Dynamics and control of tethered multi-satellites in elliptic orbits

Aerospace Science and Technology 91 (2019) 41–48

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Dynamics and control of tethered multi-satellites in elliptic orbits Gefei Shi a,b,c , Zhanxia Zhu a,b , Zheng H. Zhu c,∗ a b c

National Key Laboratory of Aerospace Flight Dynamics, PR China School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, PR China Department of Mechanical Engineering, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

a r t i c l e

i n f o

Article history: Received 12 September 2018 Received in revised form 8 April 2019 Accepted 30 April 2019 Available online 3 May 2019 Keywords: Tethered multi-satellites Elliptic orbit Nonlinear dynamics Tension control Sliding mode control

a b s t r a c t This paper studies the dynamics and libration suppression of tethered multi-satellites in elliptic orbits, where the tethered system is subject to periodic excitation resulting from the change of gravity gradient. The tethered system is modeled as a multiple dumbbell model and the relationship between periodical libration and tether length rate is studied. The outcome of study leads to a new tension control law to suppress the libration of tethered multi-satellites in a desired periodic motion by regulating tether length without thrust. The simulation results show that the libration motion of tethered multi-satellites is successfully stabilized to the desired periodic motion with the proposed tension control law. When the orbital eccentricity is small, the analytical solution of time-dependent tether lengths is derived to minimize the magnitudes of tether length rate. Furthermore, the tension control law is modified by including a sliding mode control to suppress initial external perturbations to increase the robustness of control law. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Tethered multi-satellites generally contains a main satellite and one or multiple subsatellite(s) connected by tether(s) [1–6]. They have immense potential application in orbital transportation [1, 7–9], space science and communication [6,10]. When the main satellite is moving in an elliptical orbit, the tethered subsatellites will liberate about the gravity direction periodically. If not properly controlled, the libration will become unstably due to the changes of gravity gradient, and tumbling may occur when the orbital eccentricity is sufficiently large [6,11]. This is adverse to space missions like Earth observation [8,11–14]. Thus, the critical issue associated with the tethered multi-satellites in elliptical orbits is how to keep the system in a periodically stable constellation. In the past decades, the dynamics of tethered multi-satellites has been extensively studied. For instance, Misra and his coworkers [1,8] and Lorenzini and his co-workers [2,15] investigated libration characteristics and engineering application of tethered three-satellites. Takeichi et al. [12] modeled the dynamic behavior of tethered multi-satellites in elliptic orbits using lumped massspring model. It shows that libration control is needed to ensure system’s periodical motion. Then, Kojima et al. [11] studied the dynamics and control of tethered three-satellites in elliptical orbits.

*

Corresponding author. E-mail address: [email protected] (Z.H. Zhu).

https://doi.org/10.1016/j.ast.2019.04.054 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

The study found that the orbital eccentricity causes the instability of tethered three-satellites and leads to the chaotic motion. The similar conclusion was also obtained in [6] and [16]. Aslanov and Yudintsev analyzed the motions of a tethered tug-debris system subject to perturbation of fuel residuals [17], which was modeled as tethered three-satellites. An approximate analytical solution was obtained. Jin et al. studied the nonlinear dynamics of threesatellites like tethered system [18,19], in which the tether lengths were constant. The similar systems have also been studied by Jung et al. [20] numerically. Ziegler and Cartmell [21] studied the modeling and dynamics of a symmetrical double-ended motorized spinning tethered system. Ismail and Cartmell [22] extended the aforementioned study and modeled the tethered multi-satellites as a motorized momentum exchange tether system. Both in-plan and out-plan motions were studied. Up to now, the libration stabilization of tethered multi-satellites in elliptical orbits is yet fully studied. By adjusting tether length rates, which equals to the reel velocity, Mirsa and Modi [8] developed a tether velocity control law for suppressing the libration motion of tethered multi-satellites to zero or stable periodic motions, respectively, in circular orbits. Takeichi et al. [23] studied a fundamental thrust control strategy to suppress the libration of a twobody tethered system in elliptical orbits. Although it was proposed for a two-body tethered system, the control strategy inspired the studies of tethered multi-satellites in elliptical orbits. Based on the similar control strategy, Kojima et al. [11] investigated the libration suppression of tethered multi-satellites in elliptical orbits

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G. Shi et al. / Aerospace Science and Technology 91 (2019) 41–48

