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Computational Guidance & Navigation for Bearings-Only Rendezvous – methods and outcomes of GUIBEAR

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GUIBEAR is an on-going activity supported by the European Space Agency aimed at the demonstration of the feasibility of bearings-only (B-O) far-range rendezvous in near rectilinear Halo orbits (NROs). The studied scenario is the far-range rendezvous of the HERACLES Lunar Ascent Element (LAE) with the Lunar Orbital Platform – Gateway (LOP-G) that is orbiting the Moon in an NRO. This activity advances the readiness level of the applicable technologies through a solution based on Computational GNC, structured around a Model Predictive Control framework and tools; and an approach design, development, verification and validation based on iterative prototyping which integrates high-fidelity hosting considerations into the design from a preliminary level: concurrent GNC and software architecture design with hosting, runtime assessment and optical high-fidelity simulations. This paper: presents the context – mission, system and hosting, and algorithm-level challenges and proposed concepts; details and justifies the architecture and design solutions; outlines the methodologies employed for prototyping verification and validation; provides conclusions on the obtained results.
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COMPUTATIONAL GUIDANCE & NAVIGATION FOR BEARINGS-ONLY RENDEZVOUS
– METHODS AND OUTCOMES OF GUIBEAR
P. Serra(1),(2), P. Lourenc¸o(1) , J. Branco(1), J. F. Briz Valero(3), M. Mammarella(3), T. Peters(3), D.
Mora Portela(3), S. Manara(4), D. Bernardini(4), A. Bemporad(4) , P. Foglia Manzillo(5), J.
Witteveen(5), A. Cropp(6)
(1)GMV, Alameda dos Oceanos, n.º 115, 1990-392 Lisboa, Portugal, +351 21 382 93 66,
{parroz,palourenco,jbranco}@gmv.com
(2)Lusofona University of Humanities and Technologies, COPELABS, Lisbon, Portugal
(3)GMV, C/ Isaac Newton, no. 11 – PTM 28760 Madrid, Spain, +34 91 807 21 00,
{jbriz,mmammarella,tpeters,dmora}@gmv.com
(4)ODYS, Via Augusto Passaglia, 185, 55100 Lucca LU, Italy, +39 339 187 5527,
{silvia.manara,daniele.bernardini,alberto.bemporad}@odys.it
(5)cosine Remote Sensing B.V., Oosteinde 36 2361 HE Warmond The Netherlands, +31 715 241 099,
{pfogliamanzillo,j.witteveen}@cosine.nl
(6)ESA-ESTEC, Keplerlaan 1, PO Box 299 2200 AG Noordwijk The Netherlands, +31 71 565 8797,
Alexander.Cropp@esa.int
ABSTRACT
GUIBEAR is an activity supported by the European Space Agency aimed at the demonstration of
the feasibility of bearings-only (B-O) far-range rendezvous in near rectilinear Halo orbits (NROs).
The studied scenario is the far-range rendezvous of the HERACLES Lunar Ascent Element (LAE)
with the Lunar Orbital Platform – Gateway (LOP-G) that is orbiting the Moon in an NRO. This
activity advances the readiness level of the applicable technologies through a solution based on
Computational GNC, structured around a Model Predictive Control framework and tools; and an
approach design, development, verification and validation based on iterative prototyping which in-
tegrates high-fidelity hosting considerations into the design from a preliminary level: concurrent
GNC and software architecture design with hosting, runtime assessment and optical high-fidelity
simulations. This paper: presents the context – mission, system and hosting, and algorithm-
level challenges and proposed concepts; details and justifies the architecture and design solutions;
outlines the methodologies employed for prototyping verification and validation; provides con-
clusions on the obtained results.
1 INTRODUCTION
Estimating the range to target is the first order challenge of bearings-only rendezvous [1, 2]: the rela-
tive position is not instantaneously observable and bearings measurements are nonlinear, which are a
source of inconsistency, over-conservatism or divergence of estimates. Observability analysis shows
that manoeuvring between at least three non-coplanar (or four coplanar) measurements is necessary to
resolve the full relative position and velocity observable. As a result, estimation and control/guidance
problems are inherently connected, especially in an autonomous setting. A bearings-only navigation
ESA GNC 2021 – P. Serra 1
filter for rendezvous has to cope with the non-linearity of the measurement model and the instanta-
neous non-observability [3], while including knowledge of manoeuvres through thruster acceleration
observables that introduce a significant source of uncertainty. Given all the challenges the naviga-
tion filter is exposed to, careful planning of the trajectory in order to maximize the information gain
it provides is extremely important. Effectively correcting in closed loop the results of unmodelled
dynamics, particularly manoeuvre dispersion and knowledge, favour autonomous on-board guidance
functions that present a challenge both in terms of computational load, fuel-cost, estimation perfor-
mance, and safety. NROs present significantly different orbital environments to existing Earth orbit
B-O rendezvous heritage [4, 5, 6]. The nature of the 3-body problem and the absence of complete
analytical closed-form solutions are important challenges, but extensive study of this kind of orbits
shows that, for the scenarios considered and short timescales, closed-form approximation are appli-
cable. The distances introduce significant challenges on target detection and tracking as the target
is at sub-pixel resolution, such as increased exposure times. This purely radiometric detection relies
on the capability of disentangling the target contribution to the pixel readout from the background
contribution due to cosmic background radiation (diffuse sky emission), background signals, and to
instrument noise itself. Aside from these challenges in terms of the main sensor in the considered
suite, several other hosting challenges are present. For example, effectively measuring the executed
delta-V is also a relevant challenge, given that accelerometers are perturbed by various different error
sources, from drifting biases to multiplicative cross-coupling errors. Finally, the online nature of the
GNC solution also presents significant challenges in terms of runtime constraints and processing ar-
chitectures. The on-board computational power has to be able to solve the constrained optimization
problem that underlies the guidance function in real time.
