INVERSE PROBLEM SOLUTION FOR DETERMINING SPACECRAFT ORIENTATION FROM PRESSURE MEASUREMENTS
01.05.2014 20:41
INVERSE PROBLEM SOLUTION FOR DETERMINING SPACECRAFT
ORIENTATION FROM PRESSURE MEASUREMENTS
Jochem Häuser(1), Wuye Dai(1,2), Georg Koppenwallner(3), Jean Muylaert(4)
(1)Dept. of High Performance Computing and Communications, Center of Logistics and Expert Systems (CLE)
GmbH, and University of Applied Sciences, Braunschweig-Wolfenbüttel, Germany Karl-Scharfenberg-Str. 55-57,
38229 Salzgitter, Germany, Email: J.Haeuser@cle.de
(2) Aerodynamisches Institut, RWTH Aachen, Wuellnerstr. zw. 5 u. 7,52062 Aachen, Germany
Email: w.dai@cle.de
(3)HTP Hypersonic Technology Göttingen, Germany, Max-Planck-Str. 19, 37191 Katlenburg-Lindau, Germany
Email:G.Koppenwallner@htg-hst.de
(4)Aerothermodynamics Section, ESA-ESTEC, Noordwijk, The Netherlands, Email: jmuylaer@estec.esa.nl
ABSTRACT
The goal of this paper is to develop a method of
predicting the orientation of a blunt-nosed spacecraft(e.g.
Kheops Expert) with regard to pitch and sideslip by
measuring pressure data at specified locations in the nose
region. The strategy devised here is to use analytic
sensor functions (ASF) for the prediction of angle of
attack (AoA) and yaw angle according to the local
pressure data on the vehicle surface. First, the derivation
of the sensor functions is presented. In the second step,
the range of validity of these formulas is determined with
respect to flight velocity (Mach Number), AoA, and yaw
angle by employing extensive computer simulation.
Third, the corrections of these empirical formulas is
devised for the given vehicle so that the required
accuracy (resolution better than 0.5 degrees) is
guaranteed within the range of the flight envelope.
Fourth, the impact of configuration changes on the
accuracy of these functions is also evaluated. Results
show that this methodology is effective and accurate in
the hypersonic regime, provided specific corrections
devised from numerical simulation are applied to modify
the analytic sensor functions.
1. INTRODUCTION
The ESA-ESTEC proposed air data system is supposed
to provide information of the condition on the ambient
air and on the flight state of a space vehicle. Therefore,
one wishes to relate physical quantities measured at the
vehicle surface to atmospheric free stream values as well
as to the vehicle's velocity and orientation. Atmospheric
free stream values and vehicle flow field determine
surface flow values, which can be directly computed
using the methods of computational fluid dynamics
(CFD). However, for the air data system one has to solve
the inverse problem, namely to deduce from surface
measurements the state of the atmosphere and the vehicle
orientation.
In order to do so, two empirical equations were first
developed [Koppenwallner, 2003]. These equations are
called analytic sensor functions (ASF) that are based on
empirical formulas for which pressure data are obtained
from five sensors installed in the nose of the vehicle.
Utilizing ASF, both angle of attack (AoA, a) and yaw
angle (b) are determined from this pressure distribution.
The next task is to determine the range of validity of
these formulas with respect to Mach number as well as
AoA and yaw angle. Furthermore, it is expected that
numerical computations would allow to provide
correction rules for the flight range of interest to
improve the accuracy of these formulas, meeting the
accuracy requirements of 0.5 degrees for the two angles.
2. ANALYTIC SENSOR FUNCTIONS
ASF determine AoA and yaw angle from vehicle surface
pressures, utilizing five pressures at different locations.
The form of ASF strongly depends on the five pressure
positions. Therefore, it is important to specifically select
these positions in order to simplify the form of these
functions. Although any set of locations is acceptable
for ASF, simple formulas are only obtained for special
pressure locations.
For most space vehicles it is acceptable to assume the
vehicle nose is axisymmetric. Then the position on the
surface can be defined by two angles q, and f as shown
in Fig. 1. Pressure locations are selected according to
Fig.1. These locations, with pressures denoted as p0, p1,
p2, p3, and p4, respectively, are at the stagnation point
(a = 0o, b = 0o) while the other four locations are given
by the following angles: q = 45o and circumference
angles f of 0o, 90o, 180o, and 270o.
