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Received 19 March 2013, revised 14 Ma 2013, onine pubished 19 Ju 2013

Deence Science Journa, Vo. 63, No. 4, Ju 2013, pp. 346-354 2013, DESIDOC

1. IntroDuCtIon

The primar objective o an aunch vehice is to deiver

the Paoad to the desired target within the given toerance

bounds. Since the rst use of ballistic missile in 1940s, a lot of

innovation has gone in the deveopment o more sophisticated

guidance, contro, navigation agorithms to enhance the

range, accurac, reiabiit, etc., In view o the current

working scenario there is a demand or the maneuver during

ight (trajectory reshaping) such that the mission objectivesare achieved without an compromise on the mission end

objective. A typical in-ight mid-course maneuver scenario

is shown in fig. 1, where the trajector in bue coor is the

one which is going to be oowed b the vehice i there is no

intentiona maneuver (non-maneuvering) is executed on board

and the trajector in green coor is the intentiona maneuver

trajector which is hard to predict as compared to the non-

maneuvering trajector.

A reentr vehice approaches at a ver high veocit,

tpica veocities varing rom 5 Km/s - 7 Km/s based on the

seected trajector, downrange, guidance mechanism empoed

in the design procedure1. But with growth o computing power,

more powerful and reliable estimation (prediction) and ltering

techniques are avaiabe toda b virtue o which it is possibeto predict the trajector o the reentr vehice we ahead and

take some advance corrective measures.

Optimization based trajector panning and tracking the

reerence trajector using dnamic inversion guidance aws are

proposed b Ran2, et al. In the paper the author describes the re-

ral tim Mid-cs Mav ad Gidac f a Gic ry Vicl

Avinash Chander* and I.V. Muraikrishna#

*Defence Research and Development Organisation, New Delhi-110 001, India#Jawaharlal Nehru Technological University, Hyderabad-500 085, India

*E-mail: [email protected]

AbStrACt

The aim of any mission is to accomplish the nal objective with desired accuracy and the same is valid for ageneric aunch vehice. In man missions it is necessar to execute mid-course maneuvers with an intentiona diversiontrajectory to create a counter measure or to avoid certain specic known geographical locations. The current workeaborates a nove and practica impementabe mid-course maneuver and an ascent phase guidance o a reentrvehicle executing an in-ight determined mid-course maneuver (trajectory reshaping) without compromising theaccuracy of the nal achieved target position. The robustness of the algorithm is validated with 6DoF simulationresuts b considering the dispersion o the burnout state vector conditions which arises due to variations in thrust

prole, aerodynamics characteristics of the vehicle, atmosphere, etc.

Kywds: Ascent phase guidance, mid-course maneuver, reentr vehice, termina accurac, range augmentation,trajector reshaping.

Fig 1. nmalizd ajcy f a -mavig ad mav vicl.

NormalizedAltitude

Normaized Downrange

346

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CHANDER& MURAlIKRISHNA: REAl TIME MID-COURSEMANEUVER AND GUIDANCE Of A GENERIC REENTRy VEHIClE

347

entr vehice trajector panning and guidance b considering

the path constraints ike aerodnamics heating, aerodnamic

load, etc., in r-V plane with an aero assisted conguration.

Gao Changsheng3, et al. describes a virtua dispacement

concept based reentr vehice guidance using optimization

technique and lQG based tracking o the reerence trajector.

Page & Rogers4 summarizes a ew investigations carried out

in guidance and contro o maneuvering reentr vehices b

considering cross-product, proportiona and tangent cubicguidance mechanisms having cruciform, bank to turn and xed

trim control congurations.

Expicit re-entr guidance equations or maneuvering re-

entr vehices (MaRVs) using characteristic curve approach

is deveoped Cameron5. Whie ormuating the guidance it

is ensured that termina trajector constraints on path anges

and it acceeration and its derivatives are achieved. Variabe

gain vector guidance equations are estabished b orcing

termina equation structure to be simiar to the characteristic

curve equations. But the stud doesnt consider the imitation

on aerodnamic capabiit, maximum acceeration imit nor

did an energ management requirement and it assume that this

tpe o characteristic curve cas or ess acceeration or argerange to go than that or sma range to go.

