# [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational Intelligence - Design and simulation of satellite formation flying position-keeping control method

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AbstractSatellite formation flying position-keeping control is researched. A simulation method based on STK and Matlab for the satellites formation position-keeping control is introduced. Relative dynamical models of formation flying were derived, and Hill equations were deduced based on the circular reference orbit. Linear-quadratic regulator was adopted to perform the formation flying position-keeping control. Simulations in MATLAB Simulink show the successful use of LQR control approach for satellite formation flying position-keeping control.

I. INTRODUCTION atellite formation flying is the concept that multiple

satellites can work together in a group to accomplish the objective of one larger, usually more expensive, satellite

[1]. Formation flying has emerged as an enabling technique that will be used in numerous future satellite missions. Formation flying is a new paradigm in space mission design, aimed at replacing large satellites with multiple small satellites [2]. Some of the proposed benefits of formation flying satellites are: (1) Reduced mission costs and (2) multi mission capabilities, achieved through the reconfiguration of formations. Coordinating smaller satellites has many advantages over single satellites including simpler designs, faster build times, cheaper replacement creating higher redundancy, unprecedented high resolution, and the ability to view research targets from multiple angles or at multiple times [3]. These qualities make them ideal for astronomy, communications, meteorology, and environmental uses [4]. This paper addresses the problems of maintenance of satellite formations in Earth orbits. Achieving the objectives of maintenance, with the least amount of fuel is the key to the success of the mission. Therefore, understanding and utilizing the dynamics of relative motion, is of significant importance [5]. Relative dynamical models of formation flying were introduced, and Hill equations were deduced based on the circular reference orbit. A higher order state transition matrix is developed using unit sphere approach in the mean elements space. Based on the state transition matrix analytical control laws for formation flying maintenance are proposed using linear-quadratic regulator scheme. Simulations in MATLAB Simulink show the control law works well in satellite

Manuscript received April 10, 2010. This work was supported in part by

the Scientific Research Foundation for Doctoral, Shenyang Aerospace University

Yanmei Liu is with the Shenyang Aerospace University (corresponding phone: 024-82076312; e-mail: lymlczlymlcz@163.com).

Yibo Li is with Shenyang Aerospace University. (e-mail: lymcml@126.com).

Jianhui Xi is with the Shenyang Aerospace University (e-mail: xjhui_01@163.com).

formation flying position-keeping control.

II. DYNAMICS MODELS Achieving the objectives of maintenance, with the least

amount of fuel is the key to the success of the mission. Therefore, understanding and utilizing the dynamics of relative motion, is of significant importance [6]. The simplest known model for the relative motion between two satellites is described using the Hill-Clohessy-Wiltshire (HCW) equations [7]. The HCW equations offer periodic solutions that are of particular interest to formation flying. In this paper the Hill-Clohessy-Wiltshire equations (Clohessy & Wiltshire 1960) will be presented. The equations are deduced under the assumptions that the Earth is a perfect sphere, and that the Leader satellite is in a Keplerian circular orbit [8]. For a satellite orbiting the Earth, the two-body problem applies, under the assumptions that:

1. The equations of motion are expressed in a non-inertial reference frame whose origin coincides with the center of mass of the central body.

2. Both the central body and satellite are homogenous spheres or points of equivalent mass.

3. The inverse-square gravitational force between the two bodies is the only force in action.

In our study, one satellite, or a virtual satellite, is taken as reference satellite and others as member satellites. Two Cartesian coordinates are used. As shown in figure 1, Earth Centered Inertial (ECI) coordinate is spanned by unit vectors (X,Y,Z). Local Vertical Local Horizontal (LVLH) coordinate is attached on reference satellite S0 by unit vectors (x,y,z).

Fig.1 ECI and LVLH coordinates

In the Cartesian coordinates, the HCWs equations describe the motion of a follower spacecraft relative to a leader spacecraft, corresponds to HCW equations:

Design and Simulation of Satellite Formation Flying Position-Keeping Control Method

Yanmei Liu, Yibo Li, and Jianhui Xi

S

502

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

=

=+

=+

z

y

x

Fzxz

Fyy

Fzx

2

2

32

2

1

Where is the Angular velocity of reference satellite xF yF zF are the orbit control force.

Suppose =X ( )Tyyxzyx are the state

variables ( )TzyxY = are the output variables.

According to he Eq.1the system is represented as

=+=

CXYBUAXX 2

Where A is the state matrix, B is the control matrix, C is the output matrix

=

00230000000200000100000010000001000

2

2

A ,

=

33

33

IO

B ,

[ ]3333 = OIC . Based on Hill, the equations that express general circular

reference orbit can be written as follows:

+=

+=

=

wtzwtwz

tz

wtzwtwz

ty

wtzwtwz

tx

cossin)(

cos3sin3)(

sin2cos2)(

00

00

00

3

In Eq. (3), z0 is the initial position of z. Under conditions of 5km diameter, the fly tracking for circular reference orbit is shown in figure 2

-4-2

02

4

x 104

-5

0

5

x 104

-4

-2

0

2

4

x 104

x(m)y(m)

z(m

)

Fig.2 fly tracking of 5km diameter.

III. FORMATION FLYING MODEL BASED ON STK Without lost of generality, we consider a 3-satellite system,

i.e. a free-flying reference satellite S0 (without control force) and three controlled member satellite Sj(with control force).The analysis was carried out by using the Satellite Tool Kit (STK). The STK is very useful analysis tool for space mission planning [9]. The satellites parameters are in table I.The actual separation sequence is planned as shown in the figure 3.

