Discrete-time LQ optimal control of satellite formations in elliptical orbits based on feedback linearization

Acta Astronautica 83 (2013) 125–131

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Discrete-time LQ optimal control of satellite formations in elliptical orbits based on feedback linearization Yafei Li n, Xiangdong Liu, Guangqian Xing School of Automation, Beijing Institute of Technology, Beijing 100081, PR China

a r t i c l e i n f o

abstract

Article history: Received 5 September 2011 Received in revised form 7 October 2012 Accepted 16 October 2012 Available online 20 November 2012

The sampled-data representation of the relative motion is the foundation of the discretetime LQ optimal control of spacecraft formations. The sampled-data description for the relative motion in circular orbits has been investigated in great detail. However, few are derived for the elliptical orbits. This paper will employ the discrete-time LQ optimal control theory to deal with the problem of relative orbit control of satellites in elliptic orbits. The formation vector is used to express the formation geometry, and nonlinear feedback is utilized to linearize the equations of the relative motion. An analytical state transition matrix is derived from the solutions of the linearized equations. Based on the state transition matrix, a sampled-data representation is presented for the linearized equations. The sampled-data representation is explicitly related to a sampling sequence of the true anomaly of the target satellite with constant length of the sampling intervals. In terms of the discrete-time model, a discrete-time LQ optimal controller is derived for the linearized equations. By combining the discrete-time LQ optimal control with the nonlinear feedback control, a digital controller is obtained for the satellite formations. Simulations are included to demonstrate the performance of the controller. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Relative motion State transition matrix Sampled-data representation LQ optimal control

1. Introduction Discrete-time LQ optimal control theory is favored by many researchers with its simplicity in implementation, and has been applied to the satellites formation along circular orbits in [1,2]. However, it has not been used for the satellites formation with an elliptical reference orbit. For the satellites formation between elliptical orbits, the representative model for describing the relative motion is Tschauner–Hempel (TH) equations. Carter [3,4] presented the general solutions of the TH equations by free variable transformation and Lyapunov transformation. Based on Carter’s work, Sengupta [5] derived a new form of the solutions for the TH equations. The state transition matrix of the TH equations is critical for the construction of the

n Corresponding author. Tel.: þ86 18 221326645; fax: þ 86 10 68918385. E-mail address: [email protected] (Y. Li).

0094-5765/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actaastro.2012.10.027

sampled-data representation of the relative motion. State transition matrix using time as an independent variable was developed by Melton [6] with Lagrange’s generalized expansion, but the result is in the form of series. Yamanaka [7] used the solutions of TH equations to obtain the state transition matrices of the in-plane motion and the out-ofplane motion, respectively, with the true anomaly of the target satellite as the independent variable. Instead of directly solving the complex differential equations of the relative motion, Gim [8] introduced the geometric method to derive the state transition matrix with the relationship between the relative states and the differential orbital elements, but the results are complicated. Extensive studies on the formation control along elliptical orbits have been carried out. The approaches include impulsive control [9], continuous-time LQ regulator technique [10,11], and nonlinear control [12], etc. This paper will utilize the discrete-time LQ optimal control theory to deal with the problem of relative orbit

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Y. Li et al. / Acta Astronautica 83 (2013) 125–131

control between arbitrary eccentricity elliptical orbits. By nonlinear feedback, the equations of the relative motion are linearized. The linearized equations are the same as TH equations. The solutions of the TH equations are used to derive an analytical state transition matrix of the TH equations. The state transition matrix is six-dimensional, and is valid for the eccentricities in the range 0 re o1. Then a sampled-data representation is derived in terms of the state transition matrix. The sampled-data representation is also analytical, and is directly related to the true anomaly of the target satellite. Based on the sampled-data representation, a discrete-time LQ optimal controller is obtained for the linearized equations of the relative motion. The paper is organized as follows. Section 1 is the introduction. Section 2 introduces the equations of the relative motion and linearizes the equations by nonlinear feedback. Section 3 obtains the sampled-data representation for the TH equations. Section 4 formulates the discrete-time LQ optimal controller for the TH equations. Section 5 presents the simulation results with two examples.