A practica impementabe agorithm described in the

current paper describes methodology to execute the in-ight

determined maneuver o the vehice and to guide the vehice in

the ascent phase to its predetermined target accurate with in

the desired toerance bounds. The basis o the current approach

reies on the capabiit o simuating the rea time scenario o

the vehice dnamics in the background simuation rom the

burnout point6,7 to the desired target point.

2. DeSIGn MethoDoLoGY

Most o the cassica aunch vehice guidance agorithms

re on required veocit vector concept8, which acts as basis

or hit equation6 to be soved in order to reach the desired target.

Once this required veocit vector is cacuated, the desired

burnout position and burnout ight path angle are determined9.

The innovative undering concept o the proposed agorithm

is performing an in-ight dened maneuver during the mid-

course (ater apogee i.e., decent phase) b keeping in view o

the paoad capabiities.

The duration o maneuver can be decided based ontempora or spatia means. I the duration o maneuver is based

on time then the maneuver wi be open oop orm, because the

time of ight of the vehicle will vary based on the propulsion

characteristics, range, burnout conditions. I the duration o

maneuver is a unction o atitude/range then the maneuver wi

be in cosed orm, because the aim o the paoad to impact

the desired coordinates at the predened altitude, irrespective

o time. The predetermined maneuver can be an reaizabe

unction ike sinusoida, puse, trianguar, exponentia, etc. as

shown in fig. 2. I the maneuver is o sinusoida the variabe

parameters are maneuver ampitude and requenc, i the

maneuver is puse then the variabe parameter is the puse

ampitude and i the maneuver is exponentia then the variabeparameter is the deca or rise sope o the maneuver. Maneuver

can be executed by a variable or xed thruster at center of

gravit or cose to center o gravit. Generic representation o

the maneuver unction is given beow:

Z=F(a, f, y) (1)

where

Z = Maneuver unction

a = Ampitude o the considered maneuver unction

f= frequenc o the considered maneuver unction

y = Independent variabe (time, atitude, downrange)

Vaues o the a & are decided b the vehice propusive

capabiit and the extent o dispersion panned and seection

Fig 2. Pdmid mav fcis as a fci f malizd im ad alid.

Generic Maneuver function

PredeterminedManeuver

PredeterminedM

aneuver

Normaized Atitude Normaized Atitude

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DEf. SCI. J., VOl. 63, NO. 4, JUly 2013

348

of maneuver function can be random in selection but denite

once seected. Once the determined maneuver initiation point,

duration and the maneuver function is nalized, initiate the

background simuation rom the burnout point to the target.

During the simuation, initiate the determined maneuver rom

the determined initiation point up to maneuver duration point

(time, atitude). With this maneuver, compute the dierence

in the desired and achieved atitude and ongitude at the impact

point. Augment the ascent phase target coordinates with theabove computed dierence vaues in atitude and ongitude and

sove the hit equation (initiate the ascent phase guidance) with

this augmented coordinates and repeat the above procedure

ti convergence criteria is met. Because o the mid-course

maneuver there wi be a change in the guidance soution

(burnout conditions) to reach the desired target, which can

be seen as the perturbation on the initia soution as shown

beow7.

( ) ( )( )0

2

sin1 cos

sin sin

r

a

f + f f + f= +

(2)

where

r0 = (a+h) = missie position rom the center o the eartha = equatoria radius (m), h = vehice atitude rom the

surace o the earth2

0r v

GM =

G = Universa gravitationa constant

= range angle = ight path angle at burn out

f = Augmented range ange corresponding to change innal coordinates

The steps invoved in the proposed agorithm are given

beow:

(i) With the desired burnout state vector as the initia

states, simuate the vehice trajector up to the desired

predetermined atitude, rom where determined maneuver

is panned.(ii) from predetermined atitude start o maneuver to

the termination o the maneuver, superimpose a

predetermined pseudo random maneuver (varing

ampitude and requenc with atitude as the reerence) to

the actua attitude. During this maneuver period, activate

thruster provided in the paoad (tpica caed veocit

package10) or side thrusters ocated at the center o gravit

(i known accurate) can be used.

