DETERMINING DEBRIS CHARACTERISTICS FROM OBSERVABILITY ANALYSIS OFARTIFICIAL NEAR-EARTH OBJECTS

A. M. Friedman and C. Frueh

Purdue University, School of Aeronautics and Astronautics, 701 W. Stadium Avenue, West Lafayette, IN 47907, USA,Email: friedmaa, cfrueh@purdue.edu

ABSTRACT

Observability analysis is a method for determiningwhether a chosen state can be determined from mea-surements. The state is typically composed of positionand velocity; however, including object characteristicsbeyond position and velocity can be crucial for preciseorbit propagation. For example, solar radiation pressurehas a significant impact on the orbit of high area-to-massratio objects in geosynchronous orbit. Therefore, deter-mining the time required for solar radiation pressure pa-rameters to become observable is important for under-standing debris objects. The focus of this work is the de-velopment and implementation of an extended state dis-crete time-varying observability analysis tool for debrisobjects. Previous results on incorporating measurementnoise into observability analysis are implemented to de-termine the effect of measurement noise on extended stateobservability analysis. Ten objects, five low Earth objectsand five geosynchronous objects, are simulated for sev-eral non-extended and extended state vector cases. Thetimes to become observable are compared for the differ-ent extended state vectors and the different measurementnoise cases. The impact of using an extended state vec-tor in observability analysis for determining debris objectcharacteristics is discussed.

Keywords: Observability; Extended State; Solar Radia-tion Pressure.

1. INTRODUCTION

As the space debris population continues to grow fromthe almost 60 years of humankind’s exploration of space,so does the concern for satellites and astronauts in or-bit around Earth. The current estimate of the numberof debris objects in orbit around Earth is over 100 mil-lion greater than 1 mm in size [9]. While a majority ofthe debris objects in orbit around Earth are on the sub-centimeter scale and would not appear to be much of athreat because of their small size, the high orbit energy ofobjects means that even objects on the millimeter scalehave the potential to damage assets in orbit [7]. Even

with the addition of the Space Fence, which will be ableto track up to 200,000 objects greater than one centime-ter in size, many of the debris objects on orbit will gountracked [11]. Therefore, the ground station resourcesavailable for tracking space debris objects will be insuffi-cient to completely track the debris population. With thisinformation, importance should be placed on gaining asmuch information about space debris from each observa-tion and tracking objects as efficiently as possible.

Observability analysis can be instrumental in improvingsensor tasking efficiency and determining more informa-tion about active satellites and debris objects. Poten-tial applications of applying observability analysis to thespace object problem are sensor tasking, planning of fu-ture ground stations, improved estimation methods, andobject characterization. This paper will focus on imple-mentation of observability analysis for determining ob-ject characteristics.

Previous work on the incorporation of measurement noiseinto observability analysis reviewed various applicationsof observability analysis [4, 5]. Extending the state vec-tor in observability analysis has been used in work byDianetti [3]. In the work by Dianetti, observability anal-ysis is performed to determined when a parameter shouldbe estimated in a consider filter framework. Rather thanspecifically using observability analysis as a tool for de-termining when a parameter should be estimated or not,this work looks more generally at extended state observ-ability analysis, its implications, and how adding extraparameters in the analysis can be used for determiningobject characteristics. Furthermore, the effect of extend-ing the state on the time for a system to become observ-able is investigated. The paper is organized as follows.First, the observability methods implemented and the dy-namical models used are introduced. Next, incorporationof an extended state vector into discrete time observabil-ity analysis is given. The paper concludes with numericalresults of several extended state cases and conclusions.

2. METHODS

In this section the discrete time-varying observability ma-trix is derived for a linearized system. In addition, the dy-

Proc. 7th European Conference on Space Debris, Darmstadt, Germany, 18–21 April 2017, published by the ESA Space Debris Office

Ed. T. Flohrer & F. Schmitz, (http://spacedebris2017.sdo.esoc.esa.int, June 2017)

namical model including solar radiation pressure (SRP)and the Jacobian of the dynamics with respect to the statevector are defined for several extended state cases. More-over, the measurement model used in the observabilityanalysis is defined.

2.1. Observability Analysis

Previous work on observability analysis was imple-mented with continuous time-varying methods [4]. Thecontinuous time-varying observability methods wereused as a first step to creating a observability tool fordetermining debris object characteristics. The discretetime-varying observability matrix is used in this workon extended state observability. An advantage of thediscrete-time formulation is that it can be applied withobservations spaced realistically. However, as a start-ing point with the discrete-time observability analysis andwith the extended state observability, this analysis is ap-plied without restrictions on how observations are spaced.

The system in which these debris objects operate is non-linear, given by the following equations.

