Upper Stage Launch Vehicle Servo Con

1

AbstractAn attitude control system using reaction thrusters

for the upper stage of a launch vehicle is considered. The thruster configuration (position and direction) determines control system response, fuel consumption, effective torque and system fault tolerance. We propose a procedure for finding the optimal thruster configuration with desired control effectiveness over the range of selected torque commands. An optimization technique called Particle Swarm Optimization is used for the numerical experiments. The validity of the solution is checked through computer simulations.

Index Terms PWPF, RCS, PSO, Thruster Configuration,

Optimization Technique

I. INTRODUCTION

Up-to-date launch vehicles carry multiple satellites and every satellite has its own demanding criteria for safely placing itself into orbit. The precision of the satellite orbit is determined by the attitude control efficiency of the upper-stage launch vehicle. The Reaction Control System (RCS) is plentifully used for attitude control means of the upper-stage launch vehicle in space. For three-axis control of the launch vehicle, multiple thrusters are required. In the case of the upper-stage launch vehicle, there is a space restriction. The thruster must be set up in the connection region that is between the upper part and the lower part of the launch vehicle. The number of thrusters is also limited because the pressure level of the fuel tank must be maintained.

In the case of linear thruster, several theoretical works have been done about thruster configuration design for limited fuel flow rate, maximum robustness to thruster failure, maximum margin of safety [1], [2]. On-off one-sided thruster may be hard to analyze by linear algebraic theory, because it has a lot of nonlinear characteristics. In this research, the control system is assumed to use on-off one-sided thrusters. In general, control system response, fuel consumption, effective torque and system fault tolerance are affected by the thruster configuration (position and direction).

Manuscript received January 31, 2003. Authors Address : all authors are with department of aerospace engineering ,KAIST, Taejon , Korea. Contacting Author: Min-Jea Tahk E-mail : [email protected] Taejon , youseong, guseong 373-1 KAIST, AE, FDCL

We propose a procedure for finding the optimal thruster configuration. The optimal solutions are given through optimizing a cost function related to control effectiveness. An optimization technique called Particle Swarm Optimization (PSO) is used for optimization.

Section II presents the general formulation of attitude dynamics of launch vehicles. In section III we explain the procedure of gain selection for quaternion feedback loop, the process of characteristic parameter determination for the PWPF modulator, and an example of the command distribution logic for a thruster configuration of the previous design. Section IV introduces a PSO optimization algorithm which is one of the newest optimization techniques. Section V considers the process of problem formulation and cost function definition. In section VI computer simulations are performed to show performance of the optimized configuration.

II. LAUNCH VEHICLE ATTITUDE DYNAMICS The orbit motions of upper-stage launch vehicle are

composed of translational and rotational motion. Their independent characteristics allow one to deal with two motions separately. For attitude control of a vehicle, only rotational equations are required. The rotational (attitude) motions of a rigid spacecraft in space are described by Eulers equations (1) [3].

|BdHM Hdt

= + GG GG

(1)

We deduced the rotational motion equation from (1)

J J + =G G G G (2) where,

( , , )Tp q r =G : Angular velocity of the vehicle w.r.t. inertial space 1 2 3( , , )

T =G : Total external torque J : Inertia dyadic w.r.t. the center of mass of the vehicle

III. ATTITUDE CONTROL SYSTEM DESIGN The attitude control system treated in this paper is a

quaternion feedback method. In the past, the open-loop type algorithm or the Euler angle feedback was frequently used for

Upper-Stage Launch Vehicle Servo Controller Design Considering Optimal Thruster Configuration

Tae Won Hwang, Chang-Su Park, Min-Jea Tahk, Hyochoong Bang

AIAA Guidance, Navigation, and Control Conference and Exhibit11 - 14 August 2003, Austin, Texas

AIAA 2003-5330

Copyright 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

control system in space. For the quaternion feedback system, the entire usable attitude related information is expressed in the body axis. There is no discordant coordinates problem within attitude angle and angular velocity information. Also, the attitude conversion of the shortest course through an Euler axis rotation is possible. The conversion equation from Euler angle ( , , ) to quaternion ( 0 1 2 3Q q q i q j q k= + + + ) is as follows.