equations: 2 m m x€ ¼ 2f_ y_ þ f€ yþ f_ x 3 ðx þ r C Þ þ 2 þ uxF i rFi rC

z€ ¼ 

f_ ¼

m r 3F i

r 3F i

y þ uyFi

z þuzF i

ð1Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m

r 3C

ð1 þe cos f Þ

ð2Þ

m f€ ¼ 2e 3 sin f rC

ð3Þ

where f is the true anomaly of the target satellite, e is the eccentricity, m is the gravitational parameter, and uxF i , uyFi , and uzF i are the control accelerations. Denote ugx Fi 9 ugy Fi 9 ugz Fi 9

2. Equations for the relative motion Consider an Earth-centered inertial frame with the orthonormal basis BI ¼ fix iy iz g. The vectors ix and iy lie in the equatorial plane, with ix coinciding with the line of equinoxes, and iz passing through the North pole. iy ¼ iz  ix . The analysis uses a local vertical–local horizontal frame, which is attached to centroid of the target satellite with the orthonormal basis BL ¼ fir iy ih g, where ir lies along the radius vector from the earth center to the target satellite, ih coincides with the normal to the orbital plane of target satellite, and iy ¼ ih  ir . The relative motion of two satellites (named as target satellite and chaser satellite) in elliptical orbits is depicted in Fig. 1, where r C and r F i are the position vectors of the target satellite and the chaser satellite, respectively, with r C ¼ r C ir . In order to describe the formation configuration geometrically, the formation vector li is introduced, with li ¼ lx ir þly iy þlz ih [13]. Denote r T i to be the expected position vector of the chaser satellite, so r T i ¼ r C þ li . From Fig. 1 it is easy to get that qi ¼ r F i r C and Dqi ¼ r F i r T i . Let qi ¼ xir þ yiy þzih and Dqi ¼ Dxir þ Dyiy þ Dzih . So the relative motion with the target satellite in an arbitrary elliptical orbit is modeled by the following nonlinear

m

2

_ f€ x þ f_ y y€ ¼ 2f_ x

uGx Fi 9

m r 3F i

m

r 3F i

m

r 3F i

m r 3C

uGy F i 9 uGz F i 9

ðx þ r C Þ

m r 2C

y z

ð4Þ

2Dx

m

r 3C

m

r 3C

Dy Dz

ð5Þ 2

€ __ € _ ulx F i 9l x 2f l y f ly f lx € __ € _2 uly F i 9l y þ 2f l x þ f lx f ly € ulz F i 9l z

ð6Þ

Gx lx ux 9uxF i ugx F i uF i uF i Gy ly uy 9uyFi ugy F i uF i uF i Gz lz uz 9uzF i ugz F i uF i uF i

ð7Þ

Then Eq. (1) can be rewritten by feedback linearization [19] as 2

Dx€ ¼ 2f_ Dy_ þ f€ Dy þ f_ Dx þ 2

m r 3C

_ f€ Dx þ f_ Dy Dy€ ¼ 2f_ Dx

Dz€ ¼ 

m r 3C

Dz þ uz

2Dx þux

m

r 3C

Dy þuy ð8Þ

When the true anomaly f of the target satellite is used as a free variable, and the Lyapunov transformation [3,4,16]

Fig. 1. Relative motion.

8 1 þ e cos f > > Du ¼ Dx > > > p0 > > > < 1 þ e cos f Dv ¼ Dy p0 > > > > > 1þ e cos f > > Dz > Dw ¼ : p0

ð9Þ

Y. Li et al. / Acta Astronautica 83 (2013) 125–131

is introduced, Eq. (8) become rather simpler in dimensionless form [3,4]

Du00 ¼

ð1e2 Þ3

0

Dv ¼ 2Du þ Dw00 ¼ Dw þ

p0 ð1 þ e cos f Þ3 n2 ð1e2 Þ3

p0 ð1 þ e cos f Þ3 n2

u

y

uz

ð10Þ

where Dv0 denotes dDv=df , p0 is a constant, a is the semiffi pffiffiffiffiffiffiffiffiffiffi major axis of the target satellite, and n ¼ m=a3 . Rewrite Eq. (10) in matrix form as d Uðf Þ ¼ A1 ðf ÞUðf Þ þ B1 ðf Þu df

