Thesis report

Spacecraft Attitude Dynamics and Control Using Single Gimbal Control

Moment Gyros

Abhilash M*

Prof. Hari B Hablani

Department of Aerospace Engineering

IIT Bombay

3rd National Symposium on Advances in Control & Instrumentation SACI-2014

*Engineer, ISRO Inertial Systems Unit, Trivandrum, Kerala

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Overview • Introduction

• CMG: Operation, Classification and

Construction

• Spacecraft Attitude Dynamics with SGCMGs

• Large Angle Yaw Manoeuvre

• General large angle 3-axis pointing manoeuvre

• Conclusion

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Introduction • More remote sensing data per

orbit • Tracking of moving targets on

land, in air or at sea • Space stations and Large

Payloads: Large Inertia

• Agility • Rotational Manoeuverability • High torque “Without consuming spacecraft propellant”

Agile Spacecraft

• Agile satellite moves at 3-4 0/s with 0.2-0.3 0/s2 acceleration

• Maximum torque capability of reaction wheels used in ISRO:

0.3 N.m @ 50 N.m.s

• Need 15-20 N.m torque for agile manoeuvres (>50-60 times the RW torque)

R. Berner, Control moment gyro actuator for small satellite applications. PhD thesis, Stellenbosch: University of Stellenbosch, 2005.

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CMG: Principle, Classification and Construction • Works on basis of law of conservation of angular momentum • Direction of the angular momentum changed by gimbaling • Result is in gyric torque orthogonal to the angular momentum and gimbal axis

CMG Working [Lappas, Vaios. Thesis]

Torque amplification: 25 N.m.s wheel gimbaled at 1 rad/s results in an instantaneous torque output of 25 N.m!

SGCMG [ISRO]

DGCMG [Yavozoglu, Thesis]

Instantaneous torque direction changes with time. So CMGs are used as clusters

Lappas 2002 and YAVUZOGLU,2003

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CMG cluster in Pyramidal Configuration

Lappas 2002 and Berner 2005

dt

dH

dt

dH

)()(

Minimum two-norm solution of CMG dynamics equation: Moore Penrose Steering Law

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Spacecraft Frames of Reference

http://www.mdpi.com/1424-8220/12/12/16964/htm

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Spacecraft Attitude Dynamics Equation with 4-CMG Cluster

CMGTT

CMGbICMG

bIsatbIbIsat

sat

CMGbICMG

CMGbIsatbICMGbIsatsat

CMGbIsatsat

Ib

HJJJ

HuH

uII

H

HHu

HIHIH

HIH

FF

1))()(()(

0

][

,

Thus

torque control this generate to has CMG

i.e zero; be torque external theLet

Let

derivative inertial the Taking

spacecraft the of momentum angular total the

frame inertial the and spacecraft the of frame body the be Let

Bedrossian 1987 and Wie 2008

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LARGE ANGLE YAW MANOEUVRE

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Large Angle Yaw Manoeuvre: Assumptions

• Spacecraft body is rigid • The initial orientation of the LVLH frame is taken as the inertial

frame for simplicity.

• External disturbance torques like torque due to air drag, gradient torque, solar radiation pressure, magnetic torque (values range from 10-4 to 10-7 Nm) acting on the spacecraft are assumed to be zero.

• The acceleration feed-forward torque is calculated in the

commanded body frame assuming that the angle of rotation along the roll and pitch directions are very small

Ref: Spacecraft Operations edited by Thomas Uhlig, Florian Sellmaier, Michael Schmidhuber

SACI-2014 10

0

0

100

0

0

0

0

100

0

0cF

c 03c3c

3c3c

3c

3c3c

3c3c

)cos()sin(

)sin()cos(

)cos()sin(

)sin()cos(

c

c

c

c

F

F

c

F

F

c and

3c

3c3c0

3c3c0

3c

3c0

3c0

)sin(

)cos(

)cos(

)sin(

Large Angle Yaw Manoeuvre

Spacecraft: LVLH and body frames are shown

Hari B Hablani

Sticky Note

don't need this transformation because the yaw rate is about the z-axis.

