1

Accelerometer Assisted High Bandwidth Control of Tip-Tilt Mirror for Precision Pointing Stability

Fujiwara Ken, Yasuda Susumu, Bando Nobutaka, Sakai Shin-ichiro, Tsuiki Atsuo1 Niwa Yoshito2, Hatsutori Yoichi, Yano Taihei3, Yamada Yoshiyuki4

1 Japan Aerospace Exploration Agency 3-1-1, Yoshinodai, Chuo-ku, Sagamihara

Kanagawa, Japan 252-5210 +81-50-3362-3360

fujiwara.ken@jaxa.jp 2 University of Tokyo

3 National Astronomical Observatory of Japan 4 Kyoto University

Abstract—Pointing stability that defines acceptable displacement of an observation target on the optical detector during the exposure time in a satellite telescope system largely dominates the quality of the image. 12This research proposes a new control approach to improve pointing stability using a tip-tilt mirror. While conventional control systems for a tip-tilt mirror use feedback error signals obtained from the optical detector, the signal loses enough bandwidth to compensate for high frequency disturbances when pointing at a dark reference guide star. This paper presents a control system subsidiarily using accelerometers to generate high bandwidth feedforward control references to a tip-tilt mirror by properly identifying transfer function between acceleration and pointing displacement. The numerical simulation and experiments using a disturbance source and a tip-tilt mirror verifies feasibility of this method.

TABLE OF CONTENTS

1. INTRODUCTION .................................................................1 2. TIP-TILT MIRROR CONTROL SYSTEM .............................2 3. ESTIMATING ARX MODEL FOR THE FEEDFORWARD

CONTROL .............................................................................3 4. NUMERICAL SIMULATIONS ..............................................4 5. DISTURBANCE SUPPRESSION EXPERIMENT .....................5 6. CONCLUSION ....................................................................6 REFERENCES ........................................................................6 BIOGRAPHY ..........................................................................7

1. INTRODUCTION

Among the technical challenges for control systems in optical observation satellites, pointing stability becomes a key factor to improve the quality of observation images. The pointing stability generally defines acceptable error envelope of the pointing axis around arbitrary observation targets during the exposure time. The obstacles degrading this stability include alignment between satellite attitude sensors and observation optics, thermal distortion of optical structure, and internal disturbances caused by mechanical

1 978-1-4244-7351-9/11/$26.00 ©2011 IEEE 2 IEEEAC paper#1140, Version 1, Updated 2010:10:25

components on board such as reaction wheels, integrated rate gyros, and mechanical coolers, described in Figure 1. Those sources of disturbances need to be compensated to achieve precision pointing stability.

While the alignment and thermal distortion occur at a low frequency domain where alignment measurement on orbit and thermal control compensates for those errors, the mechanical vibration requires disturbance compensation in high frequency domain over attitude control bandwidth. For example, a reaction wheel generates significant disturbance at the spin frequency that ranges approximately up to 100Hz, containing dynamic and static imbalance in the rotor. A mechanical cooler also induces disturbances at its driving frequency of compressor approximately up to 50Hz. Those disturbances induce spacecraft vibration at the frequency and its harmonic frequencies; therefore, a pointing actuator and sensor that covers high bandwidth is necessary.

Tip-tilt mirror, a mirror angle driving device, can be allocated in optical instruments and control the pointing direction accurately at a high frequency. A conventional usage of tip-tilt mirror is to feedback a pointing error signal detected by an imaging sensor, however, read-out noise of the imaging sensor, image processing time, and brightness of the target star limit the bandwidth of closed loop control systems. Japanese solar physics satellite "Hinode" launched in 2006 has achieved high pointing stability of 0.06 arcsec for 10 seconds; however, the control bandwidth of tip-tilt mirror was designed at 14 Hz because of image processing time delay to generate error signal[1]. This research proposes high bandwidth control of tip-tilt mirrors using an accelerometer to detect mechanical disturbances and compensate for the limitation of image sensor feedback control system.

