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(Preprint) AAS XX-XXX
OBSERVABILITY-AWARE NUMERICAL ALGORITHM FORANGLES-ONLY INITIAL RELATIVE ORBIT DETERMINATION
Adam W. Koenig∗ and Simone D’Amico†
This paper presents a simple numerical solution to the problem of estimating therelative orbit of a nearby resident space object and a subset of the observer’s ab-solute orbit elements using inertial bearing angle measurements from a singlemonocular camera. Unlike previous approaches, this algorithm is valid for ar-bitrary planetary orbit regimes, is robust to kilometer-level errors in the providedestimate of the observer’s orbit, and provides a covariance estimate for the com-puted relative state. A quantitative observability analysis is first conducted to char-acterize the achievable estimation accuracy of the absolute and relative orbits withrealistic sensor errors in a range of formation geometries. Leveraging the resultsof this analysis, an algorithm is proposed that estimates strongly observable statecomponents through iterative batch least squares and estimates the weakly ob-servable range through a sampling approach. The algorithm is validated throughMonte Carlo and hardware-in-the-loop simulations in various orbit regimes usinga high-fidelity numerical orbit propagator and representative error sources. Thesimulation results show that the proposed algorithm provides relative orbit esti-mates that are at least as accurate as the best approaches in literature in a widerrange of orbit regimes in the presence of larger errors.
INTRODUCTION
The success of flagship missions such as GRACE (NASA),1 TanDEM-X (DLR),2 PRISMA,3 andthe Magnetospheric Multiscale (MMS) mission (NASA)4 has attracted great interest to formation-flying and spacecraft swarm technologies. A well-known need in these fields is development oflow-cost navigation technologies that can be used in the widest possible range of mission scenarios.Vision-based sensors (VBS) hold great promise in this regard due to three advantages: 1) they haveminimal mass, power, and volume requirements, 2) they are passive with high-dynamic range, and3) they are ubiquitous on modern spacecraft in the form of star trackers.
Angles-only navigation is a particularly promising application for vision-based sensors. In thispaper, angles-only navigation refers to the general problem of estimating the orbits of objects inclose proximity using bearing angle measurements from one or more monocular cameras. The mostwell-studied variant of this problem consists of estimating the relative orbit of a single residentspace object using measurements from a single observer with a known orbit. This capability wasfirst demonstrated in flight by the ARGON experiment in the PRISMA mission in 2012.5 A more au-tonomous demonstration was conducted during the AVANTI experiment in 2016.6–8 However, bothof these experiments relied on a-priori relative state information from NORAD two-line-elements(TLE) to initialize the navigation filter. To enable navigation in more challenging scenarios (e.g.
∗Postdoctoral Scholar, Aeronautics & Astronautics Dept., Durand Building, 496 Lomita Mall, Stanford, CA, 94305.†Assistant Professor, Aeronautics & Astronautics Dept., Durand Building, 496 Lomita Mall, Stanford, CA, 94305.
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orbiting other planetary bodies), it is necessary to compute initial relative orbit estimates onboardin real-time.
This problem is challenging because angles-only navigation with a single camera is subject to awell-known range ambiguity. Specifically, Woffinden9 demonstrated that the range is unobservableif the Cartesian relative position evolves according to a homogenous linear model. This issue wasresolved for the ARGON5 and AVANTI8 experiments by regularly executing maneuvers to improveobservability. An alternative approach proposed by Geller10 was to offset the camera from theobserver center of mass to enable triangulation of the range. However, this approach does not addsufficient observability to be of practical value at kilometer-level separations. More recent literaturehas instead focused on use of accurate nonlinear dynamics models to resolve the range. In particular,studies by Lovell11 and Hu12 using Lie-derivative criteria have shown that the relative orbit ofa target is observable in Keplerian orbits if certain geometric conditions are met. Additionally,recent work by Sullivan13, 14 has demonstrated maneuver-free angles-only relative navigation usingan unscented Kalman filter with a nonlinear dynamics model based on relative orbital elements(ROE) in orbit regimes from low earth orbit (LEO) to geostationary orbit.
However, few approaches to the initial relative orbit determination (IROD) problem using bear-ing angle measurements have fully exploited available dynamics models that accurately model theeffects of perturbations (e.g.15, 16). For example, the approach proposed by Garg17 uses an unper-turbed second-order model of the relative dynamics in Cartesian space, which neglects the effectsof perturbations. Similarly, Geller18 used an unperturbed linear dynamics model in cylindrical coor-dinates. A more recent algorithm proposed by Sullivan13 instead computes a closed-form estimateof the relative state, which is then refined using iterative regularized batch least squares with ameasurement model that accounts for the earth oblateness J2 perturbation. However, the initialestimate is often subject to large errors and does not always converge to the correct range. The best-performing IROD algorithm to date was developed by Ardaens19 for low earth orbit. By exploitingthe separation between strongly and weakly observable components of the ROE, Ardaens computesa one-dimensional set of candidate state estimates with ranges within specified windows using it-erative batch least squares. The final state is selected as the candidate estimate with the smallestmeasurement residuals. This algorithm was tested on flight data from the AVANTI experiment andproduced state estimates with range errors of less than 10% in a small number of test cases. How-ever, these algorithms are still subject to two major limitations: 1) range of applicability (separationand orbit eccentricity), and 2) reliance on accurate absolute orbit knowledge.
