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Angles-only Navigation to a Non-Cooperative Satellite
using Relative Orbital Elements
G. Gaias∗
, S. D’Amico†
, and J.-S. Ardaens∗
German Aerospace Center (DLR), M¨unchner Str. 20, 82234 Wessling, Germany
This work addresses the design and implementation of a prototype relative navigation
tool that uses camera-based measurements collected by a servicer spacecraft to perform far-
range rendezvous with a non-cooperative client in low Earth orbit. The development serves
the needs of future on-orbit-servicing missions planned by the German Aerospace Center.
The focus of the paper is on the design of the navigation algorithms and the assessment
of the expected performance and robustness under real-world operational scenarios. The
tool validation is accomplished through a high-fidelity simulation environment based on
the Multi-Satellite-Simulator in combination with the experience gained from actual flight
data from the GPS and camera systems on-board the PRISMA mission.
I. Introduction
This work addresses the design and implementation of a prototype relative navigation tool that makes
use of camera-based measurements only. Its development was mainly motivated by the need to build know-
how for vision-based navigation and prepare for future on-orbit servicing missions of the German Aerospace
Center (DLR) such as DEOS (DEutsche Orbitale Servicing).1In this class of space missions, a servicer
satellite has to approach a client non-cooperative spacecraft from far range distance. The capability to
safely accomplish this with no need of a device that directly measures the range distance represents an
asset for the servicer’s system design. Although several theoretical studies have been conducted on angles-
only relative navigation, a solid assessment of the expected performance and robustness under real-world
rendezvous scenarios, being them with ground-in-the-loop or fully autonomous, is needed. This paper takes
advantage of the DLR’s contributions to the Swedish PRISMA mission2,3and makes use of actual flight data
collected during the related technology demonstrations to validate a novel non-cooperative relative navigation
prototype. Among the several sensors on-board, the servicer spacecraft of PRISMA is also equipped with
a star tracker developed by the Danish Technical University (DTU) called Vision Based System (VBS)
which is able to provide the line-of-sight (LOS) measurement to a target non-stellar object. User experience,
image data, and GPS precise orbit determination results acquired during the DLR’s experiments are a most
valuable resource for the design of the navigation approach and for the prediction of related performances.
The first part of the work deals with the filter design. Here the problem is described in terms of available
measurements and aimed state to be reconstructed. Measurements consist of azimuth and elevation of the
LOS to the client satellite. The relative dynamics is described through relative orbital elements. Such a
choice is justified by the clear geometrical insight that these elements provide to understand the relative
geometry and, therefore, the LOS behavior. Moreover, relative orbital elements allow the straightforward
inclusion of Earth oblateness effects due to the J2in the relative motion model.4Finally the un-observability
of the relative navigation problem is mostly condensed in only one component of the state: the relative mean
argument of latitude. Therefore the shape of the relative motion can be determined from the very early phases
of the approach based on the observable relative orbital elements. The estimation error of the relative mean
argument of latitude is stepwise decreased while executing a maneuvers’ profile. Nevertheless the knowledge
of the geometry allows performing safe approaches simply by setting a proper relative eccentricity and
inclination vector separation.5
∗Research Engineer, GSOC/Space Flight Technology, M¨unchner Str. 20, 82234 Wessling, Germany.
†Lead Research Engineer, GSOC/Space Flight Technology, M¨unchner Str. 20, 82234 Wessling, Germany.
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As well-known from the literature, the angles-only navigation problem is not fully observable in the
absence of maneuvers. Sufficient conditions to achieve observability are provided in Ref. 6,7for the bearing-
only target motion analysis. In the field of orbital rendezvous, they are presented in Ref. 8. According
to these contributions, a maneuver that produces a variation of the LOS natural trend makes the system
observable. In this work it is shown that relative orbital elements can support a direct physical interpretation
of the un-observability issues. Moreover results are generalized for the case in which the perturbations due
to the equatorial bulge are taken into account. It is here shown that observability properties improve for
certain families of relative orbits as the trends of the mean relative elements deviate from the one they have
within the Kepler assumptions. The exploitation of relative orbital elements is of great benefit also when
dealing with the assessment of the estimation accuracy achievable when maneuvers are performed. In the
literature different methods to quantify the observability of the problems are suggested. They span from the
definition of a range error metric,9to the exploitation of the covariance of the estimated state10 or of the
condition number of the observability matrix.11 In our work simple metrics are proposed to quantify the
effectiveness of the maneuvers performed. They are based on the immediate idea of what changes will take
place in the LOS behavior due to the relative orbital elements variation determined by a maneuver. In this
frame an optimization problem can be easily set up.
The validation of the tool has been accomplished through the Multi-Satellite Simulator employed at
DLR/GSOC to support various projects in the fields of formation flying and proximity operations.12 Specif-
ically, a high-fidelity simulation environment provides all the information usually available in the telemetry
of the servicer satellite. A camera model mimics the measurement data delivered by an image processing
unit. Its performances are tailored according to the processing of actual PRISMA VBS image data, col-
lected during the DLR’s Formation Reacquisition experiment(August 2011).13 Modeled measurement data,
servicer absolute information together with the estimated maneuver plan are used by the navigation tool to
perform relative orbit determination. As a result, the simulations provide the typical navigation accuracy
which is achievable through a specific maneuver profile under realistic operational conditions. The proposed
navigation tool contributes to the preparation to the DLR/GSOC Advanced Rendezvous Demonstration
using GPS and Optical Navigation (ARGON) experiment, scheduled for the extended phase of the PRISMA
mission.14
II. Mathematical Description
The relative orbit determination problem consists in estimating at a certain time the relative state
of a client satellite with respect to the servicer one, making use of a set of observations, given that the
initial relative state is unknown and an a-priori first guess might be available. In the case of angles-only
measurements, observations at each instant of time consist of a couple of angles, i.e. azimuth ηand elevation
ψ, which subtend the line-of-sight LOS unit-vector, uc, to the client satellite. The superscript ”c” denotes
the camera frame, i.e. the frame attached to the sensor, and angles are modeled according to the following
nonlinear functions of the relative state x:
η
ψ!= arctan(uc
x/uc
z)
arcsin(uc
y)!=z(t, x(t)) (1)
The nonlinear differential equations governing the relative dynamics simplifies to linear differential equations
under the assumptions of relative separation much smaller than the servicer absolute radius and servicer
satellite on a circular orbit.
˙
x(t) = ˆ
A(t)x(t) + ˆ
B(t)dv(t) (2)
The solution of the obtained set of linear differential equations can be expressed via the state transition
matrix that maps the relative state vector from one time to another. The nonlinear measurements equations
Eq.(1) can be linearized about a nominal reference relative state xref . This procedure leads to the following
set of functions in the presence of control inputs and measurements uncertainties:
x(t) = ˆ
Φ(t, t0)x(t0) + Rt
t0
ˆ
Φ(t, τ )ˆ
B(τ)dv(τ)dτ
y(t) = ˆ
C(t)x(t) + (t)(3)
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Concerning the relative orbit estimation problem, the complete estimation state is composed by:
x= aoδα
b!(4)
Where the relative state is parameterized in terms of relative orbital elements according to:
δα=
δa
δex
δey
δix
δiy
δu
=
δa
δe cos ϕ
δe sin ϕ
δi cos θ
δi sin θ
δu
=
(a−ao)/ao
ecos ω−eocos ωo
esin ω−eosin ωo
i−io
(Ω −Ωo) sin io
u−uo
(5)
Here a,e,i,ω, Ω, and Mdenote the classical Keplerian elements, whereas e= (ecos ω e sin ω)T, and
u=M+ωrepresent the eccentricity vector and the mean argument of latitude. Here the superscript ”o”
denotes quantities referring to the reference spacecraft which defines the origin of the orbital frame (here the
servicer). From now on, the absolute orbital elements which appear in the equations will always belong to the
servicer satellite, therefore the superscript is dropped. Under the assumptions of the Hill-Clohessy-Wilshire
equations (HCW),15 the magnitude of the relative eccentricity and inclination vectors, δeand δi, provide the
amplitudes of the in-plane and out-of-plane relative motion oscillations, whereas relative semi-major axis,
δa, and relative mean longitude, δλ =δ u +δiycot i, provide mean offsets in radial and along-track directions
respectively.5,4The orientation of the shape of the relative motion is driven by the phase angles ϕand θ,
which identify the relative perigee and relative ascending node of the relative orbit.
