spacebourne orbit determination of unknown satellites ... · orbital trajectory estimation and...

Spacebourne Orbit Determination of Unknown Satellites Usinga Stabilized-Gauss-Method, Linear Perturbation Theory and

Angle-Only Measurements.

Mark B. Hinga, Ph.D., P.E. ∗

Orbital trajectory estimation and refinement is a pressing issue within the national security and civilianspace domain. Space debris and uncorrelated tracks (UCT) all require accurate observation and trajectoryestimation to prevent potentially catastrophic collisions. The numbers and types of observers able to makemeasurements on these tracks are constrained, so utilizing a singular observer to refine an object trajectory(rather than multiple observers) would benefit a host of applications. Many of the existing state-of-the-artorbit determination applications are geared towards ground-based assets and do not solve the problem of co-planar geometries and Geosynchronous Transfer Orbits (GTOs). Assumptions for these existing applicationsrequire an assumed orbit class. For this investigation, an improved orbit determination (OD) algorithm willbe designed which will mitigate the co-planar singularity problem without the need of an a priori assumptionof either orbit class or any orbit-specific orbital regime. What is novel in this approach is the application ofa least squares batch initial state estimator that implements the linearized perturbation technique (fixed finaltime guidance) in conjunction with a Lambert solver and a stabilized Gaussian-IOD technique (to handle theco-planar singularity condition as low as 0.01) to initialize an EKF enabling it to keep hold of the uniquestabilized solution. This stabilized batch algorithm allows for accurate initial orbit estimation with a singleobserver using a limited set of (at least six) measurements and is built upon a system of co-planar observedsingularity conditions that form the normal equations of the exact values of the f and g series least squaressolution.

∗Capt, USAFR, Physicist, AFRL/RDST, U.S. Air Force Research Laboratory, Kirtland, AFB, Albuquerque, NM 87117

This document has been cleared for public release. AFMC-2018-0254

Copyright © 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) – www.amostech.com

I. Nomenclature

i = inclination

ω = periapsis

Ω = right ascension of the ascending node

ρk = kth range to target satellite

ρk = kth unit vector “line of sight” to target satellite

rk = kth inertial position vector to target satellite

vk = kth inertial velocity vector of target satellite

Rk = kth inertial position vector to observer satellite

f, g, ck, dk = Lagrange coefficients

µ = Earth’s gravitational parameter, 3.986004418 ×1014 m3

sec2

ξk = right hand side of Gaussian triplet of observations

H = “design” matrix of unit vectors in Least Squares estimation

tf = fixed time of arrival

t0 = time at which impulse is applied

δx = small perturbation in state vector

δr = small perturbation in position

δv = small perturbation in velocity

δt = small perturbation to fixed time of arrival

Φ(tf , t0) = state transition matrix, from t0 to arbitrary final time tf

A(tf , t0) = Jacobian partials matrix, from t0 to arbitrary final time tf

II. Introduction

II.A. Initial Orbit Determination (IOD)

THE problem of determining the orbit of an unknown object began with the advent of celestial mechanics seen in theworks of Laplace1 in 1780 and Gauss in 1801.2 Their angle-only techniques utilized three observations to compute

a position of a celestial object without the knowledge of range, which had to be guessed with the help of the roots of an eighth order polynomial. In 1889, Gibbs3 developed his own technique enhancing the Gauss method of position estimation to include the determination of the velocity, which thus defines an orbit in space. Later, Herrick4 improved on Gibbs’ technique (for short arcs) with the use of a Taylor series to compute the velocity at the middle position vector. These early techniques were developed for celestial applications well before the beginning of the space age and availability of the computer. In 1964, in the early years of the space age, using an IBM 1620, astronomer Paul Herget5 introduced an algorithm, that uses more than three angle-only measurements to estimate an orbit in which an iterative approach is applied through the variation of guessed geocentric distances to minimize a set of residuals in a least squares approach, using as many observations as are available.

Over the last several decades, many iterative methods to estimate the orbit of an unknown object (natural or artificial) using angle-only measurements have been developed. The Double r-iteration technique by Escobal6 (1965), iterates on an initial guess of the range between the observer and a target object via the numerical partial derivatives and a Newton-Raphson iteration to converge on the true range. The Gooding7 method (1993), using a minimum of three measurements, requires an initial “good” guess of the first and third ranges and whether the orbit is pro or retrograde. Common to all of these methods are the assumptions about a target satellite they make in order to converge to a solution for its orbit.

These algorithms are not robust during the situation when the target and observer are both in the same plane of motion, that is, when the “coplanarity of the observation” is pronounced. When this happens it is very difficult for the observer to “see” enough of the target satellite motion to figure out the geometry of its orbit. Because an orbit solution contains six unknowns, position (x,y,z) and velocity (x ,y,z), that define i ts shape, orientation and satellite position along the orbit track, it is required that, in the case of optical angle only measurements, there are a minimum of six measured angles. These are obtained by taking at least three observations of right ascension and declination at three observation times, with respect to the observer and can provide for a unique fit of the observer’s model of motion to the measurements. These three observations are taken in the form of three line of sight unit vectors, each of which



contains one right ascension and one declination. Unfortunately, the condition of observational coplanarity causesthese common algorithms to fail to converge to the correct answer, producing either a nonsensical solution or one thatis plausible, but wrong. Therefore, the current and future locations of such a target satellite remain unknown.

Finding a method to estimate a preliminary orbit suffering from this observational coplanarity problem is theprimary motivation for this study. This algorithm will provide an immediate position and velocity report for real-time analyses of unknown object trajectories with the use of at least six observations (six “line of sight” unit vectorspointing to the target) taken either under severe co-planar observational conditions or not, enabling a real-time (onboard platform) orbit estimate of the location and prediction of movement of an unknown space object. This solution,along with covariance, can be reported quickly as a tip/cue to other ground or space-based sensors.

The significance of the algorithm in this study is that there are no assumptions or guesses made about the targetsatellite’s orbit or about its scalar range from the observing platform, which could be placed in any orbit. A value ofzero is an excellent start for the unknown ranges contained in the state vector of this least squares algorithm. Thisinitialization of zero is made possible due to the geometry between the observer and the target satellite, see Equation9. There is no guessing of range or whether motion is pro/retro grade nor is there a required search for the real rootsof an eighth order polynomial as mentioned above in the original Gauss method. By substituting the expression for r2(Herget)5 during the iteration of the least squares solution, ρ can be optimally varied until convergence to a real root.If the problem is not ill-conditioned the convergence to truth is “guaranteed”, consistently throughout the scope of thisinvestigation. If the scenario is ill-conditioned, i.e. a severe observational coplanarity exists, a divergent nonsensicalanswer is inevitable. However, using the stabilization algorithm proposed in this study, the so-called EigenvalueDescent Control (EDC) method, provides for good orbit estimates when the above mentioned published algorithms allfail miserably.

Further significance of this algorithm is that this stabilized-Gaussian-Initial Orbit Determination (IOD) methodbuilds the system of co-planar-motion conditions that form the normal equation and is constructed from and basedon the analytical exact functions of the f and g coefficient series expansion of universal variables (Prussing).8 Theco-planar-motion is not to be confused with “observational coplanarity” but rather, is the important assumption thata series of satellite position vectors in inertial space form a plane in which its motion is constrained. The latter isthe relative geometry between an observer and target satellite and connotes the angular height that satellite has abovethe observer’s orbital plane of motion. But once the scalar ranges are estimated and the inertial position of the targetsatellite has been determined, its velocity remains unknown. It is common to implement a Lambert solver at this point,to compute the departure and arrival velocities at any two particular inertial position vectors (Prussing).8

To improve upon the velocity vector computed by the Lambert solver, thus improving the estimate of the overallorbit, the application of the linearized perturbation technique (fixed final time APOLLO guidance) (Battin)9 is appliedusing the output of a European Space Agency (ESA) Lambert10 solver to determine the velocities corresponding toany of the positions that are estimated. Once these initial conditions of position and velocity are determined they canbe provided to an Extended Kalman Filter (EKF) orbit estimator for continued tracking using subsequent line of sightunit vectors.

