guidance and control of a launch vehicle using a stochastic gradient projection method

qThis paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate EditorJ. Sasiadek under the direction of Editor K. Furuta.

*Corresponding author. Tel.: #55-12-345-1457; fax: #55-12-345-1723 .

E-mail address: [email protected] (F. Madeira)

Automatica 36 (2000) 427}438

Brief Paper

Guidance and control of a launch vehicle using a stochastic gradientprojection methodq

Fernando Madeira!,*, Atair Rios-Neto"

!Flight Mechanics Group, Section of Aerodynamics and Aeroelasticity, EMBRAER, SaJ o Jose& dos Campos, 12227-901, SP, Brazil"Institute of Research and Development, University of Paraiba Valley, UNIVAP, SaJ o Jose& dos Campos, 12245-720, SP, Brazil

Received 10 December 1997; revised 21 April 1999; received in "nal form 7 June 1999

Abstract

A recently developed nonlinear programming stochastic projection of the gradient-type technique is applied to get and test novelprocedures for tri-dimensional guidance and control of a satellite solid-fuel launch vehicle. Near optimal open-loop and low-frequency closed-loop solutions are obtained by parametrization of the control history and by considering the satisfaction ofboundary constraints unless of speci"ed random errors. Numerical simulations and tests show that the closed-loop solution can copewith the typical model uncertainties and environmental perturbations. A non standard situation of having a fourth stage with a freeburning time is tested. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Satellite launch vehicle; Stochastic gradient projection; Trajectory optimization

1. Introduction

Orbital injection usually requires the application of anoptimal guidance and control procedure in order tomaximize payload and accurately place it into the desiredorbit without violating mission constraints. The highcosts involved in satellite launching fully justify the highresearch e!orts and budgets in order to improve solutionmethods in terms of optimization, ease of use and robust-ness to environmental perturbations and modelingerrors.

Guidance and control procedures may be applied inboth an open or closed-loop fashion. Open-loop guid-ance and control procedures apply a control historyfound o!-line prior to the #ight. Works on this subjectdetermine the control history usually by solving a trajec-tory optimization problem (Breakwell, 1959). A widevariety of methods have been devised recently and ap-plied to di!erent purposes (Lu, 1993; Hargraves & Paris,

1987; Betts & Hu!man, 1991; Enright & Conway, 1992;Burlirsch & Chudej, 1992; Betts, 1994; Tang & Conway,1995; Seywald, 1993; Coverstone-Carroll & Williams,1994; Sachs, Drexler & Stich (1991); Smania & Rios Neto1988).

Closed-loop guidance and control procedures are con-ceived to cope with uncertainties and perturbations everpresent during the #ight. Great e!orts have been devotedto devising guidance and control procedures, fast enoughand reliable enough to be applied onboard the #yingvehicle. Research works include a broad range of ap-proaches in which their complexities are strongly depen-dent on the current onboard computer technology. Themajority consists of a trajectory optimization procedurefurther simpli"ed to meet the real time feasibility. Recentworks on this subject include the utilization of explicitguidance (Sinha, Shrivastava, Bhat & Prabhu, 1989;Vittal & Bhat, 1991), energy state methods (Corban,Calise & Flandro, 1991), dynamic programming (Feeley& Speyer, 1994), generalized projected gradient (Richard& Christophe, 1995), "nite element (Hodges, Calise,Bless & Leung, 1992; Leung & Calise, 1994), hybridanalytic/numerical approach (Leung & Calise, 1994),neighboring extremals (da Silva, 1994), geometrical ap-proach (Mease & van Buren, 1994), etc. Another emerg-ent approach consists of application of parallelprocessing techniques to speedup the overall trajectory

0005-1098/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 6 3 - 6

optimization computation time (Betts & Hu!man, 1991;Psiaki & Park, 1992; Wirthman, Park & Vadali, 1995).

This paper presents the "rst application of a recentlydeveloped stochastic gradient projection method to opti-mal design and closed-loop guidance and control ofa launch vehicle trajectory. The version of the stochasticgradient projection method used was proposed by RiosNeto and Pinto (1987). The success of its application insolving problems of spacecraft maneuvers (Prado & RiosNeto, 1994; Rios Neto & Madeira, 1997) motivated itsapplication in guidance and control of launch vehicle.A direct search method using this nonlinear program-ming technique is implemented. The trajectory is dividedinto a sequence of segments and the control history isapproximated by parametrized functions inside each tra-jectory segment. The objective is to "nd a parametrizedcontrol history to place the maximum satellite mass ina low Earth orbit. A low frequency closed-loop guidanceand control procedure intended for onboard use is for-mulated and tested.

A special feature of this method is the inclusion of theacceptable errors on the constraints into the problemresolution. Therefore, instead of searching a solution thatcorresponds to a de"nite value for every constraint com-ponent, the method searches solutions that make eachconstraint component to fall inside a given interval.