The last two components of the estimation state are represented by the biases of the sensor:
b= bη
bψ!(6)
Given Eqs.(4),(6), the matrix quantities that compare in Eqs.(2),(3) are composed by the following subparts:
ˆ
A(t) = A(t)O6×2
O2×6O2×2!,ˆ
Φ(t, t0) = Φ(t, t0)O6×2
O2×6Φb(t, t0)!
ˆ
B(t) = B(t)
O2×3!,ˆ
C(t) = C(t)I2×2(7)
Therefore, the 8 ×8 state transition matrix ˆ
Φrelates the 8-dimensional state vector x(t0) at time t0to
the state x(t) at time t. The 6 ×3 control input matrix Bexpresses the variation of the relative orbital
elements caused by an impulsive maneuver ∆v= (∆vR∆vT∆vN)Tat time tMwith components in radial,
along-track, and cross-track directions. The partial derivatives of the modeled angle measurements zwith
respect to the state are given by the 2×8 measurement sensitivity matrix ˆ
C.is the two-dimensional vector
of uncorrelated measurement errors characterized by a normal distribution with zero mean and covariance
Cov() = W=diag(σ2
η, σ2
ψ).
Regarding the system of equations Eq.(2) that lead to Eq.(3), a simple relative dynamics model which
captures the most relevant perturbations in low Earth orbit is employed.5,4Under the assumptions of Kepler
orbits the relative state of Eq.(5) is composed by the integration constants of the HCW equations. Therefore,
for orbits of equal energy (i.e., δa = 0), Φ(t, t0) = I6×6. When the servicer and client satellites have unequal
semi-major axis (e.g., δa > 0), the relative mean argument of latitude changes over time. If δa and δu are
small quantities (i.e., δa and δu << 1), its drift can be expressed to first order as A6,1=−1.5n, being n
the mean motion of the servicer. Thus the particular choice of the variable state allows having the following
simple relationship between the matrix of the dynamics of the system A(t) and the state transition matrix
Φ(t, t0):
Φ(t, t0) = I6×6+ (t−t0)·A(t) (8)
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Earth oblateness effects due to J2can be easily introduced in the relative dynamics model under the additional
assumptions of small magnitudes of the relative eccentricity/inclination vectors (i.e., δe and δi << 1) and
small eccentricity (i.e., e << 1). The resulting secular variations of the relative orbital elements can be
derived from the theory of Brower.16 These effects are proportional to the elapsed time ∆tand J2and are
expressed through to the following parameters:
˙ϕ= 1.5γn(5 cos2i−1)
γ= 0.5J2(RE/a)2(9)
where REis the Earth equatorial radius. Specifically, the Earth equatorial bulge causes a rotation of the
relative eccentricity vector, δe(i.e. A2,3and A3,2), a vertical linear drift of the relative inclination vector,
δi(i.e. A5,4), and a linear drift of δu (i.e. A6,4). The inclusion of the aforementioned effects leads to the
following complete state transition matrix:
Φ(t, t0) =
1 0 0 0 0 0
0 1 −˙ϕ∆t0 0 0
0 ˙ϕ∆t1 0 0 0
0 0 0 1 0 0
0 0 0 3γsin2(i)n∆t1 0
−1.5n∆t0 0 −12γsin(2i)n∆t0 1
(10)
Biases are modeled as esponentially correlated measurement errors, therefore their time evolution is expressed
by the following state transition matrix:
Φb(t, t0) = exp (− |∆t|/τ) 0
0 exp (− |∆t|/τ)!(11)
where τis the correlation time constant. Whenever τ=∞, then the modeling of the biases simplifies to
un-correlated measurement errors.
The variation of the relative orbital elements caused by an impulsive maneuver (or an instantaneous
velocity change) at time tMcan be modeled under the same assumptions of our linear relative dynamics. In
particular the inversion of the solution of the HCW equations expressed in terms of relative orbital elements
provides the following relationship:5,4
B(tM) = −1
n
0 2 0
sin uM2 cos uM0
−cos uM2 sin uM0
0 0 cos uM
0 0 sin uM
−2 0 −sin uMcot i
(12)
where uM=u(tM) identifies the mean argument of latitude of the servicer at the delta-v time. Eq.(12)
shows how along-track maneuvers cause instantaneous variations of δa and δe, radial maneuvers cause
instantaneous variations of δu and δe, whereas cross-track maneuvers affect δu and δi. It is noted that cross-
track maneuvers do not change the mean along-track separation, δλ =δu +δiycot i, because the effects
on δu and δiycancel out, thus the in-plane and out-of-plane relative motion remain fully decoupled in this
formulation. Given Eqs.(10),(12), the zero state transition of the system, i.e. the integral in Eq.(3), can be
immediately solved.
The partial derivatives of the angle measurements w.r.t. relative orbital elements about the reference
state are computed through the application of the following chain rule:
C(t) = ∂z
∂δαδα
=∂z
∂δrc·∂δrc
∂δrJ2000 ·∂δrJ2000
∂δrRTN ·∂δrRTN
∂δαδα
=∂η
∂δrc
∂ψ
∂δrcT
·Rc
J2000 ·RJ2000
RTN ·∂δrRTN
∂δαδα
(13)
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Here usage has been made of the fact that the derivative of the angle measurements w.r.t. the relative
velocity ∂z/∂δvcis zero. The expansion given by Eq.(13) contains the following main terms, namely 1)
the derivative of the azimuth and elevation angles w.r.t. relative position in the camera frame δrc, 2) the
absolute attitude of the sensor, that is the rotation matrix from J2000 to the camera frame, 3) the rotation
matrix from the RTN frame to the inertial frame, and 4) the derivatives of the relative position in the RTN
frame δrRTN w.r.t. the relative orbital elements δα. The mapping between relative orbital elements and
relative position in the orbital frame is given by the adopted linear model as:5,4
∂δrRTN
∂δαδα
=
1−cos u−sin u0 0 0
0 2 sin u−2 cos u0 cot i1
0 0 0 sin u−cos u0
(14)
The measurements partials w.r.t. relative position in the camera frame can be computed using the following
equivalence:17
∂δrc
∂δrc=I3x3=uc∂δr
∂δrc+δr ∂uc
∂η
∂η
∂δrc+δr ∂uc
∂ψ
∂ψ
∂δrc(15)
where appear the three orthogonal directions: LOS unit-vector, (∂uc/∂η) and (∂uc/∂ ψ). Among these, the
last two vectors are computed from Eq.(1):
∂uc
∂η =
cos ψcos η
0
−cos ψsin η
∂uc
∂ψ =
−sin ψsin η
cos ψ
−sin ψcos η
(16)
Now the derivative of the azimuth and elevation angles w.r.t. relative position in the camera frame can be
computed by alternatively pre-multiplying Eq.(15) by (∂uc/∂η)Tand (∂uc/∂ψ)T, as the contributions in
the remnant orthogonal directions vanish. As a result one obtains:
∂η
∂δrcδα
=1
δr cos2(ψ)∂uc
∂η T
∂ψ
∂δrcδα
=1
δr ∂uc
∂ψ T(17)
At this stage all the quantities of Eq.(13) are known in order to compute C(t). Finally, the last part of the
sensitivity matrix ˆ
C(t) is constituted by the identity sub-matrix that represents the partial derivatives of
the angle measurements w.r.t. the estimated biases.
