Periodic solutions and stability of a tethered satellite system

Mechanics Research Communications 44 (2012) 24–29

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Periodic solutions and stability of a tethered satellite system W. Zhang ∗ , F.B. Gao, M.H. Yao College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, PR China

a r t i c l e

i n f o

Article history: Received 11 November 2011 Received in revised form 17 May 2012 Available online 28 May 2012 Keywords: Tethered satellite Degree theory Periodic solution Lyapunov stability Barbashin–Krasovski theory

a b s t r a c t In this paper, several criteria on the existence of periodic solutions for a tethered satellite system (TSS) in an elliptical orbit, as well as the uniqueness of periodic solutions for the TSS in a circular orbit are presented on the basis of coincidence degree theory. In addition, the conditions on the global asymptotic stability of the equilibrium states for the TSS are also addressed in accordance with the Lyapunov stability theory and Barbashin–Krasovski theory. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Over the past two decades, with the development of deep space exploration, researches on tethered satellite systems (TSS) has drawn intensive attention (Misra et al., 2001; Misra, 2008; Beletskii and Levin, 1993; Beletskii and Pivovarov, 2000; Cartmell and McKenzie, 2008; Wen et al., 2008a,b; Schutte and Dooley, 2005; Yu, 2000, 2002; Rossi et al., 2004; Takeichi et al., 2004; Steindl and Troger, 2005; Abdel-aziz et al., 2006; Pasca, 1997). A great variety of TSSs were modeled as the dumbbell systems, in which two massive bodies were not necessary to have the same mass or size. Both of them were coupled by a low-mass tether, through which the momentum exchanged between each other (Cartmell and McKenzie, 2008; Wen et al., 2008a). In the aforementioned literature, Misra (2008) described various control schemes to stabilize the dynamics during retrieval of the subsatellite with the effects of aerodynamic and electrodynamic forces. Schutte and Dooley (2005) conducted a study of the periodicity of a TSS and extended it to a formation flying satellite system. With varying fixed tether lengths, they also stated that the TSS seemed to transfer from periodic motion to quasi-periodic motion. Yu (2000, 2002) not only presented a range-rate control scheme for a TSS model, but also investigated the stability of the stationary (quasi-stationary) configurations during the stationary keeping phase (deployment and retrieval phases). Based on the Leray-Schauder degree theory, Rossi et al. (2004) gave an interesting account of the likely periodic motions of a tether trailing satellite under the effects of atmospheric drag and nonspherical Earth. Takeichi et al. (2004) studied the periodic solution of the librational motion of a tethered system in elliptic orbit and clarified its mechanical characteristics from the analytical solution obtained by using the Lindstedt perturbation method. Moreover, it was also put forward and proved that the total mechanical energy of the system is minimum when the motion coincides with the periodic solution. This implies that the periodic solution is the minimum energy solution, and that the periodic solution in an elliptic orbit has the same significance as the equilibrium state in a circular orbit from the mechanical point of view. Therefore, one can easily find that periodic solutions play a very important role in the TSS. In addition, Pasca (1997) proposed two different nonlinear control laws for the stabilization of a TSS. The Lyapunov stability theory was used to show the global asymptotic stability of equilibrium states during the station-keeping phase in the TSS, subjected to either of the control laws. The major objective of this paper is to apply degree theory (see, for example, Gao and Lu, 2008; Gao et al., 2009, a,b; Gaines and Mawhin, 1977 and the references therein.) to investigate the existence of periodic solutions for a TSS in an elliptical orbit, as well as the uniqueness of periodic solutions for the TSS in a circular orbit. Several criteria on the existence and uniqueness of periodic solutions for the TSS are obtained.

∗ Corresponding author. Tel.: +86 10 6739 2867. E-mail addresses: [email protected], [email protected] (W. Zhang). 0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2012.05.004

W. Zhang et al. / Mechanics Research Communications 44 (2012) 24–29

25

Fig. 1. Schematic diagram of a tethered satellite system.