0

1

2

3

cos( ) cos( ) cos( ) sin( ) sin( ) sin( )2 2 2 2 2 2

sin( ) cos( ) cos( ) cos( ) sin( ) sin( )2 2 2 2 2 2

cos( ) cos( ) sin( ) sin( ) sin( ) cos( )2 2 2 2 2 2

sin( ) cos( ) sin( ) cos( ) sin( ) cos2 2 2 2 2

q

q

q

q

+

+

+

= ( )2

(3)

The attitude error quaternion is the coordinate

transformation quaternion from the current attitude Biq to the command attitude Ciq . The conversion equation (3) gives

Biq and

Ciq . Error quaternion eq is expressed as (4) from the

relationship of coordinate transformation. The upper script * denotes the multiplication operation of two quaternions.

0 1 2 3 0

1 0 3 2 1

2 3 0 1 2

3 2 1 0 3

*( ) ( )

c c c c b

c c c c b

c c c c b

c c c c b

C C Be B i i

q q q q q

q q q q q

q q q q q

q q q q q

q q q q

= =

=

(4)

The concept of quaternion control is to apply the quaternion

control toque to the vehicle. The quaternion control torque vector has a magnitude which is proportional to the rotary angle of the error quaternion and direction parallel to the Euler axis. The feedback error angle is given as equation (5). When the rotation angle is small, the error angle can be expressed as (6).

e = G (5)

[ ]1 2 32 , ,e e eq q q (6) where, [ ]0 1 2 3, , , cos ,(sin )2 2e e e eq q q q e

= G

102 cos ( )eq = : Rotation angle ,

[ ]1 2 3, , Te e e e=G = 1 2 3sin sin sin2 2 2

( , , )Te e eq q q : Euler Axis vector

It was proved in [4] that control command (7) makes the

closed-loop stable. p du K K = (7)

The block diagram of the RCS attitude tracking loop is

shown in Figure 1.

The attitude control system is composed of the quaternion

feedback loop, the PWPF modulator, and the command distribution logic.

A. Feedback Loop Gain Selection

The control gains ,p dK K of equation (7) are acquired from the solution of the LQR problem for which the performance index is given by (9) and the state equation (8) disregards the cross product term of the original equation (2).

State Equation:

,

0 1 0,

0 0 1

Jx Ax Bu u

A B

= + = = =

(8)

Performance index: 2221

2

(

1 0 1,0 0

)t

uc

V x d

Q Rc

= + = =

(9)

The optimal control gain and control command are expressed by (10). The matrix P of (10) is the solution of the arithmetic riccati equation. It can be solved analytically as follows: Optimal Control & Gain :

1

1

2

0

Tp d

T T

G R B P K K c c

A P PA PBR B P Qu G x

= = = + + =

= (10)

Closed characteristic transfer function : 2 (2 )s c s c+ + (11)

Control feedback gains are determined as follows.

, 2 , 1, 2,3i ii ip d

i i

J JK c K c i = = =

iJ : Moment of inertia

i : Maximum external torque

T FlightDynamics

Kd

Kp

+

-

IMU

-

PWPF Modulator

qb

qcUonUoff

1

-1m s + 1

KmQuaternionComputation

Fig. 1. Block diagram of RCS attitude tracking loop

3

B. PWPF Modulator Design Pulse Width Pulse Frequency(PWPF) modulators are used

for thruster control of numerous satellites and launch vehicles. As in Fig.1, the PWPF modulator is composed of a first order lag filter and Schmitt trigger. The pulse sequence (1,0,-1) proportional to the input command size will drive the thruster valves. The experimental results of control system which uses PWPF agree with results of the linear analysis well. The boundary limits of the input which maintains linear characteristics are given as (12). The effective dead band of a modulator is derived as (13) [5].

min

max 1

onm

offm

UK

UK

r

r

== +

(12)

min

( )ln(1 )on onoff offm mm m

U U U UK KT = (13)

There is no relation between the PWPF modulator static

characteristics and the vehicles moment of inertia. The parameters of the PWPF modulator are set as follows: The sampling frequency of the control loop is 50 Hz. The excess values of maxr is 2%. The target error of PWPF is chosen as 0.5D . The DC gain of lag filter mK is selected as 3. When the effective dead band is smaller than the minimum pulse width, the internal parameters of PWPF are given as (14).

0.02/( )

on m p PWPF

off m

m m m on off

U K KU K

K U U

= = = (14) [ , , ] 0.5 0.02 secTPWPF PWPF m = = =D

C. Command Distributon Logic The basic thruster configuration of the concept design is

illustrated in Fig. 2. There are 8 thrusters parallel to the Y-Z plane of the vehicle. Positive pitching moment is acquired by turning on thrusters number 5 and 6. Positive yawing moment is acquired by turning on thrusters number 3 and 4. Positive rolling moment is acquired by turning on thrusters number 1,3,5 and 7.