Nomenclature A e er eϑ li Li mi M Mi

Magnitude of libration Eccentricity Unit vector of vector r Unit vector normal to e r in orbital direction Dimensionless tether between the i − 1 and i-th body Tether length between the i − 1 and i-th body Mass of the i-th body Mass of main satellite Dimensionless mass of the i-th body

by tracking the periodic motion in circular orbits using a delayed feedback control law, where thrusters are assumed. For the same issue, Zhu and Shi [24] proposed a control strategy of tracking a steady state to suppress the libration of a tethered multi-satellite system in elliptical orbits with thrust control. Williams [25] presented a control scheme using only forced length variations to suppress the libration of a two-body tethered satellite system in elliptical orbits. The proposed method can be used in the libration suppression of tethered multi-satellites. So far, no work in the literature deals with the tethered multi-satellites in elliptical orbits by tension control without thrusters. The objective of this work is to develop a tension control law to suppress the libration of tethered multi-satellites in elliptic orbits by regulating the tether lengths without using thrusters. A novel control law that relates libration angles to tether length rates has been proposed to suppress the libration. Stability and robustness of the control law has been proved by Lyapunov theory. To enhance the robustness of the controller when subject to initial external perturbations, a new tension control law, actuated by regulating the reel velocity (tether length rate), is modified by a sliding mode control due to its simple and superior ability in suppressing perturbation for a nonlinear dynamic system [26,27]. Different from existing tension control laws, the new tension control law can effectively control Coriolis forces at the subsatellites to balance the libration motion due to orbital eccentricity. The proposed approach is verified by numerical simulations.

2. Mathematical formulation

Consider the tethered multi-satellites (N satellites) in the orbital plane as shown in Fig. 1, where the subsatellites are connected denoted as L 1 , L 2 , . . . , L i , . . . , L N −1 , the mass, elastic and the bending of the tether are ignored. Assuming the system is orbiting in a Keplerian orbit subject only to a central gravitational field. All other external perturbations are neglected, the main satellite and subsatellites are modeled as lumped point masses (M, m1 . . . mi . . . m N −1 ) due to the extreme large ratio of tether length over the dimensions of tethered bodies. The system is moving in an inertial coordinate system OXY with its origin at the centroid. The position of the main satellite (M) is denoted by a vector r measuring from the centroid. The subsatellites m1 relate to M by a distance of L 1 and an angle of θ1 , while the relative position of mi with respect to mi −1 is defined by L i and θi . The mass of the main satellite is assumed much greater than the masses of the climber and the end-body. Thus, mass center of the system can be assumed residing in the main satellite that moves in a Keplerian orbit. Based upon these assumptions, the positions of the tethered multi-satellites are written in the inertial coordinate system as

p r ri Ti

Focal parameter of orbit Position vector of main satellite Position vector of the i-th body Tension in the respective tether L i The i-th libration angle True anomaly Earth’s gravitational constant Frequency of the libration

θi ϑ

μ ω

Fig. 1. Schematic of tethered multi-satellites and coordinate systems.

r = rer r 1 = r − L 1 cos θ1 er − L 1 sin θ1 e ϑ

.. . r i = r i −1 − L i cos θi er − L i sin θi e ϑ .. . r N −1 = r N −2 − L N −1 cos θ N −1 er − L N −1 sin θ N −1 e ϑ

(1)

where r = p /k, k = 1 + e cos ϑ and i = 1, 2, . . . , N − 1. Differentiating Eq. (1) with respect to the time yields the velocities of the satellites,

r˙ = r ϑ˙ e ϑ + r˙ e r  ˙ sin θ1 − L˙ 1 cos θ1 e r r˙ 1 = r˙ + L 1 (θ˙1 + ϑ)  ˙ cos θ1 − L˙ 1 sin θ1 e ϑ + − L 1 (θ˙1 + ϑ)

.. .

(2)





˙ ˙ ˙ r˙ N −1 = r˙ N −  2 + L N −1 (θN + ϑ) sin θ N −1 − L N −1 cos θ N −1 e r ˙ cos θ N −1 − L˙ N −1 sin θ N −1 e ϑ + − L N −1 (θ˙N −1 + ϑ) 2 ˙ where  the orbital angular velocity of the main satellite ϑ = nk ,

and L˙ i is the relative velocity of the i-th body mi along

μ

n=

p3

the respective tether L i , relative to the (i − 1)-th body. For the Kepler orbit, n, p and e are constants. Accordingly, the kinetic energy K and potential energy U of the system can be expressed as,

K=

1 2

M r˙ · r˙ +



U = −μ

M

|r m |

1 2

+

m1 r˙ 1 · r˙ 1 + · · · + m1

|r 1 |

+ ··· +

1 2

m N −1 r˙ N −1 · r˙ N −1

m N −1

| r N −1 |



(3)

The equation of motion of the system is then derived by the Lagrange equation,

G. Shi et al. / Aerospace Science and Technology 91 (2019) 41–48

d ∂L dt ∂ q˙ i

−

∂L = Q i (i = 1, 2, . . . , 2N − 2) ∂ qi

Table 1 Simulation parameters.