The GUIBEAR study aims at addressing all these challenges, through the design of an algorithmic
or computational Guidance and Navigation architecture. The computational efficiency, accuracy and
robustness of the trajectory generation are at the foundation of the design, as the numerical generation
of guidance commands relies extensively on on-board computation. The autonomous guidance func-
tion is based on an optimization problem with the objective of 1) maximizing the observability of the
range and 2) minimizing the fuel consumption. This optimization problem bridges the gap between
estimation and control, thus approaching the potential of a truly autonomous system: the navigation
filter informs the computation of the trajectory, which in turn is optimized to improve the navigation
performance. In order to close the loop, the optimization problem is solved online, and its solution is
updated whenever the uncertainty of the navigation filter decreases. Adding the capability to enforce
input (saturation, maximum V, minimum acceleration necessary for detection) and output (safety
and final conditions) constraints to the optimization of the performance index lead to the formulation
of the multi-objective optimization problem in a model predictive control (MPC) framework [7]. Ulti-
mately, this provides a process to reduce the navigation errors systematically throughout the mission,
while taking advantage of this same reduction to improve the commanded plan. The bearings-only
estimation filter is tailored for monocular vision-based navigation, taking into account the particular-
ities of the NRO environment and the difficulties of navigating when the target is always at sub-pixel
resolution. For that purpose, a weighted least squares batch estimator is employed, improving the
accuracy and outlier rejection while dealing with such challenges as varying camera integration times
and computational cost. This filter is complemented by a higher frequency state estimator that not
only propagates the dynamics whenever measurements are not available, but also combines the higher
quality information provided by the batch filter. The resulting architecture is then a mixture of sequen-
tial and batch processing, and consequently high and low frequency processing tasks. Since accurate
navigation is only necessary before the computation of the next guidance profile by the MPC, this
exploits the structure of the mission scenario with the objective of improving the navigation perfor-
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mance, as it minimizes sensor measurement errors, saves computational time, and minimizes the real
time operational constraints. The proposed Computational G&N architecture is verified and validated
in simulation, taking as baseline the studied scenario. The LAE launches from the Moon, and ascends
to an elliptical low Moon Orbit, which is circularized into a circular low Moon orbit. This is followed
by an orbit transfer and phasing phase that place inject the LAE in free drift at which point it is able
to detect the target from a distance larger than 1000 km. The scenario tackled in this activity is the
approach and autonomous injection of the LAE in the LOP-G’s NRO, leaving the former in handover
conditions for the start of rendezvous proper, at 100 km from the latter.
The feasibility of performing far approach through these mass and cost-saving camera-only GNC
techniques is established and justified through prototyping and extensive testing in Model-In-The-
Loop high-fidelity simulations. Autocoding and Software-In-The-Loop testing are performed as a
bridging step towards the subsequent phases of the V&V process – laboratorial Hardware-In-The-
Loop campaigns. As part of the strategy a novel method employed for benchmarking of runtime of
the auto-coded software is described: preliminary non-realtime PIL assessment employed at algo-
rithm development level. This method includes a step of experimental compilation and running of
a segment with a number of GNC cycles in a flight-representative processor thus allowing adjusting
GNC parameters and runtime latency emulation blocks in the functional simulator.
2 THE GUIBEAR PROJECT
The GUIBEAR project is a natural extension of the NRO-GNC project [5], in the sense that it builds
on the DDVV process therein, and is focused on the final part of the Orbit transfer phase, particularly
up to and including the NRO insertion manoeuvre as noted in Figure 1.
Figure 1: LAE mission scenario, with the phases of interest highlighted in dark red: the as-
cent/circularization and LLO loitering phases; in red: the orbit transfer phase; and in grey: the
rendezvous phase. On the right, the very far range nominal scenario. In red the trajectory of the
chaser from the LLO to the NRO. In blue, the target trajectory.
2.1 Timeline
The baseline approach uses an orbital transfer from low-lunar orbit (100 km altitude) to the halo
orbit (approximately 7000 x 70000 km altitude) that consists of two impulsive manoeuvres plus a
mid-course correction (for a total of 3 manoeuvres). The orbit transfer from low-lunar orbit starts
ESA GNC 2021 – P. Serra 3
with a transfer injection manoeuvre and ends with an NRO insertion manoeuvre. The manoeuvres
are computed as two-impulse transfers analogous to the Lambert problem in central-body / Keplerian
dynamics. That is to say, the manoeuvres are computed deterministically, taking into account the
three-body dynamics in an ephemeris model of the Earth-Moon system. Relative observations can
start during the free drift prior to the NRO insertion manoeuvre (IM).