The objective of this paper is to develop an IROD algorithm that 1) provides estimates of therelative state and it’s uncertainty, 2) is valid in arbitrary planetary orbit regimes, and 3) is toler-ant of large errors in provided absolute orbit estimates. When included with image processing andmeasurement assignment algorithms such as those proposed by Kruger,20 such an algorithm couldenable fully autonomous angles-only navigation for novel mission architectures, some of whichwill be tested as part of the Starling Formation-Flying Optical eXperiment (StarFOX) on the NASAStarling1 mission in 2021.21 Within this context, the contributions of this paper to the state of the artare twofold. First, a quantitative observability analysis is conducted to characterize the feasibilityof estimating a subset of the absolute orbit of the observer in addition to the relative orbit of thetarget using bearing angle measurements from a single camera. Second, a new IROD algorithm in-spired by the work of Ardaens19 is proposed to estimate the relative orbit of the target as well as itsuncertainty. Leveraging the findings of the observability analysis, the proposed algorithm also esti-mates the semimajor axis of the observer orbit. The algorithm is validated through Monte Carlo and
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hardware-in-the-loop simulations using high-fidelity numerical orbit propagator and representativeerror sources in various earth and mars orbit regimes. These simulations show that the algorithmcan exceed the estimation accuracy of current state-of-the-art algorithms in more diverse and chal-lenging scenarios. Overall, these results show that the proposed algorithm can enable autonomousangles-only navigation in the StarFOX experiment and other novel mission applications.
MODELING PRELIMINARIES
Before describing the proposed algorithm, it is necessary to define the estimated state, measure-ments, and the model that relates the two.
State Definitions
As the algorithm proposed in this paper is intended for onboard implementation and requiresmultiple iterative batch least-squares solutions, it is imperative to minimize the computation cost ofevaluating both the measurement model and the measurement sensitivity matrix. To meet this need,the selected state parameterization includes the mean absolute orbit of the observer (denoted α))and mean relative orbital elements of the target resident space object (denoted δα). The mean orbitis computed from the osculating (or instantaneous) orbit by applying a transformation that removesthe short-period oscillations caused by perturbations. For clarity, osculating quantities are hereafterdenoted by the notation ( · ).
The selected absolute orbit state consists of the mean quasi-nonsingular orbit elements defined as
α =
aexeyiΩu
=
ae cos(ω)e sin(ω)
iΩ
ω +M
(1)
where a the semimajor axis, e is the eccentricity, i is the inclination, Ω is the right ascension of theascending node (RAAN), ω is the argument of perigee, and M is the mean anomaly. Correspond-ingly, ex and ey are the components of the eccentricity vector.
The relative orbit state consists of the quasi-nonsingular relative orbital elements adopted byD’Amico,22 which are defined as functions of the orbit elements of the observer (denoted by sub-script o) and target (denoted by subscript t) as given by
δα =
δaδλδexδeyδixδiy
=
(at − ao)/aout − uo + cos(io)(Ωt − Ωo)
ex,t − ex,oey,t − ey,oit − io
sin(io)(Ωt − Ωo)
(2)
These state definitions were selected for three reasons. First, it has been found that analyticaldynamics models developed for these states are among the most accurate models in literature otherthan numerical integration for a range of orbit regimes.16, 23 Accordingly, the algorithm proposedin this paper can be adapted to a wide range of orbit regimes by properly selecting the included
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perturbations. Second, the relative state definition decouples the weakly observable range fromother strongly observable state components. Accordingly, weakly and strongly observable statescan be estimated with different techniques at minimal computation cost. Third, the absolute orbitstate definition allows independent determination of parameters such as orbit period, eccentricity,and orientation through estimation of a subset of the orbit elements. The subset of estimated absoluteorbit elements can be selected to meet various mission needs.
Measurement Model
The batch estimation algorithm proposed in this paper requires a measurement model that com-putes the bearing angles y at a measurement epoch tm from the absolute and relative states at theestimation epoch te. This model can be expressed as a function of the form
y(tm) = h(αo(te), δα(te), te, tm) (3)
The algorithm also requires a measurement sensitivity model that evaluates (or approximates) thepartial derivatives of the measurements with respect to αo(te) and δα(te).
The measurement model developed in the following is divided into two parts for clarity: the dy-namics model and the geometry model. The dynamics model computes the osculating orbits of theobserver and target at tm from the mean absolute and relative orbits at te. The dynamics model forearth orbit used in this paper includes only the J2 perturbation. The dynamics model used in marsorbit includes both J2 and solar radiation pressure. The proposed algorithm can be adapted to anyother planetary orbit regimes by using a dynamics model that includes the dominant perturbations(e.g.16). The geometry model computes the bearing angle measurements from the osculating orbitsof the observer and target at tm and the attitude of the observer’s camera. This model is fixed bystate definition.
Dynamics Model
The dynamics model computes the osculating orbits of the observer and target at tm from pro-vided mean absolute and relative orbits at te. This procedure is divided into three steps. First, themean absolute orbit of the target is computed from αo and δα as given by
αt =
ao(1 + δa)ex,o + δexey,o + δeyio + δix
Ωo + csc(io)δiyuo + δλ− cot(io)δiy
(4)
Second, the mean orbits of the orbserver and target are propagated from te to tm. For the earth orbitscenarios considered in this paper, the propagation is accomplished using an analytical propagationmodel that includes the secular drifts in the mean anomaly, argument of perigee, and RAAN due toJ2, which are given in Alfriend24 asMJ2
ωJ2ΩJ2
=3
4
J2R2E
õ
a7/2η4
η(3 cos2 (i)− 1)5 cos2 (i)− 1−2 cos (i)
(5)
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where η is defined asη =
√1− e2 (6)
If ∆t = tm − te and no maneuvers are executed, then the mean orbits of a spacecraft at tm and teare related by
α(tm) =
a(te)ex(te) cos(ωJ2∆t)− ey(te) sin(ωJ2∆t)ex(te) sin(ωJ2∆t) + ey(te) cos(ωJ2∆t)
i(te)
Ω(te) + ΩJ2∆t
u(te) + (ωJ2 + MJ2 + n)∆t
(7)
where n denotes the mean motion of the orbit. It should be noted that unlike linear models,15, 16
this propagation procedure is valid for arbitrarily large separations between the observer and target(provided that both are in orbits where J2 is the dominant perturbation). The dynamics model usedfor mars orbit scenarios is provided in the Appendix.