A. Observability of the estimation problem based on angles-only measures
The problem of estimating a relative motion between two satellites making use of angles-only measurements
is not fully observable. Physically this phenomenon is produced by the fact that infinite relative orbits
generate the same LOS unit-vector trend over time. Equivalently the lack of measurements on the range
cannot allow univocally solving for the separation between the satellites. Some authors emphasize that
with a proper choice of parameters the main characteristics of the geometry of the natural motion can be
reconstructed despite this not complete observability.8,18 That means that shape and orientation of the
orbits are identified, leaving the ambiguity on a scale factor, i.e. the size of the relative motion, that is
function of the separation between the satellites. To this aspect, formulating the relative dynamics through
the relative orbital elements of Eq.(5) is particularly convenient. In this way in fact the range ambiguity is
almost condensed in only one term of the state, i.e. the relative mean argument of latitude. The magnitude
of the average separation over an orbital period is given by:
Z2π
0
δrc(u)du
=apδa2+δλ2(18)
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Therefore for bounded orbits (δa = 0) with δiy= 0, the average separation coincides with aδu. For δu >> δa,
the mean argument of latitude well approximates the average range magnitude. In addition to this property
and differently from other geometry-based parameterizations, relative orbital elements support a powerful
though simple model of the dynamics, able to capture some of the most relevant perturbations in low Earth
orbit. This topic was discussed in the previous section. Figure 1provides a sketch of the meaning of the
relative orbital elements. The relative motion of the client vehicle is mapped in the orbital frame centered
on the servicer and aligned with the radial (R, positive in Zenith direction), along-track (T, positive in
flight direction), and cross-track (N, normal to the orbital plane) directions. The along-track drift due to
the different orbit energies is neglected for visualization purposes. In the bottom-left subplot it is presented
a possible attitude configuration of the sensing instrument. According to the sketch, the client satellite is
preceding the servicer along the flight direction and the local camera frame is aligned to the orbital frame
in such a way to point its boresight zcin anti-flight direction:
Rc
J2000 =
100
001
0−1 0
·RRTN
J2000 =Γ·RRTN
J2000 (19)
Figure 1: Description of the client relative motion in
the servicer-centered orbital frame through relative or-
bital elements. The snapshot does not include the drift
corresponding to δa. The bottom-left view presents a
possible attitude of the sensor.
Being Eq.(19) the attitude of the sensing instru-
ment, within the relative orbital elements param-
eterization, the shape and orientation of the motion
is fully described by the first five elements. Whereas
the un-observable range acts through aδu influenc-
ing the size of the ellipsoid projected in the R-N
plane. It is emphasized that, this early reconstruc-
tion of the relative motion in the R-Nplane is a key
factor when performing a rendezvous. It allows, in
fact, setting a proper relative eccentricity and incli-
nation vector separation, thus ensuring a safe ap-
proach to the client satellite.5
The observability criteria of the relative state
determination based on angles-only measurements
have been widely tackled in the literature, across
different fields of applications. Mainly works focus
on observability as a structural property of the prob-
lem, thus the ideal condition of absence of noise in
the measurements is assumed. Biases of the sensor
are also neglected. Due to the presence of nonlinear
expressions of the measurements as function of the
relative state, either approaches based on nonlinear
observability techniques or local methods for the lin-
earized problem have been investigated. In the sequel we focus only on results achieved within this second
methodology.
In the field of bearing-only target motion analysis, Nardone and Aidala6and Hammel and Aidala7
respectively developed the 2D and 3D necessary and sufficient conditions to ensure observability. In their
working frame, a ship/submarine has to estimate the motion of a target which moves at constant absolute
velocity. The relative dynamics is expressed through a linear model in the Cartesian relative position and
velocity defined with respect to the active own-ship element. Nonlinear measurements equations, of a similar
form of Eq.(1), are recast as linear expression in the unknown-initial state vector. Therefore a criterion for
observability for linear time varying system is used.19 It exploits successive time derivatives of the reshaped
measurements equation, in order to obtain a set of independent equations in the unknown-initial state vector.
In the field of orbital rendezvous, Woffinden and Geller8developed an analytical form to ensure the
sufficient conditions for the observability of the relative orbit determination problem. To accomplish this,
they rewrite the nonlinear measurements equations in the form of LOS unit-vector function of the initial-
unknown Cartesian relative position and velocity. To this aim the state transition matrix of the linearized
relative orbital motion is exploited. Subsequently observability criteria are deduced through geometrical
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considerations on the comparison of different measurement profiles generated starting from same initial
conditions.
Both the criteria conclude that at least a maneuver is necessary in order to achieve the complete ob-
servability. Moreover such a maneuver has to produce a change in the relative position not aligned with
the instantaneous direction of the natural LOS profile. In developing these criteria the formulation of the
relative motion via Cartesian coordinates was exploited. Angles measurements and LOS unit-vector, in fact,
can be expressed as not too complicated functions of the initial state vector. According to our formulation,
the LOS unit-vector is given by:
δrc=Γ·∂δrRTN
∂δα·aδα
uc=δrc
kδrck
(20)
where the mapping matrix between relative orbital elements and relative position is Eq.(14) and the camera
frame is oriented according to Eq.(19).
All the results achieved for the Cartesian relative state are valid for the state of Eq.(5) as long as the
parameterization via relative orbital elements can be seen as the following change of state variable
δrRTN
δvRTN !=
∂δrRTN
∂δα
∂δvRTN
∂δα
·aδα(21)
and both zero input response and zero state response are invariant to state transformations. In addition to
this and as mentioned before, the utilization of relative orbital elements provides a direct physical interpre-
tation that separates the observable portion of the state variable from the un-observable part. Numerically
this can be proven by analyzing the behavior of the matrix of the accumulated partials of the measurements
w.r.t. the initial state defined as follows:
H=
C(t0)Φ(t0, t0)
.
.
.
C(tn)Φ(tn, t0)
(22)
In particular, if Hhas rank equal to 6, the matrix HTHis positive defined, thus invertible. Therefore the
complete initial state can be reconstructed via the measurements that contributed in building H. Let us
first consider the same case discussed in the literature, i.e. the relative dynamics is represented by the HCW
equations. Then, the transformed state transition matrix through Eq.(21) simplifies to the identity matrix
with I6×1= 1.5n∆t.
In Table 1is introduced a set of relative orbits (RO) that represents a sample of typical relative motions;
a brief description of their main characteristics is provided. They span from separations of 30 km to 100 m.
In particular RO1 represents a possible configuration for the beginning of an approach to a non-cooperative
client satellite. RO2 presents a drift of almost 1 km towards the client, while keeping a safe separation in
the R−Nplane.20 RO3 can represent the hold-point were transition between far-range and close-range
sensors can be performed. Finally RO4 can represent the starting point of a docking phase. In Table 2are
Table 1: Set of typical relative orbits (RO).
RO Elements set aδα[m] Description
1 (0,400,0,−400,0,−30000) Bounded (anti)-parallel horizontal δe/δivectors
2 (−100,300,0,−300,0,−20000) Drifting of circa 1 km per orbit towards the client
3 (0,0,−200,0,200,−3000) Bounded (anti)-parallel vertical δe/δivectors
4 (0,0,0,0,0,−100) V-bar station keeping
collected the rank and condition numbers achieved by these RO, when 6 measurements separated by the
time correspondent to 30 deg of variation of the servicer mean argument of latitude ∆uare accumulated.
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The size of His 12 ×6 and the condition number of HTHis used as a practical measurement criterion of
the observability of the estimation problem, in agreement with Ref. 11. There the value of 1016 is suggested
as the limit of observability.
In Table 2, for each i-th RO, four cases are considered. In the first one, i.e. first row of each sub-
block, the state vector is represented by δα. In the other cases different sub-sets of the state variable are
addressed: with ”-” are marked the excluded relative elements. As expected, within the natural motion
assumption, no RO achieves full rank. When the dimension of the state variable is reduced to 5, the
observability properties variate substantially depending on which component is excluded. Specifically the
best performance is obtained when the relative mean argument of latitude δu is not estimated, see RO2.
