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Attitude Control System of a Cube Satellite with

Small Solar Sail

Yeona Yoo∗, Soyeon Koo†, Gyeonghun Kim‡, Seungkeun Kim§, and Jinyoung Suk¶

Chungnam National University, Daejeon, 305-764, Republic of Korea

Jongrae Kimk

University of Glasgow, Glasgow, G12 8QQ, UK

This paper describes attitude control scheme and carries out numerical simulations for

a cube satellite that is equipped with a small solar sail. The satellite must perform Sun

pointing maneuver for testing eﬀects of the solar sail in a circular orbit of 700km altitude.

For the purpose of Sun pointing operation, the modiﬁed B-dot control scheme and PD-

like control scheme are implemented. To verify the feasibility and performance of the

proposed controller, the simulations consider aerodynamic, solar radiation pressure and

gravity gradient disturbances. The simulations show that the satellite can be detumbled

within 5 hour from tip-oﬀ rate due to ejection from the P-POD.

I. Introduction

The objective of CNUSAIL-1 project is to develop and operate a 3U-sized cube satellite. The purpose

of the mission is to successfully deploy the solar sail in a low earth orbit. The satellite is in circular

sun-synchronous orbit of 700km altitude and 98.1913 inclination. To test various eﬀects of the solar sail,

Sun-pointing maneuver will be performed by satellite attitude control. This research project is supported

by the Korea Aerospace Research Institute in terms of ﬁnance, environmental test, and arrangement for

possible launch in late 2015. If successful in the operation, the CNUSAIL-1 will be the ﬁrst cube satellite

with solar sail in Korea.

This paper aims to design and verify the attitude controller for spin stabilization and Sun pointing. To

verify the controller performance, numerical simulations are done for the cases with and without solar sail

deployment. Attitude determination is based on Extended Kalman Filter using sun sensors, magnetometers,

and gyroscopes. As control laws, modiﬁed B-dot and three-dimensional PD-like feedback controller are

applied to spin-stabilization and sun-pointing, respectively. The next section presents description of attitude

determination and control system, followed by a representation of the external disturbance with considering

solar sail area and angle, control modes, and control law. Figure 1 shows (a) the conceptual drawing of

CNUSAIL-1, and (b) the design of an operational orbit.

∗Graduate Student, Department of Aerospace Engineering, Daejeon, South Korea, yandoll2@gmail.com

†Graduate Student, Department of Aerospace Engineering, Daejeon, South Korea, esther91127@hanmail.net

‡Graduate Student, Department of Aerospace Engineering, Daejeon, South Korea, nuber007z@naver.com

§Assistant Professor, Department of Aerospace Engineering, Daejeon, South Korea, skim78@cnu.ac.kr, AIAA member.

¶Professor, Department of Aerospace Engineering, Daejeon, South Korea, jsuk@cnu.ac.kr, AIAA member.

kLecturer, Department of Aerospace Engineering, University of Glasgow, Glasgow, G12 8QQ, UK, jkim@aero.gla.ac.uk

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(a) Solar sail deployment (b) Operational orbit

Figure 1. A conceptual design and the operational orbit of the CNUSAIL-1

II. Attitude determination and control system

Attitude determination and control system (ADCS) of the CNUSAIL-1 assists a solar sail mission by

controlling the orientation of the satellite into the desired attitude. The systems consist of Sun sensors, an

IMU, a 3-axis magnetometer and three magnetic torquers. The requirement of the ADCS are to maintain

Sun pointing accuracy under 10 degrees. In addition, the satellite must be slowly spinned about Z-axis.

Figure 2 shows the feedback loop of attitude determination and control system.

Figure 2. Attitude control block diagram

III. System dynamics and disturbacne

A. Coordinate frame

The attitude control simulations and equation of motions are calculated in the body-ﬁxed frame. The

Earth-centered-inertial frame has an origin at the Earth center. The Z-axis is the perpendicular to Earth’s

equatorial plane. X-axis points towards the vernal equinox direction, Y-axis follows the right hand rule.

LVLH(Local vertical local horizontal) frame has an origin at the center of gravity of the satellite and follows

the orbit trajectory. The Z-axis points towards Earth center. Y-axis is perpendicular to the orbital plane,

which is parallel to the orbital angular momentum direction. X-axis follows the right hand rule. The body-

ﬁxed frame has origin at the center of mass of satellite. It is ﬁxed on the satellite and coincided with the

satellite axes of moments of inertia. Figure 3 shows the coordinate frames used in this study.

