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Robust Multi-objective Optimisation of a Descent Guidance Strategy for a TSTO Spaceplane

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This paper presents a novel method for multi-objective op-timisation under uncertainty developed to study a range of mission trade-offs, and the impact of uncertainties on the evaluation of launch system mission designs. A memetic multi-objective optimisation algorithm, MODHOC, which combines the Direct Finite Elements transcription method with Multi Agent Collaborative Search, is extended to account for model uncertainties. An Unscented Transformation is used to capture the first two statistical moments of the quantities of interest. A quantification model of the uncertainty was developed for the atmospheric model parameters. An optimisation under uncertainty was run for the design of descent trajectories for the Orbital-500R, a commercial semi-reusable, two-stage launch system under development by Orbital Access Ltd.
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ROBUST MULTI-OBJECTIVE OPTIMISATION OF A DESCENT
GUIDANCE STRATEGY FOR A TSTO SPACEPLANE
Lorenzo A. Ricciardia, Christie Alisa Maddocka, Massimiliano Vasilea, Tristan Stindtb, Jim Merrifieldb
Marco Fossatia, Michael Westc, Konstantinos Kontisd, Bernard Farkine, Stuart McIntyree
aUniversity of Strathclyde, Glasgow, UK
bFluid Gravity Engineering, Emsworth, UK
cBAE Systems, Prestwick, UK
dUniversity of Glasgow, Glasgow, UK
eOrbital Access, Prestwick, UK
ABSTRACT
This paper presents a novel method for multi-objective op-
timisation under uncertainty developed to study a range of
mission trade-offs, and the impact of uncertainties on the
evaluation of launch system mission designs. A memetic
multi-objective optimisation algorithm, MODHOC, which
combines the Direct Finite Elements transcription method
with Multi Agent Collaborative Search, is extended to ac-
count for model uncertainties. An Unscented Transformation
is used to capture the first two statistical moments of the
quantities of interest. A quantification model of the uncer-
tainty was developed for the atmospheric model parameters.
An optimisation under uncertainty was run for the design
of descent trajectories for the Orbital-500R, a commercial
semi-reusable, two-stage launch system under development
by Orbital Access Ltd.
Index TermsSpaceplane, Optimisation under Uncer-
tainty,
1. INTRODUCTION
This paper presents a novel method for multi-objective op-
timisation under uncertainty, developed to study a range of
mission trade-offs and the impact of uncertainties on system
models for launch systems. This is applied to the analysis and
design of descent trajectories for the Orbital-500R, a commer-
cial semi-reusable, two-stage launch system under develop-
ment by Orbital Access.
The set of Pareto-optimal solutions show the trade-off be-
tween minimising the induced acceleration limits and max-
imising the robustness of the solutions by minimising the sen-
sitivity to uncertainties.
Uncertainty quantification (UQ), the science of quantify-
ing the uncertainty in the desired performance of a system,
can be a key step in analysing the robustness of a control solu-
tion and of the whole guidance, navigation and control (GNC)
chain. Common approaches to UQ use extensive Monte Carlo
simulations to account for errors, unmodelled components
and disturbances. At system level, UQ analysis can trans-
late into the assessment of the whole system reliability or the
reliability of one or more components. An uncertainty quan-
tification analysis is, therefore, a fundamental step towards
de-risking any technological solution as it provides a quan-
tification of the variation in performance and probability of
recoverable or unrecoverable system failures, given existing
information.
Within the current preliminary design phase of the Orbital-
500R, one of the areas of study is the uncertainty related to
the aerodynamic modelling of the re-usable spaceplane.
The goal is, therefore, to design a robust guidance tra-
jectory considering the uncertainty related to the atmospheric
model which in turn affects the aero-, aerothermal and flight
dynamics. Furthermore, given the early stage of the vehicle
design and the wide scope of use required by the commer-
cial side, a number of re-entry scenarios are possible, each of
which is affected differently by the same sets of uncertainties.
