LTP dynamics and control

D. Bortoluzzi, M. Da Lio, S. Vitale 1Penn State, 20th-24th July 2002

4TH INTERNATIONALLISA SYMPOSIUM

LTP dynamics and control

D. Bortoluzzi, M. Da Lio, S. VitaleUniversity of Trento

http://spdext.estec.esa.nl/content/searchimage/searchresult.cfm?aid=27&cid=1874&oid=26496&ooid=26495



The LTP basic mode of operation



DOF Dc-force compensation

TM-stabilisation

Drag-free in measurement

bandwidth

Drag-free at low

frequency

Test-mass 1

x

y

z

Test-mass 2

x

y

z



Force noisemodel

Mq -1 1/s2

Electrostaticreadoutmodel

IS noise

Electrostaticsuspension

Laserreadoutmodel

Interferometernoise

fnoiseq*q*

..

factuation

Kh

Spacecraftmotion

Spacecraftdistortion

CSC

Mu

nSC

u..

LTP modelS/Cmodel

fho,fLTPo

Sensit. to noise(Bfhf,BfLTPf)

Fsc

Fsc

Drag free control and actuation system

KLTP

q

CLTP

LTPdistortion

nLTP

-

Bfhf

BfLTPf

q

q̂

q

LTP is an autonomous dynamical system



drag free ,x

S / C,xS / C n 2

Fx x

M

Requirements given for S/C control

DOF Readout noise Thrust noise Total relative displ.

xS/C

y2S/C

y1S/C

z1S/C

z2S/C

1S/C

TM-S/C relativedisplacement

Readout noiseExternal forces on S/C

suppressed by drag-free control loop

91.8 10 m / Hz 94.7 10 m / Hz 95 10 m / Hz

91.8 10 m / Hz 947 10 m / Hz 947 10 m / Hz

91.8 10 m / Hz 966 10 m / Hz 966 10 m / Hz

918 10 m / Hz 947 10 m / Hz 950 10 m / Hz

918 10 m / Hz 966 10 m / Hz 969 10 m / Hz

965 10 rad / Hz 9235 10 rad / Hz 9243 10 rad / Hz

3 210 Hz f 3 10 Hz



LTP Multibody model• Inertial reference frame• Spacecraft body fixed reference frame• S/C-LTP mechanical interface

reference frame• Optical bench body fixed reference

frame• Electrode housings reference frames• Test masses body fixed reference

frames

Defined frames

• Spacecraft (Inertial Frame)• Test mass 1 (S/C frame)• Test mass 1 (EH frame)• …

Coordinates

* * * * * * *1 1 1 1 1 1 1q x , y , z , , , u x, y, z, , ,

1 1 1 1 1 1 1q x , y , z , , ,



DistortionsDisplacements of H1, H2 (readout references) frames due to distortions occurring:A) within the spacecraft (S/C-mechanicalinterface frame)

(in S/C frame)

B) within the LTP (mechanical interface frame-optical bench frame-electrode housings frames)



Forces and torquesPicture of forces

fe1 fe2te1 te2

fLTP1 fLTP2

tLTP1 tLTP2

f12f12

t12

t12fh1

th1 fh2 th2

FSC

TSC

•Fsc, Tsc: total force and torque (in SC frame) applied by LTP onto the S/C•fe, te: total force and torque of environmental origin on TMs•fLTP, tLTP: total force and torque on TMs originated within the LTP or the S/C•f12, t12: force and torque between TM1 and TM2•fh, th: total force and torque on TMs caused by the interaction between TMs and their housings (it includes actuation force and torque)



Dynamics 1: test-masses

Dynamics 2: reaction on spacecraft

Readouts

TM-S/C couplingTM-S/C relative displacementTM-EH couplingTM-EH relative displacementInverse of mass matrix

TM-EH space-independent coupling

TM-EH relative displacementCapacitive readout cross-talk

Capacitive readout noise

Stray forcescapacitive actuationMass matrix

OB-S/C relative position

S/C motion

TM-S/C space-independent coupling

Optical readout cross-talk

TM-EH relative displacementOptical readout noise



Symbolic form (for instance: inertia)

Numeric form (for instance: stiffness & cross-talk)

Matrices provided



Noisea ,x

drag free ,x

S / C,x1/ 2 14 2xp,xnoise,x n 22

Fm fS 3 10 @1mHz a xm Ms Hz



Example: x(t) signal obtained from given x readout noise

spectral density

Example: force noise spectral density obtained from

sampled signal





© A s t r i u m9 0 S m a r t - 2 F i n a l P r e s e n t a t i o n 1 2 / 0 7 / 2 0 0 2