ð11Þ

where 2

0 0

0 0

6 6 6 6 0 6 A1 ðf Þ ¼ 6 6 1 þ e3cos f 6 6 0 4 0

0 0

1 0

0 1

0

0

0

0

0

0

0

2

0

0

2

0

0

1

0

0

2

3 0 7 07 7 17 7 7 07 7 07 5 0

0

0

60 6 6 60 ð1e Þ 6 B1 ðf Þ ¼ 3 2 61 p0 ð1 þ e cos f Þ n 6 6 40

0 0 1

07 7 7 07 7 07 7 7 05

0

0

1

0

2 1 þ e cos f 0 6 6 0 6 6 6 6 0 6 P2 ðf Þ ¼ 6 e sin f 6 p 6 0 6 6 0 6 4 0

2d3

þ



0

1 þ e cos f p0

0

0

0

0

0

1 þ e cos f p0

0

0

0

0

0

1 þ e cos f p0

0

0

f  e sin p

0

0

1 þ e cos f p0

0

0

f  e sin p0

0

1 þ e cos f p0

0

0



Z2

1

3e sin f ð1 þ e cos f ÞKðf Þ 2Z3

7 7 7 7 7 7 7 7 7 7 7 7 7 5



3d3

Z5

ð1 þe cos f Þ2 Kðf Þ þ d4

Dwðf Þ ¼ d5 cos f þ d6 sin f

ð12Þ

where Kðf Þ ¼ 0

0

Z

f

Z3 ð1 þe cos f Þ2

df

T

and

Z¼ Suppose that the true anomaly of the target satellite is f at time t. Denote _ _ X 1 ðtÞ ¼ ½DxðtÞ DyðtÞ DzðtÞ DxðtÞ DyðtÞ Dz_ ðtÞT X 2 ðf Þ ¼ ½Dxðf Þ Dyðf Þ Dzðf Þ Dx0 ðf Þ Dy0 ðf Þ Dz0 ðf ÞT

pffiffiffiffiffiffiffiffiffiffiffiffi 1e2

Consequently the relative velocities are

Du0 ðf Þ ¼ d1 ðsin f þ e sin 2f Þ þd2 cos f ðcos f þ e cos 2f Þ 

3ed3 3ed3 sin f ðcos f þ e cos 2f ÞKðf Þ 2 2Z5 Z 1 þ e cos f

Dv0 ðf Þ ¼ d1 ð2 cos f þ e cos 2f Þd2 ð2 sin f þ e sin 2f Þ þ

So X 2 ðf Þ ¼ P 1 ðf ÞX 1 ðtÞ

6ed3

Z5

sin f ð1 þe cos f ÞKðf Þ

3d3

Z2

Dw0 ðf Þ ¼ d5 sin f þd6 cos f

where 3

0

0

0

0

0

1

0

0

0

0 0

1 0

0

0 0

0

0

0

1 f_

0

0

0

0

07 7 7 07 7 07 7 7 7 07 7 5 1

1 f_

f_

And the Lyapunov transformation (9) also can be expressed in matrix form as Uðf Þ ¼ P2 ðf ÞX 2 ðf Þ

0

Dvðf Þ ¼ d1 sin f ð2 þe cos f Þ þ d2 cos f ð2 þe cos f Þ

u ¼ ½ux uy uz T

1 60 6 6 60 6 6 P 1 ðf Þ ¼ 6 0 6 6 60 6 4 0

0

Duðf Þ ¼ d1 cos f ð1þ e cos f Þ þ d2 sin f ð1 þe cos f Þ

Uðf Þ ¼ ½Duðf Þ Dvðf Þ Dwðf Þ Du ðf Þ Dv ðf Þ Dw ðf Þ

2

0

In order to construct the sampled-data representation for system (11), we need to obtain the state transition matrix of it. We first study the solutions of Eq. (10) as [5]

f0 0

3

0

3. The sampled-data representation

3

0

2 3

where p

3 ð1e2 Þ3 Du þ 2Dv0 þ ux 1 þe cos f p0 ð1 þ e cos f Þ3 n2

00

127

Denoted by D ¼ ½d1 d2 d3 d4 d5 d6 T . Then Uðf Þ ¼ Mðf ÞD. It is easy to get the elements of M(f). Because det ðMðf ÞÞ ¼ 1, M(f) is always invertible when the eccentricity of the target satellite is in the range 0 re o1. Denote M1 ðf Þ to be the inverse of M(f). The elements of M 1 ðf Þ are provided in Appendix A. So an analytical state transition matrix of system (11) is obtained as

Fðf ,f 0 Þ ¼ Mðf ÞM1 ðf 0 Þ

ð13Þ

Clearly, the state transition matrix is valid for the arbitrary eccentricity elliptical orbits. Rf Rf Define Uðf ,f 0 Þ9 f M 1 ðtÞB1 ðtÞ dt and Pðf ,f 0 Þ9 f F 0 0 ðf , tÞB1 ðtÞ dt. The elements of Uðf ,f 0 Þ are provided in