Although an accelerometer does not detect the pointing error itself, the signal with a transfer function model of structural dynamics can yield high bandwidth feedforward control input to a tip-tilt mirror. Vibration on a spacecraft changes the relative position of optical components such as its primary mirror and secondary mirror and degrades image

2

quality. In this point of view, by properly modeling the signal transfer function between mechanical vibration and optical misalignment, the accelerometer signal can be used to generate feedforward control input to compensate optical misaligment. Bando proposed this type of approach to suppress disturbances on a hard disk drive head[2]; however, its application to optics control had not been explored.

This research reports initial study of a high bandwidth control of tip-tilt mirror using numerical analysis and experimental demonstrations. A numerical control simulation models a simple mechanical vibration and optical control loop, to verify the transfer characteristics. The experiment uses simple optics with a tip-tilt mirror, a photo-detector, and accelerometers allocated nearby a disturbance source. The results demonstrate the high frequency vibration compensation for precision pointing stability of optical observation satellites.

2. TIP-TILT MIRROR CONTROL SYSTEM

A typical feedback control system for a tip-tilt mirror uses an image detector signal to generate reference signals. A feedback controller calculates the control input based on the goal pointing angle and the reference signals. The tip-tilt mirror directly controls the pointing angle by following the control input while external pointing disturbances degrade the final pointing angle, shown in Figure 2. Suppose the goal pointing angle equals zero, then the transfer function from the external disturbance, d, to the output angle, y, denotes disturbance suppression performance of this control system;

dCPH

y+

=1

1 (1)

where C, P, and H denote transfer functions representing the controller, the tip-tilt mirror, and the detector respectively.

The proposed control method uses additional accelerometers placed on the optical system to generate a feedforward control signal. The source of disturbance in this control system results from mechanical vibration on the spacecraft; therefore, the high-band width acceleration detected by the accelerometers yields the profile of disturbance through a certain transfer function G, shown in Figure 3. Properly estimating G and using the estimated transfer function G and accelerometer signals s, the controller generates feedforward control outputs to compensate high frequency disturbances. While the accelerometers provides the input signal to the unknown transfer function, G, the control system needs to acquire the output signals, to estimate G

Disturbance observer generally compensates modeling error and external disturbances, providing feedback signal from

the controller output and the response using nominal plant Pn. The control method in Figure 2 estimates external disturbances d using this observer, yet not providing the estimation to the controller as a feedback signal. The equation (2) denotes the observer output d using original disturbance d and the feedforward input uff;

)()1(

)1(ˆ dPuPCH

CPHQd ff

n +−−

−= (2)

where, Q denotes a low-pass filter. The output of the estimated transfer function G and nominal plant model yield uff; therefore,

ffn

ff uP

u '1= (3)

Then the output of observer becomes as follows using equation (2) and (3).

)'()1(

)1(ˆ duP

P

PCH

CPHQd ff

n

n +−−

−= (4)

Suppose the transfer function of the tip-tilt mirror can be measured properly and detector scarcely includes dynamic properties. The Pn ≈ P and H ≈ 1, therefore; the equation (4) becomes the following simple form.

'ˆ

)'(ˆ

ff

ff

QudQd

duQd

+=⇔

+−= (5)

From the equation (5), the observer output and feedforward input profile yield the disturbance with low-pass filter Q effectively.

Gyros

Reaction w

heelsC

oolers

Observation

Instruments

Image detector

Tip-tilt mirrorMechanical Vibration

Target star

Figure 1 –Overview of satellite telescope systems with a tip-tilt mirror and disturbance cources

3

Controller Tip-tilt mirror

Disturbance

Pointing AngleGoal Angle ++

+ -

Detector

C P

H

d

y(0)

Figure 2 –A typical feedback control system for tip-tilt mirror.

Accelerometer

Observer

++

+ -C P

H

d

y

G

s

1/Pn

Pn

QQ

-+

- +

Controller

Figure 3 –Proposed control system with feedforward input and disturbance observer.

3. ESTIMATING ARX MODEL FOR THE

FEEDFORWARD CONTROL

From the input signal s that the accelerometers detect and the disturbances d that the disturbance observer yields, the unknown transfer function, G, can be expressed as follows.