Simulations and tests are carried out for the satellitelaunch vehicle (VLS) of the Brazilian Complete SpaceMission (MEC-B).

The following sections present a brief review underly-ing the major features of the stochastic gradient projec-tion method, the launch vehicle open-loop trajectoryoptimization, and the closed-loop guidance and controlprocedure.

2. Stochastic gradient projection method: a review

2.1. Statement of the problem

Consider the nonlinear programming problem statedas

Minimize F(Y) (1)

subject to h(Y)"e, g(Y)4d, (2)

in which Y is a n-dimensional vector of parameters to beoptimized; F, h, and g are real-valued functions of Y ofdimensions 1, m

h, and m

g, respectively; e and d the error

vectors, representing the acceptable accuracy on the con-straints satisfaction, de"ned as

DeiD4et

i, i"1, 2,2,m

h,

DdjD4dt

j, j"1, 2,2, m

g, (3)

where eti

and dtj

are given and "xed accuracy limits ofconstraints satisfaction, de"ning the region within whichthe errors are considered tolerable, and which can beviewed as modeling the numerical zero for each con-straint satisfaction.

2.2. Solution of the problem

In a typical iteration, one searches an approximatesolution for the "rst-order increment DY in the problem:

Minimize F(YM #DY) (4)

subject to h(YM #*Y)"ah(YM )#e,

gi(YM #DY)"bg

i(YM )#d

i, (5)

in which YM is a given initial guess or a point from the lastiteration; g

i(YM )5dt

i, i"1,2,2, I

g, represent the set of

active constraints; 04a(1 and 04b(1 are thesearch step adjustment parameters chosen in order tocondition DY to be of "rst order of magnitude, i.e., fora highly nonlinear constraint, the associated search stepparameter value should be closer to the unity; on theother hand, for a fairly linear constraint, the associatedsearch step value may be selected closer to zero.

The left-hand sides of Eqs. (5) are replaced by lin-earized approximations together with a stochastic inter-pretation for the errors e and d

i, leading to

(a!1)h(YM )"dh(YM )dY

*Y#e3,

(b!1)gi(YM )"

dgi

dX*Y#d3

i.

(6)

In Eqs. (6), the error vectors were converted into theuniformly distributed unbiased noncorrelated randomerrors e3 and d3

i, modeled as

E[e3e3T]"diag[e2i, i"1, 2,2,m

h], e2

i"1

3(eti)2,

E[d3d3T]"diag[d2i, i"1, 2,2, I

g], d2

j"1

3(dt

j)2.

(7)

The condition stated by (4) is approximated by the fol-lowing a priori information:

!c$FT(YM )"*Y#g, (8)

in which c is a nonnegative search step to be adjustedsuch that the increment *Y turns out to be of "rst orderof magnitude; g is a uniformly distributed unbiased ran-dom vector modeling the a priori searching error in thenegative direction of the gradient $F(YM ), given by

E[ggT]"PM , (9)

428 F. Madeira, A. Rios-Neto / Automatica 36 (2000) 427}438

where PM is taken to be diagonal, since no a prioricorrelation is imposed among the errors in the gradientcomponents, and with variances adjusted by maximumlikelihood statistical criteria such as to have the gdispersion to be of "rst order of magnitude as comparedto the second order of magnitude of errors e3 and d3

i.

The nonlinear programming problem given by Eqs. (1)and (2) has been converted into the linearized problem ofEqs. (8) and (6). The linearized problem, in turn, charac-terizes a parameter linear estimation problem:

*YM "*Y#g,

Z"M*Y#V, (10)

in which Eq. (10) is equivalent respectively to Eqs. (8) and(6), using a shorthand notation.

Adopting a criterion of linear, minimum varianceestimation, the optimal search increment can be deter-mined using the Gauss}Markov}Kalman estimator(Jazwinski, 1970).

*YK "*YM #K(Y!M*YM ),

P"PM !KMPM ,

K"PM MT(MPM MT#R)~1, (11)

in which R is a diagonal matrix, given by R"E[VVT],and P is an approximation of the covariance matrix ofthe errors in the estimated components of *Y

P+E[(*Y!*YK )(*Y!*YK )T]. (12)

An alternative and numerically equivalent way of"nding the estimative of *Y, given by the Gauss}Markov}Kalman estimator, is to solve the deterministiccounterpart optimization problem

Minimize 12M(*Y!*YM )TPM ~1(*Y!*YM )

#(Y!M*Y)TR~1(Y!M* Y)N. (13)

Since constraints satisfaction is necessary for the solutionof the problem, the foregoing relation enables the estab-lishment of a weighting relationship between PM and R inorder to assure priority to constraint satisfaction overminimizing the objective function.

