Dynamics Modeling and First Design of Drag-Free Controller for ASTROD I Hongyin Li, W.-T. Ni Purple Mountain Observatory, Chinese Academy of Sciences S

Dynamics Modeling and First Design of Drag-Free Controller for ASTROD I

Dynamics Modeling and First Design of Drag-Free Controller

for ASTROD I

Hongyin Li, W.-T. Ni Purple Mountain Observatory, Chinese Academy of Sciences

S. Theil, L. Pettazzi,

M. S. GuilhermeZARM University of Bremen, Germany

2006/07/15

ASTROD 2006ASTROD 2006

Outline

Introduction of the Mission

Simulator Controller Development Conclusion Outlook

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Mission Introduction

Science goal Solar-system gravity mapping & relativistic gravity test

Orbit Heliocentric orbit, 0.5AU-1AU Orbit launch in 2015

SC configuration Cylinder : 2.5 (diameter)m ×2 m covered by solar panels. FEEP(Field Emission Electric Propulsion) Star Sensor, Gyroscope (??) EPS(Electrostatic Positioning/ Measurement System) 1 test mass

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

launch in 2015

Y A

xis

(A

U)

X Axis (AU)

ASTROD I Mecury Venus

Sun

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Simulator

Equations of Motion Disturbance & Environment

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Coordinates I

Heliocentric inertial frame Center of mass of sun i

Earth-fixed Center of earth e

Spacecraft body-fixed frame Center of mass of spacecraft b

Sensor frame for test mass

Geometrical center of sensorsens

Body-fixed frame for test mass

Center of mass of test mass tm

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Coordinates II

Frame Base point x direction z direction

Heliocentric inertial frame

Center of mass of sun Aries Normal of ecliptic

Spacecraft body-fixed frame

Center of mass of spacecraft

Parallel to symmetry axis of telescope

Symmetrical axis

Sensor frame for test mass

Geometrical center of sens

or

Normal of the surface of the house nearby

the telescope

Normal of the upper surface of

the house

Body-fixed frame for test mass

Center of mass of test mass

Normal of the surface nearby the telescope

Normal of the upper surface of

test mass

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Equations of MotionSC

Translation

SCRotation

TMTranslati

on

TM

Rotation

, , ,,( )i i i i i i

i b i b control dist coupl sci br g r a a a External forces

,1

,, ,( () [ )]b b b b bcontrol dist coupl

b b bi b iscb b bbi I T T T I

,

1ˆ

2b b b

i bi iq q

, , ,,

sens sens sens sens sens sensb tm control dist coupl sat coupl tmb tm

r g a a a a External forces

, , ,, ,1

,, ,( )( ) ( )[ ]tm tm tm tm tmi b sens tm i b sens tmtm

tm tm tm tm tmcoupl tm gg tmsens tm i btm

I T IT

,

1ˆ

2tm tm tmsens sens tm sensq q

,, '2 sens senssens tmi b coriolis s accelerat nr io

, ,, ( )sens sens sensb sens sens tmi b r r tamgental affiliated acceleration

, ,, ,[ ( )]sens sens sens sensb sens sens tmi b i b normal affiliated accer lerationr

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Disturbance & Environment

Gravity Field (first order--spherical)

,

3,,( )

ii i b

ii bi b

rg GM

R ,,

i i ib tmb tm

g G r

2 2

2 2

,, 2 2

3 3 3

3 3 3

3 3 3

b

bb tmi b

x r xy xz xGM

g yx y r yz yR

zx zy z r z

,, xyz

,bi bR

, ,x y z ,bb tmR

Elements of

Elements of

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Disturbance & Environment Gravity Field (second order… )One of the Science goal of ASTROD I is to test the gravity field parameter of the solar gravity. So, It must be

included in the further model.

,

, 223,

, , ,

,

1 5( )

31 ( ) 1 5( )

( ) 2

3 5( )

ii b

ii i b

i i ii bi b i b i b

ii b

z

R

r R zg GM J

R R R

z

R

,,

i i ib tmb tm

g G r

2 2

22 2 2

, , 2 , ,3 2 2 4, , , ,2

1 0 0 33 3 5

( ) 0 1 0 3 352

0 0 3 3 3 6

i i T i i T ii b i b i b i bi i i i

i b i b i b i b

x z xy xzR z

G E R R J xy y z yz R Rr R R R R

xz yz z

2J

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Disturbance & Environment

/ ,(1 )iR solar i bRF P C Ae

RC

,i be

A

Coefficient of reflection

Unit vector

Effective area of SC

Solar mean momentum flux /

Solar pressure

P

• Solar Pressure

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Solar Pressure

deterministic part

Solar pressure at 1AU

• Solar pressure at 0.5 AU is 4 times that of 1AU

• A low-pass filter is used to get the stochastic part with a white noise passed

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Coupling between SC&TM

Translation coupling

, , ,( )sens sens sens senscoupl tm sens tm offset sens tmtrans trans

F K r r D v

, , ,tm tm tmcoupl tm sens tm sens tmrot rot

T K D

Rotation coupling

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Sensors and actuators

Star-Sensor EPS-position measurement EPS-attitude measurement

EPS-attitude suspension FEEP-force FEEP-torque

White noise shaped noise Need to consider sample frequency, nonlinearity in the future model….

