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8/3/2019 S4.13_RRugescu3_ICMERA2011
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Orbital injection errors and sensor requirements for NERVA space
launchers
R. D. Rugescu 1+, Cr. E. Constantinescu 1, M. Al. Barbelian 1, Alina Bogoi 1, and C. Dumitrache 1
1Faculty of Aerospace Engineering, Univ. Politehnica of Bucharest, Romania
Abstract. The input module of the NERVA space launcher guidance system consisting of the inertial andsensor platform is responsible for the basic accuracy if the ascent trajectory and injection efficiency. The
sample rate magnitude and data filtering along the real time trajectory are the only tools available for
improving the guidance accuracy up to the level of requirements to secure admissible orbital injection error
and the subsequent flight corridor during the orbital ascent. Analysis of the NERVA-1 flight telemetry flow
from the onboard inertial platform raises the problem of the optimal selection of the onboard sample rate and
of the rate of telemetry, which are not identical. The orbit injection errors are chosen from the orbit altitude
constraints and subsequent accuracy requirements for the inertial sensors are derived. They show that the
accuracy requirements are moderate and may be covered with almost conventional sensors. To improve the
flight guidance accuracy the rocket motor chamber pressure and thrust are measured and observation of the
preflight zero drift, recording noise and of the high level of embedded noise during both powered and coast
atmospheric flight is performed. Simple filtering based on frequency Fourier analysis is delivered with
conclusions regarding the intelligent algorithm enhancement that are developed and implemented on the next
generation of flight research drone missiles RT-759M NERVA-2, right in preparation. The main rationale of
that algorithm stands in the method of discriminating between false and true information on each measuring
point immediately after the data are delivered by the sensors. Learning procedure from previous preflight
recordings and from gradual accumulation of concurrent data streams subjected to FF spectral analysis are
combined to improve data filtering, for immediate release to the next module of the autopilot. The rate of
sampling is optimized from the analysis of the previous flight, inertial data records and test stand pressure
and thrust records that show the level of noise. The behavior of the electronics under the dynamical loads of
the rocket flight, involving overloads of more than 20 g-s and the level of vibration during the real flight andother sources of measuring errors are also focused in the research. During simulated work of the sensor
platform the algorithm has been acceptably validated and prepared for real flight test performance.
Information important for the NERVA autopilot design activity is structured through the multiple variance
approach.
Keywords: signal processing, data filtering, real time data filtering, rocket flight control, orbital guidance.
1. IntroductionThe errors during satellite orbit injection were first investigated in detail and described during the 50-s,
considering the flight mechanics of the ideal two body problem [1], [2]. A lot of work was extended
thereafter [3], [4]. It is the scope of the present investigation to extract quantitative, practical expressions for
determining orbital injection requirements, in connection to the level of accuracy of the guidance equipment
during the atmospheric ascent towards orbit. Accuracy requirements are separately considered for onboard
data flow within the auto-piloting system and for data transfer to the ground by radio telemetry. The air-to-
ground telemetry is a follow-up technique of the flight guidance accuracy requirements. The results with the
concept, block diagram and physical build-up of the autopilot extend the previous guidance accuracy
investigation of the authors along the powered flight of a small, supersonic rocket vehicle [5] to the case of
high velocity, orbital flight.
+ Corresponding author. Tel.: +40723673054; fax: +40213181007.
E-mail address: [email protected].
ISBN 978-1-84626-xxx-x
Proceedings of 2011 International Conference on Mechanical Engineering, Robotics and Aerospace
(ICMERA 2011)
Bucharest,Romania, 20-22 October, 2011, pp. xxx-xxx
8/3/2019 S4.13_RRugescu3_ICMERA2011
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Means for securing the stable and accurate ascent trajectory consist of a sound method of noise removal
from the tracking data of the inertial sensors and prepare a flawless input for the executive autopilot. The
first flight test of the NERVA inertial platform, built along a proprietary scheme by the team of authors, has
reviled a high level of elastic noise due to the interaction of the internal IMU plate with the supporting
medium [6]. The regular approach is the spectral analysis by Fourier transform with identification of major
eigen frequencies of the phenomenon and the eventual filtering of high frequencies, specific to the elastic
behavior of the rocket structure [7], [8]. Obviously this ordinary approach is mainly appropriate when the
nature of the signal is either periodic or generally unknown. In the present case the evolution of acceleration
and angular velocity is predictable and a direct means to remove the blurring noise is recommended. Thesample rate and sensor accuracy is the main factor in securing the orbital injection accuracy.