2.3. Numerical implementation

The numerical implementation of this method is car-ried out iteratively in a sequence of two phases, namely,(i) phase of constraints acquisition, and (ii) the gradientphase (Rios Neto & Pinto, 1987; Madeira, 1996). Theiterative process begins in the phase (i) where from a feas-ible point a search aims to capture the equality con-straints, including the inequality constraints that become

active during the search process. When the equality con-straints satisfy the limits of tolerable error, a phase (ii)begins to further reduce the objective function value. Thisprocess is carried out by relaxing the order of magnitudeof error V (Eq. (10)) to allow priority for a non negligiblesearch step in the direction of the minimum. After com-pleting the gradient phase, phase (i) is repeated in orderto reacquire the constraints within the limits of tolerableerror. This sequential process continues until the pre-scribed convergence criterion is met.

The numerical algorithm implemented in this work isbased on the procedure proposed by Rios Neto andPinto (1987) and is summarized below.

(1) The search parameters a and b in each step aregiven by

1!a"1!b"s, (14)

in which

s"MinMsi, i"0,1, 2,2,m

h#I

gN, (15)

s0"1,

s2jh2j(YM )"3R

jqj, j"1, 2,2,m

h, q

j<1,

s2k`mh

g2k(YM )"3R

k`mhqk, k"1, 2,2.I

g, q

k<1. (16)

(2) The dispersion g is determined using Eq. (10):

Z"M(*YM #g)#V"ZM #Mg (17)

and

Z!ZM "eZ"Mg. (18)

Assuming no cross correlation in eZ components, andevaluating their variances by using a maximum likeli-hood criterion, results:

n+j/1

M2ijPM

jj"E[(eZ

i)2], i"1, 2,2, m

h#I

g. (19)

Let

E[(eZi)2]"3R

iqi. (20)

Hence

n+j/1

M2ijPM

jj"3R

iqi. (21)

In the application reported in this paper, the solution ofEq. (21) was found using the following approximationcriterion:

PMjj"Max

i

3Riqi

M2ijni

, i"1, 2,2, mh#I

gand M

ijO0,

(22)

in which niis the number of for each row i of matrix M.

F. Madeira, A. Rios-Neto / Automatica 36 (2000) 427}438 429

(3) In each step, the search parameter c was foundusing the approximation relation:

c"Mini

J3PMij

+iF(YM )

, +iF(YM )O0. (23)

3. Launch vehicle open-loop trajectory optimization

3.1. Vehicle modeling and mission description

The satellite launch vehicle (VLS) is a conventionalnonlifting rocket propelled launch vehicle, composed offour expendable solid-fuel stages. The #ights of the "rstand second stages are assumed to be carried out into thelower atmosphere and the atmosphere e!ects are neglect-ed for the #ight of the third and fourth stages. The burnout time of a given stage coincides with both the currentstage structure jettisoning and the ignition of the nextstage. A long coasting phase takes place between thethird and fourth stage #ights.

The mission objective consists of inserting the max-imum satellite mass into a circular orbit. The orbit isrequired to have 750 km of height and an inclinationof !253.

In order to assure the vehicle structural integrity,a maximum aerodynamic loading is imposed

qa!qaMAX

40 (24)

in which qaMAX

"4000 Pa rad.

3.2. Equations of motion

In the present approach the vehicle is treated as a pointmass model, and hence the rotary dynamics are notaccounted for. The governing equations are expressed ina moving coordinate system (Madeira, 1996) with theorigin at the Earth's center (r

%"6 378 145 m) and the

x-axis pointing toward the vehicle. The vehicle motion isconstrained to the xy-plane, y-axis points forward, andthe z-axis is perpendicular to the motion plane, thuscompleting the right-handed coordinate system.

The equations of motion are then expressed as

XQ1"r5"u,

XQ2"u5 "

l2r!

kr2

#

FT

Msin a

Pcos a

Y!

Dx

M,

XQ3"v5 "!

uv

r#

FT

Mcos a

Pcos a

Y!

Dy

M,

XQ4"tQ "A

FT

sin aY!D

ZMv B

sin/

sin h,

XQ5"hQ "A

FT

sin aY!D

ZMv Bcos /,

XQ6"/Q "

v

r!A

FT

sin aY!D

ZMv B,

XQ7"MQ "!b, (25)

in which

FT"C

Vb!A

SP

ATM(h), P

ATM(h)"

a2(h)o(h)

c,

M(t0)"M

S#M

E#M

P, Mach"

<

a,

Do (Dx,D

y,D

z), DDo D"1

2o(h)<2C

D(Mach, a). (26)

In Eq. (25), r, u and v are the vehicle radial distance, theradial and tangential velocities; t, h and / are Eulerangles relating the moving coordinate system to a non-rotating Earth-centered inertial system; they are namednode angle, inclination angle, and #ight path angle, re-spectively. F

Tand D are the vehicle thrust and the

aerodynamic drag; aP

and aY

are the control thrustde#ection angles in #ight plane (pitch) and out of #ightplane (yaw); and k is the Earth's gravitational constant(3.986012]1014 m3/s2).