Furthermore, the Lyapunov stability theory and Barbashin–Krasovski theory are adopted to study the global asymptotic stability of the equilibrium states for the TSS. Some sufficient conditions on the global asymptotic stability of the equilibrium states for the TSS are also achieved. 2. Equations of motion The general and reasonable model for a TSS consists of a group of massive bodies connected by massless tethers, with the attitude motion of satellite and the impact of the tether flexibility neglected, and often referred to as the dumbbell system when only two massive bodies are involved. The mathematical model obtained by using this modeling method is relatively simple. To reduce the difficulties of controller design and improve the computational efficiency, it is widely used in related researches. Therefore, in what follows, an idealized dumbbell system composed of an orbiter and a satellite with masses M and m, respectively, will be investigated. The Earth is assumed to have a spherical gravitational potential. The two bodies are connected by a straight tether of mass mt and length l, as shown in Fig. 1. The equations governing the pitch angle  and roll angle  are given by Misra et al. (2001) and Wen et al. (2008a,b)

  

¨ + ¨ + 2(˙ + ) ˙ r

  ˙ 

¨ + 2r 

˙ +

 

˙ 





− ˙ tan  + 3

˙ + ˙

2

 +3





 Rc3



sin  cos  =



cos2  sin  cos  =

Rc3

Q me 2 lc2  2 cos2  Q

me 2 lc2  2

,

,

(1a)

(1b)

where the dot denotes the derivative with respect to time t,  = l/lc is the dimensionless length of the tether with reference length lc ,  is the orbital angular velocity, and Q and Q are the generalized forces corresponding to  and , respectively. Rc and  are radial distances of the center of mass and true anomaly, respectively. me and r are the equivalent mass of the system and the mass ratio, which can be found in Misra et al. (2001) me = r=

[mM + (1/3)mt (m + M) + (1/12)m2t ] , (mt + m + M) m[M + (1/2)mt ]

[mM + (1/3)mt (m + M) + (1/12)m2t ]

.

Obviously, if the tether mass mt is small compared with the two end-masses, it is easy to verify that me = mM/(m + M) and r = 1. Since the elliptic orbit can be described as Rc = p(1 + e cos )−1 ,

˙ =

  1/2 p3

(1 + e cos )2 ,

¨ = −2

 p3

e sin (1 + e cos )3 ,

where p is the focal parameter and e is the eccentricity of the orbit, then, Rc = p, ˙ = (/p3 )1/2 , ¨ = 0 hold for the case of a circular orbit. In the case of an elliptical orbit, Eqs. (1a) and (1b) can be reduced to the following form



  − 2(  + 1)

  − 2

e sin  −r 1 + e cos 

e sin  −r 1 + e cos 

 

   

 



p3 Q 3 sin  cos  = , 1 + e cos  me 2 lc2  2 cos2 (1 + e cos )4

(2a)

p3 Q 3cos2  sin  cos  = , 1 + e cos  me 2 lc2  2 (1 + e cos )4

(2b)

+  tan  +

  + (  + 1)2 +



where the prime represents the differentiation with respect to the true anomaly .

26

W. Zhang et al. / Mechanics Research Communications 44 (2012) 24–29

Note that if the out-of-plane angle  = 0,and  the distance between the orbiter and the satellite decreases continuously in the tether retrieval phase, i.e., ˙ =   ˙ < 0, the term 2r   /   in Eq. (2a) is negative damping which can cause an instability of the motion. For that reason, we introduce the range rate control algorithm (Yu, 2000, 2002)  k1 + k2   , =  + 1 

k1 < −

3(2 − e) , 4r(1 − e)

(3)

where k1 and k2 are two choosable parameters. Substituting Eq. (3) into Eq. (2a) gives   − 2

 e sin 



1 + e cos 

− rk2   +

3 sin  cos  p3 Q =2 − 2 2 2 1 + e cos  me  lc  (1 + e cos )4

 e sin  1 + e cos 



− rk1 ,

(4)

3. Main results 3.1. Periodicity study For the case of an elliptical orbit, we are going to show that there exists at least one T-periodic (T > 0) solution of Eq. (4). For simplicity, some notations throughout the paper are introduced as follows: | · | denotes the absolute value and the Euclidean norm on R. I := [0, T ], C 0 := C 0 (I, R), C 1 := C 1 (I, R), CT := {u ∈ C 0 |u(0) = u(T )} with the norm |u|∞ = max|u()| and CT1 := {v ∈ C 1 |v(0) = v(T ), v (0) = v (T )} ∈I

with the norm v = max{|v|∞ , |v |∞ }. Obviously, CT and CT1 are two Banach spaces. Meanwhile, it is also denoted that L =   ,