Gsd = (6)

Introducing Auto-Regressive eXogeneous (ARX) model, the time history of the input and output signals estimates G using modeling parameters ai: i=1...Na and bi: i=1...Nb:

)()2()1(

)()2()1()(

21

21

bNb

aNa

Nksbksbksb

Nkdakdakdakd

−++−+−=−++−+−+

(7)

The ARX model can be also denoted using transfer function expression;

szA

zBd

)(

)(1

1

−

−

= (8)

where,

b

b

a

a

NN

NN

zbzbzbzB

zazazazA−−−−

−−−−

+++=

++++=

12

11

1

12

11

1

)(

1)(

Noting that the disturbance observer yields disturbances with low-pass filter, the time history of the input and out signals simply become simplified to;

)()( kkdQ Tφν=⋅ (9)

where,

[ ][

]Tb

a

TNN

NksQksQksQ

NkdQkdQkdQk

bbbaaaba

)()2()1(

)()2()1()(

2121

−⋅−⋅−⋅

−⋅−−⋅−−⋅−=

=

φ

ν

A simple least square method estimates the model parameter matrix v for the ARX model. The observation equation becomes as follows using error matrix ε.

εΦνd

ε

φ

φ

φ

+=⇔

+

+−−

=

⋅

+−⋅−⋅

b

a

N

N

T

T

T

b

b

a

a

k

ik

ik

kdQ

ikdQ

ikdQ

1

1

)(

)1(

)(

)(

)1(

)(

(10)

Noting that i < k-Na, from the equation (10), least square solution to model parameter v leads to;

dΦν +=~ (11)

where Φ+ denotes Moore-Penrose generalized inverse of Φ.

The control method proposed above works, first, to estimate the disturbance transfer function based on the input accelerometer signal and the output disturbance obtained from the observer, and second, to generate feedforward input to compensate the disturbance based on the transfer function and accelerometer. A technical challenge still arises when the detector is incapable of generating high-band width signal limited by the Signal to Noise ratio or the image processing time to estimate the transfer function. However, the optical system can calibrate the disturbance

4

transfer function before observing target stars, pointing to nearby brighter stars in this method. After calibrating the transfer function, the telescope system points to the actual target stars, compensating for high frequency disturbances.

4. NUMERICAL SIMULATIONS

A numerical simulation estimating an unknown disturbance transfer function and suppressing the disturbance by generating proper feedforward input verifies the feasibility of the control method proposed above. The first simulation estimates a black-box transfer function from a sinusoidal input and a disturbance observer output. The input frequency of 50Hz models mechanical vibration in a satellite. The tip-tilt mirror is modeled as a simple low pass filter assuming that tip-tilt mirror has enough driving bandwidth and tracking performance. The feedback controller covers the open loop transfer function with 0.16Hz crossover frequency, which is relatively low frequency in comparison to the input disturbances.

This simulation uses the following models according to the proposed control system in Figure 3.

110

1

110

1134 +

=+

== −− sQ

sP

sC

The other transfer function H is ignored, assuming the image detector provides enough performance to detect the disturbance for the model estimation. Then, a black-box model is designed to be a structural transfer function including one resonance frequency:

01.0,200,1

2 22

2

===++

=

ζπωωωζ

ω

K

s

KG

From the input disturbance s;

π100:sin =ΩΩ= ts

and output signal d , the simulator estimates ARX model for Ĝ with the sampling frequency of 2kHz. During this simulation, feedforward signal uff is terminated as the first verification of the model estimation.

Figure 4 shows the true transfer characteristic of G and the estimated transfer characteristic of Ĝ. Both the bode diagrams approximately match. The Ĝ becomes as follows:

21-

1

994.01.90z-1

0974.0ˆ−

−

+=

z

zG

-50

0

50

Mag

nitu

de (

dB)

101

102

103

-200

-100

0

Frequency (Hz)

Phas

e (d

eg)

G

Figure 4 – Estimation Result

0 0.1 0.2 0.3 0.4-2

-1

0

1

2

Time (seconds)

Out

put y

feedback onlyfeedback+feedforward

Figure 5 – Feedforward input result

The second simulation demonstrates feedforward control using the estimated transfer function Ĝ. The feedforward input can be generated from one-step-ahead prediction of the disturbance:

[]Tb

affffff

T

ffff

Nksksks

Nkukukuk

k

kszBkuzAku

)()2()1(

)(')2(')1(')(

)(η~)()(

~)(')}(

~1{)(' 11

−−−

−−−−−−==

+−= −−

η

ν

ν

The result of disturbance suppression using the feedforward input is shown in Figure 5. The dashed line shows the response of the simple feedback control system shown in Figure 2 with 50Hz disturbance input, and the sold line shows that of the proposed control system. The peak-to-peak amplitude of the vibration on the output y decreases from 3.67 to 0.66, which reduces approximately to 17% in this simulation.