9xptkq “ fptk,xptkqq, (1)yptkq “ hptk,xptkqq. (2)

In Equations 1 and 2, tk is the time at each measurement,xptkq is the state vector, yptkq is the output or measure-ments, fptk,xptkqq is a nonlinear function of the dynam-ics, and hptk,xptkqq is a nonlinear function of the mea-surements. The state vector in observability analysis typ-ically only contains position and velocity, but this workfocuses on extending the state vector to contain furtherdebris object characteristics.

Equations 1 and 2 can be linearized in two ways: withstate and measurement variations, as in [1, 8, 15, 16], orwith the Jacobian, as in [8]. A linearization using theJacobian given in Montenbruck [8] is used to determinethe state space equations below.

9xptkq “ Aptkq xptkq, (3)

yptkq “ rHptkq xptkq ` ν. (4)

In Equations 3 and 4, Aptkq contains the linearized dy-namics, rHptkq is the linearized measurement matrix, andν is the measurement noise. From the linearized equa-tions above, the discrete-time observability matrix can bedetermined with the following. The fundamental ques-tion of observability is whether or not the initial state x0

can be determined from the measurements yptkq. Sincethis is a linearized system, the state transition matrix(STM), Φptk, t0q, operates on the states by:

xptkq “ Φptk, t0q xpt0q, (5)

where Φpt, t0q is numerically determined from the STMdifferential equation:

9Φptk, t0q “ Aptkq Φptk, t0q, (6)

with Φpt0, t0q “ Inˆn. Substituting Equation 5 intothe measurement equation, the observability matrix isformed with matrix manipulation.

yptkq “ rHptkq Φptk, t0q xpt0q, (7)

Letting,Hptkq “ rHptkq Φptk, t0q, (8)

and multiplying both sides on the left by HptkqT ,

HptkqT yptkq “ Hptkq

T Hptkq xpt0q. (9)

A system is deemed observable when xpt0q can be deter-mined given measurements yptkq. Therefore, solving forxpt0q results in:

xpt0q “ pHptkqT Hptkqq

´1 HptkqT yptkq. (10)

In the above equation, xpt0q can be determined ifHptkq

T Hptkq is invertible. O “ HptkqT Hptkq is called

the observability matrix and has the following form:

O “

mÿ

k“1

Φptk, t0qTrHptkq

TrHptkq Φptk, t0q, (11)

where m is the number of measurements. Similar deriva-tions are found in: [2, 6, 10].

The criteria used for observability here is whether or notthe observability matrix, O, is invertible. The invertibil-ity of a matrix can be determined by checking the matrixrank. The rank of a matrix can be determined by checkingwhether the eigenvalues or singular values are non-zero.Advantages of using singular values are that they are al-ways positive and they can be computed for non-squarematrices [17]. In order to determine the invertibility nu-merically in this work, the singular values of the observ-ability matrix are computed. As introduced in [4], a tol-erance must be implemented to determine when a valueis numerically greater than zero. The tolerance is definedby:

Tol “ maxpsiq ˆmaxpsizepOqq ˆ eps, (12)

where si are the singular values of the observability ma-trix O and eps is the machine precision or epsilon. Thistolerance is one of many ways to represent the numeri-cal error in a problem [12]. As more measurements getadded to the observability matrix, the singular values willchange.

This work specifically looks at extending the state vectorbeyond position and velocity, thus increasing the dimen-sion of the observability matrix. In addition, measure-ment noise is incorporated using Cholesky decomposi-tion and pre-whitening as in [4]. The measurement noisecovariance is decomposed as follows:

K´1 “ L LT . (13)

Multiplying Equation 4 by the transpose of the lower tri-angular matrix, LT , a modified measurement equationhas the form:

LT yptkq “ LT Hptkq xpt0q. (14)

The resulting observability matrix with measurementnoise, following a similar process to Equations 9 and 10,is given by:

O “

nÿ

k“1

Φptk, t0qTrHptkq

T K´1rHptkq Φptk, t0q.

(15)Next, the dynamical model used to compute the STM isintroduced.

2.2. Dynamical Model

The state vector in this work is extended to include ob-ject characteristics which are parameters of solar radia-tion pressure (SRP): area-to-mass ratio (AMR), A

m , andthe material dependent reflection coefficient, C. There-fore, the dynamics used in formulation of the STM andlinearized measurement matrix must contain SRP pertur-bations. A simple cannonball model is used to derive theperturbing accelerations due to SRP, given by:

aSRP “ ´A

mC (AU)2

E

c

s

|s|3, (16)

where C “ 14 `

19 Cd, Cd is the diffuse coefficient, AU

is the astronomical unit, E “ 1367.0 W/m2 is the solarconstant, c “ 2.998 ˆ 108 m/s is the speed of light, ands “ rsun ´ robj is the sun-object vector, which is thedistance from the sun to the object. The position vectorof the sun to the center of the Earth is given by rsun, andthe position vector of the object from the center of theEarth is given by robj. Similar forms of the SRP perturb-ing acceleration can be found in: [8, 16]. Once methodsfor extending the observability state vector have been de-veloped, higher-fidelity SRP models can be implemented,e.g., a flat plate model and modeling of Earth’s shadow.