Towards the end of the transfer the navigation will switch from a ground-based tracking solution
to a camera based navigation solution. The handover from ground tracking to the camera sensor
based navigation is planned to occur at a distance of 1000 to 1500 km distance from the LOP-G
station. The handover conditions are mainly determined by the accuracy of the ground tracking and
the illumination conditions at the time of the handover.
2.2 Scenarios
Three scenarios are defined where the initial conditions for all scenarios are the same. The scenarios
all start 20 hours before the nominal NRO insertion manoeuvre occurs, at a distance of 1500 km from
the LOP-G. The terminal conditions for all scenarios should be the same: to arrive at the hold point
at 150 km 20 hours after the start of the scenario. Note that in all scenarios some time is included
after the insertion manoeuvre for relative observations, in order to leverage the range observation
enhancement opportunity provided by the insertion manoeuvre itself. The three scenarios considered
are shown in Table 1.
Table 1: GUIBEAR Scenarios.
Scenario 1
(Baseline)
This scenario is used to evaluate how well range can be observed without performing
additional manoeuvres. This includes a scenario with pre-planned insertion (PI) and
online-computed insertion (OI).
Scenario 2
(Insertion
untouched)
MPC based observation enhancing manoeuvres are being performed, but the inser-
tion is performed as planned in the baseline.
Scenario 3
(Full MPC)
All manoeuvres, including the insertion, are recomputed using MPC. Fuel-optimal
(F) and Observability-optimal (O) cases are addressed.
The baseline scenario is essentially the same as the Heracles LAE scenario as described in [5]. This
scenario consists of a single manoeuvre taking place, the NRO insertion manoeuvre. This manoeuvre
has a nominal Vof 20 m/s. Two observations performed before and two observations performed
after a manoeuvre make the camera based navigation problem fully observable. This means that even
if no additional manoeuvres are planned, the fact that the NRO insertion manoeuvre needs to be per-
formed would already provide an opportunity for making the problem observable (that is, by taking
at least two directional measurements before and after performing the NRO insertion manoeuvre).
This is considered the baseline scenario. In the second scenario, manoeuvres are added before the
insertion manoeuvre, but the insertion manoeuvre itself is left untouched by the MPC. This implies
that the additional manoeuvres are small (of the order of 1 m/s), and that at the end of the sequence
of additional manoeuvres the LAE is in the correct position to perform the insertion manoeuvre. The
computation of the insertion manoeuvre takes into account the improved navigation available through
the sequence of manoeuvres. In practice this means that the timing of the manoeuvre can be adjusted
on-board based on the relative navigation to ensure the LAE arrives at the hold point. Ideally, the
sequence of manoeuvres should not greatly increase the uncertainty in position with respect to the
estimated ground tracking based accuracy (that is, 6 km on all axes) if no measurements would be
taken. In this way, the observability enhancement manoeuvres could be performed without interfering
ESA GNC 2021 – P. Serra 4
with the nominal operations defined in the baseline approach. In the third scenario, the MPC gen-
erates a sequence of manoeuvres that include the insertion into the hold point at 150 km. The only
restrictions are as stated above, namely, that the scenario ends 20 hours after the nominal starting
point at a distance of 1500 km and that the Vis less than a maximum that includes the Vof
the insertion manoeuvre and a margin for observability enhancements. In this scenario the MPC has
the most freedom to design the trajectory and can work with larger manoeuvres than the other two
scenarios.
3 GUIBEAR G&N DESIGN
In a classical sequential architecture, the loop starts acquiring data from the available sensor suite;
the measurements are then processed by a sequential Navigation Filter that at each time instant pro-
vides a state vector estimation - when no measurements are available the filter will only propagate the
dynamic model implemented. Using the covariance measurement it is possible to determine if the ac-
curacy has substantially improved from the past. If the accuracy has improved substantially from the
last execution of the guidance computation, the guidance function is executed. The guidance function
can also be executed periodically and in case the Navigation accuracy has not improved from the past
execution the guidance call is aborted. The guidance function computes the entire trajectory and it is
used in the high frequency cycle. On a different flavour, the G&N architecture designed in GUIBEAR
includes a mix of sequential processing and batch processing, and consequently High frequency and
Low frequency processing tasks. The high frequency tasks consists as first task collecting data from
samples as it was described for the previous architecture. The second task consists in estimating the
state vector using a simple algorithm as for example simple propagator of the dynamics, something
that in the previous architecture is also performed when measurements are not available. Knowing
the trajectory, it is possible to know if the manoeuvre has to be applied within the current cycle, if yes
the control applies the manoeuvre. In parallel, there are Low Frequency processes running that can be
synchronized with the mission trajectory. The sensors measurements are collected in data batch and
processed within a Navigation Filtering which uses batch filtering with dynamic models and mea-
surements collected at different time. The Navigation provides an estimation of the performed arc
averaging all the measurement errors of the collected data. More accurate dynamical models can be
Figure 2: The G&N architecture designed during the activity: mixed batch and sequential processing.