When the relative orbit to be estimated has no drift, i.e. δa = 0, then 5D subsets that try to estimate δu
are not fully observable (rank equal to 4, i.e. RO 1, 3 and 4). In this case, in fact, infinite orbits produce the
same LOS trend and the relative separation cannot be determined. Finally, the reconstruction of just the
relative eccentricity and inclination vectors is always characterized by rank equal to 4, and a high accuracy
can be achieved.
Table 2: Observability properties for the relative orbits of Table 1, for the Keplerian problem.
RO δa δu rankHcond(HTH)
1 x x 5 1.619
x - 5 22.94
- x 4 1.918
- - 4 1.06
2 x x 5 2.718
x - 5 22.75
- x 5 1.26
- - 4 1.06
3 x x 5 3.718
x - 5 29.45
- x 4 1.418
- - 4 1.33
4 x x 5 ∞
x - 5 24.33
- x 4 ∞
- - 4 1
In our work the matrix of the dynamics of the system Ais not exactly the transformed of the HCW
equations: it includes also the first order effects due to the Earth equatorial bulge. As discussed before,
these produce secular variations in four components of the mean relative orbital elements set, proportional
to the J2value and the elapsed time ∆t. By including these phenomena in the description of the dynamics,
the observability properties of the problem can improve. To support this topic, in Table 3are collected
the results achieved by the same RO analyzed before, when the effects of J2are accounted for. Here two
situations are compared: the difference consists in how His generated. In the first column 6 measures
spaced by ∆u= 30 deg are used, as accomplished for Table 2. In the second column the six measurements
are separated by 60 deg, therefore spanning a double ∆t. The aim is to evaluate the improvement of the
observability when effects of J2becomes more relevant. One can note that RO1 and 2 are now characterized
by rank 6. Still the condition number is quite high, index of poor accuracy of the achievable estimation. As
expected performances improve with the passing of the time. Regarding RO3 and 4, the J2cannot help as
long as δixis zero (the satellites lay on orbits of same inclination).
To conclude this session, the parameterization through relative orbital elements reveals convenient also
when dealing with the analysis of the maneuvers needed to ensure observability. Performances are assessed
through different metrics, that are discussed in the sequel. Table 4resumes the effects of impulsive maneuvers
on the set of relative orbital elements in accordance with Eq.(12). It is straightforward to verify that whatever
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Table 3: Observability properties for the relative orbits of Table 1, including J2effects.
RO ∆t1∆t2
rank(H) cond(HTH) rank(H) cond(HTH)
1 6 6.3515 6 3.4114
2 6 3.7815 6 1.6114
3 5 2.6218 5 1.7619
4 5 ∞5∞
maneuver is performed the geometry of the relative motion varies in such a way that the natural trend of
the LOS is modified.
Table 4: Effect of maneuvers on the relative orbital elements and their derivatives and, consequently on the
angles to the LOS, given the attitude of Eq.(19). Variations in brackets stand for indirect effects via δr and
δu.
dvdirection Instantaneous effect on Effect on measures
δαδ˙
α
dvRδe,δu ∆η, (∆ψ)
dvTδa,δeδ˙u∆η, (∆ψ)
dvNδi,δu (not δλ) ∆ψ, (∆η)
B. Maneuver planning to optimize observability
Once assessed that the presence of at least a maneuver is enough to achieve the full observability of the
relative orbit estimation problem, the next step consists in quantifying the level of observability that different
maneuver profiles can provide. This topic is interesting as long as the degree of observability is related to
the accuracy achievable in the estimation. Moreover, during a rendezvous it can be useful to perform more
effective maneuvers at the beginning, when errors in the estimations are bigger and when the large separation
determines a less favorable situation. In Ref. 8the authors suggest that the greater the LOS profile change
produced by the maneuver, the higher the level of observability of the system, as the amount of change in
the angle measurements is related to the range value before the maneuver. They extend further this topic
in Ref. 9defining a metric of detectability range error to quantify the achievable range accuracy, hence the
level of observability of the problem. The optimal maneuver profile is the one that minimizes this metric.
The range error, once fixed the accuracy of the sensor, is shown to be proportional to the change in position
due to the maneuver and to the reciprocal of the sinus of the angle between the natural and modified
LOS profiles. Therefore, assuming for example a servicer satellite in a positive V-bar station keeping, i.e.
aδα= (0,0,0,0,0, aδu0) according to our formulation, the optimal maneuver is to burn in positive radial
direction (thus acting on δeand δu). Geometrically the result means to establish a relative motion that is
characterized by maximum radial displacement and minimum tangential separation, assumed a fixed delta-v
magnitude. Tangential maneuvers were not taken into account due to the a-priori practical constraint of not
changing the energy of the servicer orbit (in the paper the initial separation is of 100 m which corresponds
to the very final part of a rendezvous).
When mentioning the change in position due to a maneuver, it is meant the variation in the whole
relative motion geometry, thus in the LOS natural trend, appreciable over orbit-period time spans which
leads to a non-instantaneous observability. In this context the exploitation of relative orbital elements is of
great benefit as it allows having an immediate idea of what changes will take place in the LOS behavior
after a extended-time maneuver (see Table 4and Eq.(12)). The same effect can be mapped directly on
the measurements, by expressing azimuth and elevation as functions of the relative orbital elements. Given
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Eq.(1) and sensor attitude of Eq.(19), the measures are expressed as follows:
η= arctan δa −δexcos u−δeysin u
−2δexsin u+ 2δeycos u−δu −δiycot i
ψ= arcsin aδixsin u−aδiycosu
kδrk(23)
In absence of maneuvers, these angles are orbital-period periodic functions whose amplitudes are related to
the geometry of the motion. Whenever a maneuver takes place at uM, the relative motion changes shape
affecting the behavior of Eq.(23). This suggests a simple metric of observability based on the maximum
amount of change in the measurement angles, according to:
m1=1
κ·max
u∈[uM,uM+2π]|ξforced(u)−ξnatur al(u)|(24)
Exploiting the decoupling of the problem, ξis azimuth for in-plane maneuvers and elevation for out-of-plane
maneuvers; κis the one-pixel angular resolution of the sensor expressed in radians. The metric m1allows
assessing the minimum delta-v magnitude needed to obtain observability. It is the one that allows m1to
overcome the instrument minimum sensitivity scale.
Metric m1can be rewritten as a function of the relative orbit before the maneuver. Let the ratio N/D
represent the argument of the inverse trigonometric functions of Eq.(23) then, to the 1st order approximation,
the variation of the measurements angles is proportional to the variation of the quantity ∆(N/D). This last
term can be written as:
∆(N/D) = N
D·
∆N
N−∆D
D
1 + ∆D
D
(25)
therefore the maximum of the variation of the measurements angles is given by:
m2= max
u∈[uM,uM+2π]|∆(N/D)|(26)
Here pre-maneuver quantities, i.e. Nand D, are functions of u, whereas ƥquantities are constant finite
variations determined by the type and size of the extended-time maneuver. In particular, Nstands for δrc
x
or δrc
y;Dfor δrc
zor |δr|, respectively for in-plane and out-of-plane cases. This metric is not defined when the
pre-maneuver relative orbit either has δa =|δe|= 0 or |δi|= 0, respectively before in-plane and out-of-plane
maneuvers.
Finally, Eq.(23) can be used to express how the accuracy of the measurements, i.e. σηand σψ, influence
the achievable accuracy in δu and δr when a given relative motion is established.