(4)

where L = K − U is the Lagrangian function and (q1 . . . q N −1 , q N . . . q2N −2 ) = (θ1 . . . θ N −1 , L 1 . . . L N −1 ) are the generalized coordinates. The generalized forces Q i are given by

Q 1 = Q 2 . . . Q N −1 = 0,

Q N = − T 1 . . . Q 2N −2 = − T N −1

(5)

where T i is the tension in i-th tether. For the sake of convenience, the dynamic equations in Eqs. (4) and (5) are normalized into the dimensionless form by the following dimensionless parameters [24]

M i = mi /mtot , Tˆ i = T i /ϑ L 0 mtot

˙2

li = L i / L 0 ,

˙ dϑ/dt = ϑ,

where () = d()/dϑ , mtot = m1 + m2 + · · · + m N −1 , L 0 = L 1 + L 2 + · · · + L N −1 is the total tether length initially and Tˆ i is the dimensionless tension in the i-th tether. Using Eqs. (4) and (5), the dynamic equations are derived

θ1 = f 1 (θ1 , θ1 , . . . θ N −1 , θ N −1 , l1 , l1 , . . . l N −1 , lN −1 , Tˆ 1 , . . . Tˆ N −1 ) .. . (6a) θN −1 = f N −1 (θ1 , θ1 , . . . θ N −1 , θ N −1 , l1 , l1 , . . . l N −1 , lN −1 , Tˆ 1 , . . . Tˆ N −1 ) l1 = f N (θ1 , θ1 , . . . θ N −1 , θ N −1 , l1 , l1 , . . . l N −1 , lN −1 , Tˆ 1 , . . . Tˆ N −1 )

.. .

lN −1 = f 2N −2 (θ1 , θ1 , . . . θ N −1 , θ N −1 , l1 , l1 , . . . l N −1 , lN −1 , Tˆ 1 , . . . Tˆ N −1 )

(6b)

3. Dynamic analysis and periodic solutions 3.1. Zero state solution For some practical cases, such as the observation of the space or atmosphere, the system is required to be aligned with the gravity direction, such that, θ1 = θ1 = · · · θ N −1 = 0, θ1 = θ1 = · · · θ N −1 = 0, and θ1 = θ1 = · · · θ N −1 = 0. The states θ1 = θ1 = · · · θ N −1 = 0 and θ1 = θ1 = · · · θ N −1 = 0 are called the zero state. For a space tethered system, the libration motion can be suppressed and the attitude angles can be kept with tension control only effectively [26]. It is also noted that, the tensions in tethers do not suppress the libration motion directly [28]. Actually, it is the Coriolis force that is produced by reeling in/out the subsatellites directly working against the libration.

N

3

M 1 /M 2

0.5

l1 (0) l2 (0)

0.5 0.5

e p

0.2 8,520 km

Substituting the zero libration state into Eq. (6a) leads to the tension terms being eliminated from the dynamics equations. This is because when θ1 = θ1 = · · · θ N −1 = 0, all tension terms in Eq. (6a) become zero. The required kinematic functions li (i = 1 . . . N − 1) are obtained as the inputs to control the reeling velocities,

li =

( i = 1, 2, . . . , N − 1)

43

li e sin(ϑ) 1 + e cos(ϑ)

(7)

Substituting Eq. (7) into Eq. (6b) yields the tensions corresponding to the zero state. It should be noted that li = 0 when the main satellite is in a circular orbit, which means no reeling velocity is needed to keep the zero state when e = 0. This agrees with the equilibrium of tethered multi-satellites in circular orbits. To validate this solution, numerical simulation is conducted with parameters in Table 1, where l1 (0) and l2 (0) denote the initial dimensionless lengths of tether 1 and 2, M 1 and M 2 are the dimensionless masses of two subsatellites, respectively. The initial true anomaly is zero. The simulation results are shown in Figs. 2–4. For instance, Figs. 3 and 4 show that when the tethered three-bodies are kept aligned with the gravity direction with tension control only, both the change of tether lengths, see Fig. 3, and the active control tensions, see Fig. 4, vary periodically as expected. This is because the system is orbiting on a periodic elliptic orbit. By keeping the libration angles to zero, the chaotic motion, caused by the eccentricity, can be avoided as per [11,29], which is good for system to work stably, see Fig. 2. Furthermore, θ1 = θ2 means all satellites are collinear. As shown in Fig. 3, the lengths of two tethers are changing at the same period of the orbit. From 0–0.5 orbit, the tethers are reeled out. The lengths reach the peak length (0.75) at 0.5 orbit. Both tethers’ lengths are increased by 50%. After 0.5 orbit, the tethers begin reeling in, and reduced to their initial lengths at 1 orbit. This periodic motion is repeated after 1 orbit. 3.2. Periodic solutions The zero state solution is one special period solution for the tethered multi-satellites in elliptic orbits, where the libration is zero. If subsatellites are allowed to liberate periodically within certain magnitudes, such as sinusoidal functions, the corresponding

Fig. 2. Libration angles of tethered three-satellites.