The equations of motion using the simple two body prob-lem and SRP are given by:

:r “ ´µ

|r|3r` aSRP, (17)

where r is the position vector of the object and µ “

3.986ˆ 1014 m3/s2 is the standard gravitational parame-ter of Earth. Using the methods in Montenbruck [8], theAptkq matrix in the STM differential equation for a stateextended with AMR is given by:

Aptkq “

»

—

–

03ˆ3 I3ˆ3 03ˆ1

Gtotptkq 03ˆ3BaSRPptkqBxp7q

01ˆ3 01ˆ3 01ˆ1

fi

ffi

fl

, (18)

where Gtotptkq “ Ggrav `GSRP is the sum of the Ja-cobians of the equations of motion, when reduced to sixfirst order differential equations. The partial derivativesof the extra state element beyond position and velocity,e.g. AMR or C, are given by BaSRPptkq

Bxp7q . The Jacobian of

the accelerations due to the central body with respect tothe position components, Ggrav, is given by [8]:

Ggrav “µ

|r|5

´

3 r rT ´ |r|2 I3ˆ3

¯

. (19)

Next, the Jacobian of the SRP perturbing accelerationwith respect to the position components has a similarform:

GSRP “ ´A

mC (AU)2

E

c

1

|s|5

´

3 s sT ´ |s|2 I3ˆ3

¯

.

(20)In this work, three different state extensions are tested:AMR, pAMR¨C q, and AMR & C. The first case extendsthe state vector by one variable, area-to-mass ratio. Thesecond case extends the state by one variable, the productof AMR and C. The third case extends the state by twovariables, AMR andC, separately. The partial derivativesfor the first and second cases are given by:

BaSRPptkq

BAMR“

aSRPptkq

AMR(21)

BaSRPptkq

BpAMR¨C q“

aSRPptkq

pAMR¨C q(22)

Since the third state extension case adds two variablesto the state, Aptkq is now size 8 ˆ 8 rather than 7 ˆ 7,when only one state variable was added to the positionand velocity state vector. The Aptkq for the third stateextension case is given by:

Aptkq “

»

—

–

03ˆ3 I3ˆ3 03ˆ1 03ˆ1

Gtotptkq 03ˆ3BaSRPptkqBAMR

BaSRPptkqBC

02ˆ3 02ˆ3 02ˆ1 02ˆ1

fi

ffi

fl

,

(23)where,

BaSRPptkq

BC“

aSRPptkq

C. (24)

For each state extension case, the Aptkq is used to numer-ically determine the STM by solving Equation 6. Next,the linearized measurement matrix is formed.

2.3. Measurement Model

The next component of the observability matrix is the lin-earized measurement matrix, rHptkq, which is given by:

rHptkq “Byptkq

Bxptkq, (25)

where y are the measurements right ascension, α, decli-nation, δ, right ascension rate, 9α, and declination rate, 9δat each time tk. The measurements, for a topocentric ob-server located with latitude, φ, and sidereal time, θ, are

given by:

α “ arctan´ y

x

¯

, (26)

δ “ arctan´ za

x2 ` y2

¯

, (27)

9α “x 9y ´ y 9x

x2 ` y2, (28)

9δ “´zpx 9x` y 9yq ` px2 ` y2q 9za

x2 ` y2px2 ` y2 ` z2q, (29)

where,

x “ x´RC cosφ cos θ, (30)y “ y ´RC cosφ sin θ, (31)z “ z´RC sinφ, (32)9x “ 9x`RC

9θ cosφ sin θ, (33)9y “ 9y ´RC

9θ cosφ cos θ. (34)

RC is the radius of the Earth and 9θ is the time rate ofchange of the sidereal time, which is the angular veloc-ity of the Earth. The sidereal time is a function of theepoch and the observer longitude. The x,y, z, 9x, 9y, and 9zvariables are Earth centered inertial position and velocitycomponents. When the state is extended to include SRPparameters, the matrix defined by Equation 25, containsan extra column for each added state element. However,these columns are zero for the extended state cases givenin Section 2.2, since the measurements, y “ rα δ 9α 9δsT ,do not depend on AMR or C. The measurement noisehas been defined using the works of Sanson and Frueh[13, 14] with specific values given in Section 3.

3. RESULTS

Five low Earth orbit (LEO) objects and five geosyn-chronous (GEO) objects were used to determine the ef-fect of extending the state vector in observability analy-sis. The observer location used in this work is the ZIM-LAT telescope, Zimmerwald Observatory, Switzerland.The latitude and longitude of the telescope are 46.8670˝

and 7.4670˝, respectively. The epoch used in this analy-sis is 53159.5 MJD. The four observability analysis casessimulated were for a non-extended state and three ex-tended state cases. The three extended state cases are ex-tending the state with AMR, pAMR¨C q, and AMR & Cd.The first and second state extension cases extend the stateby one variable, and the third case extends the state bytwo variables. The diffuse coefficient for all of the objectssimulated is Cd “ 0.5, leading to a C « 0.3056. Severalof the GEO objects are HAMR objects with AMR valuesabove the typical value for a GPS satellite 0.02 m2/kg.