F r e q u e n c y A n a l y s i s – X a x i s a c c e l e r a t i o n ( 1 o f 3 )

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 0- 1 4

1 0- 1 3

1 0- 1 2

x 1

H z

ms

-2/H

z0

.5 o

r s

-2/H

z0

.5

P S D t o t a leF x 1 fo r c e n o is eF y 1 fo r c e n o is eF z 1 fo r c e n o is eT h e t a 1 fo rc e n o is eE t a 1 fo r c e n o is eP h i1 fo r c e n o is eF x 1 re a d o u t n o is eF y 1 re a d o u t n o is eF z 1 re a d o u t n o is eT h e t a 1 r e a d o u t n o is eE t a 1 re a d o u t n o is eP h i1 r e a d o u t n o is e

Simulation results



© A s t r iu m9 4 S m a r t - 2 F in a l P r e s e n t a t i o n 1 2 /0 7 /2 0 0 2

F r e q u e n c y A n a l y s i s – R e m a in i n g a x e s

• R e m a in in g l in e a r a n d R o t a t io n a l a x e s m e e t f r e q u e n c y r e q u i r e m e n t s

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 0- 1 4

1 0- 1 3

1 0- 1 2

Y 2

H z

ms

-2/H

z0

.5 o

r s-2

/Hz

0.5 P S D t o t a l e

F xF yF zT xT yT zS T t h e t aS T E t aS T p h i

1 0- 4

1 0-3

1 0- 2

1 0- 1

1 0- 1 3

1 0- 1 2

1 0- 1 1

1 0- 1 0

Z 1

H z

ms

-2/H

z0

.5 o

r s

-2/H

z0

.5

P S D t o t a leF xF yF zT xT yT zS T t h e t aS T E t aS T p h i

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 0-1 2

1 0-1 1

1 0-1 0

1 0-9

1 0-8

T h e ta 1

H z

ms

-2/H

z0

.5 o

r s

-2/H

z0

.5

P S D t o t a leF xF yF zT xT yT zS T th e t aS T E t aS T p h i

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 0 - 1 2

1 0- 1 1

1 0 - 1 0

1 0- 9

1 0- 8

E t a 1

H z

ms

-2/H

z0

.5 o

r s

-2/H

z0

.5

P S D t o t a leF xF yF zT xT yT zS T t h e t aS T E t aS T p h i

1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 11 0

- 1 2

1 0- 1 1

1 0- 1 0

1 0- 9

1 0 - 8P h i 1

H z

ms

-2/H

z0

.5 o

r s

-2/H

z0

.5 P S D to t a leF xF yF zT xT yT zS T t h e t aS T E t aS T p h i

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 0 - 1 2

1 0- 1 0

1 0- 8

1 0- 6

y S C

1 0- 4

1 0- 3

1 0-2

1 0-1

1 0- 1 1

1 0- 1 0

1 0- 9

1 0- 8

1 0- 7

z S C

1 0- 4

1 0-3

1 0- 2

1 0-1

1 0- 1 2

1 0- 1 0

1 0- 8

1 0- 6

1 0- 4

T h e t a S C

Simulation results



1. ´ 10- 6 0.00001 0.0001 0.001 0.01

10

100

1000

10000

parasitic

Gk

frequency[Hz]

Compensating negative stiffness kp= 10-7 N/m and dc forces

10-10N/10-7 N/m 1 mm

Low frequency suspension



Single input single output control laws

Actuation cross-talkLow frequency suspension laws matrixCapacitive readout



4*1y o

1yo

4*2z o

2zo

4*1 o

1o

G s sKm s s

G s sKm s s

G s sKI s s

Dc force compensation only

Parameter Value so 10-5s-1

K1y 0.0075 s-2

K2z 0.015 s-2

K1 0.0012 s-2



+ TM stabilisation

2 2 2 2 2 21 1 1 2 2 2 3 3 3

2 2 2 2 2 24 4 4 5 5 5 6 6 6

s 2 s s 2 s s 2 sG s K

s 2 s s 2 s s 2 s



Optimised control

Robust against knowledge of parameters



nk

k njk 1

nj j 1 j

jj 1

a s Ah s A

s sb s

j

tn ms t'