In Eq. (26), CV

and b are the stagewise constantvacuum exhaust gas speed and the propellant mass rate,respectively; A

Sis the engine nozzle area, P

ATMis

the atmospheric pressure, h is the geometric altitude,o is the air density, a is the speed of sound, and cis the gas constant of air ("1.40); M

S, M

Eand M

Pare payload/satellite mass, the vehicle structure mass,and the propellant mass, respectively; D

x, D

yand D

zare the aerodynamic drag components resolved intothe moving coordinate system; < is the total airspeedand a is the angle of attack. The drag coe$cient isMach number and angle of attack dependent and sincethe problem deals with a nonlifting vehicle the aerody-namic lift is not accounted for into the equation ofmotion. The geometric altitude dependent functionsa and o are given by an analytical function (Du!ek& Shau, 1975) based on the US Standard Atmosphere(Regan, 1984).

3.3. Statement of the problem

Trajectory optimization problem consists of "ndinga control history uT(t)"[a

P, a

Y], t

04t4t

&, that mini-

mizes the objective function

F(X(t&), t

&)"!X

7(t&)"!M

S(27)


subjected to the "nal time equality constraints, i.e., thespeci"ed circular orbit injection conditions

h(X(t&), t

&)"C

r(t&)!r

&u(t

&)

v(t&)!Jk

r&

h(t&)!h

&D"e, (28)

given all the initial state components but X7, since the

vehicle initial mass is not known at all once the satellitemass was not calculated yet; and subjected to the giveninitial conditions

X7(t0)"X

i0, iO7, (29)

where MSis the mass at "nal time, viz., the payload mass

to be injected into orbit, and the aerodynamic loadinginequality constraint is converted into an equality con-straint. Let X

8be an additional state variable, such that

X8"G

qa!qaMAX

if qa!qaMAX

50,

0 if qa!qaMAX

(0,

in which X8(t0)"0, and let h

5be an additional equality

constraint, such that

h5(X(t

&), t

&)"X

8(t&)"e

5. (30)

Hence, the additional equality constraint, to be met atthe "nal time, assures the inequality constraint will notbe violated along the whole vehicle #ight trajectory.

3.4. Parametrization of control history

In order to parametrize the control history, the trajec-tory is "rst divided into a sequence of segments, separ-ated from each other by meshpoints. Every discontinuityin state and control coincides with a meshpoint. Thus, foreach burning out time a meshpoint is assigned due to thestage structure jettisoning.

Using a Rayleigh}Ritz method, the control history canbe approximated by an arbitrarily parametrized function(Williamson, 1971)

u(t)+u(a, t). (31)

Once the values of the parameter vector a are deter-mined, u(a, t) depends only on time. Therefore, the con-trol in each #ight trajectory segment is approximated asarcs of a function that is usually taken to be a polynomial(e.g. Williamson, 1971; Rios Neto & Ceballos, 1979;Ceballos & Rios Neto, 1981; Ceballos, 1979). In thispaper, u(a, t) is approximated by linear functions, so thatthe control history is determined by the parameters:

aTP"[a

P0, a

P1,2, a

PN], aT

Y"[a

Y0, a

Y1,2, a

YN],

where aPi

and aYi

are the pitch and yaw control values atthe ith meshpoint, and N is the number of #ight trajec-tory segments. The control for the ith trajectory segmentis then given by

aP(t)"(a

Pi`1!a

Pi)

t!ti

ti`1

!ti

#aPi

,

aY(t)"(a

Yi`1!a

Yi)

t!ti

ti`1

!ti

#aYi

, (32)

in which ti4t4t

i`1and t

0(t

1(2(t

N,t

&.

3.5. Problem solution

The equations of motion depend only on the para-meter vectors a

Pand a

Y, the duration of coasting #ight

tC, and on the satellite mass M

S. The trajectory optimiza-

tion solution problem consists of determining an opti-mizable parameter array y, given by

yT"[aTP, aT

Y, t

C,M

S] (33)

that

Minimizes F(y) (34)

subject to h(y)"e, (35)

where F and h have already been de"ned in Eqs. (27), (28)and (30).

The Jacobian matrix

Cdh

dyDt/t&

needed in the linearized approximation (see Eq. (6)) isgiven by

Cdh

dyDt/t&

"CLh

LXDt/t&CLX

Ly Dt/t&

#

Lh

Ly. (36)

Since the constraint vector h does not depend explicitlyupon the optimizable parameter array y, the JacobianLh/Ly is a null matrix. Unlike

CLh

LXDt/t&

,

which has a straightforward analytical form, theJacobian

CLX

Ly Dt/t&

is calculated numerically using the central-di!erencemethod (Gill, Murray & Wright, 1981).