To evaluate the orbit error computed by the algorithm in this study, it is important to consider the parameters forboth orbit orientation and orbit shape. Both types are “slow orbit variables”. Inclination, i, periapsis, ω, and rightascension of the ascending node (RAAN), Ω, describe orbit orientation. Eccentricity, e, conveys the “shape” of anorbit. The fast variable, which describes where the satellite is within the orbit, can be captured with either the trueanomaly ν or eccentric anomaly E. The error in fast moving variables of true anomaly and eccentric anomaly willnot be evaluated as they are too difficult to estimate under severe observational coplanarity conditions. Because thesemi-major axes of all ill-conditioned experiments are so well estimated in this study, the period of the orbits are alsowell known. So, despite lack of knowledge of the true anomaly, a well defined upper time limit to impact (in inertialspace) for a target satellite, is still available. To provide for a “visual perspective” of a solution, the propagated initialconditions of a stabilized orbit will be overlayed with the true orbit and separate EKF estimated trajectories will bepresented.

Finally, it is important to mention that all satellite orbit and measurement simulations, algorithm development andverification were carried out inside the self-developed and validated Hinga Orbit Simulator (HOrbitSIM) (Hinga).11

II.B. Development of the Coplanar System of Equations

We define the kth inertial position vector of a target satellite for n observation times, t1, t2, ...., tn as the following

rk = ρkρk + Rk, k = 1, 2, ..., n (1)



where Rk is the known observer position, ρk the scalar range from the observer to the target satellite and ρk isthe corresponding “line of sight” unit vector. The observer can be on the Earth surface or on an orbiting satellitesomewhere in space above the Earth.

For three observations, the following standard Gaussian equation expresses the relation of three vectors in inertialspace as a linear combination summing to zero with three distinct coefficients, c1, c2 and c3 as

c1r1 + c2r2 + c3r3 = 0. (2)

This was the equation used by Gauss when he predicted the position of the first minor planet after its conjunctionwith the Sun (Herget).5 If we substitute Equation 1 into Equation 2 and separate all known quantities on the right side,we end up with

c1ρ1ρ1 + c2ρ2ρ2 + c3ρ3ρ3 = −c1R1 − c2R2 − c3R3. (3)

After setting c2 = −1 and after many algebraic manipulations, it is found that

c1 =g3

f1g3 − f3g1(4)

c3 =−g1

f1g3 − f3g1, (5)

where f1, f3, g1, g3, are the so-called “Lagrange f and g coefficients” (Curtis).12 For this three observations example,the approximation (without the velocity term) of the f and g coefficients are f1 ≈ 1 − 1

2µr32τ21 , f3 ≈ 1 − 1

2µr32τ23 ,

g1 ≈ τ1 − 16µr32τ31 , g3 ≈ τ3 − 1

6µr32τ33 where τ1, τ3 are the time intervals between successive measurements of ρ1, ρ2

and ρ3.Just like Gauss, we put Equation 3 into matrix format. Because the unknown ranges appear on both sides of the

equation no closed form solution exists, forcing us to solve the system of coplanar equations through an iterativeprocedure. By starting with an initial guess of zero for the three unknown ranges we avoid forming an eighth orderpolynomial and instead, iteratively solve for the scalar ranges of ρ1, ρ2 and ρ3.

[ρ1 ρ2 ρ3

] c1ρ1

c2ρ2c3ρ3

=

R1

R2

R3

−c1−c2−c3

. (6)

We do this by inverting the 3x3 matrix of known ρi unit vectors, pre-multiplying against the right hand side bythis inverse, avoid forming and solving the traditional eighth order polynomial for the scalar magnitude of the middleinertial position vector r2 (Vallado)13 and solve for ρ1, ρ2 and ρ3. c1ρ1

c2ρ2c3ρ3

=[ρ1 ρ2 ρ3

]−1 R1

R2

R3

−c1−c2−c3

, (7)

Where c1ρ1c2ρ2c3ρ3

(8)

is defined as the state vector to be estimated.At each iteration step, both the scalar ranges, ρi, to the target and the corresponding magnitudes, ri, of the inertial

position vectors, ri, are also computed. Both are required at each iteration step (after inversion for the state vector) forcomputing the f and g coefficients. ρi is given by the relation which describes the simple geometry of the measurementscenario.

ri =[ρi

2 + 2ρiρi ·Ri +Ri2] 1

2 (9)

Carrying this out to n observations, we still group the observations into sets of three where the kth relation isdefined as

rk = ckrk−1 + dkrk+1, k = 2, 3, ..., n− 1 (10)



Here the coefficients of ck and dk are obtained similarly above (in the three observation example) by expressing thevectors rk−1 and rk+1 in terms of position and (including) velocity vectors at time tk, rk and vk using the Lagrangecoefficients f and g in the following (Curtis)12 / (Karimi)14 format.

rk−1 = fk−1rk + gk−1vk (11)

rk+1 = fk+1rk + gk+1vk (12)

By inserting these two expressions into Equation 10, vector vk can be eliminated and a relationship between rk−1,rk, and rk+1 is completely defined giving the expression for ck and dk (Karimi)14 as

ck =gk+1

fk−1gk+1 − fk+1gk−1and dk = − gk−1

fk−1gk+1 − fk+1gk−1. (13)

Notice that by eliminating the velocity term in the Lagrange equations above we are constrained to a “triplet” of threeposition vectors to compute the f and g coefficients. We can still increase the number of observations above three,however we will group the measurements into sets of three as we take on more observations to compute an orbit. Thisis discussed below in Section II.C. The Lagrange coefficients fk and gk, appearing in Equation 13 can be expanded inseries with a time difference ∆tk = tk − tk−1 up to a fourth order series expansion approximation as (Curtis 2005)12

fk−1 ≈ 1− µ

2r3k∆t2k −

µ(rk · vk)

2r5k∆t3k +

µ

24

[−2

µ

r6k+ 3

v2kr5k− 15

(rk · vk)2

r7k

]∆t4k (14)

fk+1 ≈ 1− µ

2r3k∆t2k+1 +

µ(rk · vk)

2r5k∆t3k+1 +

µ

24

[−2

µ

r6k+ 3

v2kr5k− 15

(rk · vk)2

r7k

]∆t4k+1 (15)

gk−1 ≈ −∆tk +µ

6r3k∆t3k +

µ(rk · vk)

4r5k∆t4k (16)

gk+1 ≈ ∆tk+1 −µ

6r3k∆t3k+1 +

µ(rk · vk)

4r5k∆t4k+1 (17)

where µ = 3.986004418× 1014 m3

sec2 is the Earth’s gravitational parameter.For time intervals ∆tk that are small in comparison with the orbital period, these coefficients f and g can be

well approximated using the first two terms of the series expansion, which yields the approximate expressions (Curtis2005).12 These expressions for ck and dk will actually be used in experimentation because the velocity magnitude atthe middle position vector, vk is unknown.

ck ≈ ∆tk+1

∆tk + ∆tk+1

[1 + µ

(∆tk + ∆tk+1)2 −∆t2k+1

6r3k

](18)

dk ≈ ∆tk∆tk + ∆tk+1

[1 + µ

(∆tk + ∆tk+1)2 −∆t2k

6r3k

](19)

where k = 2, ...., n− 1. For equally space measured times(∆t = constant) it is very easy to see that

ck = dk =1

2

[1 +

µ

2r3k∆t2

](20)

II.C. Multiple Observations

To extend the number of measurements to four or more using the Lagrange f and g coefficients in the coplanar systemof equations, we must arrange them in groups of three so that the coefficients are still based on f and g being expressedas a combination of one initial position and one initial velocity vector. Since velocity was solved for in Equation 11and inserted into Equation 12, we ended up with three position vectors, resulting in the relation in which the middle




position vector is a linear combination of the first and third, seen in Equation 10. Repeating this relation into a seriesof triplet measurements for a number n > 3 observations, the corresponding indices are seen as:

r2 = c2r1 + d2r3

r3 = c3r2 + d3r4

r4 = c4r3 + d4r5...

rk = ckrk−1 + dkrk+1 (21)

As we increase the number of observations, from k = 2, 3, ..., n−1 each triplet relation is preserved by consideringthe additional line of sight unit vector as the “third” observation to form rk+1, complementing the two previous. Asthis additional position vector is included, the corresponding kth “right hand side” similar to that seen in Equation 3,called the“residual” ξk, is formed. The following logic illustrates this principle.