This paper presents an extension of the capabilities of
MODHOC to account for uncertainties. The extension is
based on an Unscented Transformation to capture the first
two statistical moments of the quantities of interest. The
result is an unscented multi-objective optimal control ap-
proach that can efficiently handle the level of uncertainty in
model parameters. Using multiple objective functions, trade-
off studies on the Orbital-500R design will be presented for
the chosen performance and vehicle characteristics of the
re-entry trajectory and the robustness of those performances
(and the associated guidance laws) against the uncertainty of
the atmospheric model.
2. OPTIMAL CONTROL
Multi-Objective Direct Hybrid Optimal Control (MODHOC)
[1, 2] is based on Direct Finite Elements Transcription
(DFET), a transcription method for nonlinear multi-phase
optimal control problems [3], with MACS, a population
based memetic multi-objective optimisation algorithm [4] and
mathematical programming solvers. The DFET transcription
method allows to treat general optimal control problems,
thus it can tackle problems with any kind of dynamic model.
The memetic multi-objective optimisation algorithm MACS
allows for a global exploration of the search space and is
able to treat problems with an arbitrary number of objectives.
The mathematical programming solvers are used to refine the
solutions obtained and guarantee the local optimality of the
solutions found while also satisfying tight constraints.
The software has been successfully used for the trajec-
tory and design optimisation of vertical and horizontal launch
systems [5, 6], deployment of constellations [7], interplane-
tary exploration missions [8] and the design of multi-debris
removal missions.
2.1. Unscented optimal control
In order to perform robust optimisation of the re-entry trajec-
tory of the Orbital 500R, an Unscented Transformation [9]
was included in the formulation of the optimal control prob-
lem.
Unscented Transformations capture the first statistical
moments, mean and covariance, of the distributions of the
states of a system subject to uncertainty and undergoing ar-
bitrary non-linear transformations by propagating a small
number of sigma points. If the system depends on Nuq un-
certain variables, whose mean and covariances are known,
the unscented transformation requires the propagation of
2Nuq + 1 samples. The first sigma point takes the mean value
for all the uncertain variables, while the others assume the
mean plus (or minus) the square root of the matrix of the
covariances of the uncertain variables. All the sigma points
are propagated simultaneously with the mean and covariance
of the final states computed as a weighted combination of the
final states of each sigma point.
The approach employed in this paper follows the one de-
scribed by [10]. Let the dynamics of the systems be given
by
˙x =F(x,u,buq, t)(1)
where xare the states of the system, uare the controls, t
is time, and buq are additional static parameters which are
here assumed to be affected by uncertainty. The dynamics of
each sigma point χi, i = 1 . . . , 2Nuq + 1 is given by ˙χi=
F(χi,u,bi, t).
This means that each sigma point has a different value for
the static variables, its dynamics evolve independently of the
other sigma points, but all sigma points are controlled by the
same control law. The idea is thus to find a single control law
that, when applied to all sigma points, allows the ensemble
to reach a desired final condition and to be optimal in some
sense. The particular values for each static variable biis de-
cided by the application of the Unscented Transformation.
A known problem of the Unscented Transformation is that
it can generate covariance matrices that are not semidefinite
positive. To avoid this problem, a more recent and stable ver-
sion of the Uncertain Transformation was employed in this
paper, called the Square Root Unscented Transformation [11].
Algorithmically it is very similar to the standard version, but
it differs in the way the samples are generated and has the ad-
vantage that the resulting covariance matrix is guaranteed to
be semidefinite positive (up to machine precision). In prac-
tice, the sigma points are thus computed from the Cholesky
factorisation of the covariance matrix, and slightly different
algebraic manipulations are performed to obtain the covari-
ance matrix of the transformed states.
Mathematically, thus, the problem can be described as fol-
lows: let Xbe a state vector of length NσNx,
X:= χ0,χ1,· · · ,χNσT(2)
where Nσis the number of sigma points and Nxis the number
of states of the original system. The dynamics of this system
is then given by
˙
X:=
f(χ0,u,b0, t)
f(χ1,u,b1, t)
.
.
.
f(χNσ,u,bNσ, t)
:= F(X,u,B, t)(3)
The multi-objective unscented optimal control problem
can thus be formulated as
min
uUJ(X,u,B, t)
s.t.
