A 3-axis Simulator for Spacecraft Attitude Control Research

Lu Dai Guang Jin Changchun Institute of Optics, Fine Mechanism and Physics Changchun Institute of Optics, Fine Mechanism and Physics

Changchun, Jilin Province, China Changchun, Jilin Province, China zjdailu@yahoo.com.cn jing@ciomp.ac.cn

Abstract - This article presents the details of a new 3-axis

simulator for spacecraft attitude control research and simulation. The air bearing allows the test bed to rotate about three axes with tiny friction. The attitude determination system consists of three fibre optical gyros, an inclinometer and a magnetometer. The actuator of the system consists of three reaction flywheels and four control moment gyros (CMG). Mathematical models of the sensors and actuators are given to help algorithm design. An extended Kalman filter was designed to provide attitude information for controller. Mathematical simulation results prove the attitude determination system can achieve high precision. A coulomb friction model of reaction wheels is given with experimental results. A friction compensation algorithm was developed to raise the pointing accuracy of the control system. The article also describes the details of the hardware structure. A PD stabilizing controller is implemented to test the validation of the whole control system at last.

Index Terms - aerospace simulation, attitude determination, attitude control, extended Kalman filter.

I. INTRODUCTION

This article presents the details of a newly constructed test bed by Changchun Institute of Optics, Fine Mechanism and Physics. Such simulators which are essential for spacecraft attitude control system design and research have been equipped in many research institutions in aerospace field [1]. The design learns from the test bed of the School of Aerospace Engineering at the Georgia Institute of Technology

[2]. The air bearing provides a full rotation freedom along Zb axis and ±30° freedom along Xb/Yb axis. The platform can host a payload of up to 500 kg. The structure of the platform is a cylinder with two plates. Such design can offer more space to place hardware of the system and helps balancing which is very important to ensure a torque-free environment for the physical simulation experiments.

The test bed contains various spacecraft components on board. The implementation of attitude determination and control algorithms is done by a PC-104 industrial computer. The computer includes an 800 MHz Pentium-III processor, 256 Mb RAM, two serial ports and a 10/100 Mbps Ethernet port. A 54Mbps wireless access point is also used for the communication with the ground supervision station. The attitude determination system consists of three fiber optical gyros, an inclinometer and a magnetometer. Three reaction wheels, four control moment gyros and six thrusters are used as actuators.

The paper presents the detail information of the hardware used on the test bed, such as mathematical models and parameters of the sensors and actuators. These are important for algorithm design. A brief discussion of the attitude determination strategy is given. A PD attitude stabilizing algorithm is also given at last. The physical simulation results are provided.

Fig. 1 3-axis test bed and its body-fixed reference

II. OVERVIEW OF HARDWARE

A. Attitude Sensors Three VG-951D type fiber optical gyros (by Fizoptika,

Corp.) provide the angular velocity of the simulator with respect to the body fixed frame. The range of angular rate is ±80°/s. The resolution of the gyro is 0.00001°/s. The gyro noise model is given in ref [3]:

=

++=

u

v

b

b

ηηωω~

(1)

( ) ( ) uuvv EE 2222 , σηση ==

ω~ is the measurement of the gyros, ω is the true angular velocity, b is the bias of the gyros, vη and uη are white noises. The bias of gyro is a random walk process.

1040978-1-4244-5704-5/10/$26.00 ©2010 IEEE

Proceedings of the 2010 IEEEInternational Conference on Information and Automation

June 20 - 23, Harbin, China

Fig. 2 Measurements of the three gyros when they are kept motionless The sample rate of the gyros is 10 Hz. An experiment was

performed to determine the parameters of the gyros. The experimental results are shown in Fig 2. By data analyzing, the parameters are shown in table 1. The earth self rotation is taken into account when calculating the bias the gyros. These parameters are important for designing recursive estimation algorithms, such as the EKF.

TABLE I PARAMETERS OF THE GYROS

axis Bias(°/s) Sample noise (°/s,1 )

x -0.00476 0.001065y -0.00434 0.001037z -0.00325 0.000733

An LE-30 type inclinometer provides the tilt angles of X and Y axis of the body fixed frame with respect to the earth surface frame. Its principle and precision are very similar with that of the earth sensor which is commonly used on spacecraft. The earth sensor needs expensive earth simulator when performing ground physical simulation. So the usage of inclinometer can decrease the cost and complexity of the ground simulation system. The angle range is ±25°, the sample rate is 2.5Hz. The observation model is as follows after calibration:

+=+=

ϕ

θ

ηϕϕηθθ

~

~ (2)

( ) ( ) 2222 , ϕϕθθ σηση == EE

θ~ and ϕ~ are measured values. θ and ϕ are true values.

θη and ϕη are white noises. The experimental results are shown in Fig 3. The parameters are shown in table 2.