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Y. Li et al. / Acta Astronautica 83 (2013) 125–131

Appendix B. So

Pðf ,f 0 Þ ¼ Mðf ÞUðf ,f 0 Þ

5. Simulations ð14Þ

For a given sequence of the true anomaly f 0 o f 1 o    of N , where the length of the sampling interval Wf k ¼ f k þ 1 f k is constant, a sampled-data description of system (11) is given by Uðf k þ 1 Þ ¼ Fðf k þ 1 ,f k ÞUðf k Þ þ Pðf k þ 1 ,f k Þuðf k Þ

ð15Þ

for k ¼ 0,1, . . . ,N1. Supposing that time is tk when the true anomaly of the target satellite is fk, a time sequence is derived as t 0 o t 1 o    o t N . From Eq. (2) we know that WT k ¼ t k þ 1 t k is time-varying for the target satellite in an arbitrary eccentricity elliptical orbit. WT k is the largest when the time of passage through apogee of the target satellite is included in ½t k þ 1 ,t k Þ, and smallest when the time of passage through perigee of the target satellite is included in it.

Consider the discrete-time system (15). Next we will find a control sequence fuðf k ÞgN k ¼ 0 such that the performance index X T 1 T 1 N1 U ðf N ÞSN Uðf N Þ þ ½U ðf k ÞQ k Uðf k Þ þ u T ðf k ÞRk uðf k Þ 2 2k¼0

is minimized, where Qk and SN are positive semidefinite, and Rk is positive definite. In terms of the discrete-time LQ optimal control theory, the following controllers are derived [17,18]: uðf k Þ ¼ ðPT ðf k þ 1 ,f k ÞSk þ 1 Pðf k þ 1 ,f k Þ þ Rk Þ1 PT ðf k þ 1 ,f k ÞSk þ 1 Fðf k þ 1 ,f k ÞUðf k Þ

Example 1. The initial conditions [15] are given in Table 1. Here the eccentricity of the target satellite is small. The simulation results are shown in Table 2 and Figs. 2–4. Example 2. The initial conditions are the same as Example 1 except for the semi-major axis and eccentricity. We present the simulation results in Table 3 and Figs. 5–7.

4. Discrete-time LQ optimal control

J¼

Following simulations demonstrate the effectiveness of the proposed approach. The periapsis of the target satellite for all examples is kept constant at r p ¼ 7100 km, so that the semi-major axis of the target satellite can be obtained for any given eccentricity of the target satellite from the relation a ¼ r p =ð1eÞ. The formation vector li is chosenpffiffiffias lx ¼ 5000n sin ðntÞ, ly ¼ 10 000 cos ðntÞ, lz ¼ 5000n 3n sin ðntÞ [14]. The parameters for the performance index are taken as SN ¼ 1016 I66 , Q k ¼ 10I66 , and Rk ¼ 5  108 I33 .

ð16Þ

where Sk satisfy the equations

The discrete-time linear quadratic regulator theory with fixed terminal time has been applied in both examples. The formation errors listed in Tables 2 and 3 are the values at the terminal time. The results show that the length of the sampling interval Wf k determines the formation errors under certain initial conditions. As the sampling interval Wf k decreases, the formation errors and fuel consumption also reduce. But the formation position errors decline more and more slowly, and the computational complexity grows greatly with the sampling interval decreasing. Therefore, the sampling interval Wf k cannot be selected to be arbitrarily small. Besides, we find that the fuel consumptions of Example 1 are larger than Example 2 by making a comparison between

Sk ¼ Q k þ FT ðf k þ 1 ,f k ÞSk þ 1 Fðf k þ 1 ,f k Þ FT ðf k þ 1 ,f k ÞSk þ 1 Pðf k þ 1 ,f k ÞðPT ðf k þ 1 ,f k ÞSk þ 1 Pðf k þ 1 ,f k Þ þ Rk Þ1 PT ðf k þ 1 ,f k ÞSk þ 1 Fðf k þ 1 ,f k Þ