In Equation 10, we substitute for each of the three inertial position vectors with the position vector given byEquation 1, rk = ρkρk + Rk, expand the terms and organize the line of sight unit vectors on the left hand side andthe position vectors of the observer on the right. (Note: the observer could be a terrestrial or spacebourne platform.)

rk = ckrk−1 + dkrk+1

Rk + ρkρk = ck(Rk−1 + ρk−1ρk−1) + dk(Rk+1 + ρk+1ρk+1)

ck(Rk−1 + ρk−1ρk−1) + dk(Rk+1 + ρk+1ρk+1) = Rk + ρkρk

ckRk−1 + ckρk−1ρk−1 + dkRk+1 + dkρk+1ρk+1 = Rk + ρkρk

ckρk−1ρk−1 − ρkρk + dkρk+1ρk+1 = Rk − ckRk−1 − dkRk+1 = ξk (22)

On the left side Equation 22 the measurement “triplet” is shown as three known unit vectors pointing to the targetsatellite multiplied by unknown scalar ranges and Lagrange coefficients. The right hand side (labeled as ξk) containsthe corresponding three known inertial position vectors of the observing (satellite) platform with the same unknownLagrange coefficients. (The Lagrange coefficients are unknown because they are a function of the scalar magnitude ofthe inertial middle position vector of the target satellite.) Notice that the Lagrange coefficient of the “middle” line ofsight unit vector remains as negative one for subsequent sets of triplet observations. Starting with the very first set ofn = 3 observations, k = 2, we have

c2ρ1ρ1 − ρ2ρ2 + d2ρ3ρ3 = R2 − c2R2 − d2R3 = ξ2. (23)

Putting Equation 23 into matrix format leads to

[c2ρ1 −ρ2 d2ρ3

] ρ1ρ2ρ3

= ξ2. (24)

Adding a fourth observation (n = 4, k= 3), we group the measurements into sets of three yielding two equations,

c2ρ1ρ1 − ρ2ρ2 + d2ρ3ρ3 = R2 − c2R1 − d2R3 = ξ2

c3ρ2ρ2 − ρ3ρ3 + d3ρ4ρ4 = R3 − c3R2 − d3R4 = ξ3 (25)

To prepare this set of equations for a least squares solution (Herget)5 we leave them set equal to their own residualand “stagger” them into matrix form in Equation 26

[c2ρ1 −ρ2 d2ρ3 0

0 c3ρ2 −ρ3 d3ρ4

]ρ1ρ2ρ3ρ4

=

[ξ2ξ3

], (26)



where 0 represents a 3× 1 vector of zeros.Writing Equation 26 in compact matrix notation , we have

Hρ = ξ. (27)

Expanding this to n observations, the system of equations is given asc2ρ1 −ρ2 d2ρ3 0 0 · · · 0

0 c3ρ2 −ρ3 d3ρ4 0 · · · 0

0 0 c4ρ3 −ρ4 d4ρ5 · · · 0...

......

......

. . ....

0 0 0 0 0 · · · dn−1ρn

ρ1ρ2ρ3...ρn

=

ξ2ξ3ξ4...

ξn−1

. (28)

Notice that H is a 3(n− 2)× n matrix with ρ and ξ having the dimensions of n× 1 and 3(n− 2)× 1, respectively.Instead of inverting matrixH on the left side and iteratively solving for the state vector of unknown scalar ranges ρ,

as seen in the similar example of Equation 7, here we are able form the normal equation, Equation 29, treating ξ as theresidual with equal weights for all measurements in matrix H , and solve for ρ in Equation 30. This iterative approachis similar to that of Herget.5 But instead of trying to guess “good” starting distances for scalar ranges, ρk, initializingall of them to zero provides for excellent starting values and have always converged to the non-trivial solution (in thisstudy) for orbiting platform observations. This result was also found to be true by Karimi.14

HTHρ = HT ξ. (29)

ρ = [HTH]−1HT ξ. (30)

By forming [HTH] we created a matrix that is n × n and is possibly of full rank n, where its column spacecompletely spans the space of ρ and is invertible when its nullity is the empty set ∅ (not including the null spacecontaining precisely one zero vector). However, if the observations (line of sight unit vectors ρk) suffer the conditionwhen successive measurements contain no or very small changing values of either relative right ascension α, ordeclination δ angles, then these measurements represent nothing more than a linear combination of the previous twounit vectors and do not span a complete dimensional space of n. This condition of “observational coplanarity” causesthe corresponding rows of H to be the same or very similar to previous rows and leads to rank deficiency in H , wherethe number of elements in the basis of its null space is not zero. This means that the matrix [HTH] is not full rankn and any attempt to invert it yields singularities in the solution for ρ. Such a stressful scenario may occur when theobservational coplanarity sinks down to angles of about 5 or less. At angles smaller than 1 the condition is generallypronounced, giving a solution of infinity, or other nonsense, for scalar ranges to the target satellite.

The next section introduces a method which “salvages” the scenarios of difficult observational coplanarities bystabilizing the inversion of matrix [HTH], via the utilization of its eigenvectors and eigenvalues, providing for usefulsolutions.

II.D. Stabilization Algorithm for the Inversion of [HTH]

It is fortunate that [HTH] is real and symmetric, because its eigenvalue decomposition produces an orthogonal matrixQ whose columns are the eigenvectors of [HTH] and are always orthogonal to one another. To compute its inverse,we take advantage of its eigendecomposition which is defined as follows: Let B = [HTH], be a square (nxn) matrixwith n linearly independent eigenvectors, qi(i = 1, ....., n). Then B can be factorized as

B = QΛQT , (31)

where Q is the square (nxn) matrix whose ith column is the eigevector qi of B and Λ is the diagonal positive definite matrix whose elements are the corresponding real eigenvalues, i.e. Λii = λi, where λ1 > λ2, ..., > λn. The inverse of B is defined as B −1 = QΛ−1QT , with diagonal entries of Λ−1 equivalent to the scalar inverse (reciprocal) of the eigenvalues (Golub).15 This monotonic descent of the sequence λi will prove helpful in Equation 32 below.



Computing B−1 can become problematic when a particular eigenvalue is very small or is drastically smaller thana previous. This is caused by the condition when the real measurements are poorly observing the elements of the statedeviation δρ correction vector to ρ in Equation 30. This will result in very large or “inflated” nonsensical estimates forthe scalar ranges, if these eigenvalues are kept during the inversion for δρ. A common mitigation strategy to deflate orstabilize the inversion ofB, is to discard the very small eigenvalues or to extend the lowest reliable eigenvalue to thosebelow it (Golub).15 Other mitigation methods such as inspection, relative error, norm-norm, and the mean square error(MSE), are commonly applied (Hinga).16

In this study, the information carried by all of the reliable eigenvalues, not just the last, is extended to thosebelow them by controlling their descent rate, via the slope change among points defined by the actual (a) valueyai = log10(λi) for i = 1, 2, ....., n. For any ith eigenvalue, its predicted (p) value is computed as

ypi =

[maver(i− 1)

η

]× i+ yai−1, (32)

where maver(i − 1) is the previous value of the running average of the slope between points. Because eachsuccessive eigen value is always smaller than the previous, maver(i − 1) is always negative and decreases linearlyas i increments. If the actual value of yai falls below a predicted value (or lower limit constraint), then the actualeigenvalue is replaced with a predicted value, yai = ypi . The parameter η is a one time adjustment, affecting theresultant slope, β = maver(i−1)

η . We can see that β and η are inversely proportional, namely

limη→∞

β −→ 0. (33)

Setting η equal to 2 or 3, provides for good stabilization for GTO and non-GTO trajectories, respectively, when theobservational coplanarity angle is below 1. If this angle is larger, then stabilization is usually not necessary.