Thruster

Main Nozzle

Thruster

Main Nozzle

Payload(satellite)

Thrusters(8 )Main

Nozzle

Final Stage

Payload(satellite)

)Main

Nozzle

Final Stage Thruster

Main Nozzle

Thruster

Main Nozzle

Payload(satellite)

Thrusters(8 )Main

Nozzle

Final Stage

Payload(satellite)

)Main

Nozzle

Final Stage

The control command are given in 3 directions; pitch, yaw,

roll. As in Fig.2, 8 thrusters are needed to control the system. Thus, we require another command distribution logic. This can be solved by deriving a minimum norm type pseudo inverse E+ from the non-dimensional thruster control matrix E given in (15). The thruster control matrix E is the contribution measure of controllability for a thruster configuration. Thruster Control matrix : E

1 1

2 22 2 0 0 2 2 0 0

13 30 0 2 2 0 0 2 2

41 1 1 1 1 1 1 1

7 7

8 8

p

y

r

E

= =

# # (15)

Command Mixing Matrix : 1( )T TE E EE+ = 1

2

3

7

8

p

y

r

E

+

=

# (16)

The possible state of one-sided thrusters is just ON or OFF.

The relation (16) is used with a threshold with the size of . When an input is over (positive small value), the output should be ON for the thruster. On the other hand, when an input is smaller than , the output should be OFF. The non-dimensional effective torque due to torque command is derived as (17)

1

2

3/ /

7

8

( ) ( )e

e

p p

y on off on off y

rr e

E f E f E

+

= =

# (17)

1 2

-1 -2

-3 3

-4 4

0.5m

0.5m

1 2

-1 -2

-3 3

-4 4

11 22

6 5

8 33

7 44

0.5m

0.5mMain Nozzle 1.3m

1 2

-1 -2

-3 3

-4 4

0.5m

0.5m

1 2

-1 -2

-3 3

-4 4

11 22

6 5

8 33

7 44

0.5m

0.5m

1 2

-1 -2

-3 3

-4 4

0.5m

0.5m

1 2

-1 -2

-3 3

-4 4

11 22

6 5

8 33

7 44

0.5m

0.5mMain Nozzle 1.3m

Fig.2. Launch vehicle modeling and thruster configuration of concept design

4

When a system adopts the linear two-sided thrusters which has a large thrust limit, the torque command agrees exactly with non-dimensional effective torque (17). However, our system uses the on-off one-sided thrusters; there happens control cross coupling phenomena. Smaller effective torque compared to the torque command is produced when the simultaneous 3 direction command are given. The torque command from the pulse modulator should be one of three values, -1,0,1. But the non-dimensional effective torque can be of any value between -1 and 1. All 27 command types can exist.

IV. OPTIMIZATION TECHNIQUE In this section, we introduce a PSO algorithm which is one of

the newest optimization techniques. The PSO was originally developed to graphically simulate the bird flock or insects. The birds and insects form groups and move in a flock. They are assumed to share all information of their group. Applying these characteristics to optimization theory, the PSO technique was born [6].

The advantage of the PSO technique is easier numerical formula and more efficient convergence quality compared to general evolutionary algorithms(GA, EP, ES..). The assumption of random population is shared with general evolutionary algorithms. But PSO does not need the original characteristics of evolutionary algorithm such as reproduction, mutation, and selection process. PSO adopt the concept of position and velocity for each population.

The random velocity term is produced to aim the best points of each population and the whole group in track. Finally, the whole group will be placed in the vicinity of the optimal solution. The risk of converging to a local minimum is avoided by adjusting the velocity terms of several populations to escape from the historically best point

The algorithm of the PSO technique has 6 steps and is briefly explained as follows:

Particle Swarm Optimization (PSO) Algorithm 1. Initialize a population of particles with random positions and velocities 2. For each particle, evaluate the desired optimization fitness function 3. Compare particle's fitness evaluation with particle's pbest. Exchange the particle's fitness value and location with pbest if it is better. 4. Compare particle's fitness evaluation with the population's overall previous best, gbest. Exchange the particle's fitness value and location with gbest if it is better. 5. Change the velocity and position of the particle according to the following equations

1 2( ) ( )id idid id best id best id

id id id

c Rand p x c Rand g x

x x

= + + = +

6. Loop to step 2 until a criterion is met.

The PSO algorithm is applied to Dejong F1 cost function in Fig.3. The randomly assigned velocity term of each population give rise to individuals converging motion to the optimal point.