L : D(L) ⊂ CT → CT , N : CT → CT ,

[N]() = 2

 e sin  1 + e cos 



− rk2   −

3 sin  cos  p3 Q +2 + 1 + e cos  me 2 lc2  2 (1 + e cos )4

 e sin  1 + e cos 



− rk1 ,

(5)

where D(L) = {| ∈ CT1 ,   ∈ C(R, R)}. Thus, it is easily verified that Eq. (4) can be converted to the abstract equation L = N. Moreover, it is clear that KerL = R, ImL = {ω|ω ∈

T

CT , 0 ω(s) ds = 0} according to the definition of L. Therefore, L is a Fredholm operator with index zero. Let the projections P : CT1 → KerL,

[P]() = (0) = (T ),

Q : CT → CT /ImL,

[Qω]() =

1 T



T

ω(s) ds. 0

Then, we have ImP = KerL, KerQ = ImL . Let L−1 represent the inverse of L|D(L)∩KerP , L−1 : ImL → D(L) ∩ KerP. Then, we have



[L−1 ω]() =

T

G(, s)ω(s) ds,

(6)

0

where

G(, s) =

⎧ s( − T ) ⎪ ⎨ , 0  s <   T. T

⎪ ⎩ (s − T ) , 0    s  T. T

It follows from relations (5) and (6) that N is L-compact on X, where X is an arbitrary open bounded subset in CT1 . Then, we have the following theorems. Theorem 3.1.1.

Assume that there are positive constants  0 and D such that the following conditions hold.

[A1 ] |Q |∞  D,

3 sin  cos  1+e cos 

[A2 ] 

|| >  0 . −

p3 Q me 2 lc2  2 (1+e cos )4

−

2e sin  1+e cos 

+ 2rk1



= / 0,

|| > 0 .

Then Eq. (4) has at least one T-periodic solution if fmax T < 1, where fmax = max

∈[0,T ]

Proof.

 

e sin  1+e cos 

Consider the one-parameter family of equations

  − 2

 e sin 

1 + e cos 



− rk2   +

3 sin  cos  p3 Q = 2 − 2 1 + e cos  me 2 lc  2 (1 + e cos )4

 .

 e sin  1 + e cos 



− rk1 ,

∈]0, 1[.

(7)

Integrating both sides of Eq. (7) on the interval [0, T] yields



2 0

T

3 e sin  d  1 + e cos  2



0

T

d p3 + 1 + e cos  me 2 lc2  2 

 0

T

|Q | 4

(1 + e cos )

d + 2 ln

1 + e cos T + 2rk1 T. 1+e

(8)

W. Zhang et al. / Mechanics Research Communications 44 (2012) 24–29

Denote + =



T





| cos  ∈ [0, 1],  ∈ [0, T ] and − =



3 e sin  d  1 + e cos  4

0

+

+



p3 T

1

1+

2me 2 lc2  2 

−



2me 2 lc2  2 

where a=

d +

p3

+ rk1 T +



(1 − e)4

1 d 1−e



|Q | d +

 b=

,







| cos  ∈ [−1, 0[,  ∈ [0, T ] , which together with Eq. (8) gives

+ ln |Q |

−

27

(1 − e)4

1 + e cos T 1+e



d

 a|Q |∞ + b,

(9)

3T (2 − e) 1 + e cos T + rk1 T. + ln 1+e 4(1 − e)

According to Eqs. (3) and (9), we can select appropriatek1 such  that there is a positive constant D satisfying |Q |∞ > D. Therefore, by the condition [A1 ], there exists a constant * ∈ [0, T] such that  (∗ )  0 , which implies that ||∞  0 +

1 2



∞

T

    () d.