˙
X=F(X,u,B, t)
g(X,u,B, t, 0
ψ(X(t0),X(tf),u(t0),u(tf),B, t0, tf)0
t[t0, tf]
(4)
In this work, since there is no uncertainty about the ini-
tial state of the system, the initial conditions are expressed as
usual. The final conditions, instead, are expressed as differ-
ence between the mean and the target value.
ψ(X(tf)) := µχx(tf) = 0 (5)
where µχis the mean of the final states of the sigma points,
and x(tf)is the target value. The first objective
J1:= Ztf
t0
E(a2)dt (6)
is the integral of the expected value of the square of the total
accelerations a, which should guarantee that the accelerations
will be minimised on average along all the trajectory.
The second objective is meant to reduce the uncertainty of
the final state. To this end, the sum of the square of all the en-
tries of the covariance matrix of the final states is minimised.
J2=X
i,j
(Covi,j (X(tf))2(7)
where Covi,j is computed using the standard algebraic ma-
nipulations employed for the Square Root Unscented trans-
formation, with the additional consideration that no update of
the Cholesky factorisation is needed since no measurement is
here performed and thus no error is present. This formula-
tion has the advantage that the quantity to compute is smooth
and differentiable, it involves all components of the covari-
ance matrix, and does not require iterative procedures like de-
composition in eigenvalues to compute the principal axes of
the ellipsoid of the uncertainty. In order to give each element
of the covariance matrix the same weight even if the quanti-
ties of interest have different scales, the state variables were
scaled by the same factors internally employed by MODHOC
to ensure that all variables assume values between 0 and 1.
3. VEHICLE SYSTEM MODELS
The Orbital-500R system is composed of a first stage reusable
spaceplane, capable of rocket-powered ascent and an unpow-
ered, glided descent, and an expendable, rocket-based upper
stage. The dry mass of the spaceplane is 20 tonnes.
The flight dynamics are modelled as a variable-mass point
with three degrees of freedom in the Earth-centered Earth-
fixed reference frame, subject to gravitational, aerodynamic
lift and drag forces. The state vector contains the transla-
tional position and velocity, and time-dependant vehicle mass,
x= [h, λ, θ, v, γ , χ, m]where his the altitude, (λ, θ)are
the geodetic latitude and longitude, vis the magnitude of the
relative velocity vector directed by the flight path angle γand
the flight heading angle χ[12]. The vehicle is controlled
through the angle of attack α, and the bank angle µof the
vehicle. The Earth is modelled as a perfect sphere of radius
RE.
The aerodynamic coefficients were modelled using an ar-
tificial neural network with given an aerodynamic database
for the training data coming from CFD simulations [13].
From previous work on the optimisation of the full ascent
and reentry trajectory, the vehicle is assumed to begin its re-
entry (at time t= 0) with the following initial conditions:
Altitude h(0) = 90 km
Latitude λ(0) = 60N
Longitude θ(0) = 12E
Velocity v(0) = 3 km/s
Flight path angle γ(0) = 5
Heading angle χ(0) = 20
The only final condition imposed was the expected value of
the altitude, with a value of 10 km. In addition, the trajec-
tories of all sigma points were required to have a flight path
angle greater or equal to 20.
4. UNCERTAINTY QUANTIFICATION
Analysis to date on the design of the Orbital 500-R has em-
ployed the US-76 Standard atmospheric model [14]. The US-
76 is a global static standard model, giving the atmospheric
pressure p, temperature Tand density ρas function of alti-
tude up to 1000 km.
In order to assess the robustness of the design against un-
certainties in the atmospheric model, it was necessary to de-
velop a model for its uncertainty. To this end the NRLMSISE-
00 [15] atmospheric model was employed. It is more recent
and sophisticated than the US76 model, and takes into ac-
count several factors like daily and seasonal effects, solar and
magnetic activity and geographical dependencies. This model
thus gives temperature and density as a function of 7 other in-
put parameters in addition to altitude.
A statistical analysis of the difference of the models was
performed treating those 7 additional input parameters as un-
certain. 100000 samples were taken from the low discrepancy
Halton sequence, and the corresponding values for Tand ρ
were computed using the NRLMSISE-00 model for all alti-
tudes in the range between 0 and 100 km.