On the right, low frequency tasks execution proposed concepts overlapped over the trajectory
used since there is not a stringent requirement on the navigation frequency but especially the batch
ESA GNC 2021 – P. Serra 5
filtering is a simple method for reducing the measurement noise while fitting the trajectory. Once the
navigation solution is computed, the information is passed to the State vector estimator of the High
frequency task in order to use as soon as possible the most update information. If the accuracy has
improved substantially from the last execution of the guidance computation, the guidance function
is called. Alternatively the guidance function can be also executed periodically and in case the Nav-
igation accuracy has not improved from the past execution the guidance call is aborted. As in the
sequential case, the Guidance when executed generates an entire trajectory that updates the existing
trajectory. This strategy fits very well in the mission scenario, since accurate navigation solution is
only needed before the execution of the manoeuvres and this time can be foreseen since the trajectory
is known up to the end. Consequently, knowing the time needed for executing the Navigation Filter-
ing and the guidance functions, it can be planned measurement data collection between 2 manoeuvres
in order to have the navigation solution and possibly a re-computation of the trajectory before the fol-
lowing manoeuvre. The described concept is depicted in the right-hand side of Figure 2 reporting the
time dedicated to collect data, Navigation Filtering Execution and Guidance Execution over the tra-
jectory. This solution provides better performance because it minimizes sensors measurement errors
through a batch filtering process also saving computational time since the on-board computer (OBC)
is extensively used only when Navigation and Guidance are executed. Furthermore, it minimizes the
real time constraints of the Navigation since the only constraints is that Navigation and guidance are
executed before the next manoeuvres. The time between manoeuvres is set a priori and drives the
prediction model used on the optimization-based guidance.
3.1 Guidance
In general, a line-of-sight measurement profile is unique when, given an observation period, it can
only be generated by one possible trajectory. For different relative trajectories, there may be periods
with the same azimuth and elevation readings, but for the entirety of the observations the overall
measurement profile must be unique. Since the measurement profile can be identified with the initial
conditions, a unique profile and the associated motion are associated with uniquely determining the
initial conditions. When that is possible, the system is observable. This is the topic of several works
on bearings-only rendezvous in LEO (see [8] and [9]). To apply this concept in the context of an
NRO, it should be noted that the last segment of the transfer from LLO to halo covers a small fraction
of the orbital period, such that the dynamics is very slow. Furthermore, the chaser moves along a
trajectory that is nearly a straight line. This means that dynamical coupling of the axes is quite weak.
By contrast, the motion in the x and z directions of the LVLH frame are dynamically coupled in
the case of LEO rendezvous. These considerations can inform a geometrical observability analysis
that itself can steer the trajectory design. Some conclusions are: (i) Maximizing observability of a
manoeuvre requires that it is performed perpendicular to the direction of the target; (ii) Observability
is fostered by changes in observation angle and the angular momentum. The former are maximized
when the manoeuvre is perpendicular to the angular momentum and the direction to the target, and
the latter when it is collinear with the angular momentum; Taken together, these recommendations
point to a guidance strategy in which manoeuvres are performed such that the trajectory resembles a
3-dimensional spiral.
By deriving the observability Gramian for the bearings-only navigation problem, it is possible to add
further considerations to the G&N design: (i) If the spacecraft moves perpendicularly to the original
direction to the target twice, three different measurements will provide observability, otherwise at
least four measurements are necessary; (ii) Some manoeuvres may not provide observability if the
measurement profile does not change, i.e., if the new position lies in the same direction to the target
as it would without manoeuvres.
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3.1.1 Observability-enhancing Guidance
In fully observable frameworks, the problem of deciding how to move is typically decoupled from the
estimation problem. However, in situations where motion can critically assist the observability, or in-
deed be a necessary condition, it is beneficial to consider the two problems together. [10], [11], [12],
and [13] explore strategies for trajectories that enhance the observability of the problem, and hence,
the performance of the related estimation algorithms. Some of these works are based on geometric
considerations, and provide particular trajectories that can improve or guarantee the observability of
the manoeuvre. For example, [12] addresses the possibility of using spiralling trajectories and hints
at their interesting properties in terms of estimation performance. However, a formal understanding
and design of trajectories is not presented. [10], on the other hand, is focused on establishing the con-
cept of “observability degree” and derive a metric for the range estimation error related to the actual
distance, the angular difference between the trajectory with and without manoeuvre, and the measure-
ment error. Their next step is to use this metric as a cost function in a minimization problem that is
solved analytically for an impulsive manoeuvre and provides interesting insight into the optimal tra-
jectories. In the context of a real mission, these approaches are very interesting as stepping stones, but
when considering operational constraints the solution of such an optimization problem becomes more
complex. Not only are nonlinear non-convex problems harder to solve, an on-board solution needs to
be solvable fast and deterministically. Therefore, the optimization problem is transcribed in order to
result in a convex QP problem, since linear or quadratic optimization problems are guaranteed to be
solvable in a finite and possibly known number of iterations [14]. The approach proposed in [13], is to
obtain a discretized trajectory in n steps, with n possible inputs, which allows for quadratic program-
ming. The resulting optimization variables are the relative states in all steps, the controls, and slack
variables used to encode the norm constraints linearly. The constraints considered are the discrete
relative motion dynamics (discretized from analytically integrating the Clohessy-Wiltshire equations,
applicable only to circular orbits), boundary conditions (initial and final state), thruster saturation,
and manoeuvre objectives (approach corridors, collision avoidance). When the optimization problem
is convex, it allows to correct the rendezvous planning multiple times along the trajectory. This is,
in fact, similar to model predictive control schemes with dynamic avoidance constraints. Bridging
this approach to the typical MPC scheme, it should be noted that the typical receding time horizon
is replaced by a fixed final set point, shrinking the time horizon accordingly. This approach to the
design of a guidance function has a natural formulation as a trajectory optimization problem with a
fixed-horizon. An online computation (on-board or ground-based) of the trajectory that is updated
every time step according to the current knowledge provided by the estimation filter and taking into
account eventual errors in previous manoeuvres. Noting that the optimization problem is updated
every step, and only the first manoeuvre is executed before a new plan is optimized, this guidance
function fits directly in the Model Predictive Control area, noting that the last step of optimization is
fixed instead of receding, and the horizon is shrinking.