Figure 1. Example: Two Stabilized Eigenvalues, 13th and 14th

In summary, by controlling the magnitudes of the quickly descending eigenvalues (let us call it Eigenvalue Descent Control (EDC)) with the influence of the s lowly descending e igenvalues, we are forcing shorter eigenvectors to be longer so that the resultant basis space of the inverted matrix B−1 = [HT H]

−1 can span the solution space more effectively, reduce the residual and provide for a solution closer to truth. If nothing is done to lengthen the very short eigenvectors by increasing their very small eigenvalues, then B−1 will not span the solution space of ρ. This will cause a portion of the residual vector ξ to map to the undesirable null space. Indeed, one may complain that we are introducing a bias, however, a solution that is coherent is preferable to one that is singular or nonsensical.

Figure 1 illustrates the simple example of all eigenvalues stabilized by Equation 32, but the last two (13th and 14th) are the ones showing the most needed improvement.

II.E. Solving for Velocity: Lambert and Fixed Time of Arrival Solution

After the vector of unknown ranges have been estimated, the series of inertial position vectors of the target satellite are calculated. To evaluate the target’s velocity, a Lambert solver is applied to the first and last inertial positions with the known fixed time of flight between them. The solver used in this study is that developed by the European Space Agency (ESA) for their ten year “Rosetta” mission.10 However robust and dependable this Lambert solver is, an improvement to its solution is obtained by implementing a Two Point Boundary Value Problem (TPBVP) shooting method to “fine tune” the departure and arrival velocity at initial and final time, repsectively. The guidance or “shooting” algorithm



is based on the Linear Peturbation Theory (Battin)9 developed for the America’s Program for Orbiting Lunar andLanding Operations (APOLLO) program during the 1960s.

Using the notation of Battin,9 the relation of the final state (position and velocity) error of the target ballistic craftto the deviation of the current state from the nominal at time t is given by Equation 35. The term Φ(tf , t0) is knownas the state transition matrix and is formed by taking the partial of the time rate of change of the state with respectto the state, Gelb.17 This derivative is a matrix of partials, commonly known as the “Jacobian”, shown as matrix Ain Equation 34. In this study the Earth’s J2 gravitation model is used to define the accelerations in this derivative.Integrating this equation produces the state transition matrix, which relates the change in state from some time t0 toanother time tf . The homogenous solution, for a fixed time of integration, is given in Equation 35, and yields a fixedtime of arrival solution.

Φ(tf , t0) = A(tf , t0)Φ(tf , t0) (34)

δx(tf ) = Φ(tf , t0)δx(t0) (35)

[δr(tf )

δv(tf )

]=

[R R

V V

][δr(t0)

δv(t0)

]. (36)

Let us expand Equation 35 in terms of position “r” and velocity “v” components and assign convenient labels to theportions of the Φ matrix. Because we know the position of where the spacecraft starts and do not want to vary it, weset δr(t0) = 0. Then the expression for the perturbation to the position and velocity at final time tf is,

δr(tf ) = Rδv(t0) (37)δv(tf ) = V δv(t0) (38)

Thus, the equation that defines how to vary (or perturb) the spacecraft velocity at time t0, based on the missed distanceat the target impact, is

δv(t0) = R−1δr(tf ), (39)

where R is the upper right 3x3 matrix of the state transition matrix Φ(tf , t0).The velocity correction defined in Equation 39 is used as the iterative correction term at initial time t0 in the search

for the optimal improvement of the initial velocity to minimize the miss distance at tf . Upon convergence of thethis shooting/guidance method, the Lambert velocity solution has been improved allowing for more accurate orbitalelements to be evaluated.

II.F. Exact f and g coefficients

The accuracy of the original Gauss method can be improved by replacing the approximation of the Lagrange f andg coefficients with the exact values of the coefficients in terms of universal anomaly χ which is solved for in theUniversal Kepler’s Equation (Prussing).8 The coefficients are written in terms of the universal anomaly as

f = 1− χ2

r0C(αχ2)

g = ∆t− 1√µχ3S(αχ2). (40)

C and S are the Stumpff (Prussing)8 functions and are defined as

S(αχ2) =

√αχ2 − sin(

√αχ2)

(√αχ2)

3

C(αχ2) =1− cos(

√αχ2)

αχ2. (41)



The quantity α is defined as

α =2

r0− v20

µ, (42)

where r0 and v0 are the magnitudes of the initial position and velocity respectively. The Universal Kepler’s Equationin terms of the Universal anomaly χ is solved for in Equation 43

√µ∆t =

r0vr0√µχ2C(αχ2) + (1− αr0)χ3S(αχ2) + r0χ. (43)

vr0 is the magnitude of the tangential component of the velocity vector. The solution for χ in Equation 43 is achieved via the “Laguerre-Newton-Raphson” method described in (Prussing 1993).8 This value of χ is inserted into Equation 41 to solve for both Stumpff functions which are then used in Equations 40 to compute exact values of f and g for any type of orbit. Convergence for χ is always guaranteed (Prussing 1993).8 Equation 40 is used to compute the values of f and g in this study.

III. Application of Non-Actively Stabilized Algorithm: Experimental Results

To demonstrate that this angle only algorithm works in the general case when the problem scenario requires no active stabilization (no eigenvalue descent control), the following section simulates two theoretical cases of placing an observing platform onto the 1963 Hughes Geosynchronous Earth Orbit (GEO) “Early Bird” satellite.

III.A. Scenario 1 and 2 (not actively stabilized): GEO Satellite Observes the Moon and an Asteroid

Simulating six observations (line of sight unit vectors) taken from the Hughes 1963 GEO satellite, provides for a well determined IOD orbit solution for that of the Moon and Asteroid 2014 DX110.

Scenario 1, Figure 2 provides for favorable viewing conditions, in that the apparent change in relative right as-cension α and declination δ is adequate for estimation of the scalar distances. The Moon appears approximately 15 degrees below the plane of observation and is travelling at roughly 90 degrees relative to the motion of the GEO bird. The line of sight scalar distances between the Early Bird GEO satellite and the Moon are about 352,000 km.

Scenario 2, Figure 4 provides for less than favorable viewing conditions, but are still adequate. The asteroid appears at about four degrees below the plane of observation (not far from “observational coplanarity”) and is travelling at about 160 deg relative to the motion of the GEO bird. Relative α and δ angles are changing enough to avoid active stabilization for solution inversion. The line of sight scalar distances between the Early Bird GEO satellite and the asteroid are about 1.14 million km.

Table 1 contains the error in the estimate of the initial conditions expressed in classical orbit parameters. Figures 2 and 4 display the estimated (red) positions of the asteroid, true (blue) positions, and known positions of the observing (green) GEO satellite, in the Earth Centered Inertial (ECI) frame. Error performance of the EKF in terms of all corresponding orbital parameters are shown in the appendix. From the EKF, standard deviation in Moon position and velocity is about 18 km and 35 m/s, respectively. For the asteroid, they are 165 km and 35 m/s. The percent error relative to truth in position and velocity, for both Scenario 1 and 2 are much lower than 1 percent. Further EKF performance, for both the Moon and asteroid scenario, is shown in the appendix and notice that it can estimate the target state quite well while the out of plane observing angle is either close to or passing directly through zero.