Finally, the osculating orbits of the observer and target are computed from the correspondingmean orbits. This is accomplished for both earth and mars orbit scenarios using Schaub’s mean toosculating transformation.25
Geometry Model
The geometry model computes the bearing angle measurements from the osculating orbits ofthe observer and target at tm and the sensor attitude. The measurements are computed in foursteps. First, the Cartesian positions of the observer and target in the earth-centered inertial (ECI)frame (denoted by rIo and rIt ) are computed from the osculating orbit elements using conventionalmethods. Second, the inertial relative position (denoted δrI) is computed as given by
δrI = rIt − rIo (8)
Third, the relative position in the sensor frame (denoted δrV ) is computed as given by
δrV = RI→VδrI (9)
whereRI→V is the matrix that rotates a vector from the ECI frame to the sensor frame. This matrixwill be provided by the sensor (if it is a star tracker camera) or by the spacecraft bus. Finally, theazimuth ψ and elevation φ angles are computed from δrV as given by
y =
(ψφ
)=
(arcsin
(δrVy /||δrV ||
)arctan
(δrVx /δr
Vz
) ) (10)
Measurement Sensitivity Model
To enable refinement of the state estimate using iterative batch least squares, it is also necessary toproduce a model of the sensitivity of the measurements to changes in the provided states. This is ac-complished by differentiating the measurement model using the chain rule. This requires evaluationof the partial derivatives of each intermediate function at the current estimate. Adopting the same
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distinction between the dynamics and geometry models from the previous section, the sensitivitiesof the measurements to changes in the provided states can be expressed as
∂y(tm)
∂αo(te)=
∂y(tm)
∂αt(tm)
∂αt(tm)
∂αo(te)+
∂y(tm)
∂αo(tm)
∂αo(tm)
∂αo(te)
∂y(tm)
∂δα(te)=
∂y(tm)
∂αt(tm)
∂αt(tm)
∂δα(te)+
∂y(tm)
∂αo(tm)
∂αo(tm)
∂δα(te)
(11)
Each of the terms on the right-hand side of Equation 11 is a product of the sensitivities of the ge-ometry model (left) and the dynamics model (right). Each of these terms are derived analytically inthe following.
Geometry Sensitivity Model
The sensitivities of the bearing angles to changes in the osculating orbits of the observer andtarget can be expanded as given by
∂y
∂αt=
∂y
∂δrV∂δrV
∂rtI∂rtI
∂αt
∂y
∂αo=
∂y
∂δrV∂δrV
∂roI∂ro
I
∂αo(12)
where all quantities are evaluated at tm. The sensitivity of the bearing angles to changes in δrV isgiven by
∂y
∂δrV=
1
||δrV ||
[− sin(ψ) sin(φ) cos(ψ) − sin(ψ) cos(φ)sec(ψ) cos(φ) 0 − sec(ψ) sin(φ)
](13)
The sensitivities of δrV to changes in rIt and rIo are given by
∂δrV
∂rIt= RI→V
∂δrV
∂rIo= −RI→V (14)
Finally, the sensitivity of rI to changes in the osculating orbit elements are given by
∂rI
∂α=
∂rI
∂αkep
∂αkep∂α
=∂rI
∂αkep
1 0 0 0 0 00 cos(ω) sin(ω) 0 0 00 0 0 1 0 00 0 0 0 1 00 − sin(ω)/e cos(ω)/e 0 0 00 sin(ω)/e − cos(ω)/e 0 0 1
(15)
where αkep denotes the osculating Keplerian orbit elements (a, e, i, Ω, ω, and M ), and ∂rI
∂αkepis
the sensitivity matrix provided in.26 While this model is undefined at the singularity (e = 0), theauthors have not found any issues using this model in orbits with eccentricities as low as 0.001.
Dynamics Sensitivity Model
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The sensitivities of the osculating orbits of the observer and target at tm to changes in the absoluteand relative orbits at te can be expanded as given by
∂αt(tm)
∂αo(te)=∂αt(tm)
∂αt(tm)
∂αt(tm)
∂αt(te)
∂αt(te)
∂αo(te)
∂αo(tm)
∂αo(te)=∂αo(tm)
∂αo(tm)
∂αo(tm)
∂αo(te)
∂αt(tm)
∂δα(te)=∂αt(tm)
∂αt(tm)
∂αt(tm)
∂αt(te)
∂αt(te)
∂δα(te)
∂αo(tm)
∂δα(te)= 0
(16)
where the last term is zero because the trajectory of observer’s orbit does not depend on the relativeorbit of the target. The individual terms of these expansions are computed in the following.
First consider the sensitivity of the osculating orbit elements to changes in the mean orbit ele-ments. Since all off-diagonal terms of the sensitivity matrix for Schaub’s mean to osculating trans-formation25 are on the order of J2 (∼0.001) or smaller, this sensitivity matrix can be approximatedas
∂α
∂α= I (17)
with minimal loss of accuracy.
Next, it is necessary to compute the sensitivity of the mean orbit at the measurement epoch tochanges in the mean orbit at the estimation epoch. This is accomplished by simply evaluating thepartial derivatives of the propagation model (Equation 7 for J2-perturbed orbits). The resultingmatrices for the dynamics models considered in this paper are provided in the Appendix.