De Jong F1 : 2 2( , )f x y x y= + , Optimal Solution:(0,0)

V. COST FUNCTION AND PARAMETER OPTIMIZATION

A. Cost Function Selection The desirable thruster configuration means that effective

torque vector has the same direction as the original 27 torque command and the torque vector has maximum magnitude in each direction. A cost function which is the summation of inner product of command torque and resultant torque with weighting factors is introduced to determine the optimal configuration. Tuning the weighting factor of each command direction make it possible to give weight to a specific torque direction.

As stated in above section, the thrusters must be set up in the connection region that is between the upper part and the lower part of the launch vehicle. The thruster must be located on a circle with a radius 0.7071m where the main nozzle is the center point. is the angle which determines the place where the thruster is located. The angle between the thruster direction vector and its projected vector in the Y-Z plane is called . The angle of the projected vector in the Y-Z plane with respect to the Y-axis is called .

Fig.4. Thruster configuration and definition of position/direction angle (3-dimensional Rear view of upper-stage launch vehicle)

Fig.3. Movement of population when PSO algorithm is working

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As you seen in the Fig.4, if a thruster location is decided in the first quarter, the location of 3 thrusters in 3 other quarter planes are decided automatically in accordance with the symmetry rules. The symmetry about the X axis, the symmetry about the Y axis, and the symmetry about the origin point will satisfy the condition of zero mean torques in all 3 directions. The assumption is frequently used in the actual thruster design. The arrangement of two thrusters in the first quarter plane accomplishes the thruster configuration.

The thruster position and direction angles for the first quarter plane are limited as in Table 1. For application to other planes, the results obtained from the first quarter must be converted according to the converting logic of Table 1. If the result of conversion exceeds 180, then it can be converted as the angle in 180. The minimum angle of is limited to 30 considering the structural limitation.

TABLE I BOUNDARY CONDITION FOR , , AND CONVERTING LOGIC

1st quarter 2nd quarter 3rd quarter 4th quarter 1

1

1

0 90

180 180

30 90

<

D D

D D

D D

2 1

2 1

12

=

= =

3 1

3 1

3 1

180

180

= += +=

D

D

4 1

4 1

4 1

180

180

===

D

D

The external torque in this system is a multiplied value of thrust of thrusters and torque arm length in pitch/yaw/roll directions. The torque equation about a thruster configuration in pitch/yaw/roll directions is expressed as follows.

1 2 3 7 8where, [ , , , , , ]2 , cos( ) , sin( )

cos( ) , sin( )cos( ) , sin( )

p y v

y x v

r x y

h v

x h y h

T dn T dyT dn T dx

T dy T dxT T T T T T

dn m dx r dy rT T T TT T T T

= = = +

== = == == =

" (18)

The thrust vector can be broken up into hT parallel to Y-Z plane and vT parallel to X-axis. The distance from C.G. of the vehicle to the lower end plane is the pitch/yaw direction torque arm for hT (2m). The ,dx dy which is dependent on the location of the thruster on a circle is the another pitch/yaw direction torque arm for vT . The rolling torque is generated on Y-Z plane. The thrust hT is the only source of rolling torque.

The effective torque (18) can be non-dimensionalized by dividing it by the maximum torque value of each axis. The non-dimensional thruster control matrix E can be written as (19). As there are 8 thruster, the dimension of E is 38 matrix.

iX is the maximum torque value in the i direction.

1

2

3

7

8

,

p

pp

yy

yr

r

r

X

E EX

X

= =

" (19)

8 8

1 1

8 8

1 1

8 8

1 1

(| | ) ( | | )max( | | , | | )

2 2(| | ) ( | | )

max( | | , | | )2 2

(| | ) ( | | )max( | | , | | )

2 2

i i i i

i i i i

i i i i

p p p pp

i i

y y y yy

i i

r r r rr

i i

X

X

X

= =

= =

= =

+ +=+ +=+ +=

The control commands from the pulse modulator are

separated into 3 axes. However, we must use this information to control the switch gear of 8 thrusters. Thus, the inverse value of E is needed to control the system. The command distributing matrix 1( )T TE E EE+ = is derived as (16). It is the minimum norm type pseudo inverse of E (minimum-power controller). X is the dimensional conversion parameter which is derived in (19). Considering the maximum torque parameter X , the dimensional resultant torque is given as follows.