2. Orbital injection parametersOrbit injection errors refer to the amount of change in elliptical orbit ephemeredes {a, e, } when the
injection hyperpoint I(r, v, ) presents deviations from its nominal value. The major semiaxis a, the
eccentricity e and the local true anomaly , the last as measured from the ellipse periapsis, may be either
considered in implicit form of the orbit equation, or in explicit representation in respect to the injection
parameters. The first method is used by previous investigators [1], [8]. We adopt here the second method, by
first resolving the orbit ephemeredes from the orbit equations
+=
=
+
=+
=
,cos1
sintan
,12
,cos1
)1(
cos1
2
2
ee
arKv
e
ea
e
pr
(1)
in respect to the injection parameters,
),,( vraa = , ),,( vree = , ),,( vr= (2)
With the notation suggested in [1] for the squared ratio of the local velocity to the local circular velocity
K
rv
v
v
c
2
2
22 = (3)
they explicitly appear as
22
1
= ra , (4)
( )2
22
)(tan1
21
+
+=e , (5)
+= tan
])(tan1[2
2tanarg
22
2
. (6)
The size of the orbit depends directly on these ephemerides and any deviation in the nominal value of
each ephemerid will affect its size. From here the admissible errors are derived.
3. Admissible orbital injection errorsThe orbit selected for launching the ADDASAT is almost minimal in size, for reasons of minimal
injection energy expense. It is known that a similarly minimal orbit was used for the orbital flights of the
Mercury capsules at the beginning of the space age and its altitude was of 100 statute miles or 160 km.
8/3/2019 S4.13_RRugescu3_ICMERA2011
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At such small altitudes the life of the satellite will last for a few weeks at most, which is long enough for
the demonstration of a full orbital flight capability of the NERVA launcher. Perigee altitudes of 120 km were
encountered too, within the Russian Cosmos program of reconnaissance satellites. This small altitude of the
first ADDASATs will additionally improve the maneuver of orbital exit and descent for the eventual
recovery of the ADDASAT re-entry capsule, which is the main goal of the research program.
These values are selected for defining the limit errors in the orbital injection of ADDASAT, which must
be kept above the minimal altitude of 120 km for the periapsis
040160
+=Pr km. (7)
No other constraints are critical for ADDASAT. Consequently the value of both apsides, perigee and
apogee, must be analyzed as being subjected to potential errors over the local variables },,{ vr ,
).1(
),1(
ear
ear
A
P
+=
=(8)
This goes on finding the matrix influence of the injection parameters },,{ vr upon ephemerides a and
e, regardless where on the orbit the injection actually takes place,
,)1(
,)1(
jjj
A
jjj
P
x
eae
x
a
x
r
x
eae
x
a
x
r
+
=
=
(9)
From the explicit relations (4) and (5) the derivatives easily result
( )2222
=
r
a,
( )222
2
2
=
v
r
v
a, 0
=
a, (10)
++
=
2)(tan1
cos11
222
22
K
rv
K
rv
K
rv
K
rv
r
re , (11)
( )2)(tan1cos12
222
22
++
=
vv
e, (12)
( )2)(tan1sin2
222
22
++
=
e. (13)
These relations are valid for any desired orbit of elliptical shape, regardless of its eccentricity. We adoptpositive values of the eccentricity e, due to the periapsis reference for measuring the true anomaly.
We say that the eccentricity must fit within the semi-closed interval [0, 1). The problem appears for
circular orbits, uncovered by the previous developments [1], [8].
3.1. The case for circular orbitsIn the case of circular orbits, the eccentricity is zero, the radius of the orbit preserves constant r= a, the
nominal injection takes place along the local horizontal direction = 0 and the velocity equals the local
circular velocity, hence = 1. It is easy to observe that under these circumstances the three partial derivatives
of the eccentricity in the above equations (11) through (13) seem to become undetermined. Due to this
behavior, this case is missing from those given in [1] that only covers the realm of elliptical orbits with 1.
8/3/2019 S4.13_RRugescu3_ICMERA2011
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To actually overcome the indetermination from above the rigorous definition of the partial derivative
must be followed. The variation of the eccentricity is dealt with in the three-dimensional space
},,{ vree = . While a partial derivative is built for one variable, the others may be kept constant.
Computing the first partial derivative in respect to rfrom (11) it is observed that both rand2 are variable,
the latter due to r, contained within vc from2 , and the elevation is the only constant. Consequently
( )
( )c
cc
c rrrr
e 1
21
11 2
22
22
=
=
+
=
, (14)
When the partial derivative in respect to v in (12) is measured, the velocity v or the ratio are the only
variables, while both the radius rand the elevation are kept constant at circular values, ar= ,0 =
. This
reads
( ) ( ) cc vvvve 2
1
12
21
12
2
22
22
22
=
=
+
=
. (15)
At first sight, the next derivative in (13) seems also undetermined (0/0). Along a similar treatment
however, the derivative in respect to in (13) preserves the dimensionless velocity indeed as constant at =
1, while is soundly variable. In this manner the partial derivative equals
1cos
)tan
sin
2===
c
e
. (16)
The partial derivatives of the major semi-axis within circular orbits do not present any problem and get
the values
2=
cr
a,
c
c
c v
r
v
a2=
, 0
=
a, (17)
The avenue is thus open for determining the error margins for the injection maneuver. The influence
matrix for the circular orbit becomes
1=
c
P
r
r, 3=
c
A
r
r,
0=
c
P
v
r,
c
c
c
A
v
r
v
r4=
, (18)
c
c
P rr
=
, c
c
A rr
=
.