Finally, it is necessary to compute the sensitivities of αt to changes in αo and δα. These sensi-tivity matrices are computed by taking the partial derivatives of Equation 4 as given by
∂αt∂αo
=
1 + δa 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 −δiy csc2(io) 0 1
∂αt∂δα
=
ao 0 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 csc(io)0 1 0 0 0 cot(io)
(18)
The analytical measurement sensitivity matrix obtained by combining Equations 11-18 is validfor arbitrarily large separations between the observer and target. Additionally, the only approxima-tion used to derive this matrix is the assumption that the mean to osculating sensitivity matrix isidentity.
OBSERVABILITY ANALYSIS
To maximize the robustness of the proposed algorithm to errors in the provided orbit estimate,it is desirable to estimate the largest possible subset of the absolute orbit elements of the observerin addition to the relative orbit of the target. With this in mind, it is first necessary to conduct anobservability analysis to determine which (if any) absolute orbits can be estimated. While angles-only observability analysis has been widely studied in literature, the problem of simultaneouslyestimating the absolute and relative orbit has received little attention. The most notable study of thisproblem was conducted by Hu,12 which concluded that the orbits of two spacecraft in Keplerianorbits are not fully observable with angles-only measurements. However, this study neglects the
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effect of perturbations (particularly J2), which may improve observability. Additionally, even if thecomplete state is not observable, there is still value in estimating a subset of the state.
The observability analysis conducted in the following is based on a formulation of the covariancematrix for the estimated state computed from the measurement sensitivity matrix and a presumedmeasurement noise matrix. This approach is preferred over other techniques such as Lie derivativesor the observability Gramian due to the ability to produce intuitive and quantitative observabilitymetrics that account for expected measurement errors. Suppose bearing angles are taken at Nepochs denoted t1, ..., tN and the complete measurement sensitivity matrix H∗ is defined as
H∗ =
∂y(t1)∂δα(te)
∂y(t1)∂α0(te)
......
∂y(tN )∂δα(te)
∂y(tN )∂α0(te)
(19)
Additionally, let H denote the matrix formed by taking a subset of the columns of H∗ correspondingto states that are to be estimated. If this state subset has a known covariance P, then the measurementnoise matrix R can be computed as given by
R = HPHT (20)
If H is full column rank (which is the case for all test cases considered in this paper), then it ispossible to solve this equation for P as given by
P = (HTH)−1HTRH(HTH)−1 (21)
For an assumed measurement noise matrix (based on the expected accuracies of the sensor anddynamics model), the matrix P can be used to directly compute the expected estimation accuracyfor each state component (assuming other state components are well-known). Additionally, bycomparing the covariance matrices for different subsets of the state components, it is possible toextract useful insights about the observability of the system such as weakly observable modes.
Using the previously described measurement model, the authors computed covariance estimatesincluding various subsets of the absolute orbit using two absolute and relative orbits. The selectedearth orbits include a LEO (representative of the StarFOX experiment and most formation-flyingmissions to date) and a GTO (the most readily accessible eccentric orbit). The relative states includea formation with passive safety guaranteed by relative eccentricity inclination vector separation anda purely in-train formation, which are representative of expected formation-flying and rendezvousgeometries and are known to differ in observability.11, 12 The absolute and relative orbits for thesetest cases are provided in Table 1. Each test case includes 50 bearing angle measurements evenlyspaced over one orbit (95min in LEO, 10.5hr in GTO). The assumed measurement noise matrix is9.4 × 10−9I, which is equivalent to 20arcsec (1-σ) uncorrelated measurement errors. This erroris representative of errors expected using current state-of-the-art star tracker cameras.27 The testcases include five subsets of the absolute orbit elements. These subsets are: 1) nothing, 2) thesemimajor axis, 3) all orbit elements except the mean argument of latitude, 4) the mean argumentof latitude, and 5) the complete absolute orbit. For each of these subsets, it is assumed that absoluteorbit elements that are not estimated are known with high accuracy. The estimation uncertainty foreach state component is computed by taking the square root of corresponding element on the maindiagonal of the covariance matrix.
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Table 1. Absolute and Relative orbits at estimation epoch for observability analysis
Earth absolute orbitsa (km) ex (-) ey (-) i (deg) Ω (deg) u (deg)
LEO 6978 0.0020 98.0 90.0 45.0 0.0GTO 24521 0.7198 30.0 90.0 0.0 180.0
Mars absolute orbitsa (km) ex (-) ey (-) i (deg) Ω (deg) u (deg)
TBD TBD TBD TBD TBD TBD TBDTBD TBD TBD TBD TBD TBD TBD
Relative orbitsaδa (m) aδλ (km) aδex (m) aδey (m) aδix (m) aδiy (m)
ROE1 0.0 50.0 0.0 2000.0 0.0 2000.0ROE2 0.0 50.0 0.0 0.0 0.0 0.0
The computed uncertainties for the LEO test cases are shown in Table 2. The uncertaintiesof all elements except the semimajor axis are scaled by a to provide a geometric interpretationof their impact on navigation performance. Several conclusions can be drawn from the resultsfor ROE1. First, it is possible to estimate aδa, aδex, and aδix with accuracy of better than 10mif no absolute orbit elements are estimated. The remaining ROE have errors of approximately4% of their nominal values, indicating that these three uncertainties are all caused by the rangeuncertainty. Second, the semimajor axis can be estimated with an uncertainty of better than 100m ata cost of a modest (∼50%) increase in the uncertainties in the ROE. Adding the eccentricity vector,inclination, and RAAN to the estimated state increases ROE uncertainties by factors of 2-10, andthese components can only be estimated with uncertainties of a few kilometers, which is not anappreciable improvement over TLE estimate accuracy. Next, attempting to estimate u increasesthe range uncertainty by a factor of ten compared to only estimating the ROE. This uncertainty is40% of the nominal separation, which is not useful for applications of interest. Finally, while themeasurement sensitivity matrix for the combined absolute and relative orbits is full rank, it is evidentthat the resulting estimation uncertainties are impractically large. The trends for ROE2 are similarto those for ROE1, with the distinction that a subset of the uncertainties are more than ten timeslarger. As such, it is clear that purely in-train formations are characterized by poor observability(particularly due to difficulty distinguishing between δλ and δa). As such, additional mitigationssuch as maneuvers or use of multiple observers will be needed in these scenarios.