Orbital Elements: Soln (diff to truth) GEO observes Asteroid GEO observes Moona (m) 0.056 11.374

e -0.000008107 0.00000006i (deg) -0.000000821 -0.00000017ω (deg) 0.000000827 0.00008819Ω (deg) -0.000000235 -0.00000029ν (deg) -0.000000906 -0.00008760E (deg) na -0.00008279u (deg) -0.000000078 0.00000059

period (hours) na 0.00002927Table 1. Solution Difference to Truth

Figure 2. 1963 Hughes GEO Observes the Moon

Figure 3. Out of Plane Look Angle: 1963 Hughes GEO Observes the Moon



Figure 4. 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 5. Out of Plane Look Angle: 1963 Hughes GEO Observes Asteroid 2014 DX110



IV. Application of Stabilized Algorithm: Experimental Results

IV.A. Coplanar Scenario 3: Geosynchronous Transfer Orbit (GTO) Satellite Rendezvous with GeosynchronousEarth Orbit (GEO) Satellite

Due to the severe coplanarity condition of this scenario, Figure 6, six observations of the approaching GTO satelliteare not enough. At least twelve simulated observations (line of sight unit vectors) taken by the Hughes 1963 GEOsatellite are required to estimate the orbit of the observed (approaching) GTO satellite. The apparent change in relativeright ascension α and declination δ, is very anemic for estimation of the scalar distances. The oncoming GTO satelliteappears briefly at 0.1 below the plane of observation and quickly descends down to 0.01, while travelling intially atabout 31 relative to the motion of the observing plaform. As it closes in behind the GEO bird, the angle between theline of sight and GTO velocity vector decreases to below one degree. The line of sight scalar distances between theEarly Bird GEO satellite and the GTO satellite are initially about 37,000 km.

Such a stressful scenario requires EDC stabilization of the eignvalues if there is any hope of a useable solution.Indeed, one can be found that provides very useful information, where there was once none. Table 2 presents theestimates in form of the classical orbital elements.

In this case, the actual resulting percent error in estimated position and velocity, relative to truth, of the oncomingGTO satellite is above one hundred percent. However, we still have a reasonable estimate of the semi-major axis,eccentricity, inclination, RAAN (Ω) and period. All of the slow moving orbital elements are observed, but not the fast.With this information the user could be notified that rendezvous is indeed a possibility within a known time frame.Without this estimate, the situation would remain a mystery. The initial standard (EKF) deviations in GTO positionand velocity are about 3 km and 35 m/s. Further EKF performance (two and half hours of observations) is shown inthe appendix, along with an illustration of the (EDC) stabilized eigenvalues. Notice that the EKF can estimate thetarget state quite well during conditions of an observational coplanar geometry very close to zero.

Orbital Elements: Soln (diff to truth) GEO observes approaching GTOa (m) 34718.961

e 0.1250738i (deg) -0.2413970ω (deg) 60.1162073Ω (deg) -0.0441609ν (deg) -10.9623304E (deg) -29.9564886u (deg) 49.1538769

period (hours) 0.0225703Table 2. Solution Difference to Truth

Figure 6. GTO Satellite Rendezvous with GEO



Figure 7. EKF Estimate of GTO Satellite Rendezvous with GEO

Figure 8. Propagation of Init Cond of GTO Satellite Rendezvous with GEO

Figure 9. Out of Plane Look Angle: GEO observes approaching GTO



IV.B. Scenario 4: GPS SVN 47 Satellite Observes Coplanar SVN 54 GPS Satellite

The coplanarity condition of this scenario, Figure 10, requires six observations from the GPS SVN 47 satellite toestimate the orbit of its co-panar sister satellite, GPS SVN 54. The apparent change in relative right ascension α anddeclination δ, is very small for estimation of the scalar distances between them. The target satellite appears at 0.01

below the plane of observation and remains at this angle, while it travels with a velocity vector that remains at about15 relative to the observing line of sight. This geometry remains roughly the same for the entirety of the experiment.The line of sight scalar distance between these two GPS satellites is initially about 13,000 km, much smaller than thatseen in the above scenarios.

This coplanar scenario also requires EDC stabilization of the eigenvalues if there is any hope of a useable solution.Indeed, one can be found that provides very useful information, where there was once none. Table 3 presents theestimates in form of the classical orbital elements. In this case, as similar in the previous, the actual resulting percenterror in estimated position and velocity, relative to truth, of the sister GPS target satellite, is above one hundred percent.However, we still have a reasonable estimate of the semi-major axis, eccentricity, inclination, RAAN (Ω) and period.All of the slow moving orbital elements are observed, but not the fast.

With the information provided in Table 3, the observer knows the shape and orientation of the target orbit withrespect to inertial space. Since the estimate of the period is accurate to within 25 seconds, the observer knows thetime frame for target arrival at any point along the trajectory. A problem scenario such as this requires stabilization,because without it, only nonsensical estimates are available. Initial standard (EKF) deviations in GPS SVN54 positionand velocity are about 2 km and 35 m/s, respectively. Further EKF performance (two and half hours of observations)is shown in the appendix, along with an illustration of the stabilized eigenvalues.

Orbital Elements: Soln (diff to truth) GPS SVN 47 Observes SVN 54a (m) 10395.222

e 0.0075220i (deg) 0.130071710ω (deg) -5.5623196Ω (deg) -0.14121363460ν (deg) 34.9935435159E (deg) 35.5337314u (deg) 29.4312239146

period (hours) 0.0070243638Table 3. Solution Difference to Truth

Figure 10. Propagated Init Cond: GPS SVN47 Observes GPS SVN 54

Notice that the EKF can estimate the target state quite well during conditions of an observational coplanar geometryvery close to zero. .



Figure 11. Angles EKF: GPS SVN47 Observes GPS SVN 54

Figure 12. Out of Plane Look Angle: GPS SVN47 Observes GPS SVN 54



V. Conclusion and Future Work

The IOD algorithm of this study avoids the traditional approach of solving for the roots of an eighth order poly-nomial for the scalar distance to the middle position vector, originally developed by Gauss. There is no guessingof any scalar ranges to initiate solution, only the simple initialization with zero is required. There is no guessing ofwhether target satellite motion is pro or retro grade. Convergence to a real root is guaranteed and within the scopeof this study, all real roots were converged to correctly. Furthermore, is was found that implementation of the linearperturbation theory proved very helpful in improving the velocity computations from the estimated positions, whichdirectly influenced the final quality of the computed orbits.

A stressful observational coplanarity scenario requires stabilization of the eigenvalues if there is any hope of auseable solution. The stabilization scheme, Eigenvalue Descent Control (EDC), when applied to moderate or verysevere observational coplanarity conditions, or for that matter, any condition, offers a very useful solution to theobserver, when otherwise only nonsensical solutions are possible. Although the fast variable of true anomaly, is notprovided, orbit orientation, shape and period are of very practical use to the observer.

The EKF developed for this study demonstrated that it can be initialized with the state estimate from the batchsolution and not only track the target satellite under non-observational-coplanar conditions, as expected, but also canestimate the target state quite well while passing directly through observational coplanar angles of zero degrees.

Now that this stabilization algorithm has been proven capable of finding solutions during difficult observationalgeometries, it will be worthwhile to investigate its performance under simulated noise conditions applied to the lineof sight unit vectors, as proposed in Section VII.F. Finally, validating its capability and performance using actualmeasurements, either in real time or postprocessing, taken on board an orbiting GEO satellite platform, or from anyorbiting satellite, will prove valuable.



VI. Appendix

VII. Extended Kalman Filter (EKF) Formulation

The extended Kalman filter (EKF), a workhorse of real time spacecraft attitude estimation,18 is a recursive non lin-ear estimator (perturbed by Gaussian noise) that is descretized in the time domain by linearizing the physical dynamicsabout the current best estimate of the parameters of interest. It is not a requirement that the model of the state dynam-ics X = F [X(t), t] and observation model G[X(t), t] be linear, only that they are differentiable. This means that theformulation of the state transition matrix Φ(t, t0) and the matrix H[X(t), t], which is the partials matrix needed tocompute the predicted measurement from the predicted state (discussed below), is defined. In a non-extended Kalmanfilter, the same is true, but the linearization occurs about some pre-computed nominal trajectory of the state. In theEKF, the current best estimate comes from an optimal combination of the state from the previous time step (thatpropagated to the current time) and the current measurement. The definition of this combination is determined by theso-called Kalman gain K, which is defined below.

Kalman filters are unusual in that most filters (i.e. Butterworth filter) are formulated in the frequency domain, thentransformed back into the time domain for application. The EKF can be considered to be an adaptive low-pass infiniteimpulse response (IIR) digital filter, meaning that its response to an impulsive input is non-zero for infinite time.17 Thefrequency response of the EKF in this study is of no interest.