Also, measurement noise is incorporated into the analy-sis to see the combined impact of extending the state vec-tor and including measurement noise. Table 1 gives thedifferent measurement noise cases tested. Observability

analysis with each of the extended state cases was pre-formed without measurement noise, with a noise ratio of1.0, and with a noise ratio of 0.25. The noise ratio is de-fined by the ratio between the angle measurements andthe ratio between the angular rate measurements.

Noise ratio “σ2α

σ2δ

“σ2

9α

σ29δ

(35)

The ten orbit cases tested are given in Tables 2 and 3.GEO 1, GEO 3, and GEO 5 in Table 2 are high-area-to-mass ratio (HAMR) objects.

Table 1. Measurement Noise Cases.

Noise Ratio Case ValueNo Noise -

Ratio = 1.0σ2α “ σ2

δ “ 1.0 arcsecσ2

9α “ σ29δ“ 20.0 arcsec/sec

Ratio = 0.25

σ2α “ 1.0 arcsec, σ2

δ “ 4.0 arcsecσ2

9α “ 20.0 arcsec/sec,σ2

9δ“ 80.0 arcsec/sec

3.1. GEO Extended State Observability

The invertibility of the observability matrix, Equation 11,is determined at each time step tk. The matrix is invert-ible when the singular values are all greater than zero.Since this is being numerically determined, a tolerance,given by Equation 12, is implemented to account for nu-merical errors in the analysis. Therefore, for each of theextended state and measurement noise observability casespresented, the singular values of the observability matrixare compared to the tolerance.

0 1 2 3 4 5 6

Time [h]

10-40

10-30

10-20

10-10

100

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6 Tol

Figure 1. GEO 4 baseline observability case with onlyposition and velocity in the state vector, where si denotesthe singular values of the observability matrix.

Table 2. GEO Test Orbits

Orbit # 1 2 3 4 5a (km) 42170.238 42190.793 42164.796 42166.668 42308.743e (-) 9.7343e-4 4.9220e-4 1.5716e-3 1.6556e-4 3.1852e-4

i (deg) 35.7448 2.6447 7.4310 0.051497 13.8121Ω (deg) 359.3036 295.4140 52.7661 123.2611 30.9790ω (deg) 124.1101 255.2388 114.5845 79.7058 346.1611ν (deg) 0.0 0.0 0.0 0.0 0.0

AMR (m2/kg) 5.00 0.02 1.00 0.02 20.0

Table 3. LEO Test Orbits

Orbit # 1 2 3 4 5a (km) 8124.9673 7464.0111 7059.5685 7868.6408 7011.9387e (-) 1.4686e-1 1.1651e-2 1.3704e-2 2.2336e-3 1.7747e-3

i (deg) 32.8687 28.3284 65.0611 74.0150 39.7500Ω (deg) 55.8261 302.1046 18.5642 107.5861 28.6732ω (deg) 53.8800 183.3909 176.8673 260.0122 58.1497ν (deg) 0.0 0.0 0.0 0.0 0.0

AMR (m2/kg) 0.02 0.02 0.02 0.02 0.02

Figure 1 shows the results of observability analysis forthe GEO 4 object with a non-extended state vector, con-taining position and velocity only. This analysis is per-formed without measurement noise. Each of the six sin-gular values of the observability matrix, given by the col-ored lines in Figure 1, correspond to the six state ele-ments. The tolerance line, given by Equation 12, is shownas a black dashed line. When all of the singular valuesgo above the tolerance line, the observability matrix isinvertible, and therefore, the system is observable. Themeasurement spacing used in this case is measurementsevery 40.0 seconds. Given this measurement spacing andthe initial conditions in Table 2, the position and veloc-ity of this object become observable after approximately4.867 hours. Since this GEO 4 orbit is close to circularand equatorial, the z-components of position and veloc-ity are difficult to determine from measurements. Thisbehavior can be seen in Figure 1, where the smallest sin-gular value in red takes longer to cross the tolerance linethan all of the other singular values. Note that in Figure1, the fifth and sixth singular values, given in yellow andbrown, respectively, are indistinguishable in the plot.

Next, the state vector is extended to include AMR. Whenthe AMR is incorporated into the observability analysis,Equation 18 is used to form the STM, and the linearizedmeasurement matrix has an extra column of zeros. Fur-thermore, the tolerance changes since the dimension ofthe observability matrix is now 7ˆ 7.