σpost−man
δu =±ση(tan η·δrc
x+δrc
z)
tan η±ση
σpost−man
δr =∓σψ· kδrk · cos ψ
sin ψ±σψcos ψ
(27)
In the development of these expressions it is assumed that the final orbital elements are perfectly known
and the only source of error is the uncertainty in the angular measurements. These measurements errors are
small angles and only 1st order contributions are kept. As expected the achievable accuracies are function of
the separation between the satellites. Similarly to Ref. 9the problem of choosing the optimal maneuvers to
improve observability can be formulated as the selection of that maneuver that brings to the minimization
of the final achievable separation accuracy:
m3=
minu∈[uM,uM+2π]σpost−man
δu
minu∈[uM,uM+2π]σpost−man
δr (28)
Let us consider again the RO collected in Table 1, where RO4 is the example discussed in Ref. 9. Table 5col-
lects the scores achieved by the three metrics when maneuvers with fixed delta-v size are performed starting
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from RO1-4 initial relative orbits. In addition to the type and magnitude of the maneuver, all the metrics
are also function of the verse and location of the maneuver. The values here presented simply refer to uM= 0
deg (i.e., ascending node of servicer orbit). The verses are chosen to provide a reduction of the separation
between the satellites. The magnitude of the maneuvers is 0.01 m/s. The instrument resolution is set equal
to 0.024 deg per pixel. It is emphasized that these computations do not take into account constraints on the
field of view of the instrument and on safety.
As expected, performances improve with the decreasing of the separation (from RO1 to RO4). At big sepa-
rations, tangential maneuvers are more effective in terms of detectability of the maneuver by the instrument
(m1and m2). Regarding the optimal value of the achievable accuracy (m3), one should investigate also
the location of the maneuver. Concerning RO4, m2is not applicable as mentioned before. The value of m1
could be rescaled as long as the instrument resolution used might not be appropriate for such a small relative
separation. The tangential maneuver is shaded due to safety constraints: it will lead to collision in one orbit.
In configuration RO4 metrics do not depend on uMand δr =aδu. Thus, in agreement with Ref. 9a positive
radial maneuver is the optimal one with respect to the achievable range accuracy. Finally, it is emphasized
that the inclusion of safety and visibility constraints in the optimization problems is straightforward when
using relative orbital elements. In Ref. 14 one can derive their explicit expressions.
Table 5: Values assumed by the metrics when maneuvers of 0.01 m/s of magnitude are performed at uM= 0
deg.
RO ∆vR= ∆v·(1; 0; 0) ∆vT= ∆v·(0; 1; 0) ∆vN= ∆v·(0; 0; 1)
m1m2σδu m1m2σδu m1m2σδr
[pixels] [ad] [m] [pixels] [ad] [m] [pixels] [ad] [m]
1 0.78 3.3e−4 186.80 2.96 1.2e−3 187.45 0.78 3.3e−4 175.48
2 1.19 0.5e−3 83.77 4.47 1.9e−3 82.65 1.17 4.9e−4 102.41
3 6.76 2.8e−3 3.87 28.92 1.1e−2 3.07 7.56 3.2e−3 2.91
4 288.07 N.A. 0.05 >3e3 N.A. 0.003 226.98 N.A. 0.09
C. Observability of the complete estimation problem
According to the definition of observability, the pair of matrixes ( ˆ
A(t), ˆ
C(t)) of the linear time variant
(LTV) system of equation Eq.(2), is observable on the time interval [t0, tn], if whatever the initial state
and the system input, one can uniquely determine x0from y(t) and ∆v(t). Moreover, the zero state
response contribution does not play any role in the observability characteristics, as it is a known term
directly computed from ∆v(t) and ˆ
Φ. The observability gramian for the pair ( ˆ
A(t), ˆ
C(t)) is defined as:
Wo(t0, tn) = Ztn
t0
ˆ
Φ(τ, t0)Tˆ
C(τ)Tˆ
C(τ)ˆ
Φ(τ, t0)dτ (29)
And the observability of the LTV system is equivalently stated by detWo(t0, tn)6= 0.
By passing in the discrete time domain, the observability gramian can be written as:
Wo(t0, tn) =
n
X
k=0 hˆ
Φ(tk, t0)Tˆ
CT
ki·hˆ
Ckˆ
Φ(tk, t0)i=HTH(30)
This corresponds to the normal matrix of the least squares solution of the initial-state determination problem
when measurements are affected by observation errors.21 Here the matrix Hrepresents the accumulated
partials of the measures w.r.t. the initial state for the system with the variable state that includes the biases.
To achieve observability, the gramian matrix has to be positive definite, i.e. Hshall be full rank. Given
Eq.(7) and Eq.(11), each sub-block associated to a i-th time has the following structure:
Hi=CiΦ(ti, t0)νiI2x2(31)
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which leads to this i-th addendum of the sum:
Wo,i(t0, ti) = Φ(ti, t0)TCT
iCiΦ(ti, t0)νiCiΦ(ti, t0)
νiΦ(ti, t0)TCT
iν2
iI2x2!(32)
Thus the introduction of the biases in the estimation state does not vary the rank of H. Moreover the
eigenvalues of the gramian matrix associated to the biases are for sure positive (ν > 0). In other words,
the observability issues are confined in the reconstruction of the relative state from the available angles-
measurements.
Whenever the observations errors have zero-mean time-uncorrelated characteristics, the observability
properties are expressed by the following information matrix Λ:21
Λ=HTf
W−1H(33)
where the weighting matrix f
W= diag(W1,· · · ,Wn) is the accumulated error covariance matrix and ac-
counts for different accuracies of the angle measurements within the same data batch. In this situation the
quantity P=Λ−1represents the covariance matrix of the estimation error associated to the least squares
estimated state. Therefore Pis related to the accuracy of the obtained estimate: the larger the elements
of the matrix, the less accurate the correspondent components of the estimated state.21 In Ref. 10 the
covariance of the relative position and velocity estimation has been used to define a metric of observability
of the problem.
The introduction of the biases of the sensors into the estimation state leads to an information matrix
that is numerically badly conditioned. Thus the estimation of the relative dynamics together with the biases
is a weakly observable problem. The availability of some a-priori information related to the reference state
used in the linearization process can aid to handle this numerical issue. The total information matrix of the
problem becomes:
Λtot =HTf
W−1H+P−1
apr (34)
A non singular a-priori information matrix is sufficient to ensure that det Λtot 6= 0.22 In particular, by giving
a small a-priori weight to the relative state (that is assuming a big standard deviation of the a-priori state)
a small quantity will be added to the diagonal of Λ. Therefore Λtot becomes a quasi-diagonal matrix, with
great improvement of its conditioning characteristics. Moreover the level of confidence related to the a-priori
state can be exploited to weight differently the effort in estimating different components. This aspect is
particularly fruitful when an independent calibration system is able to provide a good estimation of the
biases of the sensors. In that case, the accurate knowledge translates in small biases-standard deviations,
thus the estimation problem focuses more on the relative state; fully observable in the presence of maneuvers.
On the other side, if a true knowledge of the relative state is available, some a-posteriori characterization of
the sensors can be accomplished by weighting more the, almost perfectly known, δαcomponents.
III. Filter Design
The estimation problem defined by the linear relative dynamics of solution Eq.(3) and nonlinear measure-
ment equations of Eq.(1) is solved via a least-squares (LSQ) methodology. In the context of a ground-base
scenario, this choice is motivated by the possibility to process at the same time the complete history of data
thus allowing an efficient data editing. Moreover a deeper understanding of the systematic errors could be
achieved as modeling errors are not absorbed by the process noise.
The least-squares relative orbit determination aims at finding the state xlsq
0that minimizes the weighted
squared sum of the difference between the actual measurements ζ= (ζ0,· · · ζt)Tand the modeled obser-
vations zof Eq.(1).21 Thus the loss function to be minimized is the following function of the observation
residual ρ:
J(x0) = ρTρ
ρT= (ζ−z)T= (ζT
1−zT
1,· · · ,ζT
n−zT
n)(35)
Here the 2D vector measurements of the batch are indexed through i= (1,· · · , n), where nrepresents the
number of processed images taken at times t1,· · · , tn.