The large increase in the range uncertainty that is observed when u is estimated has a geometricinterpretation that can be understood by considering the eigenvector corresponding to the largesteigenvalue of the covariance matrix. These eigenvectors for state subset 4 for each considered ROEare provided in Table 3. These eigenvectors have two common properties. First, the vector formedby the ROE components is almost exactly parallel to the ROE state vectors as specified in Table 1,indicating uncertainty in range. Second, the component corresponding to u is very close to negativeone half of the δλ component. Together, these behaviors suggest that there is a poorly observablemode where any error in the knowledge of u is likely to introduce an error in δλ that is oppositein sign and twice as large. This behavior is expected given the geometry of relative motion in nearcircular orbits as notionally illustrated in Figure 1
The estimation uncertainties for the GTO orbit are shown in Table 4. The trends in the uncer-
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Table 2. Observability Results for LEO
ROE 1 1 1 1 1 2 2 2 2 2State subset 1 2 3 4 5 1 2 3 4 5aσδa (m) 6.75 8.94 13.63 6.84 13.96 145.60 247.98 513.75 147.14 568.07aσδλ (km) 2.00 3.18 5.18 21.60 26.07 41.39 69.75 139.37 249.14 274.73aσδex (m) 1.09 2.14 12.06 3.15 12.06 1.10 1.38 388.21 1.11 469.01aσδey (m) 78.80 124.51 199.95 862.57 1038.06 1.88 2.07 464.41 1.88 485.95aσδix (m) 1.12 1.28 40.61 3.25 40.62 0.97 0.97 319.93 0.98 324.11aσδiy (m) 78.22 124.49 201.84 859.90 1032.22 7.30 12.22 412.91 46.43 415.10σa (m) - 75.71 93.38 - 95.04 - 96.14 249.15 - 284.46aσex (km) - - 1.21 - 1.21 - - 64.61 - 67.56aσey (km) - - 1.70 - 1.78 - - 54.50 - 65.81aσi (km) - - 2.36 - 2.42 - - 57.43 - 57.43aσΩ (km) - - 5.63 - 5.65 - - 45.36 - 45.95aσu (km) - - - 10.92 12.30 - - - 119.81 151.17
Table 3. Maximum singular vectors when estimating u.
State component δa δλ δex δey δix δiy uROE1 0.0000 0.8919 0.0001 0.0356 -0.0001 0.0355 -0.4494ROE2 0.0002 0.9037 0.0000 0.0000 0.0000 -0.0002 -0.4737
Figure 1. Conceptual illustration of the unobservable mode when u is estimated innear-circular orbits.
tainties are identical to the LEO case except that ROE2 (the in-train formation) does not have largeruncertainties than ROE1. This suggests that the observability of the relative orbit in eccentric orbitsis invariant of the geometry of the relative motion. Additionally, the eigenvectors corresponding tothe maximum singular values of the covariance matrices for state subset 4 are nearly identical tothose in Table 3, indicating that a similar weakly observable mode in eccentric orbits.
Overall these results suggest that the mean semimajor axis of the observer orbit can be estimatedwith an uncertainty of better than 100m with minimal impact on relative navigation accuracy usingcurrent star tracker sensors. Estimating any other absolute orbit elements introduces significant rela-
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Table 4. Observability Results for GTO
ROE 1 1 1 1 1 2 2 2 2 2State subset 1 2 3 4 5 1 2 3 4 5aσδa (m) 0.76 0.80 3.58 1.98 3.58 0.48 0.48 2.85 0.48 2.85aσδλ (km) 0.75 3.52 6.87 22.69 32.58 0.64 2.73 7.65 58.80 103.11aσδex (m) 1.15 1.18 3.04 1.86 3.04 0.83 0.83 3.06 0.83 3.06aσδey (m) 29.48 140.12 272.25 898.49 1290.32 0.64 0.86 3.37 26.71 46.74aσδix (m) 1.24 1.26 5.00 2.09 5.03 1.07 1.07 3.60 1.07 3.60aσδiy (m) 29.89 141.59 275.99 911.75 1309.51 0.30 0.62 1.67 12.03 21.09σa (m) - 328.16 334.51 - 372.42 - 280.23 281.90 - 283.21aσex (km) - - 0.42 - 0.48 - - 0.40 - 0.70aσey (km) - - 2.14 - 2.37 - - 2.87 - 2.88aσi (km) - - 1.23 - 1.23 - - 1.19 - 1.19aσΩ (km) - - 3.83 - 3.90 - - 2.95 - 2.95aσu (km) - - - 10.19 16.13 - - - 29.42 51.45
tive navigation uncertainty. Additionally, it appears unlikely that the eccentricity vector, inclination,and RAAN can be estimated with errors better than TLE estimates. Finally, a mode with particu-larly weak observability is introduced when the mean argument of latitude is estimated regardlessof orbit eccentricity and relative motion geometry. It follows that achievable estimation errors in therelative range may be limited by errors in the provided estimate of the mean argument of latitude.In light of these findings, the algorithm proposed in the following section is developed to estimatethe ROE and the observer’s semimajor axis (if enabled).