(iii) Once the predetermined attitude maneuver period

competed, deactivate the thruster and simuate the vehice

trajector up to the impact point. Note the achieved

atitude and ongitude.

(iv) find the dierence between the desired and achieved

atitude and ongitude, and augment the desired coordinates

with this dierence vaues.

(v) Repeat the steps rom I to IV ti the dierence between

the desired and achieved atitude and ongitude ie within

the desired toerance bounds.

3. MAtheMAtICAL MoDeLLInG oF the

PAYLoAD VehICLe

for the current work a standard noninear 6Dof

mathematica mode with 3 orces and 3 moments is

considered11. In order to make the simuation more reaistic

a noninear aerodnamic mode is considered, where the

drag and aerodnamic orces are modeed as the unctions o

atitude, ange o attack and Mach number. The earth shape and

rotationa eects12 are incuded in the simuation as the time o

trave is variabe which, i not accounted correct, eads to tens

o kiometers range errors13. A universa earth gravit mode

up to J214 term is considered to take care o earth gravitationa

eects, which is a unction o coatitude and atitude.Reentry atmospheric effects play a signicant role during

reentr, as the veocit with which the vehice reenters is ver

high, in order to take care o this unwanted aerodnamic eects

because o atmosphere, a more eaborate atmospheric mode15

is considered or simuation. In order to take care o wind

eects during the reentr phase, a reaistic wind mode 16 is

considered or studies. To assess the perormance accurate a

reaistic inertia navigation mode17 is incuded in the simuation

taking care o rea time hardware eatures (acceerometer or

acceeration, groscope or rate). A noninear reaction contro

sstem and iquid veocit package modes are considered or

the simuation studies.

( )

( )

( )

( )( )

( )( )

( ),

fx fx

x

fy fy

y

fz fz

z

T Dg

m

T A

m

T A

m

X U

g

u

v gw

X fMp x

Ixq

r M I I przy xIy

M I I pqz x yIz

= =

+

+

(3)

where u, v, w and p, q, r are transation and rotationa

components. Tfx

, Tfy

, Tfz

andAfy

, Afz

are thrust and aerodnamic

orce components.Dfx

is the drag orce action aong the bod

axia direction. m is the mass o the pa oad, gx, g

y, g

zare

the gravitationa components, andMx, M

y, M

zandI

x, I

y, I

zare

moment and inertia components respective

for the present stud, a 2 stage soid propeed aunch

vehicle with ex nozzle actuated control system is considered.

Once the soid propeed stages are separated ater propeent

got consumed, the paoad is controed b using a reaction

contro sstem powered b iquid thrusters, enabing the

exibility of switching on and off when desired. During the

ascent phase the vehice oows a preprogrammed attitude

turn keeping in view the initia constraints ike structura oad

&contro imitation, etc., Once the vehice attains the desired

reaxed conditions usua out o atmosphere, an expicit cosed

oop guidance8 wi guide and pace the vehice at burnout on

a desired ellipse (function of burnout position, velocity, ight

path ange &earth rotation rate compensated desired target

position), b virtue o which the vehices reaches the desired

target. With these desired burnout state vector, a back ground

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349

6Dof agorithm is initiated iterative b using the proposed

agorithm, ti the 6Dof achieved impact atitude and ongitude

coincides with the desired ones as per the specied tolerance

bounds.