Figure 2 shows the extended state observability analysisresults with AMR added to the state vector. Seven singu-lar values are now compared to a tolerance line. With the

0 1 2 3 4 5 6

Time [h]

10-40

10-30

10-20

10-10

100

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6

s7 Tol

Figure 2. GEO 4 extended state observability with AMR,where si denotes the singular values of the observabilitymatrix.

addition of AMR to the state, the smallest singular valuecrosses the tolerance line in approximately 4.956 hours,with observations every 40.0 seconds. Because of the ad-dition of AMR to the state vector, more information mustbe determined from the measurements; therefore, moremeasurements, and thus more time, are required for thesystem to become observable.

The next extended state case tested considers the quantitypAMR¨C q together as one state element. When these twoquantities are added to the state vector as one element to-

0 1 2 3 4 5 6

Time [h]

10-40

10-30

10-20

10-10

100

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6

s7 Tol

Figure 3. GEO 4 extended state observability with thequantity pAMR¨C q, where si denotes the singular valuesof the observability matrix.

gether, then the system does become observable, as seenin Figure 3. The time to become observable in this casewith measurements every 40.0 seconds is approximately4.956 hours, which is the same as the first extended statecase. In this case of considering pAMR¨C q together, theobservability analysis is determining the effect of a con-stant of the SRP perturbing acceleration on the time tobecome observable. For this object, the first and secondextended state cases have the same time to become ob-servable; this may not always be the case since the pa-rameter added to extend the state is different: AMR ver-sus pAMR¨C q, as seen in the LEO results of Section 3.2.

0 1 2 3 4 5 6

Time [h]

10-50

10-40

10-30

10-20

10-10

100

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6

s7

s8 Tol

Figure 4. GEO 4 extended state observability with AMRand C, where si denotes the singular values of the ob-servability matrix.

The final extended state case tested in this work, is theextension by two variables, AMR and C. In this case, theobservability matrix uses a STM formed with Equation23, which is dimension 8 ˆ 8. When two constants usedin the perturbing acceleration are added to the state vec-

tor in observability analysis, the system is not observable.This can be seen in Figure 4, where the smallest singu-lar value in red does not cross the tolerance line. Evenif this analysis were to be extended with more measure-ments and thus more time, this system would not becomeobservable. Since AMR and C are both constants in theSRP perturbing acceleration, measurements of right as-cension, declination, and the angular rates are not suffi-cient for determining changes in the orbit due to AMRcompared to changes due to C. However, if differentmeasurement types were added to this analysis, whichcould be used to determine AMR or C, then this systemwould become observable after some time.

[r;v] Only AMR (AMR·C ) AMR & C

0

1

2

3

4

5

Tim

e to

Bec

ome

Obs

erva

ble

[h] GEO 1

GEO 2GEO 3GEO 4GEO 5

Figure 5. All five GEO objects without measurementnoise.

The times to become observable for all five of the GEOobjects without measurement noise for each of the ex-tended state cases are given in Figure 5. The increase ineach of the times to become observable as a result of ex-tending the state can be seen. Also, note that the extendedstate case with AMR & C as separate state elements isunobservable for each GEO object. In addition, for thesefive GEO objects, the time to become observable is de-pendent on inclination of the orbit. A larger inclination,as in GEO 1, results in a lower time to become observ-able. Conversely, a smaller inclination, as in GEO 4, re-sults in a longer timer to become observable. The timesfor each object to become observable are given in Table4.

Next, each of the extended state observability analysiscases for the five GEO objects are performed with mea-surement noise ratios given in Table 1. The observabilitymatrix in this analysis includes measurement noise givenby Equation 15. The times for the smallest singular valueof each noise case to cross the tolerance line are shownin Figure 6. The times to become observable are on they-axis in hours, and the different extended state cases areon the x-axis. The blue squares represent the cases with-out measurement noise. The red triangles and the greencircles represent the two different measurement noise ra-tio cases. In all of the noise cases tested, the impact ofextending the state is given by the increase in the time to

[r;v] Only AMR (AMR·C ) AMR & C

4

4.5

5

5.5

6T

ime

to B

ecom

e O

bser

vabl

e [h

]

No NoiseNoise (ratio = 1.0)Noise (ratio = 0.25)

Figure 6. GEO 4 extended state observability cases withmeasurement noise.

become observable.

The changes in the times to become observable result-ing from the measurement noise are explained in previ-ous work on incorporating measurement noise into ob-servability analysis [4]. For the GEO 4 object, which isnearly circular and equatorial, the declination measure-ments are important for determining the z-components ofthe state. Therefore, when the noise in the declinationmeasurements is four times larger than the measurementnoise in right ascension, noise ratio “ 0.25, the time tobecome observable increases. In addition, the previousresults of incorporating measurement noise into observ-ability analysis showed that when the noise ratio equalsone there is only a small change in the time to becomeobservable, which is also seen in Figure 6. Next, similarresults are presented for the five LEO objects.