Table 1Open-loop solution: constraints satisfaction and the resulting satellite mass

h1

(m) h2

(m/s) h3

(m/s) h4

(deg) h5

(Pa s) MS

(kg)

Stochastic 1st strategy !5.0582E!3 !7.0608E!6 !4.1137E!6 !2.7217E!8 1.6739E!7 182.9730Gradient 2nd strategy !6.6903E!3 !2.4927E!5 !2.2198E!5 1.8844E!8 4.5541E!8 186.1121Projection 3rd strategy !4.4446E!5 !7.1028E!8 !1.4814E!8 5.4622E!12 3.1555E!9 185.7202Multiple shooting26 $5.E!3 $5.E!3 $5.E!3 * * 187.8955

Fig. 1. Open-loop solution height vs. time.

3.6. Open-loop trajectory optimization simulations

In what follows, results obtained for a typical MEC-B(Brazilian Complete Space Mission) satellite mission,from the Alca( ntara launch site, using a solid propelledfour stages launching vehicle, to place a satellite intoa 750 km height circular orbit of !253 of inclination arepresented (Madeira, 1996).

With the aim of maximizing the satellite mass capabil-ity, three di!erent strategies have been implemented. Inthe "rst strategy, a minimum number of parameters havebeen used. The pitch and yaw steering controls varylinearly from the current stage ignition till the burn outtime/stage structure jettison. After that, the controlsswitch to another slope and so on. The second strategy issimilar to the "rst one, except that it assigns two straightline segments for the control vector components in each#ight stage; and the third, most re"ned strategy, assignsfour straight-line segments for the control vector compo-nents in each #ight stage.

Even though the "rst strategy is the least re"ned one,the results obtained and depicted in Fig. 1 and Table 1indicate it yields quite satisfactory results when com-pared to an optimal multiple shooting solution to thesame problem (Zerlotti, 1990). Moreover, the second and

third strategies add no further improvements on theresults compared to the "rst strategy. The height pro"lesfor the second and third strategies were not shown inFig. 1 since they are quite so close to the "rst one thatthey overlap for almost the entire #ight path. For thethree implemented near-optimal strategies, the controlpro"les as well as state pro"les are closely related to theiroptimal counterpart (Madeira, 1996). However, the pro-"les of the aerodynamic states a (angle of attack) and qaexhibit major di!erences compared to the optimalcounterparts. Such di!erences are due to the fact that atthe meshpoint times the control switches from one slopeto another (Madeira, 1996).

4. Launch vehicle closed-loop guidance andcontrol procedure

The solution of a trajectory optimization problemgives a parameter array y (Eq. (33)). Under ideal #ightconditions, i.e., a #ight in which the e!ects of modeluncertainties and environmental perturbations are negli-gible, the use of y throughout the #ight trajectory in anopen-loop fashion will lead to the satisfaction of thecondition:

hj(y)4De

jD, (37)

that is, each mission constraint hjwill be satis"ed with as

much accuracy ej

as our trajectory optimization proce-dure performance allows.

However, for a real #ight that is no longer true. Sincemost of the parameters required for computing y aresubjected to uncertainties, and the environmental e!ectstend to perturbate the vehicle's #ight away from theoptimal nominal trajectory, an onboard in-#ight updat-ing of parameter array y is in order, otherwise unsuccess-ful constraints satisfaction will result, leading to themission objectives ful"llment failure. Such an onboardin-#ight updating demands for a closed-loop guidanceand control procedure.

Since now one is concerned with an actual #ightmission, the vehicle initial mass is a known parameterand hence the satellite mass too. The closed-loop guid-ance and control problem does not account for the satel-lite mass optimization. Hence, this problem does notrequire the minimization of an objective function and the


left-hand side of Eq. (8) is set equal to zero, leading to thefollowing a priori information:

y0-$"y/%8#g, (38)

where y0-$ is the previously known part of the vector ofparameters, needed to be updated for the remaining partof the #ight control, and it is assumed that the vehicledeviation takes place inside a suitable neighborhood ofthe reference trajectory such that the new control historyy/%8 lies inside a region around the old control historyy0-$, bounded by the uniformly distributed unbiasedstochastic error vector g. For the closed-loop guidanceand control problem the observation like equation forconstraint satisfaction is the same as Eq. (35) of thetrajectory optimization problem. However, for the sakeof real time operation feasibility, each control historyupdating will be performed in a single iteration, implyinga equal to zero in Eq. (6).