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Large Angle Yaw Manoeuvre

ccxxyy

ccxxzz

cczzyy

czz

ccyy

ccxx

F

acc

II

II

II

I

I

I

uc

33

33

33

3

33

33

cossin)(

sin)(

cos)(

sin

cos

2

0

0

0

0

0

ω

ω

ω

ω

ω

Acceleration feed-forward Torque in commanded body frame Fc

Roll and angles are small

𝜹𝜽𝟏, 𝜹𝜽𝟐, 𝜽𝟑 𝒓𝒐𝒍𝒍, 𝒑𝒊𝒕𝒄𝒉 𝒂𝒏𝒅 𝒚𝒂𝒘 𝒂𝒏𝒈𝒍𝒆𝒔 𝝎𝟎=orbital rate of the spacecraft

XbYbZb: Body Frame X0Y0Z0: Local Vertical-Local Horizontal

𝜹𝜽𝟏, 𝜹𝜽𝟐, 𝜽𝟑 𝒓𝒐𝒍𝒍, 𝒑𝒊𝒕𝒄𝒉 𝒂𝒏𝒅 𝒚𝒂𝒘 𝒂𝒏𝒈𝒍𝒆𝒔 𝝎𝟎=orbital rate of the spacecraft XbYbZb: Body Frame X0Y0Z0: Local Vertical-Local Horizontal

Predominant rotation along roll axis

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LARGE ANGLE YAW MANOEUVRE: SIMULATION RESULTS

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Large angle Yaw: Results

Body Rate of Spacecraft: Commanded/Real

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Acceleration Feed-forward torque in commanded Fc

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CMG: Gimbal Rate & Angle

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Euler Angle of Spacecraft body: Commanded/Real

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GENERAL 3-AXIS POINTING MANOEUVRE

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Assumptions • The spacecraft body is rigid.

• Orbital motion of spacecraft and earth are not considered for

simplicity

• Earth Centered Inertial Frame FI is considered for all calculations. But quantities are expressed in convenient frames.

• External disturbance acting on the spacecraft are taken as zero.

• The instantaneous quaternion error is assumed to be small

• Moore-Penrose Pseudoinverse steering law is used for calculating gimbal rate

• Controller is assumed to act in the same speed as the solution step

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Ground Trace Calculation

Peenya(ISTRAC) near Bangalore is chosen as the ground station

Inertial latitude and longitude of the satellite out. found be can δ andα terms of comparison By

latitude or nDeclinatio δ

Longitude Inertial α

AnglenInclinatio i

Node Ascendingof AsecnsionRight Ω

AnomalyTrue ν

perigee of Argumentω

δ

δα

δα

r

iνω

iνωνω

iνωνω

r

sat

sat

)sin(

)cos()sin(

)cos()cos(

)sin()sin(

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

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

I

I

F

F

r

and

r

Montenbruck 2000 and Hablani 2013

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Spacecraft in Orbit on Epoch Date (31st August,2014)

J2000)onbased days of no. (d

d)360.9856(280.4606-α(t)

Θ(t)-α(t)λ(t)

λ(t)Θ(t)α(t)

as calculated be can longitude Inertial

00

Epoch Date is chosen as 31st August,2014 ,12:00 hrs GMT d= 5356 days Earth rate = 0.2507 degrees/min

t)0.2507(159.334-α(t)λ(t)

t)0.25072933719.334(-α(t)

t)ω5356 x360.9856(280.4606-α(t)λ(t)

0

0

earth

00

1

Montenbruck 2000 and Hablani 2013

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Ground Trace Plot for 3-orbits

The marked point is the point where Peenya(Bangalore) will be visible. The true anomaly for this satellite position was calculated as ν=19.10

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Line of sight vector in the LVLH Frame

ub

0forub

a

uba

ub1

ub

ub12

1q

1

ct

ct

cb

ˆˆsin

)sin(

ˆˆˆ

ˆˆ)sin(ˆ

ˆˆ

ˆˆ

)ˆˆ(

,ˆ

figure given the From

F in axis bore payload along vector unitb

vector unit sight of linel

lu

b

Hablani 2013

𝐛 𝐮 𝐥

𝐚

𝐅𝐈

𝝋𝒄𝒕

ISTRAC coordinates are used To calculate the target location Unit vector with respect to FI

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Reference frame and Quaternion Calculation

𝐅𝐈

𝑭𝒃𝟎

𝑭𝒃

𝑭𝒄

𝒒𝒄𝒃𝟎

𝒒𝒃𝟎𝐈 𝒒𝐂𝐈

𝒒𝐈𝐛

𝒒𝒆

qcb0= quaternion of the commanded frame with respect to the body frame at t=0 qb0I= quaternion of the body frame at t=0 with respect to the inertial frame qbI= quaternion of the instantaneous body frame with respect to the inertial frame qcI= quaternion of the instantaneous commanded frame with respect to the inertial frame