3.2. LEO Extended State Observability

Similar trends to the GEO extended state observabilityresults can be seen in the LEO extended state observabil-ity results. The five LEO objects simulated are given inTable 3. The times to become observable for the LEOobjects are shorter than the times for the GEO objects be-cause of the higher orbit energy of the LEO objects; thestates of the LEO objects change by a larger amount thanthe states of the GEO objects in the same amount of time.The extended state observability results without measure-ment noise for the LEO objects are summarized in Figure7.

For each LEO object in Figure 7, the time to become ob-servable increases when an extra state element is added tothe analysis, with changes of several orders of magnitudefor a few of the objects. The perturbing effect of SRPon a LEO object is much smaller than on a GEO object.Therefore, these large changes in the time to become ob-servable are explained by how much of an effect SRP hason the LEO objects. Since the SRP perturbation effect on

[r;v] Only AMR (AMR·C ) AMR & C

10−2

10−1

100

101

102

103

104

Tim

e to

Bec

ome

Obs

erva

ble

[h] LEO 1

LEO 2LEO 3LEO 4LEO 5

Figure 7. All five LEO objects without measurementnoise.

LEO objects is small relative to other perturbing forces,the time for the extended state parameters to become ob-servable increases. Unlike the GEO results, there is not aclear dependence on inclination for the LEO objects. Thelocation of the observer becomes more significant for theLEO objects, so dependence on inclination is not as clear.Another interesting result for the LEO observability anal-ysis is the decrease in the time to become observable fromthe extended state case with AMR to the extended statecase with pAMR¨C q. This result is seen more clearly inFigure 8.

[r;v] Only AMR (AMR·C ) AMR & C

10−2

10−1

100

101

Tim

e to

Bec

ome

Obs

erva

ble

[s]

No NoiseNoise (ratio = 1.0)Noise (ratio = 0.25)

Figure 8. LEO 1 extended state observability cases withmeasurement noise.

The Aptkq matrix used to define the STM at each timestep will be different for the two cases. The last columnof the Aptkq contains the term BaSRPptkq

Bxp7q . When xp7q

is the combined term pAMR¨C q, the three terms in thelast column of Aptkq are smaller values or more nega-tive because C « 0.3056. These smaller values in theAptkq matrix result in a larger value of the smallest sin-gular value of the observability matrix. A rigorous proofof this behavior is yet to be determined, but this behavior

accounts for the decrease in the time to become observ-able when extending the state with the combined termpAMR¨C q compared to extending the state with AMRonly. In Figure 7, it appears that only some objects ex-hibit this behavior of decreased time to become observ-able between the two extended state cases, but the timesgiven in Table 5 show that the times to become observ-able in the combined extend state case are less than thetimes to become observable than the AMR only extendedstate case. This behavior is not seen with the five GEOobjects tested; this could be due to the differences in theeffect of SRP on LEO and GEO objects, or it could alsobe due to the large time steps used in the GEO analysis.

Figure 8 also depicts the impact of measurement noise onthe time to become observable. Similar to the findingsof previous work on measurement noise in observabilityanalysis, for the LEO 1 object, greater measurement noiseon either of the measurements than the other will increasethe time to become observable [4]. Furthermore, even thenoise case where the noise ratio is one is not necessarilythe shortest time to become observable. In addition, theanalysis performed in previous work on this topic lookedat angles-only measurements, so these results are not ex-actly comparable, but some similar trends are apparent.

3.3. Times To Become Observable

Tables 4 and 5 summarize the times to become observablefor each of the extended state cases without measurementnoise. The GEO times are given in hours and the LEOtimes are given in seconds. In Section 3.2 and specif-ically, Figure 7, an interesting behavior was seen withseveral of the times to become observable being lower forthe combined pAMR¨C q extended state case compared tothe AMR extended state case. Even though the behavioris not apparent for all of the LEO objects, Table 5 showsthat the times to become observable for the pAMR¨C qcase are less than the AMR case. This trend is due to thesize of the BaSRPptkq

Bxp7q terms in the formulation of the STM.A similar behavior is suspected for the GEO objects, butthe larger time steps may be obscuring the differences inthe times to become observable between the two extendedstate cases.

Table 4. Times to become observable for the GEO objectswith extended states and without measurement noise,times given in hours.

Orbit # rr; vs AMR pAMR¨C q AMR & C

1 0.467 0.511 0.511 –2 4.2 4.289 4.289 –3 2.967 3.078 3.078 –4 4.867 4.956 4.956 –5 1.389 1.478 1.478 –

Next, Table 6 gives the same times to become observable

Table 5. Times to become observable for the LEO objectswith extended states and without measurement noise,times given in seconds.

Orbit # rr; vs AMR pAMR¨C q AMR & C

1 0.04 0.824 0.536 –2 1.552 6.08 5.92 –3 0.536 0.808 0.768 –4 0.40 904.0 864.0 –5 0.02 1.60 0.80 –

as Table 4, but with the time the seventh singular valueof the AMR & C extended state case crosses the toler-ance line. For all of the GEO objects, the time for thisseventh singular value of the AMR & C extended statecase to cross the tolerance line is greater than the timesto become observable for the AMR and pAMR¨C q cases.Therefore, this indicates that the times to become observ-able for seven of the state parameters increases when aneighth parameter of the state is unobservable.