Ideally, if the model of the state and measurements are complete and accurate and perpetrate no acts of erroromission or commission, then the covariance P(t) of the estimate state, will accurately reflect the confidence of theestimated state vector, and those parameters will have a mean error of zero. Stated a different way, the variance andcovariance of the estimated state parameters will have a distribution about the true state. Invoking the expectationoperator E[f(ξ)] =

∫∞−∞ f(ξ)dξ, where f(ξ) is the function of interest, if there are no biases in the estimate X and

residual y = (Yktruth −G[Xk, k]) then:

E[Xktruth − Xk] = 0

E[Yktruth −G[Xk, k]] = 0, (44)

and the covariance matrices for the state estimate and residual, defined as,

Pk,k = E[(Xk − E[Xk])(Xk − E[Xk])T

]

Sk,k = E[(y − E[y])(y − E[y])T

] (45)

will have zero bias. However, since the filter of this study is intentionally mechanized as a suboptimal filter, smallbiases will be present and the Equations of 44 will be close to zero.

The state of the filter is represented by Xk, the estimated state at time k, and the error covariance matrix Pk, whichis a measure of the confidence in that state estimate. The EKF has two separate phases, which are called predictionand update. In the prediction phase, the estimate from the previous time step (k-1), both the state and covariancematrix, are propagated forward to the current time step (k). Then during the update phase, the state is refined withmeasurement information from the current time step. It is intended that after this refinement, the new estimate of thestate is more accurate, i.e. closer to the truth. In this study (note: there is neither a control model nor a control inputvector), the equations for these two phases are as follows.19

Predict Phase

Xk = Φk,k−1Xk−1 (predicted state estimate) (46)

Pk,k−1 = Φk,k−1Pk−1,k−1Φk,k−1T + Qk (predicted covariance of estimate) (47)

where Φk,k−1 and Qk are the state transition and process noise matrices respectively. In this investigation a Runge-Kutta 4 integration scheme is used to propagate both Φk,k−1 and Xk−1 forward one time step to give Xk, using theJacobian matrix (a matrix of partial derivatives) A defined in Equation VII.B in Section VII.B.



Update Phase

yk = Yk −G[Xk, k] (form measurement residual or innovation) (48)

Sk = HkPk,k−1HkT + Rk (covariance of residual or innovation) (49)

Kk = Pk,k−1HkT [HkPk,k−1Hk

T + Rk]−1

(optimal Kalman gain) (50)

xk = Kkyk (optimal update for the state estimate) (51)

Xk = Xk + xk (update the state estimate) (52)

Pk,k = (I−KkHk)Pk,k−1(I−KkHk)T

+ KkRkKkT (update the estimate covariance) (53)

The terms G[Xk, k] and Hk are the measurement model and the partials of the measurement model with respect tothe state, and are defined in Section VII.D. Yk is the actual measurement taken by a camera to produce a line of sightunit vector and is discussed in Section II.B.

VII.A. F Vector - The Error State Equations

The state estimate of a navigation filter is an error state. That is, we are estimating the error committed by a navigator.We are not estimating the state of a target (vehicle) itself, directly, i.e. position, velocity, attitude, gyro bias, etc.We are estimating error in these parameters, then updating the parameters themselves with the error, to improve theknowledge of them. In other words, this error state is used to update the knowledge of the state vector.

This error state is represented by δXk, at time k, and the error state covariance matrix Pk, which is a measure ofthe confidence in that error state estimate. By applying the estimate of the state error to the target vehicle state itself,the knowledge (indicated by an IMU, for example) of the state is improved and reconstructed at every time step of thetrajectory.

Let’s begin with the expression of the “true” target vehicle state followed by that for the error state. Dropping thesubscript k for clarity we have

X(t)TRUE = X(t)INS + δX(t)ERROR

δX(t)ERROR = X(t)TRUE −X(t)INS (54)

By differentiating this equation with respect to time, we obtain the differential equation for the error state,

δX(t)ERROR = X(t)TRUE − X(t)INS (55)

Because we do not know what truth is, we approximate it with the Taylor series expansion

f(y + δy) = f(y) + f ′(y)δy +1

2!f ′′(y)δy2 + H.O.T. (56)

and ignore second order terms and higher. Thus,

f(X(t)TRUE) = f(X(t)INS + δX(t)ERROR)

X(t)INS + δX(t)ERROR = X(t)TRUE

= X(t)INS +∂X(t)INS

∂X(t)∂X(t) (57)

If, for example, we were to include a measurement state U , of an inertial navigation system (INS), defined as

U(t)TRUE = U(t)INS + δU(t)ERROR, (58)

where,

U(t)INS = funct(wgyro, faccel), (59)

as a costate into the error state equation we re-define error state equation as

δX(t)ERROR =∂X(t)INS

∂X(t)∂X(t) +

∂X(t)INS

∂U(t)∂U(t) + H.O.T., (60)



(notice that the INS term X(t)INS cancels out as it appears on both sides of Equation 57).Therefore, the INS “state” now includes measurement vector U. The time rate of change of this state is

X(t)INS = X(t)INS(X,U, t) = F(X,U, t). (61)

This is the “state” that refers to the state of the vehicle (position, velocity, attitude) and the state of the sensors, thatis the biases, scale factors and misalignments of both the accelerometers and gyros. This is not the (estimated) statevector of the EKF. The state vector of the EKF is an “error state” that is based upon the partial of vehicle dynamicsand sensor dynamics in Equation 61 with respect to the elements of the INS state and sensor states. The time rate ofchange of the EKF state vector is the following.

[δX(t)ERROR

δU(t)ERROR

]=

∂F(X,U, t)INS

∂X(t)

[δX(t)

δU(t)

]. (62)

Clearly Equation 61 requires defining the vehicle state vector and state dynamics of an INS before we can takethe partial of F with respect to every element included in that “state”, to eventually obtain the “error state” equations.Equation 66 illustrates an INS EKF State vector which includes the error in accelerometer and gyro measurements:

X(t)EKF =

[δX(t)ERROR

δU(t)ERROR

]=

∂re

∂ve

∂ωeb∂f bB∂f bSF∂f bMA

∂G bB

∂G bSF

∂G bMA

(63)

Therefore F(t) is defined as (in the following equations let’s denote a vector with boldface, i.e. ve.)

F(X,U, t) =

re

ve

φebf bBf bSFf bMA

G bB

G bSF

G bMA

=

ve

Ceb fb + ge(re)− 2(ωe × ve)− ωe × ωe × re

ωb′

+ 12 (φ× ωb

′

) +× 112φ× (φ× ωb

′

)

0

0

0

0

0

0

(64)

But because we have no INS in this investigation, we have no INS state measurement vector, the above equationsare simplified down to the following:

[δX(t)ERROR

]=

∂F(X, t)

∂X(t)

[δX(t)

]. (65)

X(t)EKF =[δX(t)ERROR

]=

[∂re

∂ve

](66)

F(X, t) =

[re

ve

]=

[ve

ge(re)

](67)



VII.B. The A Matrix - The Jacobian of the System Dynamics

The A matrix, which is the Jacobian ∂F(X,t)

∂X(t) from Equation 65, is a 2x2 matrix, where each element is 3x3. Thus wedisplay a 6x6 matrix where each element Aij is 3x3.

A(t) =∂F(X, t)

∂X(t)=

∂F1

∂X1

∂F1

∂X2· · · ∂F1

∂Xn

∂F2

∂X1

∂F2

∂X2· · · ∂F2

∂Xn

......

. . ....

∂Fn

∂X1

∂Fn

∂X2· · · ∂Fn

∂Xn

(nxn matrix) (68)

A(t) =

[A11 A12

A21 A22

](69)

Let us define each 3x3 element.

