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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: rugescu@yahoo.com.

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

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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.

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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.

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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:

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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.