Table 6. Seventh singular values of the AMR & C ex-tended state case for the GEO objects.

Orbit # rr; vs AMR pAMR¨C qAMR & C

(7th s)1 0.467 0.511 0.511 0.5442 4.2 4.289 4.289 4.3673 2.967 3.078 3.078 3.1784 4.867 4.956 4.956 5.0445 1.389 1.478 1.478 1.567

All of the LEO objects for the AMR & C extended state,given in Table 7, exhibit the same trend as found in Table6, where the time for the second smallest singular valueof the AMR & C extended state case to go above the tol-erance line is greater than the times for the other extendedstate cases to become observable.

Table 7. Seventh singular values of the AMR & C ex-tended state case for the LEO objects.

Orbit # rr; vs AMR pAMR¨C qAMR & C

(7th s)1 0.04 0.824 0.536 0.8562 1.552 6.08 5.92 8.563 0.536 0.808 0.768 0.8324 0.40 904.0 864.0 1052.05 0.02 1.60 0.80 1.672

3.4. Observations Spacing for Sensor Tasking

Next, more realistic observation spacing is implementedin the discrete-time observability analysis. In addition,propagation time is extended to determine whether thereis a point at which the singular values are at a maximumover an orbit.

The following results are for discrete-time observabilityanalyses, where observations are taken in three batches,with five observations per batch and 100 seconds betweeneach observation. The time between the observationsbatches is varied to determine if there is a point wherethe information from the observations is maximized. In-tuitively, and from initial orbit determination methods,spreading observations farther apart should provide moreinformation about an orbit. However, the geometry of theorbit and observer complicates how observations shouldbe spaced to maximize observability.

0 2 4 6 8 10 12 14

Time Between Batches [h]

10-25

10-20

10-15

10-10

10-5

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6 Tol

Figure 9. GEO 4 rr; vs observability with three batchesof observations, where si denotes the singular values ofthe observability matrix.

Figure 9 shows the singular values of the observabilitymatrix for the GEO 4 object with only r and v in thestate vector. The y-axis contains the six singular valuesof the observability matrix corresponding to the six stateelements, and the x-axis is the spacing between observa-tion batches. For example, at the eight hour spacing mark,batches of observations would be taken at t “ 0.0 h, t “8.0 h p0.334 periodq, and t “ 16.0 h p0.668 periodq. Notethat the system becomes observable when three batchesof observations are spread by approximately 3.24 hours.Changing the number of observation batches and thenumber of observations per batch will change the spac-ing between batches required for the system to becomeobservable. In addition to the point where this system be-comes observable, another point of interest is where thesmallest singular value, given in red in Figure 9, is at amaximum. When this point is at a maximum, given itis above the tolerance line, this point could be consid-ered as the best time to observe the object. In the pres-ence of measurement noise and uncertainty in the stateused to calculate the STM and linearized measurement

matrix, the singular value curve given in red may changeslightly; therefore, the best time to observe this object toensure observability of the state elements is to space theobservation batches so the singular values will be furthestfrom the tolerance line. For Figure 9, this point occurswhen observations batches are spread by approximately8.49 hours.

0 2 4 6 8 10 12 14Time Between Batches [h]

10-23

10-22

10-21

10-20

10-19

10-18

Sm

alle

st S

ingu

lar

Val

ue

[r;v] OnlyAMR(AMR·C )Tol

Figure 10. Smallest singular values of the different GEO4 observability cases with three batches of observations.

Next, the same simulations were performed for two ofthe extended state cases: AMR and pAMR¨C q. Figure10 shows the smallest singular value of each observabil-ity run against the spacing between observation batches.Note there are two tolerance lines because the extendedstate analysis has an observability matrix of dimensionseven rather than dimension six of the observability ma-trix for the analysis with position and velocity only. Onthe scales given, the points where these curves cross thetolerance lines appears to be the same, but upon closerinspection the crossing points are slightly different.

The addition of the SRP parameters in the state wasexpected to change the best time to observe an object,similar to how the time to become observable, in Ta-ble 4, changed with addition of the extra state parame-ter. However, the point of maximum observability didnot change for the GEO 4 object; the maximum point ofeach observability run in Figure 10 is the same, approx-imately 8.49 hours. Differences in the maximum pointcould be obscured by the time step used in this analysis,40.0 seconds. However, the reason for performing thisanalysis of finding the maximum point is to determinewhether or not extending the state will drastically changethis maximum point; therefore, changes smaller than thetime step of 40.0 seconds would not be significant.

This analysis is repeated for the GEO 1 object whichhas a larger AMR “ 5.0 m2/kg. Figure 11 shows theobservability analysis results for the GEO 1 object withchanging the spacing between observation batches. Thesmallest singular value curve, given in red, has a differentshape than the corresponding GEO 4 curve.