Cost function The cost function to be considered has two main terms, an actuation cost and an
observability cost. The former is
Jf=
N(t)1
X
k=0
1T|uk|(1)
where 1R3is a vector of ones used to sum the absolute values of each input. The observability cost
is based on the work in [9] and [13]. These costs proposed are not convex, and, as such, are convex-
ified by means of slack variables and previous trajectory information. Furthermore, it became clear
during the activity that the above observability cost does not correlate perfectly with the observability
ESA GNC 2021 – P. Serra 7
of the range, instead being replaced by the angle between the would-be trajectory and the actual one
(or its cosine, in fact):
Jo=
N(t)1
X
k=0
xkΦT(tk+1, tk)STS(Φ(tk+1, tk)xk+G(tk+1, tk)uk))
kΦ(tk+1, tk)xkk kS(Φ(tk+1, tk)xk+G(tk+1, tk)ukk(2)
where Sselects the relative position component.
Dynamic model The propagation methods analysed for GUIBEAR can be separated into numerical
non-linear propagation and linearized propagation. The baseline trajectories of chaser and target have
been obtained through a numerical integration of the equations of motion. These equations include
central forces from the gravity of the Earth and the Moon:
aSC =GMEarth
krSC Earth k3rSCEarth GMM oon
krSC Moon k3rSC Moon (3)
Expressed as the 6 first order differential equations
˙
rSC
˙
vSC =f(vS C ,aSC ) = vSC
aSC .(4)
This propagation method considers an inertial frame centred at the Earth-Moon barycentre that is
parallel to J2000. The position of the Earth and the Moon is obtained through an ephemeris model of
the two bodies (from JPL ephemeris). The initial conditions of the Target (which follows an NRO)
is obtained through a process of differential correction to obtain a semi-stable trajectory, whereas the
chaser trajectory is obtained through mission analysis. Both the differential correction process and the
mission analysis are detailed in Error! Reference source not found.. The trajectories obtained through
this numerical integration are considered the reference trajectories, and the different linearized meth-
ods are compared to these references. The trajectory of the target is considered as reference, and the
trajectory of the chaser is a deviation with respect to the reference. The method chosen is based on a
computation of the state transition matrix (STM), which maps an initial state 0 to a final state f as:
xf=Φ(tf, t0)x0=Φrr (tf, t0)Φrv(tf, t0)
Φvr (tf, t0)Φvv (tf, t0)r0
v0(5)
which can be obtained as a solution to the system
˙
Φ(t, t0) = f
x
|{z}
A(t)
Φ(t, t0),Φ(t0, t0) = I(6)
The approach consists on integrating the system (6) numerically. The matrix A(t)R6×6is evaluated
at each point in time using the reference trajectory of the target.
While the numerical propagation of the STM seems expensive at first, the computation of the STM
depends only on the trajectory of the target, the Moon and the Earth. These three elements are known
a priori on ground, so it is possible to propagate the STM numerically on ground for an arbitrarily
large interval (e.g. one NRO period, or a few days around the aposelene), perform a fitting of the
STM coefficients on ground and then sending/storing them on the OBSW. With this approach, the
STM would be obtained in a similar way as planetary ephemeris are used, requiring a small number
of operations to evaluate a polynomial. Analysis results show that a fitting using polynomials of order
6 to 8 provide results that are good for the applications proposed. Furthermore, it could be possible to
obtain the STM between any given interval by applying STM composition rules (generally requiring
the evaluation of 2 STM and inverting one of them).
ESA GNC 2021 – P. Serra 8
Constraints The (convex) constraints considered in the guidance design are (i) maximum delta-
V per manoeuvre; (ii) total delta-V budget; (iii) target rendezvous sphere avoidance; (iv) escape
trajectory avoidance; and (v) passive safe trajectory enforcement.