ALGORITHM DESCRIPTION
The algorithm proposed in this paper proceeds in two steps. First, an estimate of the ROE (andsemimajor axis if enabled) is computed from an a-priori estimate of the observer’s absolute orbit(nominally provided by the ground using TLE or the Deep Space Network) and a set of bearing anglemeasurements. Second, the covariance is computed using the converged measurement residualsand measurement sensitivity model for the final state estimate. The procedures for each of thesecomputations are described in the following.
State Estimation
The observability analysis in the previous section demonstrated that the observer’s semimajoraxis and all ROE except for δλ are strongly observable, while the range (mainly δλ) is weaklyobservable. Leveraging this result, the state estimate is computed using a practical approach inspiredby the work of Ardaens19 that decouples estimation of weakly and strongly observable components.First, the user selects a set of samples of δλ based on the mission scenario (i.e. expected minimumand maximum separations). For each δλ sample, a complete state estimate is computed by usingiterative batch least squares to estimate the other state components. For the first sample, δα isinitialized with as a zero vector with the exception of δλ and αo is initialized with the a-priori orbitestimate. For subsequent range samples, δα is initialized as the previous converged solution scaledby the ratio of the current sample of δλ to the previous sample to reduce the number of requirediterations. Similarly, the semimajor axis is initialized to the previous converged value after the first
11
solution is computed. More explicitly, the new initial values of the ROE (δα+) and semimajor axis(a+) are computed from the previous converged estimates δα−f and a−f as given by
δα+ = δα−fδλ+
δλ−a+ = a−f (22)
where δλ+ denotes the new sample and δλ− denotes the previous sample.
Each iteration of the refinement procedure includes two steps. First, the increment for each statecomponent in the kth iteration are computed as given by
∆δak∆δex,k∆δey,k∆δix,k∆δiy,k∆ak
=(HTk−1Hk−1
)−1HTk−1
ytrue(t1)− h(αk−1, δαk−1, te, t1)...
ytrue(tN )− h(αk−1, δαk−1, te, tN )
(23)
where H only includes the columns of the measurement sensitivity matrix corresponding to esti-mated states. Second, the state increment is applied to the estimates as given by
δαk = δαk−1 +
∆δak0
∆δex,k∆δey,k∆δix,k∆δiy,k
αk = αk−1 +
∆ak00000
(24)
This iteration procedure repeats until either a user-specified number of iterations is exceeded orthe state increment is smaller than a user-specified threshold. After the batch least squares refine-ment terminates, the norm of the converged measurement residual vector is computed and saved forcomparison.
Once state estimates for each value of δλ are computed, the final state estimate is selected asthe candidate with the smallest measurement residual vector. As demonstrated by Ardaens,19 thenorm of the measurement residual is normally a convex function of the range, but the slope of thismetric in the vicinity of the minimum is a function of measurement errors, dynamics model errors,formation geometry, whether maneuvers are performed, and other considerations.
Uncertainty Estimation
This is a work in progress to be included in the final paper.
VALIDATION
The proposed estimation algorithm is validated through high-fidelity Monte Carlo and hardware-in-the-loop simulations of scenarios in both the earth and mars orbits in Table 1. The test cases areselected to demonstrate that the algorithm provides accurate state estimates across a wide range oforbit regimes. To ensure a fair comparison across orbit regimes, the measurement rates for eachsimulation are selected to ensure that 50 bearing angles are provided per orbit. Bearing angle mea-surements are provided every 120s in LEO, 750sec in GTO, and TBDs in mars orbit. Additionally,
12
the algorithm is run with and without estimation of the semimajor axis and with measurement arcsof one and two orbits. These solutions are used to quantitatively assess the impact of semimajoraxis estimation on relative navigation accuracy and the sensitivity of the accuracy to the duration ofthe measurement arc.
The algorithm parameters used throughout these simulations are as follows. The sampling of aδλis from 1km to 200km in 1km intervals. The maximum number of refinement iterations for eachcandidate solution is five and the state increment termination threshold is set at 10−9.
Monte Carlo Simulations
The Monte Carlo simulations are intended to characterize the statistical characterization of theestimation errors and quantitatively assess the benefits of including the absolute orbit in the esti-mated state. For each simulation, the algorithm inputs (initial observer orbit estimate and bearingangle measurement batch) are computed using the following procedure.
1) The initial estimate of the observer’s orbit at the estimation epoch is computed by corruptingthe user-specified ground truth mean orbit with noise sampled from a distribution with 1-σerrors of 2km in a and 2km/a in all other orbit elements (to be representative of errors fromTLE estimates).
2) The user-specified ground-truth mean absolute and relative orbits are used to compute the os-culating orbits of the observer and target using Schaub’s mean to osculating transformation.25
3) The osculating orbits of the observer and target are propagated to each measurement epochusing a high-fidelity numerical orbit propagator.
4) Bearing angle measurements are computed for each measurement epoch by corrupting thetrue bearing angles for the user-specified sensor attitude with zero-mean Gaussion noise with1-σ errors of 20arcsec.
The key parameters and models for the numerical orbit propagator are provided in Table 5.
The relative orbits for each test case are randomly sampled from a distribution with the followingproperties: 1) aδλ is uniformly distributed between 50km and 150km, 2) the magnitudes of therelative eccentricity and inclination vectors are uniformly distributed between 2000m and 4000m toensure uniform observability between all test cases, 3) the phase angles of the relative eccentricityand inclination vectors are uniformly distributed, and 4) aδa is uniformly distributed between -100mand 100m. This distribution was selected to be representative of scenarios expected in the StarFOXexperiment.21
The performance of each test case is assessed by four error metrics. The first metric of interestis the range error εrange, which is the weakly observable part of the relative state. However, due tothe presence of the mode with particularly weak observability, it is expected that the range cannotbe estimated with 1-sigma errors of less than 4km due to the errors applied to the mean argumentof latitude. With this in mind, the second error metric is called the compensated range error εcomp,which removes the component of the range error due to errors in the provided estimate of u. Thethird error metric is the unit vector error εunit, which is the distance between the unit vectors parallelto the true and estimated ROE. This metric captures the error in the relative motion excluding the
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Table 5. Numerical orbit propagator models and parameters.