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(a) Earth-centered inertial frame (b) LVLH frame (c) Body-ﬁxed frme

Figure 3. Coordinate Frames

B. Dynamics and kinematics

The attitude dynamics of rigid satellite is as follows1

I ˙ω=Tdis +Tmag −ω×Iω (1)

where Iis the inertia matrix, ωis the angular velocity of the body frame, Tmag denotes the magnetic control

torque and Tdis denotes the disturbance torques which include aerodynamic torque, solar radiation pressure

torque, and gravity-gradient torque. The attitude kinematics in terms of quaternion are2

˙=1

2η1 + ×ω(2)

˙η=−1

2Tω(3)

where is the vector part of the quaternion, ηis the scalar part of the quaternion and

×=

03−2

−301

2−10

.(4)

Magnetic control torque law is

Tmag =M×B.(5)

This magnetic torque eﬀects the satellite perpendicularly to the Earth magnetic ﬁeld by magnetic torquers.

Magnetic coils have saturation on the maximum magnetic moment and coil current. The maximum current

is restricted to 0.1A, and the maximum magnetic moment is 0.2Am2.

C. Disturbance

The gravity gradient torques result from the Earth’s gravitational force which is not constant with the

distance and position from the Earth’s center. The gravity gradient torques of the body frame are expressed

as1:

Tgg =3µ

r3

o

ue×Iue(6)

where µ= 3.986 ×1014m3/s2is the earth’s gravity constant, rois the distance from the earth’s center, ueis

the unit vector towards the nadir direction of body frame and I is the inertia matrix.

In low earth orbits, there will be residual earth’s atmosphere, and this causes a drag force on the satellite.

The aerodynamic force acting on solar sail of satellite is

Faero=−1

2CdρV 2As(7)

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where Faero is the aerodynamic force, Cdis the drag coeﬃcient, ρis the atmosphere density, Vis the satellite

velocity, Ais the projected srea, and sis the direction unit vector. The aerodynamic torque occurs because

of oﬀset between the center of pressure and center of mass rcp. It can be expressed as :

Taero =rcp ×Faero (8)

The SRP (solar radiation pressure) force occurs due to photons acting on a satellite and sail surface in

space. The SRP force acting on a ﬂat sail surface with optical properties of sail material is as3:

Fsrp =−P Acos α(1 + rs) cos α+Bfr(1 −s) + efbf−ebBb

ef+eb

(1 −r)n+ ((1 −rs) sin α)t(9)

where Fsrp is the SRP force, P= 4.563 ×10−6N/m2is the nominal solar radiation pressure constant at 1AU

from the sun, ris the reﬂectivity of surface, sis the specular reﬂection coeﬃcient, αis the sun angle, ef&

ebis front and back surface emission coeﬃcients and Bf&Bbis non-Lambertian coeﬃcients for front and

back surfaces. The solar radiation pressure torque gives similar to the aerodynamics torque as :

Tsrp =rcp ×Fsrp (10)

Figure 4. Disturbacne as an altitude

Figure 4 shows maximum torques with respect to altitude change. The solar radiation pressure torque,

and the aerodynamic torque get maximum when the solar sail is the perpendicular to each force vector.

Figure 5 shows the disturbance torques in the Sun pointing mode operation. Aerodynamic torque ﬂuctuates

according to the variation of angle between the sail and velocity vector.

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0 5000 10000 15000

−5

0

5x 10−6

Time(Period)

Torque(N/m

Disturbance

Gravity Gradient

SRP

Aero Drag

0 5000 10000 15000

−2

0

2

x 10−6

Time(Period)

Torque(N/m

Gravity Gradient

SRP

Aero Drag

0 5000 10000 15000

−1

−0.5

0

0.5

1x 10−5

Time(Period)

Torque(N/m

Gravity Gradient

SRP

Aero Drag

Figure 5. Disturbance in the Sun pointing mode

IV. Control law

The CNUSAIL-1 has two control modes. First control mode, detumbling mode, will use a ‘modiﬁed

B-dot controller’ to damp out the initial tip-oﬀ rate and establish a desired spin rate about Z-axis using

magnetic torquers. The control law is given as4:

M=−K( ˙

B+ωd×B) (11)

where K is a positive gain, B is the magnetic ﬁeld in the body frame, ˙

Bis the derivative of magnetic ﬁeld,

ωdis the desired spin rates, and M is magnetic dipole.