/, ( )e e e

e e e

p p p p p

y y y y on off y

rr e r e r r e

X

X E f E

X

+

= = (20)

The of function /on offf is set to 0.3 in this research.

The 27 commands/responses against three pitch/yaw/roll

Radius=1Sphere

Normalized Command Torque n

Dimensional Resultant Torque Tr

Radius=1Sphere

Normalized Command Torque n

Dimensional Resultant Torque Tr

Fig.5. Normalized torque command and dimensional resultant torque

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directions are possible to get from equation (20). The cost function for optimization is set as (21). It is a summation of inner product between normalized command torque vectors and weighted dimensional resultant toque vectors. Minus sign is applied for minimization. In this paper, we emphasized the margin of rolling torque. Thus, we set higher weightings in rolling torque direction. The optimal thruster configuration is accomplished by searching for the location/direction angle which minimizes the cost function.

27

1kk k r

kJ n w T

=

= GG i (21)

,

[ , , ] ,e e e

Tr p y r k

where

T w weighting factor = =G To verify the selected cost function, the cost function is

evaluated for 3 different thruster configurations with 0 = . 3 cases are shown in Fig 6. The cost function value for each case is -2259.8, -2417.8, -2110.4. Case C is previous configuration design. The smallest cost function is acquired in case B. Thus, case C is a near optimal configuration.

B. Parameter Optimization As mentioned in the previous section, the uncertain

parameters are the 6 position/direction angles of the two thrusters in the first quarter plane. PSO technique in Section IV is applied to find the optimal solution. Boundary conditions are as follows.

1 1 10 90 180 180 30 90 < D D D D D D

Case 1) Thruster establishment location is fixed ( 1,2 0 = ) For some cases, the thruster establishment locations are

limited due to the fuel tank and valve locations. The thruster nozzle positions are fixed and the nozzle directions are freely chosen by the optimization process. Fig.7. is the optimal thruster configuration with that case.

Case 2) Thruster establishment location is free ( 1,2 is free) If there are no constraints on the position of two thrusters in

first quarter plane, the optimal thruster configuration by PSO is

plotted as Fig. 8. It is configured to have maximum torque quantity in all directions.

0 500 1000 1500 2000

-150

-100

-50

0

50

100

150

200 Swarm Optimization ResultsThruster Position/Direction Angle

Estim

ated

Opt

imal

Pos

ition

/Dire

ctio

n (D

eg)

Iteration Number

1 1 1 2 2 2

Fig.9. Variation of estimated parameters (angle) when PSO is applied

Fig.8. The optimal thruster configuration ( 1,2 is free)

Fig .7. The optimal thruster configuration ( 1,2 0 = )

(case A) (case B) (case C) Fig.6. Configuration example for cost function validity test

7

Fig.9 and Fig 10 shows that it converges after about 1000

steps. PSO gives robust convergence irrespective of initial proposition of the population. The sequential quadratic programming (SQP), optimization technique provided MATLAB, is applied to same cost function to confirm the reliability of solution [7]. SQP could not match the PSO in convergence ability.

VI. NUMERICAL SIMULATION The assumed upper-stage launch vehicle specification and

thrust level in numerical simulations are as follows.

TABLE II UPPER-STAGE LAUNCH VEHICLE SPECIFICATION

Mass (Kg)

C.G. (m)

length (m)

radius (m)

Ixx (Kg-m2)

Iyy (Kg-m2)

Izz (Kg-m2)

1500 2.0 4.0 1.3 1267.5 2633.7 2633.7

TABLE III

THRUSTER SPECIFICATION Thruster level 20N No. of available Thruster 8 EA Fuel limit MAXIMUM OPERATING TIME : 500 SEC

Numerical simulations are performed using Simulink in

MATLAB. The validity of the proposed optimal thruster configuration is checked with the RCS attitude control loop response and fuel consumption.

Fig 11 and Fig 12 shows the responses of quaternion feed back loop and accumulated fuel consumption when using normal operating thrusters. It shows that the proposed optimal configuration is little bit superior to the existing configuration from the view point of response characteristics. However, the proposed configuration requires slightly more fuel consumption than the previous one.

To check the response differences in respective

configuration in case of abnormal operating thruster case, we assumed a situation where one of 8 thrusters went out. Fig.13 and Fig.14 shows the maximum overshoot case and the minimum undershoot case in simulation results. Fig.13 is for the previous concept design configuration and Fig.14 is for the proposed optimal thruster configuration. It shows the optimal thruster configuration can achieve smaller overshoot and undershoot error in case of a trouble. Securing more effective torque in every command direction made it possible to gain more robustness about a thruster failure.