Considering these coefficients of influence the total error in the perigee (k= P) or in apogee (k = A)
appears as
ck
ck
ck
k rvvrr
rrr
+
+
= , (19)
where a even distribution of errors in primary variables is assumed. Provided the primary errors act
separately, the following admissible errors result
3/40= cr km, for a variation in apogee; (20)
95.114//40 == ccc rvv m/s in apogee; (21)
=== 35.0006118.06538/40R
c . (22)
In computing the results the following values for Earth parameters were used:
8/3/2019 S4.13_RRugescu3_ICMERA2011
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4457.398762=K km3/s
2, 6378=R km. (23)
For kinematical purposes only the radius of the Earth at the equator is used, were the minimal Earth orbit
could only exist. As far as the errors in injection parameters may add with different weights, the real
restrictions in the accuracy of the guidance system may be evaluated in respect to the guidance procedure
itself. When for example a gravity turn trajectory is approximately followed, the main error is in the altitude
of flight. The velocity is much better controlled and the injection angle of elevation will be automatically
achieved. Under these conditions of flight the error margin of -13 km is an easy to achieve task.
4. Filtering rule for guiding sensorsReal flight tests were first performed in June of 2010 with the proprietary inertial platform on a
specialized drone missile with similar dynamic characteristics to the eventual space carrier NERVA. The
first stage consisting of a solid propellant heavy booster stands for the propulsion system, in order to
materialize the high acceleration during the lift-off of the vehicle. The longitudinal acceleration ranges from
20 to 29 g-s at the launch temperature of 46C. When dealing with previously unknown signals and noise
sources the commonly adopted filtering technology is by Fourier, harmonic analysis of the information and
by subsequent filtering decision in respect to the frequency of the components. For the NERVA rocket flight
there is an acceptable knowledge of the low frequency component.
5. ConclusionsOrbital injection errors and their influence upon the inertial sensor requirements were investigated in a
proprietary manner, with application to quite circular orbits, specifically at very low altitude. The results are
compared with known developments existing in the literature and the differences are commented.
The results of the first in-flight measurement and telemetry of data regarding the rotational velocities and
linear accelerations on-board the first stage of the future Romanian space launcher NERVA are also
presented, analyzed and interpreted. They show that a number of problems must be solved regarding the
noise processing within the data and this aspect of the research became suddenly very tough. The NERVA-1
experiment has surpassed the provisions of the research and development contract and offered a massive data
quantity for analysis and development of the entire auto-pilot of the space launcher. The results are opening
the way for the development of the entire Flight Controller and the given experimental data are recorded as
the first real flight, telemetry achievement in Romania.
6. AcknowledgementsThe research was sponsored through the contract NERVA 82076 by the Romanian Authority for
Research and Development in the Superior Education UEFISCDI of the Romanian Ministry of Education.
7. References[1] K. A. Ehricke, Space Flight, New York, D. Van Nostrand, 1960.[2] S. Herrick, "Accurate Navigation of Intercontinental and Satellite Vehicles in the Earth's Gravitational Field", The
Astronautics Symposium, San Diego, CA, February 18 to 20, 1957.
[3]
A. E. Roy, Orbital Motion. Adam Hilger, Bristol, 3rd edition, 1988.[4] J. E. Marsden, S. D. Ross, K. Grubits, O. Junge, Wang-sang Koon, F. Lekien, M. Lo, S. Ober-blbaum, K.Padberg, R. Preis, and B. Thiere, New Methods in Celestial Mechanics and Mission Design, Bulletin (New
Series) of the American Mathematical Society, Volume 43, Number 0, 2005, pp. 000000.
[5] R. D. Rugescu, and Cr. E. Constantinescu, Accuracy Requirements for On-Board Drag Measurement Systems,Sc. Bulletin U.P.B., series D, Mech. Eng., 62, 2(2000), pp. 63-72.
[6] Cr. E. Constantinescu, R. D. Rugescu, S. Ciochin, and R. C. Cacoveanu, First flight experiment with theNERVA-1 inertial platform, Proceedings of 2010 International Conference on Mechanical Engineering, Robotics
and Aerospace ICMERA-2010, ISBN 978-1-4244-8867-4, IEEE Catalog Number CFP1057L-PRT, December 2-4,
Bucharest, Romania, 2010, pp. 312-316.
[7] W. von Braun, H. H. Koelle (ed.), Handbook of Astronautical Engineering, McGraw Hill, New York, 1961.[8] Shawn Beaudette, Satellite Mission Analysis, Carleton University Spacecraft Design Project, 2004.