Due to the nonlinearity of the measurement equations, it is not possible to locate the absolute minimum
of Eq.(35). A local minimum however can be found in the proximity of a reference state xref
0=xapr
0,
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normally available within the application context. In addition to that, information on the accuracy of this
reference value is usually also available, in the form of the a-priori covariance Papr of the state. Such further
piece of information can be incorporated into the least-squares estimation which, finally, aims at minimizing
the following performance index:
J(x0) = ρTρ+ (x0−xapr
0)TΛapr (x0−xapr
0) (36)
As a result, the estimation is accomplished through an iterative dynamics batch least-squares estimator
with a-priori information.22 The estimation state which best fits the observations in a least-squares of the
residuals sense, or in other words, which minimizes the loss function defined by Eq.(36), is given by the
iterations
xlsq
j+1 =xlsq
j+Λapr +HT
jf
W−1Hj−1hΛapr(xlsq
j−xapr
0) + HT
jf
W−1ρji
Plsq
j+1 =Λapr +HT
jf
W−1Hj−1(37)
which are started from xlsq
0=xapr
0and continued until convergence, typically after 4-5 iterations.
It is emphasized that the estimation time is the time at which the filter is initialized. Thus it could
correspond either to the earliest time of the batch (t1) or to the latest one (tn). As a consequence the
propagation of the state has to be accomplished coherently with the direction of changing of the time. This
is achieved by design in Eqs.(10) and (11).
IV. Validation via Simulations
In order to validate this relative navigation tool a modeling and simulation environment has been devel-
oped making use of the libraries and functionalities of the Multi-Satellite-Simulator developed at DLR/GSOC
to support various projects in the field of multi-satellites applications.12 The purpose of this module is to
emulate the behavior of the space segment of a two-satellites mission. Meanwhile the estimated delta-v of
the performed maneuvers and the absolute state and attitude of the servicer satellite are logged. According
to our filter design, these information are used to propagate the estimated relative state and to model the
measurement observations zof Eq.(1).
The inertial orbits of both satellites are numerically integrated subjected to the gravity potential of a
non homogeneous mass distribution of the 30th order and degree, together with the main orbit perturbations
acting in low energy orbits. The attitude mode of the servicer satellite is emulated. Maneuvers are intro-
duced as time-tagged delta-v which are afterwards translated into extended-time thrust burns. Regarding
the simulations here discussed, the space systems are customized on the PRISMA mission. Therefore the
satellites possess all the main characteristics of the Mango and Tango satellites, that respectively play the
roles of servicer and client spacecraft. The model of the camera instrument also emulates the behavior of the
Visual Based System (VBS) camera mounted on Mango, in terms of resolution, field of view, blinding angles,
measurement noise and biases. Its output are the x-ycoordinates on the image plane according to Eq.(1).
That coincides with the output that would be provided by an image processing unit elaborating real images.
It is emphasized that the characteristics of the camera lens are not taken into account. The maneuvers
profile provided to the simulation module is produced by a maneuver planning tool, which computes the
ideal impulsive maneuvers that are needed to achieve a given relative formation, starting from the latest
estimated relative state.
The main goal of the validation process is to assess the closed-loop-control accuracies achievable when
performing a rendezvous (RdV) from far-range distance to a non-cooperative client satellite. To accomplish
this the following realistic operational conditions are considered:
•The whole simulation is settled on the true orbit passes plan generated from the Tango orbit information
available at the time. In particular the far-range RdV here discussed was scheduled between the 23-
Apr-2012 and the 27-Apr-2012. Eventual Moon and Sun eclipses are accounted for.
•Constraints related to typical operational day are included. They involve the location of the delta-v
opportunities and the presence of data gaps in the measurements. These gaps can be caused either by
mass memory and TM downlink limitations or by eclipses and instrument temporary malfunctioning.
Regarding the RdV here presented, a data gap of about 7 hours, ending approximately around 14:00
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UTC, is introduced daily. It could represent the effect of a repositioning of the mass memory reading
pointer so that the latest available TM refers to the current state of the space segment.
•The camera model provides measurements every 30 s. Such sampling interval represents typical time
frequencies of the image processing unit output, when exploiting relative target and stars movements
from picture to picture.
•Measurements biases are included.
•When computing the extended-time thrust profile correspondent to the ideal delta-v provided by
the maneuver planning tool, errors in the execution of the maneuver are introduced. Moreover the
estimated delta-v that are logged during the simulation include also the errors accomplished by the
estimation process. Aa a result the information used within the filter to model the measurement
observations is affected by both maneuver execution and estimation uncertainties.
In addition to these operational constraints, in order to emulate an approach to a non-cooperative client
satellite, the whole simulation is structured as follows. Each main task, i.e. maneuver planning, simulation
and relative orbit determination, is accomplished by independent teams. In the sequel the relative orbit
determination task will be referred as ROD. The simulation-specialist plays the role of the administrator of
the RdV rehearsal. He is the only person aware of the true state of the client satellite, of the actual instrument
biases and of the magnitude of maneuver execution and estimation errors. The simulation-specialist provides
to the other teams an initial relative state and sets the aimed final hold-point of the rendezvous, e.g. Table 6.
The error between true and provided initial conditions (IC) is of order of magnitude of the error committed
when using NORAD TLE to compute the first-guess relative state. In the current simulation, biases were
set of magnitude of about 1/8 of pixel ( ≈10 arcsecs).
The rehearsal of the whole RdV develops through several steps, i.e. iterations, each composed by an exe-
cution of maneuver planning, simulation and relative orbit determination tasks. The objective is to refine
step-wise the estimation of the relative state between the satellites. The process starts from the provided
IC. Afterwards, during each iteration, given the latest estimation of the relative state, the maneuver plan-
ning module provides the next maneuvers to be accomplished. These are commanded to the simulation
environment, that applies their effect on the space segment. The new acquired TM data together with the
maneuver profile are subsequently used to perform a ROD run on the new data batch to update the estima-
tion of the relative state, thus completing an iteration. As a result the errors committed by the relative orbit
determination influence the performances of the maneuver plan. Errors in execution of the maneuvers affect
the result of the aimed maneuvers. Therefore the complete functional chain allows assessing the realistic
closed-loop-control performances of the rendezvous.
Table 6: Initial conditions (IC) provided to the ROD-team. They refer to MOS initial time: 23-Apr-2012
14:30:14).
Elements’ set aδα[m]
Provided IC -1.00 -41.16 -377.60 19.53 246.50 -30658.14
(Unknown) true relative state -9.32 -79.24 -395.29 46.42 557.99 -30179.70
(Unknown) error at initial time 8.33 38.07 17.69 -26.89 -311.48 -478.44
Aimed final hold-point 0.0 0.0 -200.0 0.0 200.0 -3730.0
During the RdV, the maneuver planning module computes step by step the ideal delta-v needed to
track a specific guidance profile that is designed to bring the servicer from the provided IC to the aimed final
conditions. Such profile satisfies the eccentricity/inclination vector separation criteria suggested in Ref. 20. In
addition to that, the guidance profile shall account for visibility constraints of the camera, safety constraints
of the formation, delta-v budget limitations, delta-v execution opportunities and maneuvers selection to
improve the observability of the relative navigation problem. All these topics and the results discussed in
section Blead to the choice to perform couples of along-track maneuvers and cross-track corrections during
the most far apart phase of the approach. Combinations of radial–along-track maneuvers and cross-track
burns to refine the out-of-plane motion are performed in the final part of the approach. Regarding the
distribution of delta-v opportunities, it is here assumed that two windows are daily available. Therefore, at
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least two ROD runs per day can profit of the presence of new maneuvers. Figure 2reports all the delta-v
recorded in the TM for the RdV defined in Table 6. The values nearer to zero are due to the errors in the
execution and estimation of the maneuvers. The total delta-v spent is of ≈0.87 m/s.
12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
∆V [m/s]
04/24 04/25 04/26 04/27
R
T
N
Figure 2: Estimated values of the maneuvers performed during the RdV of Table 6.
A. Rendezvous simulation results
In Table 7are described all the ROD runs performed during the RdV from the initial conditions to the
aimed final hold-point reported in Table 6. Runs are tagged with labels from R1 to R9. To each of them it
is associated the latest active orbit contact, the time span covered by the measurements and the number of
camera outputs used for the relative orbit determination. Due to how the measurements are modeled in the
simulation environment, there is no need to perform any data editing. Therefore all the measures outside
data gaps can be used by the filter. Runs R1-6 exploit all the available data till the latest time; whereas R7-9
use just the most recent 24 hours of data. The choice of which data batch to use comes from some analysis
accomplished during the ROD module development. Moreover, the strategy was finally refined by tailoring
it on the results gained by different simulation scenarios, though sharing the same guidance approach during
the rendezvous.