Newtonian pressure distribution on axisymmetric bodies
was used to derive ASF. The procedure of the derivation
is not complex and given in [Koppenwallner, 2003].
Here only the results are listed as Eqs. 1 And 2.
=tan−1 1
4 sin2/4
q3−q1
q0
(1)
=tan−1
cos
4 sin2/4
q4−q2
q0
(2)
where q is the local pressure difference (with respect to
freestream pressure). Hence these measured values can
be directly inserted into Eqs.1 and 2.
a) Lateral View (c) Projection against x axis
Figure 1: Sketch of sensor locations.
3. METHODOLOGY
The ASF (Eqs. 1 and 2) derived above rely on the
Newtonian flow assumption for axisymmetric bodies.
Since there are stringent requirements on the accuracy of
the orientation of the vehicle along its trajectory, namely
angles a and b need to be predicted with an error of less
than 0.5 degrees, the simple form of Eqs. 1 and 2 needs
to be corrected to account for geometrical effects as well
as flow viscosity and non-equilibrium phenomena. To
this end, the proper flow database has to generated by
computer simulation. These data then are used to obtain
corrected formulas from Eqs. 1 and 2.
The KHEOPS model (Expert program) proposed by J.
Muylaert, ESA and computed by [Walpot, 2002] is the
model selected in the present study(referring Fig. 2).
This model comprises a body of revolution and an
ellipsoid-clothoid-cone, obtained from a twodimensional
longitudinal profile. Its nose, which is of an
ellipsoidal shape, has second order smoothness when
combined with the cone, to avoid any geometry induced
pressure jumps. The grid for KHEOPS, shown in Fig. 3,
was generated by GridPro using box technique [Häuser,
2004].
Two solvers were used in the course of the simulations,
namely the CFD++ solver from Metacomp, U.S.A. and
the ESA Lore code. The CFD++ code is based on a
unified grid, unified-physics, and unified-computing
framework. CFD++ uses a multi-dimensional secondorder
total variation diminishing scheme to avoid
spurious numerical oscillations in the computed flow
field, along with an approximate Riemann (HLLC)
solver to guarantee correct signal propagation of
convective flow terms. The multi-grid technique is used
to accelerate convergence along with a second order
accurate point implicit scheme.
The ESA developed Lore code was employed to
validate the numerical results obtained from CFD++.
The Lore code is a multi-block structured code which
covers the subsonic up to hypersonic flow regime. This
flow solver is based on a finite volume formulation in
which fluxes are computed with a modified AUSM
scheme. It incorporates several multi-temperature, finite
rate chemistry models. Several algebraic and 2-equation
turbulent models are also available. The system of
equations is solved fully implicit using a line Gauss-
Seidel relaxation method. The Lore code provides an
additional feature in form of a boundary condition for a
fully catalytic wall, not available in CFD++.
A large variety of examples were used to validate the
CFD++ code, with validation runs in two- and threedimensions,
using both perfect and real gas, steady flow
Figure 2: KHEOPS configuration with surface temperature
solution
Figure 3 Mesh for the KHEOPS revision 4.2 generated by
GridPro using the BOX technique.
and transient flow, inviscid and viscous flow, as well as
non-reactive flow and chemically reactive flow[Häuser,
2004] [Chakravarthy, 2002]. The Lore code was also
widely used and tested in ESA [Muylaert, 2001]. In the
present study, the two solvers were used to solve the
same cases and their results were compared. The results
obtained from the CFD++ and Lore codes are quite close
despite their completely different numerical solution
techniques.
The following strategy to study the range of validity of
ASF with CFD was used: First, a study of the impact of
flow physics (see Sec. 4.1) on the orientation angles at
two specified freestream conditions, namely M∞ = 12.92
and 25.0 was carried out. As a result, it was found that
Euler computations were sufficient to achieve the
required accuracy. Second, numerous computations were
performed at Mach numbers 4.98 and 12.92,
investigating effects of angle of attack and yaw angle.
Moreover, the influence of freestream Mach number on
the analytic sensor functions was studied by varying the
Mach number from 1.6 to 25 for angle of attack 10o and
yaw angle 5o. Finally, simulating flow past a sphere , the
impact of geometry is discussed.