0 2 4 6 8 10 12 14

Time Between Batches [h]

10-25

10-20

10-15

10-10

10-5

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6 Tol

Figure 11. GEO 1 rr; vs observability with three batchesof observations, where si denotes the singular values ofthe observability matrix.

0 2 4 6 8 10 12 14

Time Between Batches [h]

10-25

10-20

10-15

10-10

10-5

Sin

gula

r V

alue

s, s

s1

s2

s3

s4

s5

s6

s7 Tol

Figure 12. GEO 1 AMR extended state observability withthree batches of observations, where si denotes the sin-gular values of the observability matrix.

Figure 12 shows the observability analysis results for theGEO 1 object with a state extended by AMR. The small-est singular value curve appears to be bounded by the sec-ond smallest singular value. The red curve never actuallyis the same as the blue curve in Figure 12, but it doeschange the shape of the red curve when compared to theresults for only position and velocity in the state. Thesmallest singular value curves for the different extendedstate cases are compared in Figure 13.

In Figure 13, the smallest singular value curves match ex-cept for the region where observation batches are spacedbetween approximately 9.5 and 12.0 hours. However, thevalue of the maximum for the blue and green curves isclose to the value of the red curve at the same observa-tion spacing. Therefore, the differences in the singularvalue curves will not affect the maximum observabilitypoint significantly.

0 2 4 6 8 10 12 14Time Between Batches [h]

10-22

10-21

10-20

10-19

10-18

10-17

Sm

alle

st S

ingu

lar

Val

ue

[r;v] OnlyAMR(AMR·C )Tol

Figure 13. Smallest singular values of the different GEO1 observability cases with three batches of observations.

4. CONCLUSIONS AND FUTURE WORK

This work focused on the implementation and effectsof extending the state vector in observability analysis.Discrete-time observability methods use an observabilitymatrix formed with the state transition matrix (STM) andthe linearized measurement matrix. This analysis was ex-tended beyond the position and velocity state by includ-ing parameters of solar radiation pressure (SRP): area-to-mass ratio (AMR) and the material dependent reflectioncoefficient, C. In order to incorporate these parametersin the observability analysis, the SRP perturbing acceler-ation is included with the two body dynamics. Therefore,the Jacobian of the SRP acceleration with respect to thestate is used in the formulation of the STM differentialequation. Also, when SRP parameters are added to theextended state, extra columns are added to the linearizedmeasurement matrix.

Discrete time-varying observability analysis was per-formed for ten objects, five low Earth orbit (LEO) ob-jects and five geosynchronous (GEO) objects. Each ofthe ten objects were simulated for four different cases:position and velocity only and three extended state cases.The three extended state cases were AMR, pAMR¨C q,and AMR & C, where C “ 1

4 `19Cd. Furthermore, mea-

surement noise is added to the observability analysis tosee the combined effect of extending the state and mea-surement noise.

The system is considered observable when the smallestsingular value of the observability matrix is greater thana tolerance line defined by the largest singular values, sizeof the observability matrix, and the machine epsilon. Theeffects of extending the state vector in the observabilityanalysis can be seen in the times to become observable.The times to become observable increased when solar ra-diation pressure parameters were added to the positionand velocity state. Furthermore, one of the extended statecases, where AMR & C are considered as separate pa-

rameters, does not become observable for the angle andangular rate measurements used in this analysis. If dif-ferent measurement types were to be added, the effectof AMR could be differentiated from the effect of C,thus making this case observable after some time. Eventhough this extended state case did not become observ-able in this analysis, the times for the second smallestsingular value to become greater than the tolerance in-creased. Therefore, extending the state beyond what canbecome observable is detrimental to the other observablevalues in the system.

If specific object characteristics are desired, similar anal-yses to the methods implemented in this work couldbe implemented to determine what object characteristicsare observable with specific measurement types and overspecific time periods. If an object characteristic is in-cluded in the analysis and the characteristic is not observ-able, the rest of the system could be impacted as seen bythe time to become observable in this work.

In addition, more realistic observation spacing was im-plemented, and the observability results looking at thetime to become observable were extended to determinewhen the smallest singular value of the observability ma-trix was is maximized. This maximum point of the small-est singular value could be considered as the observationtime where the system is most observable.

Future work on extended state observability will focuson including different measurement types to resolve theAMR and C SRP parameters. Furthermore, work on theobservation spacing will focus on determining the differ-ent trends seen in the GEO objects and why the differ-ences in the extended state cases are occurring. In addi-tion, realistic constraints will be placed on the times toobserve an object and the measurement noise will be de-fined for specific sensor systems.

ACKNOWLEDGMENTS

This material is based on research sponsored by Air ForceOffice of Scientific Research under agreement numberFA9550-14-1-0348DEF. The author would also like toacknowledge the Indiana Space Grant Consortium fortheir support of this research.

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