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Rate and Angle Profile along Eigen Axis The peak rate is found from maximum angular momentum capability of CMG. The acceleration was fixed as 0.20 /s2

0.t @ to respect with of

quaternion initialtheiswhere

Ib

boI

boIcbcI

ct

ct

cbo

FF

q

qqq

aq

0

)2cos(

)2sin(ˆ

IbcIe

bIbI

qqq

qq

ErrorQuaternion

)(2

1

Only 70% of the angular momentum capacity was used to account for gyric terms in the attitude dynamics equation

SACI-2014 25 )(

)()(

.

_

cIsatcIcIsatfwd

fwd

fwdecvecec

ωIωωIu

u

uq2q2u

114

torque forward Feedwhere

qKKqK

form the of is controller PID The

Hz 0.071/t is bandwidth Controller n.calculatio from secondst

esiDe_scalarp

peakpeak

Block Diagram

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PID Control Design: Pitch Axis

ca along rate maximum attaining for required timet

where,t

2π ω

λωa

λως2ωa

ς and 0.1δwhereωςδλandλως2a

λ)(s)ωsως2(s(s)C

I

Ka

I

Ka

I

Kawhere 0asasas(s)C

:is polynomial sticcharacteri loop closed ideal The

(s)TKsKsKIs

s(s)θ

KsKsKIs

KsKsKθ(s)

:as written be can function transfer The

peak

peak

c

2

ci

cc

2

cp

cccccD

2

ccc

2

ideal

i

i

p

p

D

Dip

2

D

3

ideal

d

ip

2

D

3c

ip

2

D

3

ip

2

D

1,

,,

SACI-2014 27

PID Control Design

csatbIsatbIsatbI

cIsatcIcIsatfwd

fwd

fwdecvecc

uIII

IωIu

u

uqqu

11 ][)ω(ω][ω

)ω(ω

)2()(2

9.86000

06.12520

006.1146

16.400

005.60

005.5

7.20500

03.2990

00274

94300

013720

001256

1.14

equation dynamics spacecraft the gintegratin by obtained be can rate spacecraft The

torque forward Feedwhere

qKKqK

form the of is controller PID The

= K= K= K

:as obtained are gains controller The

matrix inertia satellite the isI

Hz 0.07 is bandwidth Controller n.calculatio from secondst

esiDe_scalarp

Dip

sat

peak

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Simulation Results: Total Control Torque

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Torque Commanded by PID Controller

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4-CMG Cluster Gimbal Rate and Angle

11/23/2014 MTP Ist Stage 31

Spacecraft Body Rates

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Error in Spacecraft Body Rates

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Spacecraft Euler Angle

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Error in Spacecraft Euler Angle: General : 3-axis pointing manoeuvre

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Conclusion • CMG is a very useful actuator for agile manoeuvres when

compared to reaction wheels.

• Numerical simulation of a large angle yaw manoeuvre and general 3-axis pointing manoeuvre are performed using 4-CMG cluster

• The large angle attitude manoeuvres performed are within the

torque capability of the 4-CMG cluster and does not cause singularity

• A high torque agile manoeuvre is illustrated without using thrusters

• Different types of spacecraft manoeuvres have to be simulated with CMGs and their singularity avoidance strategies is to be studied thoroughly as future extension of the work.