Model Predictive Control Within the GUIBEAR project, the software tool MPCSofT developed
during the project “ROBMPC – Robust model predictive control of space constrained systems” (2010-
2012) has been modified to support the features needed to formulate and solve the MPC problem
proposed above to control far-range bearings-only rendezvous in NRO. The original implementation
of the toolbox MPCSofT lacked some of the features needed to handle the peculiarities of the MPC
problems arising in the context of the GUIBEAR scenarios. Specifically, such peculiarities consist in
the presence in the cost function of: (i) a linear term, aimed at minimizing fuel consumption, (ii) a
positive semidefinite quadratic term, aimed at enhancing the observability of the trajectory, that may
render the problem not strictly convex. In order to overcome these limitations, some improvements
to the toolbox have been carried out and the current MPC formulation supported by the toolbox has
been modified to enable the user to minimize linear terms in the cost function.
The LTV-MPC toolbox MPCSofT2.0 relies on a Simulink Library that includes a block for LTV-MPC,
implementing the controller in Embedded MATLAB (EML). MPCSofT used to be interfaced to the
QP solver that was available from the MPC Toolbox for MATLAB in 2010. Such QP solver was based
on rather classical active-set methods for strictly convex QPs , and therefore appeared not tailored to
address the solution of QP problems such as those arising in the GUIBEAR scenarios, in which, due
to the observability-enhancing cost, the Hessian matrix is positive semidefinite. In order to overcome
this issue, MPCSofT2.0 was interfaced to two more recent QP solvers, both based on the alternating
direction method of multipliers (ADMM) [15] namely: (i) a QP solver internally developed at ODYS,
that implements the ADMM in EML and is therefore tailored for code generation, (ii) the open-source
solver osQP [16], which is called by MPCSoFT2.0 through its MATLAB interface.
3.2 Navigation
The developed Navigation algorithm is based on batch filtering that processes Image Processing mea-
surements and, in between filter updates, it propagates the translational states based on the dynamics
model. The approach followed has resulted in a very efficient method since the batch filtering is less
sensible to measurement noise due to the fact that a large number of measurements are used to fit
the relative dynamics. On the other hand, as the Image Processing is the only relative sensor, it is
certainly understandable that the Navigation performance is highly connected to the Image Process-
ing performances and measurement availability. Within GUIBEAR, the approach followed has been
to implement an Image Processing Performance Model to mimic the behaviour of the IP algorithms
both in the measurements performance and the availability in function of the distance and the angu-
lar velocity of the object within the camera frame. Indeed, based on a camera model implemented
in PANGU [17] and tuned with respect to the parameters of the selected camera, different synthetic
images have been generated. The different synthetic images are used to build an Image dataset, which
then serves as input to the real Image Processing algorithm (IP Functional Model). The outputs of
the IP functional models have been statistically analysed in order to derive the performance of the
measurements and the probability of providing measurements of the IP algorithms. The estimated
statistical quantities have been used to derive the IP performance model that is integrated in the Func-
tional Engineering Simulator (FES) together with the GNC algorithms.
This section includes a summary of the batch processor algorithm used in the state estimation. The
summary includes a general description of the algorithm, plus specific points that apply to the GUI-
BEAR scenario. A more detailed mathematical description and derivation of the algorithm is included
ESA GNC 2021 – P. Serra 9
in [18].
3.2.1 Batch filter
The objective is to estimate a deviation vector x0at a reference time t0. The information provided is
the initial condition x(t0), an a priori estimate of the initial condition ¯
x0and its associated covariance
matrix P
0. An estimate of the deviation vector ˆ
x0is obtained solving the normal equation
Λ0ˆ
x0=HTR1y+¯
P1
0¯
x0,(7)
where yi=h(x(ti), ti)is the measurement vector, Ris the measurement covariance, Λ0is the
information matrix, given by
Λ0=HTR1H+¯
P1
0,(8)
and His the observation-state mapping matrix, evaluated at the reference trajectory mapped to the
reference time, as in
Hi=h(x(ti), ti)
xΦ(ti, t0).(9)
In this activity, the observation transformation h(x(ti), ti)is defined as a projection of the target state
vector in camera frame, scaled with the camera focal length.
The measurement noise covariance is considered as a diagonal matrix for each measurement, with
value R=σ2
pxI, where σ2
px is the variance of the measurement noise, considered constant for every
measurement.
The state propagation is performed by linearly mapping a state vector from a reference time to a time
iusing the state transition matrix. The state vector at a time iis a linear combination of the state
vector at the reference time, plus the effect of each manoeuvre, mapped from manoeuvre time to time
i:
xi=Φ(ti, t0)x0+
m
X
n=1
Φ(ti, tn)∆vn(10)
The state transition matrix defined depends on the trajectory of the target, it is not modified by the
manoeuvres and only depends on the initial and final times. Because of that, the state propagation can
be arranged into a single matrix product with an augmented state transition matrix and an augmented
state vector, xi=Φ(ti, t0)X
0=Φ(ti, t0)Φ(ti, tm)· · · Φ(ti, t1)xT
0vT
m. . . vT
1T.
Since the reference trajectory is propagated linearly, both the reference and the deviation with respect
to the reference are propagated using the same approach. Finally, the a priori covariance is obtained
for the augmented state vector using the ground tracking estimate and an estimation of the manoeuvre
execution. ¯
P0= diag σ2
x0I, σ2
vI.