The closed-loop guidance and control then consists ofupdating the part left of control history yT"[aT

P, aT

Y, t

C]

at some prescribed time instants during the #ight, solvingthe "ltering problem:

y0-$"y/%8#g, h(y/%8)"e. (39)

After linear perturbation this problem is converted intoa linear optimal "ltering problem solved with theGauss}Markov}Kalman estimator (Eq. (11)) providingthe optimum correction increment *y( , leading to

y( /%8"y0-$#*y( . (40)

For the onboard in-#ight control, the actual stateX(q

k) at some assigned meshpoint time q

k(k"1, 2,2,¸,

q0,t

0, q

&,t

&) is not, to a lesser extent, equal to the

nominal reference state XH(qk). The nominal reference

state was found earlier, by the o!-line trajectory optim-ization procedure and afterwards by the most recentupdate pf the closed-loop guidance and control. Theactual state is externally supplied by some navigationsensor output, e.g., GPS, INS, a combination GPS/INS,etc. The stochastic gradient projection method takes theinformation provided by the actual state X(q

k) as start-

point to update the control history which will actually beused in the time interval (q

k, q

k`1). The time meshpoints

assignment follows the same guidelines stated for par-ametrization of control history.

At beginning of the #ight (q0), the actual control his-

tory to be used until q1

is

y0"[a00, a0

1,2, a0

N, t0

C]T (41)

which has been previously computed in the open loopsolution.

In order to simulate the real #ight conditions, dynamicmodel parameter uncertainties are arti"cially introduced

and Eq. (38) are numerically integrated. The perturbedvalue of a parameter p

jdue to model typical uncertain-

ties *pj

is de"ned as:

p6j"p

j#rand

j*p

j, (42)

in which

!14randj41 (43)

is an unbiased uniformly distributed random number.It must be stressed that the calculation for updating

the control history y are performed using the unpert-urbed parameters p

j. The perturbed parameters p6

jare

used ad hoc to drift the vehicle away from the nominalreference trajectory through the introduction of modeluncertainties in an attempt to simulate a real #ight.Therefore, in order to accomplish the "ltering task, theperturbed values are not known a priori, only their e!ectsare observed a posteriori.

Prior to the kth updating, the actual control history is

yk~1"[ak~1j

, ak~1j`1

,2, ak~1N

, tk~1C

]T, qk~1

(t(qk,

(44a)

in which tj(q

k~1(t

j`1. The actual state Xk~1(q

k) is

found by taking the perturbed parameters p6j

and theactual control history yk~1 into the dynamic Eq. (38) andintegrating them from q

k~1to q

kand taking Xk~1(q

k~1)

as initial condition.If q

k(t

j`1, then ak~1

jis still the "rst actual vector

component of y. Thus,

ak~1j

(qk)"(ak~1

j`1!ak~1

j(q

k~1))

qk!t

jtj`1

!tj

#ak~1j

(qk~1

)

(45a)

and ak~1j

(qk) replaces ak~1

j(q

k~1) into Eq. (44a).

If, qk't

j`1, then a

jmay be dropped from the control

history, leading to

yk~1"[ak~1j`1

, ak~1j`2

,2, ak~1N

, tk~1C

]T (44b)

and thus

ak~1j`1

(qk)"(ak~1

j`2!ak~1

j`1(q

k~1))

qk!t

j`1tj`2

!tj`1

#ak~1j`1

(qk~1

)

(45b)

replaces ak~1j`1

(qk~1

) into Eq. (44b).Once the "rst actual vector component has been suit-

ably updated for t"qk

(Eq. (45)), the control historyyk~1 (Eq. (44)) is taken to be the a priori informationy0$$ (Eq. (39)) and, from an initial state Xk~1(q

k), the

updated control history y( /%8"yk is next calculated usingthe stochastic gradient projection method. The new con-trol history yk will be the actual control history to be usedin the time interval q

k(t(q

k`1.


Table 2Parameter uncertainties

Parameter Uncertainty (%) Parameter Uncertainty (%)

CD0

10 CV

5C

La 10 b 0.5M

E1 a 0.1

MP

1.5 c 1o 10

Fig. 2. Control vs. time for the standard case (case study 1).

4.1. Closed-loop solution simulation

The numerical simulations of the closed-loop guidanceand control procedure were performed using typicalparameter uncertainties for the MEC-B launch vehicle(Madeira, 1996), as depicted in Table 2.

The control history updating was performed threetimes during each booster phase and once during thecoasting phase (¸"13, number of meshpoints q

k). This

approach leads approximately to one updating at every20 s during the booster phases. Such a low updatingfrequency is part of an endeavor to achieve real timeoperation feasibility.

The real time operation will not be addressed. For thepurposes of this phase of exploratory study it is assumedthat closed-loop guidance and control procedure consistsof a reactive system operating under the hypothesis ofsynchronous approach (Benveniste & Berry, 1991).

The synchronous approach hypothesis considers anidealized system which produces its output synchronous-ly with its input. Therefore, at time q

k, when the kth

updating takes place, the actual state X(qk) (input) is

determined and hence the new control history yk (output)is synchronously found, then replacing at the same timethe old control history yk~1. In other words, it is assumedthat the elapsed time for computing yk is negligible and att"q

kthe actual control history switches from yk~1 to yk.