4. SIMuLAtIon reSuLtS

To vaidate the proposed agorithm, dierent burnout

conditions are considered or a given target as shown in Tabe

1. for the stud a sinusoid (quaternion

18

) with an ampitudeand requenc o 0.0001 Hz and 0.15 Hz is considered or

determined attitude maneuver. Here it shoud be noted that

the maneuver activation is based on atitude not on time, since

the trajector varies with burnout conditions and the guidance

probem considered or simuation is a ree time probem (i.e.,

the aim is to reach the target without an constraint on the time

of ight). The input amplitude and frequency are same for all

the three cases considered or simuation, but the trajector

parameters var based on burnout conditions i.e., veocit,

position, ight path angle, etc.

The thrust orce can be provided b a sma propusionpackage with respect to atitude. During maneuver phase, a

thruster with constant thrust orce o 20 KN is considered. Once

Cas

Sa vc a sa f

proposed algorithm (fight

pa agl, vlciy &

psii)

Dsid mial

cdiis

lc d

0.01o (dg)

Acivd mial cdiis

(wi mid-cs mav

dcpi algim) dcpi

ajcy (psd ag) (dg)

Acivd mial cdiis

(wi mid-cs mav

dcpi algim) acal

ajcy (acal ag) (dg)

BO

(D)

VBO

(m/s)

PBO

(m)latitude longitude latitude longitude latitude longitude

1 13.85 4658 6514670

41.488637 87.088686 41.366969 87.115274 41.482585 87.090507

(Desired -Achieved)

latitude & longitude0.121668 -0.026588 0.006051 -0.001821

2 16.07 4492 6519650

41.488637 87.088686 41.367018 87.111770 41.480964 87.088383

(Desired -Achieved)

latitude & longitude0.121619 -0.023084 0.007672 0.000303

3 13.55 4612 6526009

41.488637 87.088686 41.395181 87.115627 41.496412 87.089302

(Desired -Achieved)

latitude & longitude0.093455 -0.026941 -0.007775 -0.000616

tal 1. Diff cdiis a csidd f a giv ag

Fig 3. Cas 1 Alid vs laid ad lgid ajcy.

longitude (D)

longitude (D)latitude (D)

latitude (D)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Note: The trajectory (nal achieved coordinates) is sensitive to the burnout conditions and the band for burnout conditions considered for the

simuation is seected considering some variations o soid propusion.

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350

the maneuver period is competed the thruster gets deactivated

and the vehice oows the baistic path there ater.

The execution o the determined maneuver or case 1

is shown in fig. 3. The trajector shown in bue coor is the

one which is generated b the paoad without an deception

maneuver and the green one is the one which is generated b

the paoad with a predetermined maneuver execution. from

the fig. 3 it is evident that the deception maneuver started at

150 km with a deviation rom the predicted trajector (buetrajector). The trajector shown in red coour is the background

6Dof trajector which provides the reerence or rea time

deception trajector (green trajector). The background and

rea time trajector are in tight agreement because o which

it is not possibe to see the dierence between red and green

trajectory in the gure(s). Finally the realtime trajectory

achieves the desired atitude and ongitude with in the given

toerance bounds. figure 4 shows the atitude variation or

with and without maneuver with respect to time. from the

data markings in the gure, it is clear that the difference in

the atitude between the non maneuvering and maneuvering

trajector is varing rom 0 km to 9 km rom 150 km atitude

point to impact point. This magnitude can be increased b an

additiona impuse in the maneuvering vehice.

The working o the agorithm or case 2 and case 3 areshown in figs 5 and 6. figures 5 and 6 shows the atitude and

ongitude variation with respect to atitude and it is cear rom

this gures that the algorithm drives the payload towards the

desired target rom the start o the deception point.

The robustness o the proposed agorithm under mode

uncertaint is studied b perturbing the wind, atmosphere

Fig 4. Cas 1 : tim vs alid.

Altitude(Km)

Altitude(Km)

Time (s)

Time (s) Time(s) Time(s)

Fig 5. Cas 2 : Alid vs laid ad lgid ajcy.

latitude (D)

latitude (D)

longitude (D)

longitude (D)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

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351

Fig 8. Cas 4 : Alid vs laid ad lgid.