(10)

0

Multiplying both sides of Eq. (7) by () and integrating from 0 to T, we arrive at



  2   d = −2

T



0

T

e  sin  d + 3 1 + e cos 

0



T

0

 sin  cos  p3 d − 1 + e cos  me 2 lc2  2 

 0

T

(1 + e cos )4

Substituting inequality (10) into Eq. (11) yields





T



T

|  |2 d  fmax (

|  |d)2 + 2fmax 0 +

0

0

+

3T (2 − e) + a(D + M0 ) + (fmax + rk1 )T 4(1 − e)

3T0 (2 − e) + 2a 0 (D + M0 ) + 2T0 (fmax + rk1 )  fmax T 2(1 − e)





Q



T

d − 2

 e sin 

0

1 + e cos 



− rk1  d. (11)

T

|  |d

0

T



T

|  |2 d + c T 1/2 ( 0

1

|  |d) 2 + d,

(12)

0

where c

= 2fmax 0 +

d

=

M0

= max |Q |.

3T (2 − e) + a(D + M0 ) + (fmax + rk1 )T, 4(1 − e)

3T0 (2 − e) + 2 a 0 (D + M0 ) + 2 T0 (fmax + rk1 ), 2(1 − e) ||0

Therefore, it can be shown from inequality (12) and fmax T < 1 that there exists a constant M1 > 0 such that this with inequality (10), we have ||∞  0 +

T   2   d  M1 . Combining 0

1 M1 T  M2 . 2

Moreover, in terms of () ∈ CT1 , i.e., (0) = (T), then there is a * ∈]0, T[ such that   (∗ ) = 0, which together with Eq. (7) implies that







|  | = |

  (s) ds|  2(fmax + rk2 ) ∗

T

|  |d + 0

3T (2 − e) + 2 aM Q + 2 T (fmax + rk1 ), 2(1 − e)

  where MQ = max Q . Hence, there is a constant M3 > 0 such that |  |∞  M3 . ||M2

Let X : = { : ||∞ < M2 + 1, |  |∞ < M3 + 1}. Then, for ∀ ∈]0, 1[, Eq. (7) has no solution on ∂X with ∀  ∈ KerL ∩ ∂X and





T

T



[N]() d = − 0

0



2e sin  p3 Q 3 sin  cos  − / 0. − + 2rk1 d = 1 + e cos  1 + e cos  me 2 lc2  2 (1 + e cos )4

Thus, L = / N, for ∀ (, ) ∈ (D(L) ∩ ∂X) ×]0, 1[ and N ∈ / ImL, for ∀  ∈ KerL ∩ ∂X . / 0, where J : ImQ → KerL is an isomorphism. To this end, we define J() : = . Without loss of Now, we claim that deg{JQN, X ∩ KerL, 0} = generality, from the condition [A2 ], we assume





3 sin  cos  2e sin  p3 Q − − + 2rk1 1 + e cos  1 + e cos  me 2 lc2  2 (1 + e cos )4

and denote 1− H( , ) =  − T

 0

T





< 0,

for || > 0 ,



2e sin  3 sin  cos  p3 Q − − + 2rk1 d, 4 2 2 2 1 + e cos  1 + e cos  me  lc  (1 + e cos )

28

W. Zhang et al. / Mechanics Research Communications 44 (2012) 24–29

for ∀ ∈ [0, 1]. While  ∈ ∂X ∩ R, in terms of the assumptive condition, we have H( , ) =  2 −

1− T





T

 0



2e sin  p3 Q 3 sin  cos  − − + 2rk1 d > 0. 1 + e cos  1 + e cos  me 2 lc2  2 (1 + e cos )4

Therefore, H( , ) is homotopy and deg{JQN, X ∩ KerL, 0}

= deg{H( · , 0), X ∩ KerL, 0} = deg{H( · , 1), X ∩ KerL, 0} / 0. = deg{I, X ∩ KerL, 0} =

By applying the continuation theorem (Gaines and Mawhin, 1977), we conclude that equation L = N has at least one T-periodic solution () on X with ||∞  M2 . This completes the proof of Theorem 3.1.1. For the case of a circular orbit, e ≡ 0, Eq. (4) can be reduced to the following form   + 2 rk2   + 3 sin  cos  −

p3 Q me 2 lc2  2 

= −2 rk1 .