A11 =δre

δre=

δrexδrex

δrexδrey

δrexδrez

δreyδrex

δreyδrey

δreyδrez

δrezδrex

δrezδrey

δrezδrez

=

0 0 0

0 0 0

0 0 0

(70)

A12 =δre

δve=

δrexδve

x

δrexδve

y

δrexδve

zδreyδve

x

δreyδve

y

δreyδve

z

δrezδve

x

δrezδve

y

δrezδve

z

=

1 0 0

0 1 0

0 0 1

(71)

A21 =δve

δre=δg(re)− (ωe × ωe × re)

δre (72)

δg(r)

δre =

δgxδx

δgxδy

δgxδz

δgyδx

δgyδy

δgyδz

δgzδx

δgzδy

δgzδz

(73)

δ(ωe × ωe × re)

δre =

−ωe2 0 0

0 −ωe2 0

0 0 0

(74)

Thus,

A21 =

( δgxδx + ωe2) δgxδy

δgxδz

δgyδx (

δgyδy + ωe2)

δgyδz

δgzδx

δgzδy

δgzδz

. (75)

A22 =δve

δve=δ(−2ωe × ve)

δv=

0 2ωe 0

−2ωe 0 0

0 0 0

(76)

All of these above partials in this section are defined in the next section, Section VII.C.

VII.C. Earth J2 Gravity Field Model

In this investigation, a gravitation field i s m odelled u sing t he s pherical h armonic expansion o f E quation 7 7. This expression describes a three dimensional gravitational potential, U, in the free space (zero density) above the Earth



(Tapley, Born, Schutz20),

U =GM

r+ U

′

U′

= − GM∗

r

∞∑l=1

(aer

)lPl(sinφ)Jl

+GM∗

r

∞∑l=1

l∑m=1

(aer

)lPl,m(sinφ)[Cl,mcosmλ+ Sl,msinmλ], (77)

where mass distribution is expressed in the spherical coordinates (r, φ, λ), with φ and λ representing geocentric latitudeand longitude, respectively. The scale factors M? and reference distance ae nondimensionalize the mass propertycoefficients Cl,m and Sl,m. The term Pl,m is Legendre’s Associated Function of degree ` and order m. If only the Jterm is considered with a degree of 2, we call this a J2 gravitation field. This is of high enough fidelity for the purposesof the HASP mission. For the Earth, 95 percent of the non-spherical mass distribution can be accounted for by theinclusion of only the J2 term. Not including higher order terms in the gravitation potential causes about 100 meters ofposition error over the time length of the flight.

U =GM

r+GM∗

r

∞∑l=1

(aer

)lP2(sinφ)J2. (78)

Expanding the double summation in this equation with m = 2 and using it to derive the partials seen in Equationsbelow (along with the observation-state partials of Equation 96), results in a system of equations of 3 unknowns in therepresentation of the dynamic equations for velocity, i.e. acceleration.

By letting M = M∗ (Earth mass), sinφ = zr , r = (x2 + y2 + z2)

12 and µ = GM , we can express P2(sinφ) =

32sin

2(φ)− 12 , and simplifying, we obtain the gravitation potential in Cartesian coordinates as

U =GM

r− GMae

2J22

[3z2

r2− 1

r3

](79)

The gravitation gradient is obtained by applying the gradient operator ∇ on the potential U to compute the accel-eration due to gravitation in the three cardinal directions in the navigation frame, namely

−→g =−→∇U =

δU

δx

−→i +

δU

δy

−→j +

δU

δz

−→k . (80)

Thus gx = δUδx , gy = δU

δy , and gz = δUδz .

After several steps of derivation we obtain the following expression for the acceleration in a J2 gravitation field ineach of the three cardinal directions as

gx = −µxr3− 3µae

2J2x

2r5

[1− 5z2

r2

](81)

gy = −µyr3− 3µae

2J2y

2r5

[1− 5z2

r2

](82)

gz = −µzr3− 3µae

2J2z

2r5

[3− 5z2

r2

](83)

For purposes for determining the Jacobian partials needed to compute the time rate of change of the velocity due togravitation effects we need to compute the partials of each of the above terms with respect to, again, all three cardinaldirections in the navigation frame, namely,

δgxδx

δgxδy

δgxδz

δgyδx

δgyδy

δgyδz

δgzδx

δgzδy

δgzδz

(84)



By letting K = 3µae2J2

2 and after several steps of derivation we obtain the following expressions for all nine termsin Equation 84 for a J2 gravitation field.

δgxδx

=µ

r3

[3x2

r2− 1

]− K

r5

1− 5

(x2 + z2

r2

)+ 35

x2z2

r4

(85)

δgxδy

=3µxy

r5− 5Kxy

r5

7z2

r4− 1

r2

(86)

δgxδz

=3µxz

r5− 5Kxz

r5

7z2

r4− 3

r2

(87)

δgyδx

=3µxy

r5− 5Kxy

r5

7z2

r4− 1

r2

(88)

δgyδy

=µ

r3

[3y2

r2− 1

]− K

r5

1− 5

(y2 + z2

r2

)+ 35

y2z2

r4

(89)

δgyδz

=3µyz

r5− 5Kyz

r5

7z2

r4− 3

r2

(90)

δgzδx

=3µxz

r5− 5Kxz

r5

7z2

r4− 3

r2

(91)

δgzδy

=3µyz

r5− 5Kyz

r5

7z2

r4− 3

r2

(92)

δgzδz

=µ

r3

[3z2

r2− 1

]− K

r5

3− 30

z2

r2+ 35

z4

r4

(93)

VII.D. Observation Model G and the H Matrix: Angle Only

The observation model G[Xk, k], a function of right ascension and declination, f(α, δ), is based on the differencebetween the inertial position vector to the observing platform Rk (P) and the inertial position vector to the targetsatellite rk (T), Eqn. 94,

G[Xk, k] = ‖rk −Rk‖= (xT i + yT j + zTk)− (xP i + yP j + zPk)

= (xT − xP)i + (yT − yP)j + (zT − zP)k

= Lx i + Ly j + Lz k. (94)

Because the target satellite rk (T) is unknown, xT, yT, yT is unknown, we must compute the line of sight unit vectorof L = lx i + ly j + lz k, which is based on the known angles of right ascension α and declination δ, namely

lx = cos(δ)cos(α)

ly = cos(δ)sin(α)

lz = sin(δ). (95)

To relate the elements of the state vector to the target satellite X(t)satellite, we take the partial of L with respect tothe elements of the state vector



δlxδxT

=−(xT − xP)

2+ L2

L3

δlxδyT

=−(xT − xP)(yT − yP)

L3

δlxδzT

=−(xT − xP)(zT − zP)

L3

δlyδxT

=−(yT − yP)(xT − xP)

L3

δlyδyT

=−(yT − yP)

2+ L2

L3

δlyδzT

=−(yT − yP)(zT − zP)

L3

δlzδxT

=−(zT − zP)(xT − xP)

L3

δlzδyT

=−(zT − zP)(yT − yP)

L3

δlzδzT

=L2 − (zT − zP)

2

L3

δlxδxT

= 0

δlxδyT

= 0

δlxδzT

= 0

δlyδxT

= 0

δlyδyT

= 0

δlyδzT

= 0

δlzδxT

= 0

δlzδyT

= 0

δlzδzT

= 0

(96)

Now, we can define the H matrix and relate the change in a measurement to a change in the state vector.

H =

δlxδxT

δlxδyT

δlxδzT

δlxδxT

δlxδyT

δlxδzT

δlyδxT

δlyδyT

δlyδzT

δlyδxT

δlyδyT

δlyδzT

δlzδxT

δlzδyT

δlzδzT

δlzδxT

δlzδyT

δlzδzT

. (97)

Notice that the last three columns are all zero. This shows that velocity is not observable in the line of sight unit vectorand there is no relation between the change in the elements of L, lx, ly, lz to the change in velocity.



H =

δlxδxT

δlxδyT

δlxδzT

0 0 0δlyδxT

δlyδyT

δlyδzT

0 0 0δlzδxT

δlzδyT

δlzδzT

0 0 0

. (98)



VII.E. Stabilization under Noise Conditions: Sensitivity Analysis

TBD (Coming soon)

VII.F. Small Angle Noise Simulation: Small Angle DCM Rotation

Because no measurement is perfect, the measured line of sight obtained from a non-perfect camera is simulated witha small angle rotation about each axis of the sensor frame. This results in a small angle rotation about some resultantaxis that is pointed at a small angle away from the ideal.