Table 7: Summary of the ROD runs performed during the RdV.
ROD Pass # Data time span # measures
[ad] from [GPS time] to [GPS time] [ad]
R1 9767 23-Apr-2012 14:30:14 23-Apr-2012 19:28:44 598
R2 9778 23-Apr-2012 14:30:14 24-Apr-2012 13:49:15 1991
R3 9782 23-Apr-2012 14:30:14 24-Apr-2012 20:26:45 2786
R4 9793 23-Apr-2012 14:30:14 25-Apr-2012 14:50:45 4185
R5 9797 23-Apr-2012 14:30:14 25-Apr-2012 21:24:45 4973
R6 9807 23-Apr-2012 14:30:14 26-Apr-2012 14:09:15 6175
R7 9811 25-Apr-2012 19:47:54 26-Apr-2012 20:45:24 2189
R8 9821 26-Apr-2012 14:09:23 27-Apr-2012 13:26:54 1956
R9 9826 26-Apr-2012 19:08:43 27-Apr-2012 21:39:54 2321
As explained before in the filter design section, the LSQ filter can be initialized either at the initial or
at the final time of the data batch (marked by a star in Figure 3). Depending on the verse of the time
propagation, the filter can be labeled as forward (FW) or backward (BW). As the dynamics is modeled by
a linear system and the LSQ approach fits the whole data set at once, the accuracy of the estimated state is
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the same, disregarding the verse of time propagation. In the case of sequential relative orbit determinations
during the rendezvous, the filter is set to BW , as its output directly feeds the maneuver planning tool
(structure depicted in the bottom part of the sketch in Figure 3).
Figure 3: Schematic view of the LSQ approach and of
the concatenations of successive ROD runs.
Concerning how successive RODs are linked one
can focus on what data batches they use. Let bold
arrows of Figure 3symbolize each ROD run, i.e. es-
timation based on a given data batch. A i-th ROD,
i.e. solid arrow, can be followed by one that uses
just new data, i.e. sequential link marked with dot-
ted arrow, or by one that overlaps new and old data
in a nested way, i.e. dash-dotted arrow. For BW
LSQ, the estimation time will be the same, the data
fitted by the nested link, however, will be more and
distributed on a wider time span. Numerical sim-
ulation of different guidance profiles showed that a
nested link should be preferred for the following rea-
sons. The trends of estimation errors over consecu-
tive ROD runs are less irregular for the nested link
case. This aspect is important for the closed-loop
control of the rendezvous, as ROD output can be
considered reliable since the first steps. Unexpected
bigger errors influence the MAP profile and, conse-
quently, the ∆Vbudget to achieve the final target
hold-point. Furthermore, when a nested approach
is used, each ROD run can contain a greater equal number of maneuvers than the sequential run terminating
with same estimation time. This aspect improves the degree of observability of the estimation problem.
Finally, taking into account the ground-in-the-loop architecture, it is convenient to exploit at maximum the
amount of available data; coherently with the choice to employ a LSQ approach.
The last issue involved in the definition of time spans in Table 7is related to how big the nested data
batch can become. The rule employed in the RdV was determined by comparing the magnitude of the
estimation errors after each run when either all available data or most recent 24 hours were used, during
previous rendezvous rehearsals. Within the simulation characteristics and the modeling limitations of this
application, it turned out to be convenient to switch to shorter data batches from the second part of day-4.
Regarding the investigated guidance profiles, at that time the average along-track separation was less than
4 km. The maximum time span covered by the data batch should be traded off considering when the benefit
of having an extensive sample of measurements is countered by the deterioration of the capability to model
the measurements due to un-modeled phenomena over time. To this topic contribute the accuracy of the
model of the relative dynamics and the behavior of the true observations over time. Our model of the relative
dynamics does not include the differential drag effects (see Eq.(10)). In the simulation frame, the reality is
represented by the numerical integration of the absolute dynamics of the two satellites. True observations
are modeled disregarding many aspects of the instrument functioning. Therefore camera lens characteristics,
different lighting conditions, dimension of the client spot and behavior of the centroiding function are not
taken into account. As a consequence the true-measurements are always of the same quality during the RdV,
thus favoring the choice of data batches of wide time spans.
The initial condition for the biases was set to 0 radians. Due to the bad conditioning of the information
matrix of the complete LSQ problem it is set that the filter cannot substantially modify the values of the
estimated biases. This is accomplished by attributing an a-priori standard deviation to the biases significantly
smaller than the one of the other parameters. In particular the following values were used during the RdV:
σb,η = 1e−5[deg], σb,ψ = 1e−7[deg] (38)
The standard deviation of the measurements, i.e. ση=σψ, was set to 1/2 pixel circa (≈0.012 deg),
coherently with the instrument noise employed in the MOS simulation. The consequent weight matrix W
was kept constant throughout the whole simulation.
The remaining filter parameter of each run is the a-priori covariance matrix Papr computed from the
a-priori standard deviation set for the relative orbital elements σδα. Table 8lists all the sets employed
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Table 8: A-priori standard deviation of the relative orbital elements at each run.
ROD σδα=diag(Papr) [m]
R1 20.00 100.00 100.00 100.00 100.00 10.00
R2 1.34 1.88 1.95 1.88 2.10 50.00
R3 0.25 0.32 0.59 0.32 0.82 50.00
R4 0.28 0.34 0.59 0.34 0.77 50.00
R5 0.36 0.35 0.65 0.34 0.79 50.00
R6 0.01 0.01 0.01 100.00 100.00 0.01
R7 47.80 48.87 48.84 229.25 232.14 50.00
R8 0.64 0.78 2.46 0.77 2.58 50.00
R9 0.32 0.37 2.87 0.35 3.00 50.00
during each run of the RdV. In particular, in R1 σδαrepresents the typical confidence that the provided IC,
computed via TLE, can guarantee. In addition to that it was artificially set σδu smaller than the one of the
other elements, since the filter should not modify too much the related element of the initial guess state.
This is motivated by the fact that during the flight time of R1 no maneuvers were executed, which reduced
the degree of observability of the problem. By fixing the average along-track separation, the filter can act
only on the remaining parameters which define the shape of the projection of the relative motion on the
radial–cross-track (R-N) plane. Given Eq.(19), camera boresight direction and mean tangential separation
at the beginning of the rendezvous, the R-Nplane almost coincides with the plane of the image of the
camera. Thus the fitting leads to the identification of the family of the relative motion, with an error in
the scale factor related to the ambiguity on aδa,aδu and bη. Nevertheless, the less accurate components
of the initial guess from TLE, i.e. aδeand aδi, can be efficiently adjusted [c.f. the third line of Table 6
and the R1 abscissa of Figure 5]. Such an early correction, accomplished without need of maneuvers, brings
great benefit to the successive MAP planning. σδαsettings in R2-5 R7-9 are related to the choice of how
to link the estimation information from run to run. The LSQ filter provides the covariance matrix of the
least-squares solution according to Eq.(37). This quantity is function of the a-priori confidence and of the
number of observations used. Therefore the magnitude of the elements of the obtained Plsq, i.e. the formal
standard deviation of the estimated state, are not representative of the estimation error committed, depicted
in Figure 5. Moreover it has been observed that by simply letting Plsq to become the a-priori confidence of
the initial conditions of next run, the filter reaches in few runs a saturation point. In particular the estimated
state is trusted more than the fresh information brought by new measurements, leading the filter to diverge.