3.2.2 Analysis
During the activity, an analysis and trade-off of bearings-only navigation approaches based on the
batch least-squares processor described above was performed. The first objective was to define a mea-
surement strategy, i.e. number and distribution of measurements, considering the relevant parameters
of the problem, which are enumerated and defined below. The second objective was to understand
the impact of each parameter in the solution obtained, which can be used as an iteration with the
requirements. In summary, the navigation strategy should consider the following conclusions:
Both manoeuvre size and manoeuvre estimation are important parameters when improving the
radial component estimation. A budget of the order of 20 m/s could provide solutions with
ESA GNC 2021 – P. Serra 10
a standard deviation of the initial condition of 3 km, but this budget is only accessible if the
MPC has control over the insertion manoeuvre. To reduce the number of manoeuvres, each
manoeuvre needs to have a value of at least 1 m/s. The manoeuvre estimation error needs to
be of the order of 1 cm/s, but the combination of larger manoeuvres and smaller errors has not
been analysed in detail, a preliminary limit of manoeuvre estimation error is set to 1%.
It would be necessary to rely on a more accurate propagation and estimation of the state tran-
sition matrix to improve the solutions and benefit from smaller manoeuvre estimation errors.
Considering the current navigation errors and the navigation objective, this update does not
seem necessary at this moment.
Several successive measurements can be used to improve the solution, which have the effect
of reducing the measurement noise. If the measurement noise was reduced, the number of
measurements could be reduced. However, to be compatible with conservative values (1 px
3σ), 3 to 5 measurements seem to be enough.
Distancing the measurement to increase the change in angle optimizes the measurement place-
ment.
The current values for ground tracking estimation errors (6 km, 12 cm/s, 1σ), in combination
with the rest of uncertainties using conservative values, produce solutions that have a position
error of 0.4 % with respect to the distance and a velocity error of 0.4% with respect to the
relative velocity. It was expected to maintain the percentage over the entire scenario, obtaining
Navigation error improved of 1 order of magnitude with respect to the error considered without
GUIBEAR.
The current combination of noise and parameters appears to be robust to large uncertainties.
4 SIMULATION RESULTS
The main objective of the GUIBEAR G&N system is to ensure that the mission vehicle arrives at
the hold point (HP) within the specified area and performs the insertion manoeuvre successfully.
The G&N must tackle the optimization objectives and constraints while dealing with uncertainty in
some parameters in the system, sensor noise, and external disturbances. Given the closed-loop nature
of the guidance function the DDVV process has some similarities to the design of a control system,
including the synthesis and first analysis of the G&N functions. However, the performance assessment
is done by means of a SW Functional Engineering Simulator (FES). The FES for detailed performance
assessment contains high fidelity models that are non-linear, in general. Additionally, other aspects of
the mission such as the Mission & Vehicle Management functions and the mode transition logic can be
considered. In this phase, the G&N performance was evaluated against specific mission requirements,
defined in the mission domain (e.g. fulfilment of HP requirement or fuel consumed).
Figures 3 and 4 show the nominal navigation performance of the two MPC-based scenarios, and table
2 allows to compare the three main scenarios statistically (and which can be directly compared, as
the insertion is based on on-board navigation for all three). From this analysis, it is clear that the
introduction of observability-enhancing manoeuvres has a beneficial effect. Furthermore, the fully
optimized Scenario 3 shows the best performance, as expected.
The comparison in tables 3 and 4 again shows the good performance of the overall G&N, as it allows
reducing the position and velocity dispersion when compared to the online planned insertion with no
guidance. This reduction has a significant impact, because it will not only allow a better planning
ESA GNC 2021 – P. Serra 11
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in X-Axis
Error
3
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in Y-Axis
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in Z-Axis
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-4
-2
0
2
4
Range Error [km]
-0.6
-0.3
0
0.3
0.6
Range Error [%]
Range Estimation Error
Real
Relative
REQ
Figure 3: Navigation results for Scenario 2.
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in X-Axis
Error
3
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in Y-Axis
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-15
-7.5
0
7.5
15
Position Error [km]
Position Estimation Error in Z-Axis
0 2 4 6 8 10 12 14 16 18 20
Time [h]
-3
-1.5
0
1.5
3
Range Error [km]
-2.5
-1.25
0
1.25
2.5
Range Error [%]
Range Estimation Error
Real
Relative
REQ
Figure 4: Navigation results for Scenario 3.
of the rendezvous, but also a reduction on delta-V spent to correct for it. This fact, allowed with the
significantly improved knowledge of the relative state can lead to a less strict mission analysis plan,
allowing the rendezvous to start closer to the target, again reducing delta-V expenditure as seen in
table 5.
The delta-V budget allocated for the GUIBEAR approach is, naturally, constrained by the mission
analysis and the end-to-end results of NRO-GNC [5]. It is clear that the considerations of that activity
did not take into account the possibility of performing an on-board optimized trajectory to approach
the hold point for preparation of the rendezvous, nor the existence of a long range navigation function
that could improve on the costly and infrequent ground estimates. For that reason, the insertion
manoeuvre was planned with a large dispersion in mind, stemming from the TCM. With GUIBEAR,
it is not only possible to have a bearings-only navigation that provides timely estimates of the relative
position and velocity of the target, but also to plan the trajectories taking into account these. This
comes, naturally, at a cost of delta-V but not of extra sensors. The analysis present in this paper shows
that a very small delta-V budget dedicated for observability-enhancing manoeuvres does help the
navigation performance and the observability, but its influence is limited. However, larger manoeuvres
ESA GNC 2021 – P. Serra 12
Table 2: Statistical results of the Range Estimation Error before the NIM for the main cases.