(13)

Then, we have the following results. Theorem 3.1.2. [C1 ] Q  > 0,

Assume that there is a constant  0 > 0 such that the following conditions hold. || >  0 .

[C2 ]  3 sin  cos  −

p3 Q me 2 lc2  2 

+ 2rk1



|| >  0 .

= / 0,

Then Eq. (13) has at least one T-periodic solution. Proof.

The same proof presented above also works for this theorem.

Remark. Obviously, condition [C1 ] in Theorem 3.1.2 is weaker than [A1 ] in Theorem 3.1.1 while e ≡ 0. This is due to the fact that the coefficient of damping term in Eq. (4) is a function of the true anomaly, which makes the equation of motion nonlinear and non-autonomous, and more complicated than the nonlinear and autonomous system (13). Theorem 3.1.3.



Assume that Q > 3 me 2 lc2  2  p3 . Then, Eq. (13) has at most one T-periodic solution.

Proof. We give the proof by contradiction. Assume that  1 () and  2 () are two arbitrary distinct T-periodic solutions of Eq. (13), then, it can be easily verified that





1 − 2 + 2 rk2 1 − 2 + J(1 ) − J(2 ) = 0,

(14)

where J() := 3 sin  cos  − (p3 Q )/(me 2 lc2  2 ). Multiplying both sides of Eq. (14) by  1 −  2 and integrating on the interval [0, T] yield



T

   1 − 2 2 d =



0

T





J(1 ) − J(2 ) (1 − 2 ) d.

0

(15)

T

2

Since J  () = 3 cos 2 − (p3 Q )/(me 2 lc2  2  < 0), which together with Eq. (15) implies that 0 1 − 2  d < 0. This is a contradiction, which came about by assuming  1 = /  2 . Therefore,  1 ≡  2 . This completes the proof of Theorem 3.1.3. Theorem 3.1.4. Proof.

Assume that [C1 ] and [C2 ] hold and Q > 3 me 2 lc2  2 /p3 . Then, Eq. (13) has a unique T-periodic solution.

Obviously, this result can be derived directly from the previous results.

3.2. Stability analysis In order to investigate the dynamic behavior for equilibrium states of Eq. (13), it is supposed that Q ∈ [(4rk1 − 3)me 2 lc2  2 /(2p3 ), (4rk1 + 3)me 2 lc2  2 /(2p3 )]  . Otherwise, there will be no equilibrium states of Eq. (13) if Q ∈ / . Theorem 3.2.1.



[H1 ]

Assume that the following conditions hold



Q1 − Q2 (1 − 2 ) = / 0,





for ∀ 1 = / 2 .

/ 0 and lim [H2 ] theta J() + 2 rk1 > 0,  =

||→+∞

 0

[J(s) + 2 rk1 ] ds = +∞.

Then, the equilibrium states of Eq. (13) are globally asymptotically stable. Proof.

Consider the associated equivalent system of Eq. (13) as follows

1 = 2 ,

(16a)

W. Zhang et al. / Mechanics Research Communications 44 (2012) 24–29

2 = −2 rk2 2 − 3 sin 1 cos 1 +

p3 Q1 me 2 lc2  2 

− 2rk1 .

29

(16b)

Choose applicable k1 such that Q |=0 = 2 rk1 me 2 lc2  2 /p3 . Then, it follows from the condition [H1 ], Eqs. (16a) and (16b) that the equilibrium points of system can be translated to the point (0, 0). Thus, the stability of equilibrium states of Eq. (13) reduces to the stability of the zero solution of Eq. (13). Denote



V (1 , 2 ) =

1

[J(s) + 2 rk1 ] ds + 0

1 2  . 2 2

(17)