ð17Þ

for k ¼ 0,1, . . . ,N1. Define uðt k Þ ¼ ½uxF i ðt k Þ uyFi ðt k Þ uzF i ðt k ÞT

Satellite

By combining the discrete-time LQ optimal control with the nonlinear feedback control, a digital controller for system (1) is given as follows: Gx lx uxF i ðt k Þ ¼ ux ðt k Þ þ ugx F i ðt k Þ þ uF i ðt k Þ þ uF i ðt k Þ Gy ly uyFi ðt k Þ ¼ uy ðt k Þ þ ugy F i ðt k Þ þ uF i ðt k Þ þ uF i ðt k Þ Gz lz uzF i ðt k Þ ¼ uz ðt k Þ þ ugz F i ðt k Þ þ uF i ðt k Þ þuF i ðt k Þ

Table 1 Initial conditions for the target and chaser satellites.

ð18Þ

a, km e i, deg O, deg o, deg f0, deg fN, rad

Example 1

Example 2

Target

Chaser

Target

Chaser

7171.717 0.01 95 30 0 105 16.8326

7000 0.01 96 30 0 100

10 142.857 0.3 95 30 0 105 16.8326

10 000 0.3 96 30 0 100

where uðt k Þ ¼ ½ux ðt k Þ uy ðt k Þ uz ðt k ÞT

ð19Þ

uðt k Þ ¼ uðf k Þ

ð20Þ

uðf k Þ ¼ ðPT ðf k þ 1 ,f k ÞSk þ 1 Pðf k þ 1 ,f k Þ þ Rk Þ1 PT ðf k þ 1 ,f k ÞSk þ 1 Fðf k þ 1 ,f k ÞP 2 ðf k ÞP 1 ðf k ÞX 1 ðt k Þ for t 2 ½t k ,t k þ 1 Þ, k ¼ 0,1, . . . ,N1.

ð21Þ

Table 2 Formation errors and fuel consumption, e ¼0.01. Performance indices

Df k ¼ 0:1 rad

JDqi J2 , m JDq_ i J2 , m/s JuJ1 , m/s

Df k ¼ 0:01 rad

Df k ¼ 0:001 rad

0.2284

0.0055

5:3862  104

0.0015

1:4197  105 320.4565

1:3287  107 320.1959

325.8296

Y. Li et al. / Acta Astronautica 83 (2013) 125–131

target chaser

x 106

8 6 4

z[m]

2 0 −2

129

Table 3 Formation errors and fuel consumption, e¼ 0.3. Performance indices

Df k ¼ 0:1 rad

Df k ¼ 0:01 rad

Df k ¼ 0:001 rad

JDqi J2 , m JDq_ i J2 , m/s

7.1872

0.0068

8:7228  105

0.0308

JuJ1 , m/s

264.3157

3:1921  104 254.3460

2:9236  106 253.4992

−4 −6

x 10

target chaser

−8 6 4

x 107

2

1

y[m] 0 0.5

−2 −8

−4

−6

0

−2

2

4

8

6

z[m]

−4

x 106

x[m]

0 −0.5

Fig. 2. Relative trajectory under conditions e ¼ 0:01, Df k ¼ 0:01 rad.

1 −1 4

x 106 Position error[m]

2

2

x 105

0

Δx Δy Δz

−2 −4

2 4000

6000

8000

10000

12000

14000

Time[s] 400 dΔx/dt dΔy/dt dΔz/dt

300 200

Position error[m]

2000

−2

−0.5 −4

−6

100

−1.5

]

x[m

x 105

Δx Δy Δz

0 −2 −4 −6 −8 −10 0

0

0.5

1

1.5

2

−200 2000

4000

6000

8000

10000

12000

14000

Time[s] Fig. 3. Tracking error under conditions e ¼ 0:01, Df k ¼ 0:01 rad.

0.05

uxFi uyFi uzFi

0.04 0.03

300

Velocity error[m/s]

0

2.5

x 104

Time[s]

−100

dΔx/dt dΔy/dt dΔz/dt

250 200 150 100 50 0 −50 0

0.5

1

1.5

Time[s]

2

2.5

x 104

Fig. 6. Tracking error under conditions e ¼ 0:3, Df k ¼ 0:01 rad.

0.02

u [m/s2]

−1 −8

x 107

Fig. 5. Relative trajectory under conditions e ¼ 0:3, Df k ¼ 0:01 rad.

−6 0

0 0

y[m ]

−8

Velocity error[m/s]

0.5

0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 0

2000

4000

6000

8000 10000 12000 14000

Time[s] Fig. 4. Control acceleration under conditions e ¼ 0:01, Df k ¼ 0:01 rad.