0 10 20 30 40 50 60 70 80 90 100-20

0

20

40

60

80

100

120

140

160

180Euler Angle : PWPF Coarse Control

Time

Quaternion Feedback Result

PhiThetaPsiCommand

0 10 20 30 40 50 60 70 80 90 100

-60

-40

-20

0

20

40

60


Time


PhiThetaPsiCommand

Fig.13. Existing thruster configuration: (One of 8 thrusters is not operating) Euler angle response (Maximum overshoot, Minimum undershoot case)

0 10 20 30 40 50 60 70 80 90 100-10

0

10

20

30

40

50

60

70


Time


PhiThetaPsiCommand

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40int (1/2 u2) : PWPF Coarse Control

Time


Control Force

Fig.12. Proposed thruster configuration: (Normal Operation) Euler/Quaternion response and accumulated fuel consumption

0 10 20 30 40 50 60 70 80 90 100-10

0

10

20

30

40

50

60

70


Time


PhiThetaPsiCommand

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35int (1/2 u2) : PWPF Coarse Control

Time


Control Force

Fig.11. Existing thruster Configuration: (Normal Operation) Euler/Quaternion Response and accumulated fuel consumption 0 500 1000 1500 2000

-2600

-2500

-2400

-2300

-2200

-2100

-2000Swarm Optimization Results Cost Function

Cos

t

Iteration Number

J

Fig.10. Variation of cost function when PSO is applied

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2Quaternion Angle : PWPF Coarse Control

Time

q0q1q2q3q0cq1cq2cq3c

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2Quaternion Angle : PWPF Coarse Control

Time

q0q1q2q3q0cq1cq2cq3c

8

VII. CONCLUSION This paper considers an attitude control system using

reaction thrusters for the upper stage of a launch vehicle. The attitude control loop is designed based on error quaternion feedback algorithm. We introduce a gain selection procedure for quaternion feedback loop, a characteristic parameter determination process for PWPF modulator and a way to set the command distribution logic. The thruster configuration has a big influence on control system response, fuel consumption, effective torque and system fault tolerance. We proposed a procedure to find the optimal thruster configuration that guarantees an effective torque vector which has similar direction to original torque command with maximum possible magnitude. The proposed procedure can find optimal thruster configuration which improves control loop efficiency even when the position and direction of the thruster configuration are limited.

The control gain determination process is demonstrated. Computer simulation result show that the optimal thruster configuration has more desirable control characteristics than that of the previous thruster configuration.

ACKNOWLEDGMENT The authors would like to acknowledge support from the

Korea Ministry of Science & Technology (MOST) under Project Number M1-0138-00-0005, the space technology development program.

REFERENCES [1] R.S.Pena, R.Alonso, P.Anigstein,"Robust Optimal Soltuion to the

Attitude/Force Control Problem ", IEEE Transation on Aerospace and Electronic System, Vol. 36, No.3 July 2000

[2] H.P.Jin, P.Wiktor, D.B.Debra, "An Optimal Thruster Configuration Design and Evaluation for Quick Step", Control Eng. Practice., Vol. 3, No.8 1995, pp. 1113-1118

[3] A.E. Bryson, Control of Space craft and Aircraft , Princeton University Press, 1994

[4] B.Wie, J.Lu, "Feedback control logic for spacecraft eigenaxis rotations under slew rate and control constraints", Journal of Guidance, Control, and Dynamics V.18,1995, pp. 1372-9.

[5] B.N.Agrawal, R.S.Mcclelland, G.Song, "Attitude Control of Flexible spacecraft Using Pulse-Width Pulse-Frequency Modulated Thrusters", Space Technology, Vol. 17, No. 1, 1997, pp. 15-34

[6] Kennedy and Eberhart, Particle swarm optimization" Proc IEEE Int. Conf Neural Networks, Nov 1995

[7] T.F.Coleman, Y.Jhang, "Optimization Tollbox User's Guide", Mathworks, July. 2002

0 10 20 30 40 50 60 70 80 90 100-20

0

20

40

60

80

100


Time


PhiThetaPsiCommand

0 10 20 30 40 50 60 70 80 90 100-20

-10

0

10

20

30

40

50

60

70


Time


PhiThetaPsiCommand

Fig.14. Proposed thruster configuration: (One of 8 thrusters is not operating)Euler angle response (Maximum overshoot, Minimum undershoot

Upper Stage Launch Vehicle Servo Con

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