Fig. 3 Measurements of the inclinometer after calibration

TABLE 2

PARAMETERS OF THE INCLINOMETER

axis Sample noise ( ° ,1 )

x 0.0010y 0.0008

A CXM-539 type magnetometer (Crossbow, Corp.) measures the local magnetic field and thus gives the orientation of the test bed. The earth magnetic field is easily disturbed by environments. So the accuracy of the orientation information is low compared to that of the inclinometer. The range of the magnetic field strength is ±1 Gauss. The accuracy of the magnetometer is ±3nT. The sample rate is programmable from 1Hz to 150 Hz. we adopt the same frequency as the inclinometer of 2.5 Hz. The experimental results are shown in Fig 4. The results show that the magnetic field in the laboratory changes slightly with time.

Fig. 4 Measurements of the magnetometer on three axes

B. Actuators

Three reaction wheels are installed orthogonally along the three body frame axes to provide torque for attitude control. The maxim output torque is 0.1Nm and the maxim momentum is 1Nms. The chief disturbance of the reaction wheels is their coulomb friction. The friction will cause attitude steady-state error without compensation. So the mathematical model of the friction is essential for compensation algorithm design. The experimental results of the reaction wheels are shown in Fig 6. The speed of the reaction wheels were accelerated to ±6000 rpm, and decelerated freely. So the friction can be calculated.

Fig. 5 Reaction wheels on the test bed The model of the coulomb friction is : nKTnT ff ⋅+= 0)( (3)

0fT is static friction that can be determined by tests. K can be

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determined by the least-square method with the experiment data. The chosen loss function is:

2

1))(()(

=−= m

j jjkj nTTKL

The chosen K should minimize )(KL . And so we have:

==

=

−=

=⋅−−−=

m

j jm

j fjj

m

j jfjj

nTTnK

nKTTndK

KdL

12

1 0

1 0

)(

0)(2)( (4)

So we can achieve the 0fT and K of the reaction wheels:

(5a) (5b) (5c)

Formula (5) is the coulomb friction model of the reaction wheels. It’s the basis of the friction compensation algorithm.

Fig. 6 Torque–speed results of three reaction wheels calculated from the

experiment data.

Fig. 7 Control moment gyro installed on the test bed

Four principle prototypes of CMG are equipped on the test bed to evaluate the algorithms of spacecraft large angle maneuver control with CMGs. The Fig 7 shows the structure of a CMG. The 1 part is a reaction wheel working in fixed speed mode. The momentum is 1Nms. The 2 part is a gimlet with gear reducer. The 3 part is a stepper motor that drive the gimble. The 4 part is an encoder which measures the rotation angle of the gimble. The maximal speed of the gimble is 90°/s.

The maximal torque of the CMG is 1.57Nm. The four CMGs are installed in pyramid-structure.

III. ATTITUDE DETERMINATION ALGORITHM

The EKF algorithm used for attitude estimation was developed by E.J.Leffers, F.L.Markley and M.D.Shuster [4]. The filter works in three steps: Prediction:

+−+

++−+ =⋅Ω= kkkkk bbqq 11

ˆ,ˆ)ˆ(ˆ ω

QPP Tkkkk +ΦΦ= +−

+1 Filtering:

1111111 )( −

+−+++

−++ += RHPHHPK T

kkkTkkk

))ˆ((ˆ 1111 rqAyKx kkkk ⋅−=Δ −+++

++

Updating: −+++

++ −= 1111 )( kkkk PHKIP

−++

++ ⊗Δ= 111 ˆˆˆ kkk qqq

11

11ˆˆ

++++ Δ+= kkk bbb

+++

++ −= 111

ˆ~ˆ kkk bωω Please refer to ref [4] for more details of the algorithm. The observation equations and R need to be determined here. The gravity vector is:

[ ]TG 100 −= The observed vector from the inclinometer measurement is:

[ ]TG )~(sin)~(sin1)~sin()~sin(~

22 ϕθϕθ −−−=

For xG~ :

θθθ ηθηθηθθ sincoscossin)sin()~sin(~ +=+==xG

θθ

θ ηθηθ sincos)2

1(sin2

+−=

θηθ ⋅+= cosxG

So as for yG~ :

ϕηϕ ⋅+= cos~yy GG

For zG~ :

−−⋅=

−−⋅=

⋅+⋅+=

2222 1cossin,

1cossin

~

yxyx

zz

GGv

GGu

vuGGϕϕθθ

ηη ϕθ

So the observation equation of the inclinometer is:

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⋅+⋅=00

0cos000cos

)(~

ϕ

θ

ηη

ϕθ

vuGqAG

⋅+⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

=2222

222

222

coscoscoscos0cos0cos

ϕθϕθ

ϕϕ

θθ

σσσϕσθσϕσϕσθσθ

vuvuvu

RG

0008.0,001.0 == ϕθ σσ

So as for the observation equation of the magnetometer:

+⋅=

Z

y

x

MqAMηηη

)(~

=2

2

2

000000

z

y

x

MRσ

σσ

GGG zyx444 1052.5,1068.3,109.6 −−− ×=×=×= σσσ

The R is:

=×

×

m

G

RR

R33

33

00

The Q is:

=2221

1211

QQQQ

Q

Δ⋅+Δ⋅

Δ⋅+Δ⋅

Δ⋅+Δ⋅

⋅= ×

322

322

322

3311

313131

tt

tt

tt

IQ

uzvz

uyvy

uxvx

σσ

σσ

σσ

Δ⋅

Δ⋅

Δ⋅

⋅−= ×

22

22

22

3312

212121

t

t

t

IQ

uz

uy

ux

σ

σ

σ

2112 QQ =

Δ⋅Δ⋅Δ⋅

⋅= ×

ttt

IQ

uz

uy

ux

2

2

2

3322

σσσ

st 4.0=Δ

TABLE 3

PARAMETERS OF THE Q

axis vσ uσ

x 0.000337°/s 3 °/s y 0.000334°/s 3 °/s

z 0.000312°/s 3 °/s Now we have all the necessary parameters and equations for the EKF algorithm. The mathematical simulation results are shown in Fig 8.

Fig. 8 Mathematical simulation results of the EKF algorithm. The left column

is attitude error of the three axes. The right column is bias estimation of the gyros.

The attitude estimation error is less than 0.0012° (3 ) on X/Y axes and less than 0.02° (3 ) on Z axis, because the inclinometer is more accurate than the magnetometer. The bias of the gyros can be well estimated too.

IV. ATTITUDE CONTROL SIMULATION AND RESULTS

The attitude control system consists of five attitude sensors and at least three actuators. The sensors and reaction wheels all receive and send out massages via an RS-232 serial port. The encoder on the CMG sends out measurements via an RS-485 serial port. The stepper motors on the CMGs are controlled by 8 I/O ports. A PC-104 computer can hardly have so many peripherals. A CAN bus has no such limit. So a smart CAN node is designed to connect all the attitude hardware to a CAN bus. The PC-104 computer receives measurements of sensors and sends out orders to actuators through the CAN bus via a COM-1273 card (Eurotech, corp.). A DSP28335 helps the data process and attitude determination algorithm

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implementation. The real time information of the system is transferred via an access point down to the monitor computer through wireless Ethernet connection. The structure of the system is shown in Fig 10.

Fig. 10 structure of the hardware on the test bed

The attitude control algorithm design is not the main objective of the article and is introduced briefly here. The equations for the motion of the test bed controlled by reaction wheels are [5]:

qq

MhIhI

⋅Ω=

=+⋅⊗++⋅

)(21

)()(

ω

ωωω

A PD controller with friction compensation is designed to control the test bed to stabilize to a horizontal level. The AEKF algorithm provides the estimated q and �. The controller is:

)(nTDqPu f−⋅−⋅−= ω

)(nTf is calculated by the equation (5). n is the speed of the

reaction wheels. The stability of the algorithm is proved by Lyapunov function. The experimental results are shown in Fig 11.

Fig. 11 Physical simulation results of the simulator controlled by reaction wheels with A PD controller. The left column is the attitude error. The right

column is the angular rate. The system stabilizes after 100 seconds from the beginning

of the simulation. The steady-error of the three axes is shown in Fig 12. The attitude error is less than 0.02° on X/Y axes and 0.05° on Z axis. So the pointing accuracy and stability is enough for attitude control algorithms and strategies evaluation.

Fig. 12 Steady-state error of the three axes.

V. CONCLUSION

We have presented the details of the spacecraft simulator for attitude control research. The system incorporates a variety of attitude sensors and actuators for different configurations. Mathematical simulation proves the simulator can achieve high attitude determination accuracy. A PD stabilizing controller was implemented to test the validation of the whole control system. The simulator will be essential for attitude control algorithm and strategy evaluation in future research.

REFERENCES [1] J. Schwartz, M. Peck, and C. Hall, “Historical Survey of Air-Bearing

Spacecraft Simulators,” Journal of Guidance, Control, and Dynamics, 2003,vol. 26, 513–522.

[2] D. Jung, P. Tsiotras, “A 3-DoF Experimental Test-Bed for Integrated Attitude Dynamics and Control Research”, AIAA Paper 03-5331, AIAA Guidance, Navigation and Control Conference, Austin, TX, 2003.

[3] J. Crassidis, J. Junkins, Optimal Estimation of Dynamic Systems, Chapman & Hall/CRC, New York, 2004.

[4] E. Leffers, F. Markley, M. Shuster, “Kalman Filtering for Spacecraft Attitude Estimation,” Journal of Guidance, Control and Dynamics, 1982, VOL.5, No.5: 417-428.

[5] Bang. Wei Space Vehicle Dynamics and Control, American Institute of Aeronautics & Astronautics 1998.

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