Fig 6. Cas 3 : Alid vs laid ad lgid ajcy.

longitude (D)


latitude (D)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Fig 7. Cas 4 : Alid vs wid vlciy, dsiy ad dag.

Zona Wind Veocit (m/s)Meridian Wind Veocit (m/s)

Deceeration (m/s2)Densit (kg/m3)

Altitude(Km

)

Altitude(Km)

Altitude(Km

)

Altitude(Km)


latitude (D) longitude (D)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

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352

BO

(D)

VBO

(m/s)

PBO

(m)

13.85(D) 4658(m/s) 6514670(m)

tal 3. b sa vc csidig f sss sdis

Fig 9. Cas 5 : Alid vs wid vlciy, dsiy, ad dag.

Zona Wind Veocit(m/s)Meridian Wind Veocit(m/s)

Densit(kg/m3) Deceeration(m/s2)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Fig 10. Cas 5 : Alid vs laid ad lgid.

longitude(D)

longitude(D)latitude(D)

latitude(D)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Altitude(Km)

Cas Pcag f vaiai mial wid,

dsiy ad dag (assmd mdl)

Wind Densit Drag coefcient

4 50 5 2

5 -50 -5 -2

tal 2. Pai ads csidd f sss sdis

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353

tal 4. rsss s cas simlai sls

CasDsid mial

cdiis (dg)

Acivd mial cdiis

(wi mid-cs mav

dcpi algim) psd

ag (dg)

Acivd mial cdiis

(wi idal mdl mid-cs

mav dcpi algim)

acal ag (dg)

Acivd mial cdiis

(wi pd mdl mid-

cs mav dcpi

algim) (dg)

latitude longitude latitude longitude latitude longitude latitude longitude

4

41.488637 87.088686 41.348213 87.112340 41.485034 87.086906 41.480771 87.085815

(Desired -Achieved)

latitude & longitude0.1404 -0.0237 0.0036 0.0018 0.0079 0.0029

5

41.488637 87.088686 41.348213 87.112340 41.485034 87.086906 41.491567 87.089976

(Desired -Achieved)

latitude & longitude0.1404 -0.0237 0.0036 0.0018 -0.0029 -0.0013

and aero modes. The case studies are isted in Tabe 2. The

burnout state vector consider or the simuation studies shown

in Tabe 3.

Tabe 4 shows the desired target point ocation, achieved

termina point ocation without (predictabe trajector) and

with (deception trajector) reentr maneuver.

figure 7 & 9 shows the wind, atmospheric densit

and drag variation with respect to the atitude. The curves in

green shows the mode considered or background trajector

simuation, whie the red one is considered or ore ground

simuation. figures 8 & 10 shows the atitude and ongitude

variation with respect to the atitude and it is cear rom the

gures that the nal impact is achieved with in the prescribed

toerance bound, under mode perturbations.

5. ConCLuSIonA practica working and impementabe agorithm or

rea time mid-course maneuver with pre-corrected ascent

phase guidance is described in the current paper. The work

describes in detai the impementation o the deception

maneuver and guidance agorithm b means o a 6Dof

simuation rom burnout point to impact. With the practicabe

avaiabe subsstems the paper shows the robustness o the

agorithm b means o some simuation cases b considering

a wide band o burnout state vector vaues at the burnout. One

of the exibility of the proposed work is in selecting online

the start and end point o maneuver with the variation o the

ampitude and requenc o the maneuver, b keeping in view

o the thrust orce avaiabiit and capabiit. The present workcan be extended to ascent phase guidance b virtue o which a

wide band o dispersion at reentr can be taken care and it is

aso possibe to use side thruster during the maneuver phase to

decrease the paoad response time or maneuver.

reFerenCeS

George, N. lewis & Theodore A. Posto, Deense1.

and Arms Contro Studies. Massachusetts Institute o

Technoog, future Chaenges to the Baistic missie

deense.IEEE Spectrum, 1997, 34(9), 60-68.