Range Estimation Scenario 1-OI Scenario 2 Scenario 3
Metric: 5% 5% 5%
Mean: 9.7462% 1.2624% 1.1874%
Std. Deviation: 17.708% 1.2424% 1.2362%
Success Rate: 65.5% 97.9021% 99%
Table 3: Statistical results of the position dispersion at Hold Point for the main cases
Position disp. (km) Scenario 1-OI Scenario 2 Scenario 3
Metric (worst axis): 17 (Y axis) 17 (Z axis) 17 (Z axis)
Mean: 10.7046 7.7466 4.4589
Std. Deviation: 11.4983 4.7605 2.2293
Success Rate: 81.5% 98.6014% 100%
have a significant effect.
The test results indicate that the GUIBEAR G&N can perform with good results the approach and
insertion to the near-rectilinear orbit of the LOP-G, indeed improving on the nominal scenario both in
state dispersion and knowledge with a small delta-V cost. In particular: (i) The on-board G&N is ca-
pable of performing the insertion manoeuvre in the nominal terms, i.e., by considering fuel-optimality
and replacing the ground navigation and a priori computed insertion manoeuvre. (ii) Generated tra-
jectories are inherently safe through optimization constraints. (iii) Small manoeuvres do improve the
navigation performance to a limited extent. (iv) Letting the on-board optimization take care of the
insertion, with the increased delta-V budget associated, is a good option in terms of better use of the
manoeuvres for observability. (v) Having an off-pointing of 5 degrees reduces the detection range to
a little above 1000 km. This is within the requirements but can be significantly improved with a small
pointing error. However, that has thermal and power implications.
5 CONCLUSIONS AND FUTURE WORK
The GUIBEAR project has demonstrated that bearings-only navigation for far-range rendezvous is
feasible from the point of view of the G&N. The current G&N design improves on the previously
analysed approach to the end of the orbit transfer phase of the LAE when rendezvousing the LOP-
G at the small added cost of a delta-V expenditure of few meters per second. In particular: (i) the
post-insertion dispersion can be about 4x below (3σ) what is achieved with the pre-planned insertion
nominally considered in NRO-GNC, and up to 6x below in terms of velocity dispersion; (ii) the
navigation estimates have an average accuracy of one order of magnitude better than that achievable
without GUIBEAR; and (iii) the propellant consumption required for the good results above is on
the order of 5 m/s above that of the pre-planned insertion, and, with as low as 1 m/s in average it is
possible to achieve performance improvements, by following Scenario 2. The MIL tests have shown
that the GNC software is capable of fulfilling all functions and with adequate performance for the
current phase of the activity. The GNC is compliant with the requirements defined at the start of the
activity, an aspect that is aided by the optimization-based design of the guidance function since the
requirements can be translated into optimization constraints, thus resulting in correct trajectories by
design. The sensor models that were used in the real world were relatively simple, mostly consisting
of behavioural models of sensors and actuators. The interfaces of these models do not correspond to
the interfaces of any real, existing hardware. To further increase the TRL of the GNC, representative
ESA GNC 2021 – P. Serra 13
Table 4: Statistical results of the velocity dispersion at Hold Point for the main cases
Velocity disp. (m/s) Scenario 1-OI Scenario 2 Scenario 3
Worst axis: Y axis Y axis Y axis
Mean: 1.1238 0.19221 0.29872
Std. Deviation: 1.7218 0.16201 0.20381
Table 5: Statistical results of the delta-V consumption for the main cases
Delta-V (m/s) Scenario 1-PI Scenario 1-OI Scenario 2 Scenario 3
Planned Extra delta-V compared to the planned insertion
Mean: 20.4144 -1.4523 1.0893 5.4418
Std. Deviation: 6.7249 1.9717 1.0781 2.3319
interfaces with the sensors and actuators, as well as representative sensor pre-processing functions
will need to be added to the GNC. This implies that more representative models of the sensors and
actuators will need to be developed and included in the real-world model.
As mentioned throughout the paper, the GUIBEAR activity has been focused on the development of
G&N functions for the HERACLES mission scenario with the LAE vehicle. Within the duration of
the activity, the HERACLES mission definition has changed due to the programmatic change of the
Lunar Gateway. The mission has evolved in 2 different activities: the ESA-EL3 (European Logistic
Lunar Lander) activity and the ESA CLTV (Cis-Lunar Transfer Vehicle). At the current time, a
sample-return is not being considered for EL3, and therefore the HERACLES assumptions are not
currently valid. However, CLTV (Cis-Lunar Transfer Vehicle) will require to dock with the Gateway,
and could therefore benefit from the design and the results obtained within the GUIBEAR activity. A
dedicated step to adapt the design to the CLTV mission scenario and vehicle has been consequently
envisaged as natural continuation of the GUIBEAR activity.
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