In this paper the numerical results for two case studiesare presented. The "rst case study considers the standardMEC-B launch vehicle con"guration. The second casestudy considers the upper stage burning time as beingfree.

4.1.1. Case study 1: the standard caseThe standard case considers the same launch vehicle

con"guration used for open-loop trajectory optimizationand all assumptions done up to this point remains stillapplicable. Two realizations, named A and B, found intests (Madeira, 1996) to be representative of the range ofbehavior due to typical perturbations were selected forpresentation.

The Fig. 2 illustrates the di!erences in the pitch andyaw control histories for realizations A and B, respective-ly, as compared to the open-loop control histories foundby the trajectory optimization procedure. Though the

entire left control history is calculated for every updating,only the control history actually used is shown. Duringthe coasting phase, between the third and fourth stage#ights, there is no control action and therefore no controlhistory is shown. Table 3 shows the resulting orbitcharacteristics.


Table 3Orbit characteristics for the closed loop standard case (case study 1)

Realization Apogee height Orbit Orbit inclination(m) eccentricity (deg)

A Closed-loop 750051.3786 7.0763E!6 !25.000159Open-loop 955135.2059 2.5867E!2 !25.312379

B Closed-loop 731829.9049 2.5564E!3 !24.999736Open-loop 687526.6107 1.3953E!2 !24.952062Nominal orbit 750000 0 !25

The discontinuities in the closed-loop control histories(Fig. 2) are due to the switching from one control historyto the next, at the updating times. Moreover, the upperstage #ight demands larger variations on the closed-loopcontrol histories once the time-to-go is even more closeto zero and the upper stage burning time is "xed.

4.1.2. Case study 2: free upper stage burning timeThe MEC-B launch vehicle is composed by four stages

propelled by solid-fuel rocket engines. Solid-fuel rocketengines have two features which distinguish them fromtheir liquid-fuel counterparts, namely, nonthrottleabilityand nonshuto! capability (burning time "xed). Forclosed-loop guidance and control purposes, the solid-fuelfeatures become a drawback. In order to cope with theuncertainties and perturbations, the additional controldegree of freedom assured by the throttleability andshuto! capability of a liquid-fuel propelled rocket enginewould improve the resulting orbit characteristics. There-fore, a liquid rocket engine would be desirable at least forupper stages propulsion.

In order to investigate the performance of a liquid-fuelpropelled upper stage during a real #ight mission, thetime of burning of the fourth stage was left to be free, i.e.,it was assumed that the fourth stage exhibits the shuto!capability. This approach is equivalent to assume thatthe fourth stage is propelled by a liquid-fuel rocket engineat full-throttle, displaying the same characteristics of itssolid-fuel counterpart, except the burning time which,due to the shuto! capability, is free.

Initially, a nominal reference time of burning forthe fourth stage equal to the original fourth stagesolid-fuel time of burning was assumed. Such a timewas then appended to the control history y and washence updated accordingly. Also, even the resulting massof the fourth stage propellant and satellite was known, itwas assumed that their masses are a priori unknownapart.

Note that a fraction of the satellite mass calculated bythe trajectory optimization problem should be allocatedto an additional propellant mass, leading to a smalleractual satellite mass. With this approach after performing

some simulations, one should be able to establish anadditional propellant mass which could safely ac-complish the mission. On the other hand, one couldexplore the other liquid-fuel rocket feature by assumingone can control the mass #ow rate (throttleability). How-ever, the implementation of such a feature is not sostraightforward.

Fig. 3 presents the closed-loop control histories behav-ior obtained in the simulation study for realizationsA and B as compared to the open-loop control history. Itcan be noticed that the additional degree of freedomprovided by the shuto! capability turned out to demandsmaller variations on the closed-loop control histories forthe upper stage #ight as compared to the standard case.Table 4 shows the resulting orbit characteristics and thefourth stage time of burning.

4.2. Closed-loop results analysis

For the standard case, the results presented show thatfor a real #ight, the closed-loop guidance and controlprocedure yields smaller orbital injection errors. For theMEC-B data collection class satellite, the tolerable errorsfor orbital injection are (Madeira, 1996):

f apogee height: $50 km,f orbit eccentricity: $0.05,f orbit inclination: 3.

Accordingly to Table 3, for all realizations the closed-loop procedure could inject successfully the satelliteinto the desired orbit; the orbital injection errors areinside the tolerable interval. However, the applicationof the control history in an open-loop fashion couldnot achieve the apogee height error inside the tolerableinterval.