In order to avoid this phenomenon it has been established a rule to link the Plsq to the a-priori confidence
of the successive ROD run. A minimum-reference confidence σref −min is set, coherently with the expected
order of magnitude of the estimation error in the final phases. During the simulation such quantity was fixed
to:
σref −min = (3,10,10,10,10,50) [m] (39)
Then, whenever any element was satisfying σδα< σref −min , the whole set was rescaled so that its most
violating term would be equal to its correspondent in σref −min. Thus, values reported in Table 8are
computed via the above mentioned rule, starting from the solution of Eq.(37) (elevation defined w.r.t. the
local cross-track direction). In R6 a different decision was employed: the idea was to exploit the execution of
the first radial maneuvers to refine the estimation of the biases. A radial maneuver in fact inserts a variation
in the aδu, thus adding information to resolve the ambiguity on the scale factor of the projection of the
motion on the R-Nplane. The bias in azimuth, is perceived as a shift in the radial direction, thus it couples
itself with the estimation of aδa. The bias in elevation has no justification w.r.t. the solution of the relative
dynamics, given the attitude of Eq.(19).
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123456789
−150
−100
−50
0
50
aδa [m] ref, o est
123456789
−80
−55
−30
−5
20
aδex [m] ref, o est
123456789
−500
−400
−300
−200
−100
aδey [m] ref, o est
123456789
−20
−5
10
25
40
aδix [m] ref, o est
ROD run 123456789
100
225
350
475
600
aδiy [m] ref, o est
ROD run 123456789
−40
−30
−20
−10
0
aδu [km] ref, o est
ROD run
Figure 4: Estimated and reference relative orbital elements at the estimation time of each run.
1 2 3 4 5 6 7 8 9
−10
−5
0
5
10
e aδa [m]
1 2 3 4 5 6 7 8 9
−4
−2
0
2
4
e aδex [m]
1 2 3 4 5 6 7 8 9
−20
−10
0
10
20
e aδey [m]
1 2 3 4 5 6 7 8 9
−2
−1
−1
0
1
e aδix [m]
ROD run 1 2 3 4 5 6 7 8 9
−30
−20
−10
0
10
e aδiy [m]
ROD run 1 2 3 4 5 6 7 8 9
−1.5
−0.9
−0.3
0.4
1.0
e aδu [km]
ROD run
Figure 5: Estimation error at the estimation time of each run.
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To this aim a loop of runs were performed in which on turn only some elements were allowed to be adjusted.
In particular the following sequence was executed:
1. aδu was fixed (similarly to R1), in order to estimate shape and orientation of the relative motion;
2. all relative elements were fixed except aδu and bη, in order to refine the azimuth bias estimation;
3. aδa and bwere fixed, to refine shape and orientation w.r.t. updated values of the biases;
4. all fixed except aδi; to refine the estimation of the relative inclination vector.
These last settings correspond to the ones reported in Table 8. For each bias, the true values set in the
MOS simulator were of 10 arcsecs; in the current simulation the process just described did not improve
the accuracy of the estimation of b. It should be noted, however, that R6 achieved the minimum value of
estimation error in aδu (abscissa R6 in Figure 5). The effectiveness of this loop of runs varied across the
different simulations performed. It revealed more efficient when both the biases in the MOS simulation and
the error between biases initial-guess and true-value were bigger.
Figures 4and 5, report the accuracies achieved component by component for each run of the RdV.
The difference between the estimated state and the true one defines the estimation error committed at each
step. These figures allow respectively understanding what was the rendezvous guidance strategy and with
what accuracy it was achieved. Note that features regarding aδu are plotted in km. By observing Figure 5,
one can note that the error committed in aδa stays in ±8 m during the rendezvous and decreases with the
proceeding of the runs. Both aδexand aδixare well estimated. Their y-components are estimated with an
error of ±30 m. The mean along-track separation is always estimated within an error minor than the 7.5%
of the current true mean along-track separation, which spans from circa 30 to 3 km.
Finally, Figure 6and Table 9report the characteristics of the observation residuals during the RdV.
One can note that the residual in elevation, i.e. cross-track direction, is biased throughout the simulation.
Regarding our experience, during all the simulations performed that shared the same sensor attitude, the
bias in elevation has been estimated with less accuracy. Together with this, the residuals present a periodical
pattern superposed to a not zero mean value that clearly is an evidence of some not modeled phenomenon
that ultimately ends into the residuals. This is related to how the instrument is modeled in the simulation:
information from absolute positions and attitude are used to reproduce the x-y coordinates of Tango on the
image plane, therefore true measurements are not completely independent information. As a confirmation,
if biases are set to 0 during the simulation, the behavior of the residuals obtained during a relative orbit
determination phase, presents a mean trend around zero, and less marked oscillations.
ROD Residuals mean±σ
[arcsecs]
ρηρψ
R1 0±66 131±31
R2 4±53 148±32
R3 −11±70 140±37
R4 −1±70 102±72
R5 1±68 100±80
R6 1±59 97±59
R7 14±84 86±64
R8 2±56 78±38
R9 −3±126 101±96
Table 9: Merit features of the
residuals for each run of the
RdV.
−400
−300
−200
−100
0
100
200
300
400
ρψ [arcsecs]
R1 R2 R3 R4 R5 R6 R7 R8 R9
23/04, 12:00
24/04, 00:00
24/04, 12:00
25/04, 00:00
25/04, 12:00
26/04, 00:00
26/04, 12:00
27/04, 00:00
27/04, 12:00
28/04, 00:00
−400
−300
−200
−100
0
100
200
300
400
ρη [arcsecs]
R1 R2 R3 R4 R5 R6 R7 R8 R9
23/04, 12:00
24/04, 00:00
24/04, 12:00
25/04, 00:00
25/04, 12:00
26/04, 00:00
26/04, 12:00
27/04, 00:00
27/04, 12:00
28/04, 00:00
Figure 6: Behavior of the observation residuals over time, during each
run of the RdV.
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V. Conclusion
This work addressed the design and the implementation of a relative navigation tool that makes use of
camera-based angles-only measurements.
The first part of the paper dealt with the theoretical background of the estimation problem and with the
subsequent filter design. The relative dynamics has been parameterized through relative orbital elements.
This choice allowed exploiting a simple model of the dynamics which includes the effect of the Earth oblate-
ness, that is one of the main perturbations in low energy orbits. Moreover the parameterization allowed
having a direct physical interpretation of the range un-observability of the relative orbit determination using
line-of-sight measurements. Observability properties were discussed for the Kepler and the perturbed prob-
lems. Given the need to perform a maneuver in order to achieve the full observability, in the paper has been
introduced some possible metrics to quantify the subsequent level of observability. Such metrics exploit the
simple relations between maneuvers and changes in the relative orbital elements. Finally it is discussed how
the observability properties behave when sensor biases are included in the variable state.
The estimation problem of a far-range orbital rendezvous was solved with an iterative least-squares
approach that makes use of a-priori information. The proposed relative orbit navigation tool has been
validated within a high-fidelity simulation environment. There several realistic operational conditions have
been included. Despite this, still some topics cannot be fully addressed through a simulation environment.
The choice of the size of the data batches, for example, could be not optimal for a real-world application.
This is related to the difference between the reality and the simplified simulation environment. Moreover
the behavior of the true measurements over time, due to different illumination conditions and decreasing
relative separation was not included in the simulation. Again this could lead to prefer shorter/longer data
batches. Finally, in the simulation environment the behavior of the observation residuals revealed artificial
and provided very little information compared to what it provides when real measurements coming from
flight data are used.
The set up of a realistic rehearsal of a rendezvous from far-range distance to a non-cooperative client
satellite allowed assessing the achievable closed-loop-control performances. It was demonstrated the feasibil-
ity of a rendezvous guidance profile based on (anti-)parallel eccentricity/inclination vectors whose magnitude
shrinks with the passing of the time. Moreover it was assessed the robustness of the relative navigation tool
with respect to realistic errors in execution and estimation of the maneuvers.
The development of this tool happened in the frame of the preparation to the DLR/GSOC Advanced
Rendezvous Demonstration using GPS and Optical Navigation (ARGON) experiment. Future refinement
will take advantage of the flight data collected during the execution of this experiment.
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