G

5

5. DISTURBANCE SUPPRESSION EXPERIMENT

The mathematical description and the simple simulation show the feasibility of the control method. The controversy remains how the method works in an actual optical system. The transfer function denoted as G depends on structure of the optical system and the place to measure acceleration, and all sensors include process noises; therefore, the experimental setup shown in figure 6 demonstrates its behaviors in an actual application.

The experimental setup consists of two modules; the laser and alignment adjustment module and the tip-tilt mirror and detector module. The 1064nm laser emits the light source of 40mW, passes through light intensity adjustment, and aligns to a mirror in tip-tilt stage. The tip-tilt stage with three piezoelectric actuators drives a 1-inch mirror up to 1kHz without significant phase delay, which gives enough response performance for this experiment. The beam reflected on the tip-tilt mirror points to a quadrant photodiode that detects center of the light spot on the detector. A mechanical shaker actuated by coils and permanent magnets is located on the tip-tilt mirror, and detector module creates arbitrary disturbances on the optical system. One three-axis accelerometer is also placed on the module to detect the mechanical vibration. Both modules are implemented on an optical table surrounded by a plastic curtain to eliminate external vibration and air flow. Signals from the photodetector and the accelerometer are connected to A/D board and processed with Matlab/xPC Target, and D/A board on the same computer outputs driving signals to the tip-tilt mirror and the shaker. The computer handles output and input signals with 2kHz sampling frequency.

Figure 7 and 8 show frequency domain responses of the acceleration and displacements of the spot center on photodetector when 50Hz sinusoidal vibration is generated from the shaker.

Alignment mirrorsAccelerometerTip-tilt mirror

Shaker Photodiodes Laser Optical path

Alignment mirrorsAccelerometerTip-tilt mirror

Shaker Photodiodes Laser Optical path

Figure 6 – Experiment setups

100

101

102

103

0.01

0.0001

1

Frequency (Hz)

Acc

eler

atio

n (m

/s2 )

xyz

Figure 7 – Accelerometer output

100 101 102 103

10-9

10-5

10-7

Frequency (Hz)

Spot

dis

plac

emen

t (m

)

XY

Figure 8 – Displacement

10-6

10-4

Gai

n

101

102

103

-100

0

100

Frequency (Hz)

Phas

e (d

egre

e)

raw dataARX model

Figure 9 – ARX model and Actual Transfer function X

6

10-6

10-5G

ain

101 102 103

-100

0

100

Frequency (Hz)

Phas

e (d

egre

e)

raw dataARX estimation

Figure 10-ARX model and Actual Transfer function Y

0 0.1 0.2 0.3 0.4-6

-4

-2

0

2

4

6x 10

-6

Time (seconds)

Spot

dis

plac

emen

t x (

m)

Feedback onlyFeedback+Feedforward

Figure-11 Feedforward control result x

0 0.1 0.2 0.3 0.4-1.5

-1

-0.5

0

0.5

1

1.5x 10-6

Time (seconds)

Spot

dis

plac

emen

t y (

m)

Feedback onlyFeedback+Feedforward

Figure-12 Feedforward control result y

In figure 9, the solid line shows the gain and phase profile between accelerometer x and displacement of spot x. The dashed line shows estimated ARX model obtained from the input and output signal. Figure 10 shows the profile figures between input accelerometer signal x and output displacement signal y. The ARX models single input single output transfer function; therefore, signal choices are among three axis input signals and output signals. In this

experiment, accelerometer x to both displacement x and y resulted in a precise model of the transfer function where ARX model coincides with the mode at 50Hz. At the other frequency, the model and the actual transfer function does not match because the evaluation function minimizes the error based on amplitude of the vibration. The significant vibration amplitude occurs at disturbance frequency; therefore, the ARX model matches the gain and phase at the vibration frequency. The discrete transfer functions below show the estimated ARX models for X and Y.