Common ParametersIntegrator Runge-Kutta Dormand-PrinceStep size 30sInitial Epoch January 1, 2018 00:00:00
Earth orbit modelGeopotential GRACE Gravity Model 01S 120× 1201
Atmospheric density NRLMSISE-00 Model28
Solar radiation pressure Flat plate with conical Earth shadowThird-body gravity Lunar and solar point massesLunisolar Ephemerides Analytic
Mars orbit modelMars gravity TBDThird-body gravity TBDSolar radiation pressure TBD
Ballistic PropertiesObserver ballistic coefficient 0.001 m2/kgDifferential ballistic coefficient 20%
range, which should be small in all simulations. The final error metric is the error in the estimate ofthe semimajor axis. These error metrics are formally defined as
εrange =∣∣∣||δαest|| − ||δαtrue||∣∣∣ εmode = |(δλest − δλtrue) + 2a(uest − utrue)|
εunit =∣∣∣∣∣∣ δαest||δαest||
− δαtrue||δαtrue||
∣∣∣∣∣∣ εa = |aest − atrue|(25)
where ground truth quantities are denoted by the subscript true and estimated quantities are denotedby the subscript est.
LEO Results The converged measurement residuals for an example simulation are shown as afunction of aδλ in Figure 2. It is clear from this plot that the range is observable due to the clearconvexity of the measurement residual curve. In this example, the range is estimated within 2km(3%) of the true value. Additionally, it can be seen that the final measurement residuals are compa-rable to the noise added in the simulation environment. If the measurement noise was larger or thedynamics model were less accurate, range estimation errors could be significantly larger
The cumulative distribution functions (CDFs) for the error metrics for all LEO simulations areprovided in Figure 3. It is evident from the behaviors of εrange and εunit that including the semima-jor axis in the estimated state reduces relative navigation errors by at least a factor of four. Whennormalized by the separation, 90% of simulations have range errors of under 15% with one orbit ofmeasurements and under 8% with two orbits of measurements, which are comparable to the accu-racy demonstrated by Ardaens19 using more accurate orbit estimates. Additionally, it is noteworthythat CDF for εrange is consistent with the predicted minimum 1-σ error of 4km with 2 orbits ofmeasurements. Since the distrubition of εmode is signficantly smaller than the distribution of εrange(about 12km less in the worst-case values), it is evident that a signficiant portion of the range er-ror is due to the unobservable mode. This validates the previous hypothesis that achievable rangeaccuracy is limited by expected errors in the provided estimate of u. Finally, the algortihm is able
14
Figure 2. Behavior of converged measurement residuals vs. aδλ for example oneorbit simulation in LEO showing estimated value (green) and true value (red).
to reduce the error in the provided semimajor axis estimate by at least a factor of ten. Overall,these results follow predicted trends and provide estimates that are at least as accurate as the bestapproaches in literature with larger absolute orbit errors.
Figure 3. Cumulative distribution functions for error metrics from Monte Carlosimulations in LEO with one orbit of measurements (blue) and two orbits of mea-surements (red) with and without (dashed) estimation of the semimajor axis. Theuncorrected semimajor axis error CDF is shown in black.
GTO Results The CDFs for the error metrics for all GTO simulations are provided in Figure 4.The first trend that can be observed from this plot is that inclusion of the semimajor axis in the esti-mated state reduces εrange by at least a factor of two. When normalized by the true separation, the
15
range error is less than 15% in 90% of all conducted simulations including absolute orbit estimation.However, this metric shows little sensitivity to the duration of the observation arc. Since the distri-butions of εrange and εmode are nearly identical, it is likely that this error arises from modeling error(e.g. neglected effects of solar radiation pressure and third body gravity). Unlike the LEO results,εunit is estimated to better than 0.001 in all test simulations, even when the semimajor axis is not es-timated. Finally, εa exhibits a strong dependence on the duration of the observation arc. Specificallythe error for two orbit simulations is ten times less than the error in one orbit simulations.
Figure 4. Cumulative distribution functions for error metrics from Monte Carlosimulations in GTO with one orbit of measurements (blue) and two orbits of mea-surements (red) with and without (dashed) estimation of the semimajor axis. Theuncorrected semimajor axis error CDF is shown in black.
Mars Orbit Results This is a work in progress that will be included in the final paper.
Summary Overall, these simulations show that the proposed algorithm is able to accurately es-timate the relative orbit of a resident space object in a wide range of orbit regimes with large errorsin the provided absolute orbit estimates. As such, it is expected that this algorithm will enable forthe first time autonomous onboard initialization of an angles-only navigation filter in the StarFOXexperiment on the Starling1 mission. Additionally, the error trends show strong agreement withpredicted behaviors from a quantitative observability analysis, validating an intuitive geometric in-terpretation of a discovered weakly observable mode.
Hardware-in-the-Loop Simulations
This section is a work in progress that will be included in the final paper.