Tables 2 and 3. And the results in Figs. 4 and 7 indicate that JuJ1 of Example 2 is less than Example 1. On the one hand, the weighting matrices may cause this. Although we choose the same weighting matrices for the dimensionless state Uðf k Þ in both examples, the weighting matrices for the real system state X 1 ðt k Þ are equal to P T1 ðf k ÞPT2 ðf k ÞQ k P 2 ðf k ÞP 1 ðf k Þ. The weighting matrices are time-varying and they are dependent on the eccentricity and the semi-major axis of the target satellite. Due to the different weighting matrices, the distribution of the fuel

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Y. Li et al. / Acta Astronautica 83 (2013) 125–131

0.02

uxFi uyFi uzFi

0.015 0.01

M 1 11 ðf Þ ¼ 

3

Z2

ðe þ cos f Þ

M 1 12 ðf Þ ¼ 0

0.005

u [m/s2]

Appendix A. Elements of M 1 ðf Þ

M 1 13 ðf Þ ¼ 0

0

M 1 14 ðf Þ ¼ 

−0.005

M 1 15 ðf Þ ¼

−0.01 −0.015

1

Z2

1

Z2

sin f ð1þ e cos f Þ

ðe cos2 f 2 cos f eÞ

M 1 16 ðf Þ ¼ 0

−0.02

M 1 21 ðf Þ ¼

3eð2þ 3e cos f þ e2 Þ

Z5

−0.025 0

0.5

1

1.5

Time[s]

2

x 104

Fig. 7. Control acceleration under conditions e ¼ 0:3, Df k ¼ 0:01 rad.

Kðf Þ

e 2 3 e sin f þ 2sin 2f þ sin f  2 1 þ e cos f Z

2.5

M 1 22 ðf Þ ¼ 0 M 1 23 ðf Þ ¼ 0 M 1 24 ðf Þ ¼

and formation errors in the performance index will be different. On the other hand, the norm of B1 ðf Þ in Example 1 is much smaller than Example 2, which means that Example 1 will need more u than Example 2 for the same increment of Uðf k Þ. Except for the above results, we also see that u is not zero at the terminal time from Figs. 4 and 7. There are two reasons. Firstly, here the linear quadratic regulator theory with fixed terminal time is proposed, so the coefficient matrix which times the system state in u is timevarying. Although the system state is close to zero, the coefficient matrix is not close to zero. So u is not close to zero, too. Secondly, the nonlinear feedback terms (4)–(6) are not zero, especially for Example 2. Therefore, u is not zero at the terminal time. In a word, if we want to construct a formation with fewer energy, the selection of the sampling interval, weighting matrices, and formation geometry are important.

3e2

Z

5

sin f ð1 þ e cos f ÞKðf Þ þ

e cos2 f þ cos f 2e

Z2

e 2 sin f  sin 2f 3e 2 2 M 1 ðf Þ ¼ ð1 þ e cos f Þ Kðf Þ þ 25 5 2

Z

Z

M 1 26 ðf Þ ¼ 0 2 M 1 31 ðf Þ ¼ 2þ 3e cos f þe

M 1 32 ðf Þ ¼ 0 M 1 33 ðf Þ ¼ 0 M 1 34 ðf Þ ¼ e sin f ð1 þ e cos f Þ 2 M 1 35 ðf Þ ¼ ð1 þ e cos f Þ

M 1 36 ðf Þ ¼ 0 M 1 41 ðf Þ ¼

3ð2þ 3e cos f þ e2 Þ

Z5 

3

Z2

e sin f

Kðf Þ

2 þe cos f 1 þe cos f

M 1 42 ðf Þ ¼ 1 M 1 43 ðf Þ ¼ 0 3e ð2 þe cos f Þð1e cos f Þ M 1 44 ðf Þ ¼ 5 sin f ð1þ e cos f ÞKðf Þ 2

Z

6. Conclusions The nonlinear differential equations for the relative motion are first linearized by nonlinear feedback. The independent variable transformation and Lyapunov transformation are introduced to simplify the linearized equations. In terms of the solutions of the linearized equations, a state transition matrix is obtained, based on which an analytical sampled-data description of the linearized equations is presented. The sampled-data representation is explicitly related to a sampling sequence of the true anomaly of the target satellite with constant length of the sampling intervals. With the discrete-time model, a digital controller is developed for the relative motion. Simulation results show that the selection of the sampling interval has an effect on the formation errors, so an appropriate sampling interval needs to be chosen. Besides, the selection of the weighting matrices and formation geometry are also important for a formation.