To account for the possible differences between the mod-
els, the relative average difference between the values of the
two model was computed for all the thermodynamic quanti-
ties. These relative errors were then treated as random fluctu-
ations, for which averages and covariances were computed as
a function of altitude. These relative errors and covariances
deviations are plotted in Figures 1, 2 and 3.
Fig. 1. Relative error between models, density
Fig. 2. Relative error between models, pressure
Fig. 3. Relative error between models, temperature
It can be observed that for the temperature the mean rel-
ative errors are very low, with a 1σrelative error around 5%
for altitudes below 80 km. The mean relative errors for pres-
sure and density are also low for relatively low altitudes, but
they tend to drift systematically at higher altitudes, and the 1σ
bands are much wider. This happens above 40km where pres-
sure and density have a very low absolute value, so a large
relative error still means a low absolute error. It is thus ex-
pected that the impact of these relatively high differences in
density between the two models will not affect significantly
the first part of the trajectory.
Since the approach employed is that of the Square Root
Unscented Transformation, different models were generated:
one employing the mean relative error, and the others adding
to the mean the Cholesky factorisation of the covariance ma-
trix of the uncertain quantities at each altitude. Figure 4 shows
the five generated temperature profiles, and their comparison
with respect to the US76 model. A similar approach was
followed for the density. As it can be seen, the model re-
ferred to as Sigma point 0 is quite close to the US76 model.
Sigma ponts 1 and 2 add the standard deviation to this mean,
while Sigma ponts 3 and 4 also include the correlation be-
tween variations in temperature and variations in density. The
correlation between temperature and density changes sign re-
peatedly as the altitude changes, thus the profiles assume val-
ues lower than one standard deviation only to cross the mean
and assume values higher than one standard deviation else-
where. Among the 100000 samples generated by the Halton
sequence on the NRLMSISE-00 model, several profiles do in-
deed have this kind of shape, which is significantly different
to the US76 model. The temperature affects the computation
of the Mach number, on which the aerodynamic coefficients
depend. The dependence of the aerodynamic coefficients on
the Mach number is stronger around Mach 1 and weaker for
high Mach numbers, thus it is not easy to foresee the effect of
these variations. Including these sigma points in the design of
a robust guidance law is thus expected to make it more robust.
Fig. 4. Temperature profiles for the models of each sigma
point, and comparison with the US76 model
5. RESULTS
The equations were discretised employing the DFET tran-
scription included in MODHOC, using 3 elements of order
7 for all the states and the controls. Since all the sigma points
are propagated simultaneously, this means that there are 30
states and 2 controls, making it a rather large problem. In or-
der to have a good approximation of the Pareto Front, MOD-
HOC was run for a total of 30000 function evaluations, keep-
ing 10 solutions in the archive. The computed Pareto front is
shown in Fig. 5, which confirms a trade-off between the ex-
pected total acceleration load and the total uncertainty of the
final state.
Fig. 5. Pareto Front
Figure 6 show the time histories of the altitude for all the
solutions in the archive, 7 shows the flight path angle and 8
the velocity, with the dashed and dotted lines indicating the
1σuncertainty.
Fig. 6. Altitude vs time
In all these plots it is possible to see that solutions in
green, light blue and blue (Solutions 1 to 4) have lower uncer-
tainty for the final state. This can also be seen from Figures
9, 10 and 11, which show the standard deviation of altitude,
Fig. 7. Flight path angle vs time
Fig. 8. Velocity vs time
flight path and velocity as a function of time. As previously
stated, even if the uncertainty on density is relatively high at
high altitudes, its effect is quite limited for the first part of
the trajectory. However, it becomes more important as the
altitudes get to approximately 40 km.
This as the Pareto front was showing, solutions associated
with lower uncertainty in final states have higher acceleration
loads, as shown in 12.
Figure 13 shows the time history for the control α. In all
cases, the angle of attack starts with the maximum possible
value of 45 deg, and then progressively decreases to a more
moderate value around 510 deg. Solutions with lower ac-
celerations stay in this regime for a while, and finally con-
clude with a value around 0 deg. Solutions with lower fi-
nal uncertainty instead have a progressive decrease in αuntil
Fig. 9. Altitude vs time, standard deviations
small negative values, then a sharp increase to values around
10 15 deg, and finally stabilise around 0 deg like the other
trajectories.