TABLE I SATELLITES PARAMETERS

RAAN Orbit

sat1 0deg Circular sat2 120deg Circular

sat3 240deg Circular

Fig.3 3-satellite system formation flying described by STK

IV. DESIGN OF OPTIMAL CONTROLLER Formation flying can be defined that a relative position

between several spacecraft is maintained within some specified range. The accuracy of keeping cluster formation depends on mission requirement. Because Linear Quadratic Regulator (LQR) control leads to linear control laws that are easy to implement and analyze, LQR was adopted to perform the formation flying position-keeping control.

The Linear Quadratic Regulator (LQR) is a special optimal controller whose cost function, measure of performance, is a quadratic function of states and controls. Two desirable

503

properties of the LQR are good stability margins and sensitivity properties.

The system is represented in Eq.(2). Suppose [ ]rrrrrrr zyxzyxX = is the desired states

of member satellites, and [ ]ddddddd zyxzyxX = is actual state of

the member satellites, the relative motion state is

=+=

XCYBUXAX 4

Where dr XXX = . Pontriagins minimum principle is used to solve the optimal control problem. A cost function can be defined as

( )

+=0

21

t

TT dtRUUQXXJ

Where J is minimized with respect to the control input u(t). J represents the weighted sum of energy of the state and control. Q and R are weighting matrices, or design parameters, where the state-cost matrix, Q, weights the states while the performance index matrix, R, weights the control effort. If Q is increased while R remains constant, the settling time will be reduced as the states approach zero. LQR controller is designed as shown in figure 4.

Fig.4 LQR controller

According to the Riccati equation

01 =++ QPBPBRPAPA TT 5 The feedback gain matrix

PBRK T1= 6 Then the formation keeping control force effecting to the member satellites

XKF = 7

Then

=

f

f

f

f

f

f

z

y

x

zzyyxxzzyyxx

Kuuu

*

V. SIMULATION RESULTS Considering member satellite1, the report data of angular

velocity can be got from STK: w(x)=-0.03, w(y)=-0.01, w(z)=0.03,and the synthesized Angular velocity is w(1)=0.04. The feedback gain matrix K calculated based on Eq.(6) is K =

[ 0.0384 -0.0000 0.0114 0.2806 -0.0000 0.0024;

0.0000 0.0384 0.0000 -0.0000 0.2801 -0.0000;

-0.0113 -0.0000 0.0434 0.0024 -0.0000 0.2968]

In this section, the designed controller is tested in the missions as follows. The ode45 function in MATLAB is used for numerical integrations. The station keeping problem is studied.

In order to compare the simulation results with other literature, a circular reference orbit case is simulated. The leader is in the circular orbit with an altitude of 800 km. The follower is in circular orbits with the same altitude. This represents a typical formation keeping problem. figure 5 show the relative position error in the LVLH coordinate system.

Fig.5 (a) Position error for satellite S1

Fig.5 (b) Position error for satellite S2

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Fig.5 (c) Position error for satellite S3

figure 5 demonstrated that for this challenge mission the designed controller can achieve a much good control precision. The LQR controller is stable for this simulated mission.

VI. CONCLUSION In this paper, a linear model for the relative position

dynamics of a Leader/Follower spacecraft formation, called the Hill-Clohessy-Wiltshire equations, has been presented. Linear-quadratic regulator controller was used for each satellite. Simulations were performed in MATLAB Simulink environment .Simulations results show the successful use of LQR control approach.

REFERENCES [1] Cai, C. and Teel, A. R., eds, Results on Input-to-State Stability for

Hybrid Systems, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain.2005.

[2] Cai, C., Teel, A. R. and Goebel, R., eds, Converse Lyapunov Theorems and Robust Asymptotic Stability for Hybrid Systems, Proceedings of the American Control Conference,Portland, OR, USA. 2005.

[3] Laila Mireille Elias, PhD thesis: Dynamics of Multi-Body Space Interferometers Including Reaction Wheel Gyroscopic Stiffening Effects: Structurally Connected and Electromagnetic Formation Flying Architectures, Massachusetts Institute of Technology, March, 2004.

[4] Guangyan Xu, A Novel Explicit Satellite Dynamics Under J2 Perturbation, UK-Singapore Workshop on AEROSPACE ENGINEERING: CHALLENGES & OPPORTUNITIES, pp.29-30, May.2006.

[5] Xibin Cao and Donglei He. Relative motion equation for perturbed ellipitical reference orbit formation. Journal of Jilin University (Engineering and Technology Edition). vol.39,no.1, pp.234-239,2009,

[6] Wong H, Pan H, Kapila V. Output feedback control for spacecraft formation flying with coupled translation and attitude dynamics, Proceedings of the American Control Conference, Portland, OR,pp.24192426, June.2005.

[7] Litao Li, Yang Xu, Shunli Li, Mid-range relative navigation method for non cooperative target satellite, Journal of Jilin University (Engineering and Technology Edition). vol.38,no.4, pp.986-990,2008.

[8] Guangyan Xu. Pongvuthithum R, Veres SM, Gabriel SB, Rogers E. Universal adaptive control of satellite formation flying, IEEE Aerospace Conference 2009.

[9] Analytical Graphics Inc. Satellite Tool Kit webpage; http://www.stk.com/

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