4 RESULTS AND DISCUSSION
4.1. Effects of Flow Models
The flow models considered include
• perfect gas Euler flow (EU),
• perfect gas viscous flow (PG, NS),
• real gas viscous flow with adiabatic wall boundary
condition (RG, NS),
• real gas viscous flow with fixed wall temperature
(Tw=1,000 K) (RG, NS, TW),
• real gas with chemically reactive, viscous flow with
fixed wall temperature (Tw=1,000K) (RG, NS,TW,
NE), and
• real gas with chemically reactive, viscous flow with
fixed wall temperature (Tw=1,000K) and full catalytic
wall (RG, NS, TW, NE, Fullcat).
Most viscous flows in CFD++ are modeled by employing
the two-equation k-e turbulence while in the Lore code
the algebraic Baldwin-Lomax model was used. High
temperature effects were modeled by the twotemperature
chemical non-equilibrium assumption. The
reaction model was set up to the standard 5 species and
34 reactions model of Dunn and Kang [Gnoffo,1989].
Effects of different flow models upon pressure
coefficient along the wall at Ma =12.92 are shown in Fig.
4. For Ma =15.78 similar results are obtained.
Axisymmetric flow simulations were performed, justified
by rotational symmetry of the nose and body of
KHEOPS, except for the rear part containing the flaps.
Neither a difference in the flow model nor a change in
the wall boundary condition leads to a significant
change in the pressure coefficient.
Heat flux profiles for Ma=25 and a = 10o, b = 5o along
different lines along the surface are plotted in Fig. 5.
These are results from three dimensional simulations. It
is remarkable that both CFD++ and Lore codes provide
very close and physically reasonable results.
Figure 4: Effect of different flow models upon pressure
coefficient along the wall at Ma =12.92.(axis-symmetric
simulations, a = 0, b = 0 ). The results labeled Lore were
computed by the Lore code, the others by the CFD++ solver.
The comparisons of absolute errors of the predicted
AoA and yaw angle between different flow models and
solvers for Ma=12.92 are presented in Figs. 6. Again
results from both codes are almost the same. The error
of predicted AoA and yaw angle resulting from different
flow models, wall boundary conditions, and turbulence
models is less than 0.5o , but there is a systematic error.
It is concluded that an inviscid flowfield simulation is
sufficient to determining the range of validity of ASF.
4.2. Effects of Angle of Attack and Yaw Angle
The predicted AoA using ASF versus actual AoA for
different angles of attack and yaw angles are shown in
Fig.7. It is observed that the predicted AoA deviates
from the actual AoA, but it is interesting to note that the
deviation is linear and independent on yaw angle. The
deviation between predicted AoA and actual AoA is
caused mainly by three factors:
1) The original sensor functions, given in Eqs. 1 and 2,
hold for hypersonic flow only, because of the
Newtonian flow assumption that leads to a
systematic error in the pressure distribution when
compared to the CFD solutions.
2) The geometry studied is that of a three-dimensional
vehicle, so axisymmetric flow assumption for ASF
causes some error.
3) The CFD method itself also produces numerical
errors. Since errors are present, ASF need to be
modified accordingly.
Predicted yaw angles are displayed in Fig.8. Like
predicted AoAs, the calculated yaw angles deviate from
the actual yaw angles. The deviation is approximately
linear, but is a function of both AoA and yaw angle. This
should be expected from Eq. 2, since the predicted yaw
angle is determined by both the pressure relation
q4−q2
q0
and the predicted AoA.
The requirement is to provide an accuracy in angle
Figure 7: Predicted AoA from surface pressure distribution
using ASF versus actual AoA.
resolution better than 0.5 degrees for the complete
trajectory. To this end, a simple correction was found,
since all errors are linear or approximately linear. The
modified results are shown in Figs. 9 and 10 that were
obtained using the following modified formulas:
(a) windward side
(b) leeward side
Figure 5: Heat flux profiles for Ma=25 (3D Simulation, a =
10o and b = 5o), shown for the (a) windward and (b) side
leeward side in the symmetry plane y =0.
(a) absolute errors of AoA
(b) Absolute errors of yaw angle
Figure 6: Comparisons of absolute errors of (a) AoA (b) yaw
angle obtained from CFD++ and Lore by employing different
flow models for M∞=12.92, a =10o, b=5o.
modified=1.055 cal (3)
modified=cal 1.071.04cal
2 (4)
Where the unit of angle is radian, and subscript cal
indicates the values got from Eqs.1 and 2.