j jj 1 j 10

y t A x t A e x t t ' dt ' A x t y t

x(s) h(s) y(s)

j j j

ts t s t ' s t

j j j j j0

y t t e y t A e x t t ' dt ' e y t x t A t

jj

1for t<<s

Numerical implementation of control laws



1

1

1

1

1

1

s t1 1 1

s t2 2 2

s t3 3 3

s t4 4 4

s t5 5 5

s t6 6 6

y n 1 y n Ae 0 0 0 0 0y n 1 y n A0 e 0 0 0 0y n 1 y n A0 0 e 0 0 0 x[n 1] ty n 1 y n A0 0 0 e 0 0y n 1 y n A0 0 0 0 e 0y n 1 y n A0 0 0 0 0 e

ARMA basic step

m

jj 1

y n 1 y n 1 A x n 1



TM motion after a force step 10-11 N

Low frequency suspension control

Numerical implementation of control is good

Very long damping time

Stiffer TM actuation needed for emergency



Limits of the low-frequency actuation:

•Low damping

•Low maximum force

x omax,x

K df2

Limit to stiffness

Limit to maximum force

Different operational mode (accelerometer mode) is defined in which larger maximum force can be exerted with larger stiffness (up to hardware limit)



x1,…,x6

Transfer functions

parameters upload

Threshold detector

DC force offsets

Fd1,…, Fd6

DC comp. a

DC comp. b

TM stabiliz a

TM stabiliz b

SCIENCE MODE

Suspension c

Suspension d

ACCELEROMETER MODE

LARGE AMPLITUDE MODE

TBC

Transfer functions selection

External command

Low frequency sine wave

Charge measurement

dither

Poles, zeroes, gain

Suspension switch

Caging command

Charge measurement

command

Channels combinator

Ch1, ch2,…

Calibration parameters

upload

Capacitive actuation functional block diagram



x1,…,x6

Transfer functions

parameters upload

Threshold detector

DC force offsets

Fd1,…, Fd6


DC comp. a

DC comp. b

TM stabiliz a

TM stabiliz b

SCIENCE MODE

Suspension c

Suspension d

ACCELEROMETER MODE


TBC


External command


Charge measurement

dither

Poles, zeroes, gain

Suspension switch

Caging command

Charge measurement

command

Channels combinator

Ch1, ch2,…


upload




x1,…,x6

Transfer functions

parameters upload

Threshold detector

DC force offsets

Fd1,…, Fd6


DC comp. a

DC comp. b

TM stabiliz a

TM stabiliz b

SCIENCE MODE

Suspension c

Suspension d

ACCELEROMETER MODE


TBC


External command


Charge measurement

dither

Poles, zeroes, gain

Suspension switch

Caging command

Charge measurement

command

Channels combinator

Ch1, ch2,…


upload




Accelerometer mode: high damping, large force

2 2

122 2 2 1a o 2

2 212

2

sss ss

max2

F m0.15m s

TM motion after a force step 10-7 N

Accelerometer mode control

damping time shorter



Transition to-from accelerometer mode requested to damp long term transitory

Needs adjustment of long term behaviour

TM subjected toforce step

Accelerometermode threshold

TM motion arrestedTM moved towards EH center

Passage to low frequency suspension law



Dc-force dominate

Low frequency suspension and accelerometer mode asyntotic behaviour must be the same to avoid overshoot at handover

lfslfsj

lfsj

accaccj

accj

lfsns t jlfs lfs lfs

j j j dc s tj 1

accns t jacc acc acc

j j j dc s tj 1

A ty e y A tx f x

1 eA t

y e y A tx f x1 e

lfs acc

lfs accj j

lfs accn nj j acc lfs 2 2

s t s t s 0 s 0j 1 j 1

A t A tLimh s Limh s 10 s

1 e 1 e

Adjust low frequency gain



Capacitive actuation

1

Carrier waveform synthesis

3

ISFEE DAC

4

TM

Fd1,…, Fd6V1,…, V12

V1(t)..V12(t)

Capacitive model equation

parameters upload

Carrier waveform parameters upload

x1,…,x6

F to V conversion

2

F to V conversion

2

Switch

Optical metrology

Capacitivesensing

Switch

forces

ADC

ADC

Com

man

d

Command



Summary:

•LTP dynamics mathematical model •Capacitive actuation control laws: low frequency suspension and accelerometer mode•Noise models•Simulations results of a drag-free and attitude control system matching the requirements given

Documents

LTP dynamics and control