Second control mode, Sun pointing mode, will maintain the Sun pointing orientation within 10 degrees

which faces normal to the solar sail in order to investigate solar radiation pressure eﬀect. The control torque

is designed as the PD-like control law2:

Tdes=−kpe−kdωe(12)

where kdis the positive derivative gain, kpis the positive proportional gain, eis the attitude error by

quaternion, and ωeis a rate error. The required magnetic moment is calculated as :

M=Tdes ×B

|B|2.(13)

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V. Simulation

The numerical simulations are done with considering aerodynamic, solar radiation pressure, and gravity

gradient disturbances in a MATLAB Simulink environment to verify the feasibility and performance of the

proposed controller. The satellite is assumed to be rigid. The optical properties of metalized mylar is

considered for solar sail material. Assumptions and initial conditions are given as follow.

Table 1. Simulation parameters

parameter value

Satellite mass 4 kg

Sail un-deployed inertias Ixx=0.0506, Iyy=0.0506, Iz z=0.010

Sail deployed inertias Ixx=0.6, Iyy=0.6, Iz z=1.2

Orbit 700km, Sun Synchronous Orbit

Initial attitude quaternion [0 0 0 1]T

Initial body rate [10 10 10]Tdeg/sec

A. Modiﬁed B-dot contol scheme simulation

Figure 6. Detumbling mode; body rates are damped out from initial rotational velocity within 3 orbits.

The simulations are carried out with the modiﬁed B-dot control scheme. Desired spin rates ωdare [0 0

5]Tdeg/sec. Figure 6 shows that body rates are damped by magnetic torquers in about 3 period(5 hours).

Also, Z-axis body rate converges to the desired spin rate. Figure 7 shows that body rates are not damped

out by torquers in about 3 orbit periods and Z-axis spin rate diverges. Once the solar sail of satellite is

deployed, inertia matrix is large so that body rates cant be damped out by the magnetic torquer only. For

the purpose of the Sun pointing mode operation, spin rate must be damped out. Thus, more control torque

will be required by using reaction wheels.

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Figure 7. Sail deployed. The body rates are not damped out from initial rotational velocity.

B. PD-like control scheme simulation

Figure 8. PD-like control scheme. The initial pointing error is 90 degree.

Figure 8 shows the result of Sun pointing simulation using the PD-like control scheme with allowable

torques are unlimited. The Sun pointing error is conversed to the zero. More results and detailed analysis

using the PD-like control scheme will be updated in the ﬁnal manuscript.

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VI. Conclusions

This paper presented the attitude control system for a cube satellite with solar sail. The two modes

were designed to stabilize the satellite and maintain Sun pointing orientation. External disturbances were

calculated and compared with the worst case. In the Sun pointing mode operation, the solar radiation

pressure torque was maintained constantly, but aerodynamic torque was changed according to the angle

between tangential vector of sail and velocity vector. The modiﬁed B-dot control scheme damped out body

rates from initial tip-oﬀ rate when the sail was not deployed. However, it can’t stabilize body rates when

the sail was deployed. The simulation of PD-like control scheme was performed for the situations when the

sail is deployed and undeployed. The PD-like control scheme simulation results and detailed analysis will be

updated in the ﬁnal manuscript.

Acknowledgments

This research was supported by Cubesat Contest and Developing Program through the National Research

Foundation of Korea (NRF) and the Korea Aerospace Research Institute (KARI) funded by the Korea

government (Ministry of Science, ICT and Future Planning)(No. NRF-2013M1A3A4A01075962).

References

1Wertz, J.R, Spacecraft attitude determination and control, D. Reidel publishing Company, 1986.

2Anton H. de Ruiter, Christopher Damaren, James R. Forbes , Spacecraft Dynamics and Control: An Introduction, Wiley,

2013.

3Bong Wie, “Solar Sail Attitude Control and Dynamics, Part 1,” Journal of guidance, control, and dynamics, Vol. 27,

No. 4, 2004, pp. 526–535.

4Glenn Creamer, “The HESSI magnetic attitude control system,” AIAA Guidance, Navigation, and Control Conference ,

1999.

5Julie Thienel, Robert Bruninga, Robert Stevens, Cory Ridge, and Chad Healy, “The Magnetic Attitude Control System

for the Parkinson Satellite (PSAT) A US Naval Academy Designed CubeSat,” AIAA Guidance, Navigation, and Control

Conference, 2009.

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