21-

16

935.01.92z-1

1024.1ˆ−

−−

+×=

z

zGx

21-

17

857.01.84z-1

104.96ˆ−

−−

+×=

z

zGy

The feedforward signals generated from the estimated ARX model are implemented in the control system. Figure 11 and 12 show the displacement profile obtained from control experiments with and without feedforward control. The dashed line shows the one uses only feedback control, and the solid line shows the one used feedback and feedforward control. In both x and y cases, the proposed method increased the stability of spot on the detector.

6. CONCLUSION

Mechanical vibration on a satellite degrades pointing stability of the optical system on board. A tip-tilt mirror directly controls its optical axis and increases the pointing stability compensating the vibration; however, a simple feedback control using an image detector fails to compensate high frequency vibration when targeting dark stars. This paper presents a new control method of tip-tilt mirror using accelerometer to generate high frequency feedforward input to suppress the vibration. The mathematical description, numerical simulation, and experiments demonstrate the effectiveness of this method.

REFERENCES

[1] Kazuhide Kodeki, et. al, "Development of a Correlation Tracker and a Tip-Tilt Mirror System for SOLAR-B," Journal of the Japan Society for Aeronautical and Space Sciences, Vo.55, No. 637, 2007

[2] Bando Nobutaka, "Design of High Precision Servo Control System Based on Time Series of Data," University of Tokyo, Dissertation, 2004

7

BIOGRAPHY

Fujiwara Ken is an Engineer of Navigation Guidance Control Group at the Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency. He received BEng in 2005 and MEng in 2008 from Tokyo Institute of Technology. He had worked on a small satellite project, "Cute-1.7+APDII" in his graduate school.

Since 2008, he has joined ISAS and focused on high agility control using control moment gyros and high precision control using tip-tilt mirror for astrometry satellites.

Yasuda Susumu is an Associate Senior Engineer of Spacecraft Structures and Mechanisms Group at Aerospace R&D Directorate, Japan Aerospace Exploration Agency (JAXA). He received B. Eng. in 1989 and M. Eng. in 1991 from University of Tokyo. From 1991 to 2006 he had worked at Canon Inc., where he had done research for

MEMS devices such as atomic force microscope probes and micro optical scanners. He joined JAXA in 2006 and one of his interests is vibration control of spacecrafts. He is a registered professional engineer of Japan (mechanical engineering).

Bando Nobutaka is an Engineer of Navigation Guidance Control Group at the Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency. He received BEng in 2000, MEng in 2002 and Deng in 2005 from University of Tokyo. He is interested in high precision and high speed control of satellite and industrial applications. He has also

joined ASTRO-H project as a member of Attitude Control System team..

Shin-ichiro Sakai received the B.S., M.S. and Ph.D. degrees in electrical engineering from the University of Tokyo, in 1995, 1997 and 2000, respectively. In 2001 he joined The Institute of Space and Astronautical Science / JAXA, and in 2005, he became associate professor. His research fields are control theory and its industrial applications, typically, motion

control of electric vehicles, control of satellite in space, mechatronics, etc.

Tsuiki Atsuo is an Associate Senior Engineer of Mission Design Support Group at Systems Engineering Office, Japan Aerospace Exploration Agency. He received BEng in 1982 and MEng in 1984 from Tokyo Denki University. He has studied and assisted the upstream design of Spacecrafts (e.g. “Small-

JASMINE” for the astrometry satellite mission, “GCOM-W1 / C1” for the global change observation mission, and “ALOS-2” for the land observing mission. ).

Yoshiyuki Yamada is an Assistant Professor of Theoretical Astrophysics Group, Department of Physics, Kyoto University, He received B Sci in 1986, M Sci in 1988, and D Sci in 1992 from Kyoto University. He had worked on Cosmic Ray, Fluid dynamics, Method of Numerical Simulation, and Astrometry. Since 2003, he has joined JASMINE collaboration and

is working on astrometric data analysis and on-board detection of stellar images.