Finally, Figures 14, 15 and 16 show the time history of
altitude, flight path angle and velocity of all sigma points for
Solution 1 and 10. As it is evident, the green lines have a
much lower scattering at the final time than the black lines,
indicating that Solution 1 (green) is subject to less uncertainty
than solution 10 (black). This figures also give an idea of the
complexity of the problem tackled by this approach, where
the same control law is applied to multiple independent sigma
points (lines with the same colour) and is able to steer the sys-
tem to a given expected final state while also reducing the un-
certainty associated to the final state, or reducing the expected
acceleration load.
6. CONCLUSIONS
This paper presented and extension of MODHOC to perform
robust multi-objective optimisation of the reentry trajectories
of the Orbital 500R. Employing the Unscented Transforma-
tion, a different atmospheric model was developed for each
of the sigma points. All the sigma points share the same con-
trol law, thus making the trajectory robust against model un-
certainty. Albeit only the first two statistical moments of the
uncertain values were considered for this work, it is possible
to account for higher order moments, making the resulting
trajectory even more robust. To this end, however, a larger
number of sigma points will be required, and the resulting op-
timal control problem becomes progressively larger, requiring
the use of large scale optimisation code.
Fig. 10. Flight path angle vs time, standard deviations
7. ACKNOWLEDGEMENTS
This work has been partially funded by the UK Space Agency
and European Space Agency (ESA) General Support Tech-
nology Programme (GSTP).
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Fig. 13. Angle of attack vs time
Fig. 14. Time history of the altitude for all sigma points of
Solutions 1 and 10
Fig. 15. Time history of the flight path angle for all sigma
points of Solutions 1 and 10
Fig. 16. Time history of the velocity for all sigma points of
Solutions 1 and 10
... It should be easier to approximate a given distribution than it is to approximate an arbitrary nonlinear, in particular, to determine an upper bound on the actual squared error of the final state and costate of function. We note that in recent optimal state estimation theory and robust trajectory optimization of aerospace nonlinear dynamics, the derivative-free covariance propagation and estimation, such as unscented transformation are analyzed and usedRicciardi et al. (2019);Ricciardi and Vasile (2020). With sigma points of unscented transformation, and corresponding weights, covariance of the initial guess can be obtained using final states and costate. ...
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Modeling and Simulation of Aerospace Vehicle Dynamics, Third Edition unifies all aspects of flight dynamics for the efficient development of aerospace vehicle simulations. It provides the reader with a complete set of tools to build, program, and execute simulations. Unlike other books, it uses tensors for modeling flight dynamics in a form invariant under coordinate transformations. For implementation, the tensors are converted into matrices, resulting in compact computer code. In this third edition, the emphasis shifts from FORTRAN to C++, to give fealty to the upsurge of object oriented programming in engineering simulations. A new appendix spotlights the C++ architecture of the CADAC++ simulation framework. To aid in the new focus, the CADAC4 software package provides—in addition to the FORTRAN programs—eight C++ simulations, which range from UAVs, aircraft, missiles, and boosters, to hypersonic aircraft with transfer vehicles for satellite rendezvous. CADAC4, including CADAC Studio for plotting, may be downloaded for free by visiting the Modeling and Simulation of Aerospace Vehicle Dynamics, Third Edition book page at arc.aiaa.org and entering the Supporting Materials password. You need only a Windows based PC (32 or 64 bit) and a Microsoft C++ compiler. This book also serves as an anchor for three revised and updated self-study courses, Building Aerospace Simulations in C++, Third Edition; Fundamentals of Six Degrees of Freedom Aerospace Simulation and Analysis in C++, Second Edition; and Advanced Six Degrees of Freedom Aerospace Simulation and Analysis in C++, Second Edition, which are based on M&S courses in C++ previously taught at the University of Florida (self-study courses are available for purchase at arc.aiaa.org). Amply illustrated, this text may be used for advanced undergraduate and graduate instruction or for self-study. Seventy eight problems and nine projects amplify the topics and develop the material further. Qualified instructors may obtain a complimentary solutions manual from AIAA.
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