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References [1]R. Berner, Control moment gyro actuator for small satellite applications. PhD thesis, Stellenbosch: University of Stellenbosch, 2005. [2] \http://www.sstl.co.uk/missions/bilsat-1{launched-2003/bilsat-1/bilsat-1{themission." [3] \http://www.spaceight101.com/soyuz-vs-04-pleiades-1b-launch-updates.html." [4] V. J. Lappas, A control moment gyro (CMG) based attitude control system (ACS)for agile small satellites. PhD thesis, University of Surrey, 2002. [5] E. YAVUZOGLU, Steering laws for control moment gyroscope systems used in space- crafts attitude control. PhD thesis, MIDDLE EAST TECHNICAL UNIVERSITY,2003. [6] Isro internal reports.“ [7] H. Yoon and P. Tsiotras, Singularity analysis of variable speed control moment gyros," Journal of Guidance, Control, and Dynamics, vol. 27, no. 3, pp. 374{386, 2004. [8] G. Margulies and J. Aubrun, Geometric theory of single-gimbal control moment gyro systems," AIAA Guidance and Control Conference, San Diego, California. [9] H. B. Hablani, AE:626 spacecraft attitude dynamics and control, course material,"2013. [10] M. Oliver and G. Eberhard, Satellite orbits:. models, methods and applications,"2000. [11] A. Defendini, P. Faucheux, P. Guay, K. Bangert, H. Heimel, M. Privat, and R. Seiler, Control moment gyro cmg 15-45 s:Acompact cmg product for agile satellites in the one ton class," in 10th European Space Mechanisms and Tribology Symposium, vol. 524, pp. 27{31, 2003. [12] D. Brown and M. A. Peck, Scissored-pair control-moment gyros: A mechanical constraint saves power," Journal of guidance, control, and dynamics, vol. 31, no. 6, pp. 1823{1826, 2008. [13] N. S. Bedrossian, Steering law design for redundant single gimbal control moment gyro systems," M.S Thesis,Charles Stark Draper Laboratory, 1987. [14] H. Schaub, S. R. Vadali, J. L. Junkins, et al., Feedback control law for variable speed control moment gyros," Journal of the Astronautical Sciences, vol. 46, no. 3, pp. 307{328, 1998. [15] B. Wie, Space vehicle dynamics and control. Aiaa, 2008. [16] Y. Nakamura and H. Hanafusa, Inverse kinematic solutions with singularity robustness for robot manipulator control," Journal of dynamic systems, measurement, and control, vol. 108, no. 3, pp. 163{171, 1986.

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THANK YOU

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ADDITIONAL MATERIAL

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RW vs CMG

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LARGE ANGLE YAW MANOEUVRE

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Parameters Used for Simulation Sl

No: Parameters for Simulation Value

1 Satellite Inertia diag [1256 1372 943] kgm2

2 Angular momentum per CMG 15 N.m.s

3 Orbital rate of the satellite ωo

0.060/s

4 Skew Angle of 4 SGCMG cluster β 54.730

5 Initial gimbal angle [0 0 0 0]T degrees

6 Initial gimbal rate [0 0 0 0]T degrees/s

7 Initial spacecraft attitude [0 0 0]T degrees

8 Total Manoeuvre Time 6 minutes

9 Simulation time step 40 ms

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LARGE ANGLE YAW MANOEUVRE: SIMULATION RESULTS

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Body Rate of Spacecraft: Commanded/Real

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Error in Euler Angle of Spacecraft body

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GENERAL 3-AXIS POINTING MANOEUVRE

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General Large Angle 3-axis pointing Manoeuvre

• The spacecraft us assumed to at an initial orientation. The payload bore axis on satellite has to be pointed to a predetermined target location on earth.

• The final commanded quaternion is calculated from the bore axis unit vector and target location unit vector in the LVLH frame.

• The manoeuvre is also called an Eigen axis manoeuvre as we rotate through the smallest angle along the Eigen axis to reach the final orientation.

• A rate profile is planned along the Eigen axis which when tracked points the payload axis to the target location.

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Parameters Used in Numerical Simulation Sl No:

Parameters for Simulation Value

1 Satellite Inertia diag [1256 1372 943] kgm2

2 Angular momentum per CMG

25 N.m.s

3 Orbital rate of the satellite [0 0 0]T

4 Skew Angle of 4 SGCMG cluster β

54.73 degrees

5 Initial gimbal angle [0 0 0 0]T degrees

6 Initial gimbal rate [0 0 0 0]T degrees/s

7 Initial spacecraft attitude [80 0 0]T degrees

8 Payload bore axis in Fb [0 0 1]T

9 Radius of earth 6371km

10 Altitude of satellite 450km

Sl No: Parameters for Simulation Value

11 Inclination Angle 98.28 0

12 Right Ascension of Ascending Node

3110

13 True Anomaly 19.1 0

14 Earth Rate 0.2507 0/min

15 Orbital Period 100 min

16 Latitude of target location 130 North

17 Longitude of target location 77.5 0E

18 Inertial Longitude @ 31st Aug, 12:00 hrs. GMT[8]

(77.5+159.5)=2370

19 Total Manoeuvre Time 42s

20 Simulation time step 40 ms

11/23/2014 MTP Ist Stage 49

Incremental Quaternion: Given to Controller

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