M 1 45 ðf Þ ¼ M 1 46 ðf Þ ¼ M 1 51 ðf Þ ¼ M 1 52 ðf Þ ¼ M 1 53 ðf Þ ¼ M 1 54 ðf Þ ¼ M 1 55 ðf Þ ¼ M 1 56 ðf Þ ¼ M 1 61 ðf Þ ¼ M 1 62 ðf Þ ¼ M 1 63 ðf Þ ¼ M 1 64 ðf Þ ¼ M 1 65 ðf Þ ¼ M 1 66 ðf Þ ¼

3

Z5

Z

2

ð1 þ e cos f Þ Kðf Þ þ

0 0 0 cos f 0 0 sin f 0 0 sin f 0 0 cos f

ð2 þ e cos f Þðe sin f Þ

Z2

Y. Li et al. / Acta Astronautica 83 (2013) 125–131

U13 ðf ,f 0 Þ ¼ 0 U23 ðf ,f 0 Þ ¼ 0 U33 ðf ,f 0 Þ ¼ 0 U43 ðf ,f 0 Þ ¼ 0

Appendix B. Elements of ! ðf f 0 Þ f e þ cos f  n2 p0 1þ e cos f f 0

Z2

U11 ðf ,f 0 Þ ¼ U21 ðf ,f 0 Þ ¼

" #f 1 Z4 eþ cos f eZ2 ðe þ cos f Þ2  U53 ðf ,f 0 Þ ¼ þ  p0 n2 ð1þ e cos f Þ2 2n2 ð1 þ e cos f Þ2  f0 " #f  4 1 3eZ Z sin f ð2 þe cos f Þ  U63 ðf ,f 0 Þ ¼  2 Kðf Þ þ 2  p0 2n 2n ð1 þe cos f Þ2 



1 Z 3eKðf Þ 6e 3e3  Kðf Þ 2 Kðf Þ p0 n2 1 þ e cos f n2 Z 2n Z Z2 ð3e2 þ 2Þ sin f ð2 þ e cos f Þ þ 2n2 ð1 þ e cos f Þ2 #f  Z4 sin f   2  2 n ð1 þ e cos f Þ f

f0

0

References



U31 ðf ,f 0 Þ ¼ 

f Z4 e eþ cos f 

n2 p0

6

U41 ðf ,f 0 Þ ¼

1 þe cos f 

f0



Z 1 3 9e2 3 Kðf Þ 5 Kðf Þ 7 Kðf Þ 7 Kðf Þ n2 p0 Z3 Z 2Z Z 3 Kðf Þ e sin f þ 5 þ Z 1 þe cos f Z2 ð1 þ e cos f Þ þ

5e sin f ð2 þe cos f Þ 2Z4 ð1 þe cos f Þ2

#f    

f0

U51 ðf ,f 0 Þ ¼ 0 U61 ðf ,f 0 Þ ¼ 0 " #f 1 3eZ Z4 sin f ð4 þ 3e cos f Þ  U12 ðf ,f 0 Þ ¼ Kðf Þ 2   p0 2n2 2n ð1þ e cos f Þ2

f0

"

U22 ðf ,f 0 Þ ¼

1 3e 2 3e2 Z sin f 3eZ4 1 K ðf Þ þ 2 Kðf Þ p0 2n2 n 1 þ e cos f 2n2 ð1þ e cos f Þ2

þ

U32 ðf ,f 0 Þ ¼

Z2 e þ cos f n2 1 þ e cos f

þ

Z2

e þ cos f

n2 ð1 þ e cos f Þ2

2

þ

e ðe þ cos f Þ 2n2 ð1 þ e cos f Þ2

 f 1 Z5 Z6 e sin f  Kðf Þ þ p0 n2 n2 1 þ e cos f f 0

 1 3 2 3eZ sin f Kðf Þ K ðf Þ þ 2 p0 2n2 n 1 þ e cos f #f  Z4 2 Z4 1   2  2  2 n ð1 þe cos f Þ n 1 þ e cos f 

U42 ðf ,f 0 Þ ¼

f0

U52 ðf ,f 0 Þ ¼ 0 U62 ðf ,f 0 Þ ¼ 0

131

#f    

f0

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