Hence, we have



dV  ∂V  ∂V   +  = −2 rk2 22  0.  = d (17) ∂ 1 1 ∂ 2 2 Then, it can be easily verified from condition [H2 ] that V is positive definite and

lim

 2 + 2 →+∞ 1

sup V (1 , 2 ) = +∞. In addition, it can also be

2

shown that the set {( 1 ,  2 )|V ( 1 ,  2 ) = 0} \ (0, 0) contains no complete trajectory of Eq. (13). Therefore, the equilibrium states of Eq. (13) are globally asymptotically stable according to the Barbashin–Krasovski theorem. 4. Conclusions In this paper, coincidence degree theory is applied to establish the criteria to guarantee the existence of periodic solutions for a TSS in an elliptical orbit, as well as the uniqueness of periodic solutions for the TSS in a circular orbit, respectively. Moreover, the Lyapunov stability theory and Barbashin–Krasovski theory are also adopted to study the global asymptotic stability of the equilibrium states for the TSS. According to the results presented in Section 3, the existence and uniqueness of the periodic solutions of the TSS is mainly influenced by the pitch angle  in the case of a circular orbit as can be easily demonstrated. However, the existence of periodic solutions of the TSS is predominantly effected by the eccentricity e and the true anomaly  in the case of an elliptical orbit. In addition, the existence of periodic solutions as well as the stability of the equilibrium states of the TSS are also associated with a number of factors, including the control algorithm, the generalized force which corresponds to the pitch angle , the mass of the system, the length of the tether, the focal parameter of the orbit and the orbital angular velocity. Acknowledgments The authors are very grateful for the support of the National Natural Science Foundation of China (NNSFC) through grant Nos. 10732020, 11072008, 11172009 and 10872010, and the reviewer’s useful comments and suggestions on the language of our manuscript. References Abdel-aziz, Y.A., et al., 2006. Periodic motions of spinning rigid spacecraft under magnetic fields. Applied Mathematics and Mechanics 27, 1061–1069. Beletskii, V.V., Levin, E.M., 1993. Dynamics of space tether systems. Advances in the Astronautical Sciences 83, 443–464. Beletskii, V.V., Pivovarov, M.L., 2000. The effect of the atmosphere on the attitude motion of a dumb-bell-shaped artificial satellite. Journal of Applied Mathematics and Mechanics 64, 691–700. Cartmell, M.P., McKenzie, D.J., 2008. A review of space tether research. Progress in Aerospace Sciences 44, 1–21. Gaines, R.E., Mawhin, J.L., 1977. Coincidence Degree and Nonlinear Differential Equations. Springer-Verlag, Berlin, pp. 26-35. Gao, F.B., Lu, S.P., 2008. New results on the existence and uniqueness of periodic solutions for Liénard type p-Laplacian equation. Journal of The Franklin Institute 345, 374–381. Gao, F.B., et al., 2009a. Periodic solutions for p-Laplacian neutral Liénard equation with a sign-variable coefficient. Journal of The Franklin Institute 346, 57–64. Gao, F.B., et al., 2009b. Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument. Nonlinear Analysis 70, 3567–3574. Misra, A.K., et al., 2001. Nonlinear dynamics of two-body tethered satellite systems: constant length case. Journal of the Astronautical Sciences 49, 219–236. Misra, A.K., 2008. Dynamics and control of tethered satellite systems. Acta Astronautica 63, 1169–1177. Pasca, M., 1997. Nonlinear control of tethered satellite system oscillations. Nonlinear Analysis 30, 3867–3878. Rossi, E.V., et al., 2004. Existence of periodic motions of a tether trailing satellite. Applied Mathematics and Computation 155, 269–281. Schutte, A.D., Dooley, B.A., 2005. Constrained motion of tethered satellites. Journal of Aerospace Engineering 18, 242–250. Steindl, A., Troger, H., 2005. Is the Sky-Hook configuration stable. Nonlinear Dynamics 40, 419–431. Takeichi, N., et al., 2004. Periodic solutions and controls of tethered systems in elliptic orbits. Journal of Vibration and Control 10, 1393–1413. Wen, H., et al., 2008a. Advances in dynamics and control of tethered satellite systems. Acta Mechanica Sinica 24, 229–241. Wen, H., et al., 2008b. Optimal feedback control of the deployment of a tethered subsatellite subject to perturbations. Nonlinear Dynamics 51, 501–514. Yu, S.H., 2000. Tethered satellite system analysis (1)—Two-dimensional case and regular dynamics. Acta Astronautica 47, 849–858. Yu, S.H., 2002. Dynamic model and control of mass-distributed tether satellite system. Journal of Spacecraft and Rockets 39, 213–218.