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Simulator

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Controller Design

Structure of Controller Requirements of controller Synthesis Closed loop Analysis

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Structure of Controller

DFACS 3 Sub-system

Sensor Actuator

SC-Attitude

(3DOF)

Star-Sensor

Gyroscope (??)

FEEP

Drag-Free

(3DOF-translation)

EPS FEEP

TM-Suspension

(3DOF-rotation)

EPS（ Electrostatic Positioning/

Measurement System ）

EPS

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Requirements of Controller

SC-Attitude

(3DOF)

Drag-Free

(3DOF-translation)

TM-Suspension

(3DOF-rotation)

(????)It depends on the cross-talking between rotation and translation DOFs of test mass. It also need to meet the requirements of laser interferometer(not too big anglar velocity).

1/2 14 2 2 1/ 2a

0.3S ( ) 3 10 [ 30 ( ) ]

3

mHz ff ms Hz

f mHz

2 rad （3 ）

1/2 9 2 1/ 20.3S ( ) 5 10 [ 30 ( ) ]

3p

mHz ff mHz

f mHz

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Synthesis I

With the LQR method we can derive the optimal gain matrix K such that the state-feedback law minimizes the quadratic cost function

0( 2 )T T TJ x Qx u Ru x Nu dt

x Ax Bu Given following linear system :

Just as the kalman filter we can get the minimum steady-state error covariance

LQR: Linear-Quadratic-Regulator

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Synthesis II

LQG ＝ LQR ＋ KF Linear-Quadratic-Gaussian ＝ Linear-quadratic-regulator + kalman filter(Linear-quadratic-filter)

With DC disturbance need feed forward

DC disturbance

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LQG Design Plant

3 3 3 3 3 3 3 3 3 3 3 3,

3 3 3 3 3 3 3 3 3 3,

3 3 3 3 3 3 3 3,

, 3 3 3 3

3 3

3 3 3 3 3 3 3

,3 3 3 3 3

,

3

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0

1 1

0

0 0 0

trans tr

bi b

b

i b

sensb tm

sensb tm

tmsens tm

tm

s

an

e

s

ns tm

tm tmv

r

E

D KM M

E

1 1

3 3

,

,

,

,

,3 3 3

,3 3 3 3 3 3 3 3 3 3

0

0 0 0 0 0

bi b

bi b

sensb tm

sensb tm

tmsens tt m

tm

m

sen

tm

tm rot tm r

s t

t

m

oI D I K

E

v

r

3 3 3 3

3 3 3 3 3 3

_

3 3

_

3 3 3 3 3 3_

3 3

3 3

1

1

3 3

1

3 3

,

3

1

3

0 0

0 0 0

0

0 0 0

0

0 0 0

1

sc

sc

sens sc

b sctm

sens sc tm

bb cosens

b sen

ntrol

sens

tm control

tmtm contr

b sc tm

s

ol

T

f

T

I

A EM

I

r I

A I

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Controller

,

,

_ 3 3

,

_ _,

_ 3 3,

,

0

0

bi b

bi bb

b control sensb tmsens senstm

senstm control sc solarscb tmtm

tmtm controls

LQR

ens tm

tmsens tm

Tv M

u f fMr

K

T

Standard LQR used here is MIMO PD controller. It’s feedback is proportion (position, angle ) and derivative (velocity and angular velocity) of outputs of system. Feed forward part can cancel the static error.

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Synthesis III

What does feed forward do to the frequency Response of the controller? Improve the performance in low frequency range.(integral )

but reduce the stability of the closed loop ,need to trade-off …

Stability of the closed loop system? with some weight gains, there is still some poles on the right phase because the

negative stiffness. So we need to chose the weight gain carefully. Try and tuning to get better performance as well as a stable system.

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Closed loop analysis

Transfer function & PSD of Simulation Data

Pointing accuracy :

7 622.3 10 10

3rad rad

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Test mass position

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Transfer functionanalysis

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PSD analysis

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Conclusion

A model for ASTROD I is built. However it is simple, the structure is established and each subsystem can be extended to get a more detailed model.

LQG (LQR+KF) was developed which can meet the requirements of the mission.

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Outlook

Detailed model Based on SC design– Inertial sensor ： Cross-coupling between DOFs– Star sensor ： Sample frequency ， data fusion of two

sensors. – Gyroscope to help the attitude estimate.– FEEP nonlinearity ， frequency response.Location of

FEEP clusters. Advanced control strategy

– Loop-Shaping with weighted LQR– Decoupling of controllers and DOFs– Robust Control method

Documents

Dynamics Modeling and First Design of Drag-Free Controller for ASTROD I Hongyin Li, W.-T. Ni Purple Mountain Observatory, Chinese Academy of Sciences S