One can see from Figs. 9 and 10 that as long as
4.98Ma25.0, 0< a < 30o, and 0< b < 10o the
errors in the orientation angles are less than 0.5o.. It
should be noticed that the range of validity listed is the
range for which computations have been performed, and
thus is confirmed to be effective for the sensor functions.
In practice, the range of validity could be extended even
further.
Figure 8: Predicted yaw angle from surface pressure
distribution using ASF versus actual Yaw angle.
4.3. Effects of Mach Number
Figure 11 presents the predicted AoA and yaw angle,
obtained from the modified formulas Eqs. (3) and (4),
versus freestream Mach number for the actual AoA is 10o
and Yaw 5o. Result shows the sensor functions holds
indeed only for Hypersonic flow.
4.4. Impacts of Configuration Changes
From the aforementioned discussion on the factors
responsible for the error in angle determination, it can be
seen that configuration change gives rise to variation of
the error in predicted angles using the original ASF.
Substituting a unit sphere for the KHEOPS vehicle,
analogous computations were performed to see the effect
of configuration changes. Results are compared in Fig
12. Predicted angles for the sphere are more accurate
Figure 9: Predicted AoA angle using the correction as
stated in Eq. (3).. The modified formula achieves the
required precision of 0.5o.
Figure 10: Predicted yaw angle using the correction as
stated in Eq.(4). The modified formula achieves the
required precision of 0.5o.
Figure 11: Variations of modified predicted AoA and Yaw
from pressure distribution versus free stream Mach
numbers.
than for KHEOPS. Predicted AoAs almost meet the
accuracy requirement without any corrections. This is
expected since a sphere is closer to the model from
which ASF was obtained. Results indicate that ASF work
for different configurations only with proper corrections
that depend on the geometry. Hence, corrections Eqs.
(3, 4) is not universal applicable, but are valid for
KHEOPS only. But it is worth noting that the pressure
distribution measured in the nose region of the vehicle
accurately predicts its orientation. Therefore, it is the
configuration of the nose that matters. The body
geometry and the base have little effect.
(a) predicted AoA versus Actual AoA
(b) predicted Yaw
Figure 12: Comparisons predicted AoA and Yaw for different
of Geometry .
5 CONCLUSIONS
A method of predicting from measuring pressure data at
specified locations in the nose region of a space vehicle,
its orientation with regard to pitch and sideslip was
developed. The strategy is to use ASF for the prediction
of AoA and yaw angle using pressure data on specific
locations on the vehicle surface. A large number of
numerical simulations were performed to study the
range of validity of these formulas with regard to flight
velocity (Mach Number), AoA, yaw angle, and
geometry change. The following conclusions can be
drawn: first, the calculated AoA using ASF deviates
from the actual AoA and the deviation is almost linear
and independent on yaw angle. While, the calculated
yaw angle also deviates from the actual yaw angle and
the deviation is approximately linear but it is a function
of both AoA and yaw angle. Second, Both predicted
AoA and yaw angle can be modified using simple
corrections to the required precision of 0.5ofor the given
space vehicle. Third, When 4.98Ma25.0, 0< a <
30o and 0< b < 10o, the sensor functions are verified to
be effective. In practice, the range of validity could be
extended even further. The last, a different nose
geometry requires different corrections.
6. REFERENCES
Chakravarthy S., Peroomian O., Goldberg U.,
Palanishwamy S., and Batten P., The CFD++
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Gnoffo P.A., Gupta R.N., Shinn J,K,; Conservation
Equations and Physical Models for Hypersonic air
Flows in Thermal and Chemical Nonequilibrium,
NASA TP 89-2867.1989.
Häuser J. and Dai W., Numerical simulation for Flushand
Laser Air Data System(FADS).(final report),
Department of High Performance Computing Center
of Logistics and Expert Systems GmbH, Salzgitter,
Germany, 2004.
Koppenwallner, G., Definition of requirements and
operational specifications for FADS, Technical note
WP1: Flush and Laser Air Data System, HTG TN-
03-6, 2003.
Muylaert J., Kordulla W., Giordano D., Marraffa L.,
SchwaneR. Spel M., Walpot L., Wong H.,
Aerothermodynamic Analysis of Space-Vehicle
Phenomena, Bulletin 105. february 2001.
Walpot, L., Ottens, H., FESART/EXPERT Aerodynamic
and Aerothermodynamic analysis of the REV and
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