields of other planets and so solar pressure), so the covariance of the process noise w was considered zero, Q = 0. Also, the initial values for P0 (covariance of the state estimate error) was considered as P0 = 2 ∗ 106 I6×6 using units of km and km/s for position and velocity, respectively. The covariance P0 is defined as ˜ 0 )(x ˜ 0 )T ], where E is the Expectation and x ˜ 0 is the initial state estimate error. Figures E[(x 3 and 4 show the history of position and velocity error and the associated 3-σ bounds and the true and estimated position and velocity respectively. The final estimated position and velocity relative error are rerror = 0.0004 and verror = 0.0015 respectively. The relative error |true − estimates| . was computed as Errorrelative = true

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Figure 3. Estimated Position and Velocity Error with the 3-σ Bounds

Figure 4. Estimated and True Position and Velocity Components

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VII.

Position Estimation Using Multiple Visible Planets

Up to now, we have assumed that only one visible planet can be observed. It is possible that two visible planets can be observed by the same star tracker or using two distinct star trackers. The presence of two star trackers onboard interplanetary missions is quite commonly justified for redundancy attitude determination system.b . In fact, by observing, for instance two planets, the positioning estimation becomes a simple triangulation problem. With reference to Fig. 5, describing the geometry of the problem, we can write

Figure 5. Geometry of two-planets observation scenario

r = R1 − ρ1 ρˆ1 = R2 − ρ2 ρˆ2 .

(30)

Performing the scalar product of Eq. (30) by ρˆ1 and ρˆ2 we obtain   ρˆT R − ρ = ρˆT R − (ρˆT ρˆ ) ρ 2 1 1 2 1 1 1 2  ρˆT R1 − (ρˆT ρˆ1 ) ρ1 = ρˆT R2 − ρ2 2

2

(31)

2

consisting of two equations in two unknowns, ρ1 and ρ2 . This system can be written in matrix form         T T ρˆ1 ρˆ2  ρ1 ρˆ1 (R2 − R1 )  −1 = , (32) ρ2  ρˆT (R2 − R1 ) −ρˆT2 ρˆ1 1 2 where ρˆT1 ρˆ2 = cos ϑ12 . This linear system admits a solution as long as cos2 ϑ12 6= 1, that is, when ϑ12 6= 0 and when ϑ12 6= π. Once ρ1 and ρ2 have been computed the spacecraft position vector r is simply obtained using Eq. (30). b

Star trackers provide directions estimation more accurate along the optical axis than orthogonal to it. 13 of 14

VIII.

Conclusions

An interplanetary autonomous navigation system using only onboard star tracker is proposed and numerically tested. The idea is pointing the star tracker to a visible known planet and using an angle-only space-based Initial Orbit Determination (IOD) method to provide the initial guess to an Extended Kalman Filter (EKF) for subsequent more accurate, filtered, orbit estimation. Method Pn has been selected as the space-based IOD method. Due to large distances and velocities involved, Pn and EKF estimations include two corrections: one due to the finite velocity of light, known as space-time correction, and another due to the observer’s velocity, known as starlight aberration. Using planet Earth as visible planet, numerical simulations have been included to validate the proposed method. The interplanetary trajectory of the observer was selected few million kilometers from Earth. Position and velocity estimation convergence has been provided. The algorithm was mainly developed for using one visible planet, whereas using multiple planets would result in more observability and the problem can be solved by triangulation.

References 1

Seidelmann, P.K., Explanatory Supplement to the Astronomical Almanac, University Science Books, Mill Valley, CA, 1992, Ch. 9. 2

Green, R.M., Spherical Astronomy, Cambrdige University Press, 1993, Ch. 8.

3

Schaffner, K.F.,Nineteenth-century aether theories, Oxford: Pergamon Press, 1972, pp. 99-117.

4

Liebscher, D.E., and Brosche, P., “Aberration and Relativity,” Astronomische Nachrrichten, Vol. 319, No. 5, 1998, pp.309–318. 5

Gjurchinovski, A., “Aberration of Light in a Uniformly Moving Optical Medium,” American Journal of Physics, Vol. 72, No. 7, 2004, pp. 934–940. 6

Crassidis, J.L., and Junkins, J.L., Optimal Estimation of Dynamic Systems, 2nd Ed., CRC Press, Boca Raton, FL, 2011, pg. 289. 7

Crassidis, J.L., and Junkins, J.L., Optimal Estimation of Dynamic Systems, 2nd Ed., CRC Press, Boca Raton, FL, 2011, pp. 243–342. 8

Karimi, R.R. and Mortari, D. “Orbit Determination Using Prescribed Orbits,” AAS 10-236, AAS/AIAA Space Flight Mechanics Meeting Conference, San Diego, CA, February 14-18, 2010. 9

Karimi, R.R. and Mortari, D. “A Performance Based Comparison of Angle-only Initial Orbit Determination Methods,” AAS 13-823, The 2013 AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, SC, August 11-15, 2013.

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