To define this small angle Direction Cosine Matrix (DCM) rotation, the following expression describes a so-called“3-2-1” or a small angle rotation about the “z-y-x” of the sensor frame. Note: it is irrelevant in which order these threerotations are carried out. The final resulting skew symmetric matrix seen in Equation 100 will always occur.

Φ =

1 0 0

0 cosφx sinφx0 −sinφx cosφx

cosφy 0 −sinφy

0 1 0

sinφy 0 cosφy

cosφz sinφz 0

−sinφz cosφz 0

0 0 1

(99)

=

cosφycosφz cosφysinφz −sinφy(sinφxsinφycosφz − cosφysinφz) (sinφxsinφysinφz + cosφxcosφz) sinφxcosφy(cosφysinφycosφz + sinφxsinφz) (−sinφxsinφysinφz − sinφxcosφz) cosφxcosφy

.Because the angles φx, φy and φz are all “small”, we can make the approximation that cosφ ≈ 1 and sinφ ≈ φ.

This leads to the simplification of Equation 99

Φ =

1 · 1 1 · φz −φy(φxφy · 1− 1 · φz) (φxφyφz + 1 · 1) φx · 1(1 · φy · 1 + φxφz) (−φxφyφz − φx · 1) 1 · 1

but φ · φ ≈ 0 and φ · φ · φ ≈ 0 thus

Φ ≈

1 φz −φy−φz 1 φx

φy −φx 1

Φ ≈ I −

0 −φz φyφz 0 −φx−φy φx 0

E =

0 −φz φyφz 0 −φx−φy φx 0

Φ = [I − E]

(100)

Each small angle in matrix E is sampled from a mean zero Gaussian distribution of small angles. The resulting Φmatrix will be the rotation applied to the ideal line of sight unit vector ρideal to simulate a real world measurement.

ρcorrupted = Φρideal (101)

Although noisy measurements have been introduced in this study and their effect on the stabilization has be seen,it is not the goal to thoroughly study the sensitivity of the stabilization method to noise, but rather to examine sometrends of effectiveness of the stabilization as a proof of concept in the presence of noise.



VII.G. Scenario 1 (not actively stabilized): 1963 Hughes GEO Observes the Moon

Figure 13. Semi-Major Axis Error: 1963 Hughes GEO Observes the Moon

Figure 14. Eccentricity Error: 1963 Hughes GEO Observes the Moon

Figure 15. Inclination Error: 1963 Hughes GEO Observes the Moon



Figure 16. Periapsis Error: 1963 Hughes GEO Observes the Moon

Figure 17. RAAN Error: 1963 Hughes GEO Observes the Moon

Figure 18. True Anomaly Error: 1963 Hughes GEO Observes the Moon



Figure 19. Eccentric Anomaly Error: 1963 Hughes GEO Observes the Moon

Figure 20. Argument of Latitude Anomaly Error: 1963 Hughes GEO Observes the Moon

Figure 21. Period Error: 1963 Hughes GEO Observes the Moon



VII.H. Scenario 2 (not actively stabilized): 1963 Hughes GEO Observes an Asteroid

Figure 22. Semi-Major Axis Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 23. Eccentricity Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 24. Inclination Error: 1963 Hughes GEO Observes Asteroid 2014 DX110



Figure 25. Periapsis Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 26. RAAN Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 27. True Anomaly Error: 1963 Hughes GEO Observes Asteroid 2014 DX110



Figure 28. Eccentric Anomaly Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 29. Argument of Latitude Anomaly Error: 1963 Hughes GEO Observes Asteroid 2014 DX110

Figure 30. Period Error: 1963 Hughes GEO Observes Asteroid 2014 DX110



VII.I. Coplanar Scenario 3: Geosynchronous Transfer Orbit (GTO) Satellite Rendezvous with GeosynchronousEarth Orbit (GEO) Satellite

Figure 31. EDC Stabilized Eigenvalues: 1963 Hughes GEO observes GTO Satellite

Figure 32. Semi-Major Axis EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

Figure 33. Eccentricity EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite



Figure 34. Inclination EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

Figure 35. Periapsis EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

Figure 36. RAAN EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite



Figure 37. True Anomaly EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

Figure 38. Eccentric Anomaly EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite




Figure 39. Argument of Latitude EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

Figure 40. Period EKF Estimate Error: 1963 Hughes GEO observes GTO Satellite

VII.J. Coplanar Scenario 4: GPS SVN 47 Satellite Observes Coplanar SVN 54 GPS Satellite

Figure 41. EDC Stabilized Eigenvalues: GPS SVN47 Observes GPS SVN 54



Figure 42. Semi-Major Axis EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 43. Eccentricity EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 44. Inclination EKF Estimate Error: GPS SVN47 Observes GPS SVN 54



Figure 45. Periapsis EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 46. RAAN EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 47. True Anomaly EKF Estimate Error: GPS SVN47 Observes GPS SVN 54



Figure 48. Eccentric Anomaly EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 49. Argument of Latitude EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

Figure 50. Period EKF Estimate Error: GPS SVN47 Observes GPS SVN 54

References1Laplace, P. S., “Memoires sur la determination des orbites des cometes.” Memoires de lAcademie Royale des Sciences de Paris, 1780.2Gauss, C. F., Theoria Motus: Theory of the Motion of the Heavenly Bodies Moving About the Sun in Conic Sections, Boston: Little Brown,

1857.3Bate, Muller, W., Fundamentals of Astrodynamics, Dover, 1971.4HERRICK, S., Astrodynamics: Orbit Determination, Space Navigation, Celestial Mechanics, Vol. 1, Van Nostrand Reinhold Co., London,

1971.5Herget, P., The Computation of Orbits. Edwards Brothers Inc., Ann Arbor, published by the author himself (1948), The M.I.T. Press, 1974.6Escobal, P., Methods of Orbit Determination, John Wiley and Sons, Inc., New York, 1965.7Gooding, R., A New Procedure for the Solution of the Classical Problem of Minimal Orbit Determination from Three Lines of Sight, Vol.

66, No. 4, 1997, pp. 387-423, Celestial Mechanics and Dynamical Astronomy, 1997.



8Prussing, J. E. and Conway, B. A., Orbital Mechanics, Oxford Press, New York, 1993.9Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, American Insitute of Aeronautics and Astrodynamics,

1987.10(ESA), E. S. A., Advanced Concepts Team (ACT), European Space Agency (ESA), The PaGMO development team, 2004-2015.11Hinga, M. B., “Hinga Orbit Simulator (HOrbitSim),” Sandia national laboratories, future sand report no. sand 2018-0720, 2018.12Curtis, H., Orbital Mechanics for Engineering Students, Elsevier Butterworth-Heinemann, Amsterdam/Boston, 2005.13Vallado, D. A., Fundamentals of Astrodynamics and Applications, Microcosm Press, El Segundo, CA, 2001.14Karimi, R. and Mortari, D., “Initial orbit determination using multiple observations,” Celest Mech Dyn Astr, Vol. 0018-9235/97/, IEEE,

2011, pp. 109:167–180.15Golub, G. H. and Van Loan, C. F., Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, 1996.16Hinga, M. B., “Using Parallel Computation to Apply the Singular Value Decomposition (SVD) in Solving Large Earth Gravity Fields Based

on Satellite Data,” Ph.d. dissertation, university of texas at austin, 2004.17Gelb, A., Kasper, J. F., Nash, R. A., Price, C. F., and Sutherland, A. A., Applied Optimal Estimation, The M.I.T. Press, 1974.18Markley, F. L., Crassidis, J. L., and Cheng, Y., “Survey of Nonlinear Attitude Estimation Methods,” AIAA Guidance, Navigation and Control

Conference and Exhibit, 2005, AIAA Paper 2005-5927, 15-18 August 2005.19Kalman, R., “A New Approach to Linear Filtering and Predicition Problems,” Journal of of Basic Engineering, , No. pp. 35-45, 1960.20Tapley, Born, and Schutz, Statistical Orbit Determination, Elsevier Academic Press Inc., 2004.



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