Ran, Zhang; Huieng, li & Xudong, Cao. A new approach2.

or re-entr vehice trajector panning and guidance,

ICAS, 2012-5.7.5, 4, pp. 3158-3164.

Changsheng, Gao; Wuxing, Jing; & Chaoong, li.3.Optima guidance aw design or reentr vehice using

virtua dispacement concept. In the Proceedings o the

26th Chinese Contro Conerence, Zhangjiajie, Hunan,

China, Ju 26-31, 2007, pp. 507-510.

Page, J.A. & Rogers, R.O. Guidance and contro o4.

maneuvering reentr vehices, Decision and Contro

incuding the 16th Smposium on Adaptive Processes and

a Specia smposium on fuzz set theor and appications,

1977 IEEE Conerence, Dec-1977, 16, pp. 659-664.

Cameron, J.D.M. Expicit guidance equations or5.

maneuvering reentr vehices, re-entr and environmenta

sstems divisions. Genera Eectric Compan,

Phiadephia, Pa 19101, TP3-3:30.

George; R. & Pitman, Jr, Inertia Guidance. John Wie6.

& Sons, INC. Nework, london, 1962.

Wheelon, Albert D. Free ight of a ballistic missile.7. ARS

Journal, 1959, 29(12), 915-926.

Siouris, George M. Missie guidance and contro sstems.8.

Springer, 2004

Thomas, Tess. Guidance scheme or soid propeed9.

vehice during atmospheric phase, Def. Sci. J., 2005,

55(3), 253-264.

Wike, John W. Veocit package. United States Patent10.

No. 3,260,204. Ju 12, 1966.

Kadam, N.V. Practical design of ight control systems for11.aunch vehices and missies. Aied Pubishers Pvt ltd.

2009.

Department o deence word geodetic sstem 1984.12.

Nationa Imager and Mapping Agenc, 3 Januar 2000.

Corneisse, J.W.; Schoer, H.f.R. & Wakker, K.f. Rocket13.

propulsion and space ight dynamics. Pitman Publishing

limited, 1979.

Kenneth R. Britting, inertia navigation sstems anasis,14.

Wie-Interscience, 1971.

Ananthasaanam, M.R. & Narasimha R. Standard15.

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354

atmosphere or aerospace appications in India. Dept. o

Aerospace Engg. Indian Institute o Science, Bangaore.

Report No. 79 fM 5.

Narasimha, R. The wind environment in India.16. NAl.

Technica Memorandum Du 8501, 1985.

Titterton, David & Weston, John. Strapdown inertia17.

navigation technoog, IET, 2ndEd, 2004.

Kuipers, Jack B. Quaternions and rotation sequences. A18.

primer with appication to Orbits. Aerospace, and Virtuareait. Princeton Universit Press, Princeton, New Jerse.

2002.

Cis

M Avias Cad received his BE

(Eectrica Eng.) rom IIT Dehi, in 1972 and

MS (Spatia Inormation Technoog) rom

Jawahara Nehru Technoogica Universit,

Hderabad. He is present working as

Distinguished Scientist in DRDO. He

received man Awards ike,DRDO Scientist

of the Year-1989, Astronautical Society

of India Award-1997, DRDO AGNI Selfreliance Award-1999, Dr Biren Roy Space award-2000, DRDO

Award for Path Breaking Research/Outs tanding Technology

Development-2007, Outstanding Technologis t Award-2008 b

Punjab Technica Universit. His areas o interest incude:

Navigation, guidance, mission des ign.

D Iyyaki V. Mali Kisa received

his MTech rom IIT Madras and PhD rom

IISc, Bangaore in 1977. He is present

working as Director, Institute o Science

and Technoog and Coord inator o Centre

or Atmospheric Sciences and Weather

Modiication Technoogies at Jawahara

Nehru Technoogica Universit, Hderabad.His areas o interest incudes: Geospatia

technoog and data mining, sot computing technoogies, disaster

management, sateite meteoroog and weather inormatics.

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