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CONCLUSIONS
This paper proposes a numerical solution methodology for the problem of estimating the relativeorbit of a resident space object using angles-only measurements from a single monocular cam-era. To meet this objective, a quantitative observability analysis is conducted to determine whatcomponents of the absolute orbit can be estimated alongside the relative state using angles-onlymeasurments. This analysis resulted in three key findings. First, the semimajor axis of the observerorbit can be estimated with an accuracy of approximately one hundred meters using current startrackers in orbits of arbitrary eccentricity. Second, when the mean argument of latitude is includedin the estimated state, there is a mode that cannot be estimated with any useful accuracy. This sharpincrease in the uncertainty is due to an unobservable mode that includes the range and the meanargument of latitude. Third, for in-train formations in near-circular orbits, the range is nearly in-distinguishable from the relative semimajor axis. However, in-train formation geometry does notcompromise observability in eccentric orbits.
Next, an algorithm was proposed to estimate the mean relative orbit of a resident space object,the mean semimajor axis of the observer orbit, and corresponding uncertainties. The algorithmestimates the weakly observable range through a sampling approach and the remaining stronglyobservable state components using iterative batch least squares refinement. The algorithm wasvalidated through high-fidelity simulations of scenarios representative of formation-flying missionsin both low earth orbit and geostationary transfer orbit with short measurement arcs of one to twoorbits. It was found that with 2km 1-σ errors in the provided absolute orbit estimate, the algorithmis able to estimate the relative range to within 15% in 90% of conducted simulations when thesemimajor axis is estimated. This reduces the range error by a factor of two to ten as compared tosimulations where the semimajor axis is not estimated. Additionally, it was demonstrated that therange error for simulations in low earth orbit is dominated by the discovered unobservable mode.
Overall, these results show that the proposed algorithm is able to accurately estimate the relativeorbit of a target using bearing angle measurements from a single camera and a coarse estimate of theabsolute orbit in a wide range of earth orbit regimes. The algorithm can be adapted to any planetaryorbit regime by simply modifying the dynamics model to include the dominant perturbations. Withthese capabilities, the proposed algorithm can enable fully autonomous angles-only navigation inthe StarFOX experiment and future missions orbiting other planetary bodies.
ACKNOWLEDGMENTS
This work was supported by the NASA Small Spacecraft Technology Program (cooperativeagreement number 80NSSC18M0058) for contribution to the Starling-1/StarFOX efforts.
DYNAMICS MODELS FOR MULTIPLE ORBIT REGIMES
Absolute Orbit Propagation Sensitivity Matrix for Earth Orbit
The absolute orbit propagation sensitivity matrix is computed by taking the partial derivativesof Equation 7 with respect to each component of the absolute orbit. To simplify the followingderivations, let κ be defined as
κ = 0.75J2R2E
õE (26)
where J2 is the earth oblateness coefficient, RE is earth’s radius, and µE is earth’s gravitationalparameter. Additionally, orbit elements at te are denoted by the subscript e and orbit elements
17
evaluated at tm are denoted by the subscript m The nonzero partial derivatives of the drift rates inEquation 5 with respect to the orbit elements are given by
∂ΩJ2
∂ae=
7κ cos(ie)
a9/2e η4
e
∂ΩJ2
∂ex,e= −8κex,e cos(ie)
a7/2e η6
e
∂ΩJ2
∂ey= −8κey,e cos(ie)
a7/2e η6
e
∂ΩJ2
∂i=
2κ sin(ie)
a7/2e η4
e
∂ωJ2∂ae
= −7κ(5 cos2(ie)− 1)
a9/2e η4
e
∂ωJ2∂ex,e
=4κex,e(5 cos2(ie)− 1)
a7/2e η6
e
∂ωJ2∂ey
=4κey,e(5 cos2(ie)− 1)
a7/2e η6
e
∂ωJ2∂i
= −10κ sin(ie) cos(ie)
a7/2e η4
e
∂MJ2
∂ae= −7κ(3 cos2(ie)− 1)
a9/2e η3
e
∂MJ2
∂ex,e=
3κex,e(3 cos2(ie)− 1)
a7/2e η5
e
∂MJ2
∂ey=
3κey,e(5 cos2(ie)− 1)
a7/2e η5
e
∂MJ2
∂i= −6κ sin(ie) cos(ie)
a7/2e η3
e
If Φ denotes ∂αm/∂αe, then the nonzero terms of this matrix are given by
Φ11 = 1 Φ21 = −ey,m∂ωJ2∂ae
∆t Φ22 = cos(ωJ2∆t)− ey,m∂ωJ2∂ex,e
∆t
Φ23 = − sin(ωJ2∆t)− ey,m∂ωJ2∂ey,e
∆tΦ24 = −ey,m∂ωJ2∂i
∆t Φ31 = ex,m∂ωJ2∂ae
∆t
Φ32 = sin(ωJ2∆t)+ex,m∂ωJ2∂ex,e
∆t Φ33 = cos(ωJ2∆t)+ex,m∂ωJ2∂ey,e
∆t Φ34 = ex,m∂ωJ2∂i
∆t
Φ44 = 1 Φ51 =∂ΩJ2
∂ae∆t Φ52 =
∂ΩJ2
∂ex,e∆t Φ53 =
∂ΩJ2
∂ey,e∆t Φ54 =
∂ΩJ2
∂ie∆t
Φ55 = 1 Φ61 =(∂MJ2
∂ae+∂ωJ2∂ae
− 3n
ae
)∆t Φ62 =
(∂MJ2
∂ex,e+∂ωJ2∂ex,e
)∆t
Φ63 =(∂MJ2
∂ey,e+∂ωJ2∂ey,e
)∆t Φ64 =
(∂MJ2
∂ie+∂ωJ2∂ie
− 3n
ae
)∆t Φ66 = 1
Absolute Orbit Propagation Model for Mars Orbit
This section is a work in progress that will be included in the final paper.
Absolute Orbit Propagation Sensitivity Matrix for Mars Orbit
This section is a work in progress that will be included in the final paper.
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