For the free upper stage burning time, the resultspresented suggest that a liquid-fuel propelled rocketengine for the upper stage propulsion would achievesmaller orbit injection errors. Beyond, an upper stagecon"guration with a burning time additional capabilityof about 2 s seems to be suitable for complete the missionsafely. Such a con"guration demands approximately for


Table 4Orbit characteristics for the closed loop and free upper stage burning time

Realization Apogee height Orbit Orbit 4th stage(m) eccentricity inclination (deg) burning time (s)

A 757450.8277 1.1301E!3 !25.001406 70.100870B 746865.1286 4.4664E!4 !24.998467 73.157844Nominal orbit 750000 0 !25 72

Fig. 3. Control vs. time for the free upper stage burning time (casestudy 2).

a satellite mass fraction in excess of 23 kg to be allocatedas additional propellant mass. The amount of fuelwhich could eventually be saved may be further used forexecution of orbital maneuvers in order to improve thesatellite lifetime.

5. Conclusions

The use of a stochastic gradient projection method fortrajectory optimization and low frequency closed-loopsatellite launch vehicle guidance and control waspresented and tested.

The trajectory optimization results obtained show thepotential of the stochastic gradient projection method foraccurate orbital injection of payloads. Even the poorestapproach for the control history parametrization (the"rst strategy) achieved tight constraints satisfaction. Themethod was also successful in maximizing the payloadcapability. The satellite mass obtained using the secondstrategy was only 0.95% below the solution reported byZerlotti (1990) who used an optimal indirect multipleshooting method.

The proposed closed-loop guidance and control pro-cedure was able to cope with the uncertainties and per-turbations typically found during a real #ight missionand inject satisfactorily a data collection class satelliteinto orbit. The computational workload demanded andreliability indicate the feasibility of real-time application.

Acknowledgements

This research was supported by CNPq * NationalResearch Council. It was developed while the "rst authorwas graduate student at the Technological Institute ofAeronautics (ITA) and the second author was seniorresearcher at the National Space Research Institute(INPE).

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Fernando Madeira was born in Brazil in1966. He received his B.S. degree in Phys-ics from Federal University of SantaCatarina*UFSC in 1987, his M.S. degreein Physics from State University of Cam-pinas * UNICAMP in 1990, and hisPh.D. degree in Aeronautical Engineeringfrom Technological Institute of Aeronaut-ics* ITA in 1996. Since 1996 he has beenwith the Flight Mechanics Group atEMBRAER, where he has worked on thedevelopment of regional jets ERJ-145,

ERJ-135, and ERJ-170, and on the light attack airplane EMB-314/ALX.He is member of the Brazilian Society of Automatica * SBA andBrazilian Society of Mechanical Sciences * ABCM.

Atair Rios Neto was born in Brazil onOctober 31, 1942. He Received his B.S.degree in Mechanical Engineering fromthe University of Sa8 o Paulo * USP in1966 and a Ph.D. degree from the Univer-sity of Texas at Austin in 1973.

From 1967 to 1979 he held the positionof Professor at the Polytechnic School ofEngineering of the University of Sa8 o Paulodeveloping teaching and research activitiesin applied mechanics and optimal control.During this period while attending his doc-

toral program he was a Research Assistant at the Department ofAerospace Engineering of the University of Texas at Austin (1969 to1971). Still during this period he helped to organize: the "rst graduatecourse in the area of systems control of the University of Sa8 o Paulo(1973}1974); the "rst Orbital Dynamics and Control group of Brazil, at


the National Space Research Institute * INPE (1976}1978); and theSchool of Engineering of Ilha Solteira of the Paulista State University*UNESP (1979). From 1980 to 1996 he held the position of researcherat the National Space Research Institute* INPE, developing researchand graduate teaching activities in space mechanics and control. Dur-ing this period he organized, and was the "rst head of, the SpaceMechanics and Control Department of INPE (1982}1985); the presi-dent of the Brazilian Society of Automatica (1985}1987); the generalmanager of the corporate university of Empresa brasileira de Aero-na& utica* EMBRAER (on leave, 1988}1991); the head of the Center ofSpecial Technologies of INPE (1995}1996). Since 1997, he is a fullprofessor and researcher at the University of the Paraiba Valley* UNIVAP, in the Institute of Research and Development* IPD, as

the coordinator of the Neural Networks in Signal Processing andOptimal Control of Dynamic Systems research group.

His scienti"c and technical publications have been mainly in theareas of statellite launching, attitude and orbit control and appliedoptimal control. His current research interest is on neural networksapplied to control.

Dr. Rios Neto is a Sigma Gamma Tau member; a cofounder memberof the Brazilian Society of Automatica * SBA (a$liated to IFAC);a cofounder member of the Brazilian Society of Mechanical Sciences* ABCM. He has served on many national and international commit-tees, has been for many years an Ad Hoc consultant for Brazilianresearch agencies; and presently is also a member of the editorial boardof the Journal of the Brazilian Society of Mechanical Sciences.


guidance and control of a launch vehicle using a stochastic gradient projection method

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