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Fig. 3. Dimensionless lengths of tethers.

Fig. 4. Dimensionless tensions in tethers.

active control inputs can be obtained to track the given functions. It should be noting that from the practical aspect, it is desired to keep all satellites inline and liberating periodically, such that, θ1 = · · · θ N = θ . Then, all tension sections in Eq. (6a) are eliminated. Accordingly, the desired liberation of the tethered multi-satellites is designed as

θ = A sin(ωϑ)

(8)

Thus, the angular velocity of the subsatellites can be written as

θ  = A ω cos(ωϑ)

This implies the desired reeling magnitude of tethers can be reduced significantly for easy implementation by properly selecting the magnitude of the periodic solution. Assume the desired A is small ( A < 0.5), sin( A sin ϑ) ≈ A sin ϑ . Substituting Eqs. (8) and (9) into Eq. (10), the desired length of each tether can be derived from Eq. (10) by integration, such that,

 ln(li ) =

=

(9)

dϑ 4k(1 + A cos ϑ) (2 A + e ) ln(1 + A cos ϑ) + (2e − 5 A ) ln(1 + e cos ϑ) 2( A − e ) (2 A + e ) ln(1 + A ) + (2e − 5 A ) ln(1 + e ) − 2( A − e )  + ln li (0)

Substituting Eq. (9) into Eq. (6) yields the required reeling in/out velocity of each tether li ,

li =

li [4e sin(ϑ)(1 + θ  ) + 2 Akω2 sin(ωϑ) − 3 sin(θ)] 4k(1 + θ  )

(10)

Eq. (10) shows the magnitude and period of the required tether velocity depends on the desired libration motion of subsatellites. To make the analysis simple, we design the period of the libration equals to the orbit period, such that, ω = 1. Fig. 5 shows the impact of the desired magnitude A. The simulation parameters are shown in Table 1. For different desired magnitudes, the required tether velocities are obviously different. For the tethered threesatellites orbiting in an elliptic orbit with e = 0.2 and A = 0.2, the desired magnitude for each tether reeling in/out is 0.01. This means to keep the desired periodic libration, the maximum length of reeled out tether is only 2% of the initial length, which is much less than 24% and 18% for A = 0.1 and 0.3, respectively.

2( A + 2e + 3 Ae cos ϑ) sin ϑ − 6 A sin ϑ

(11a)

and

lim

A →e

d ln(li ) dA



= lim

3e [ln(1 + e cos ϑ) − ln(1 + A cos ϑ)](1 + A cos ϑ)

( A − e )2

A →e

− +

+e ( A − e )(2 A + e ) + 3e (1 + A ) ln 11+ A

2(1 + A )( A − e )2

(2 A − Ae − e 2 ) cos ϑ ( A − e )2 2



=0

(11b)

Equation (11b) shows the tether length li reaches minimum at A = e. This explains the dynamic phenomenon in Fig. 5. It should

G. Shi et al. / Aerospace Science and Technology 91 (2019) 41–48

45

Fig. 5. Tether lengths required to keep the periodic solutions.

Fig. 6. Tether length change vs. designed magnitude in one orbit when e = 0.7.

be noted that the conclusion is only applicable when e is small to satisfy the assumption of small A. For greater e, a large magnitude of li is needed to suppress the libration due to the orbital eccentricity. Fig. 6 shows the change of tether lengths relative to periodic liberation solutions with different magnitudes with respect to true anomaly for e = 0.7 in one period. It should be noted that, according to Eq. (11), the tether lengths with different designed magnitudes change periodically with respect to the true anomaly, and the changing period is one orbit. Therefore, only one orbit period is presented in Fig. 6. The simulation parameters are the same as those in Table 1, p = 7100(1 + e ) km. As shown in Fig. 6, the magnitudes of function li increase as the desired magnitudes of the periodic solution θi = A sin(ωϑ) increases. The magnitude of tether lengths is 0.1 when A = 0.5, and it increases continuously to 0.3 as A increases to 0.9. The minimum magnitude of li does not appear at A = e = 0.7. This is because in this condition, the assumption of sin( A sin ϑ) ≈ A sin ϑ is no longer satisfied. Moreover, the negative tether tension appears numerically, which means the tether slack, and the dynamic equation of the tethered satellites become singular. In summary, for missions orbiting in elliptic orbits with orbit eccentricity up to 0.5, the desired li can be designed by A = e to minimize its magnitude. When e is large, the regulation by changing tether lengths is not effective. Furthermore, to avoid the tether slacking, tensions in tethers should always be greater than zero. Substituting Eqs. (8), (9) and (10) into Eq. (6b), the tension in tether (N − 1), which is the smallest tension in all tethers [24], can be obtained. To ensure Tˆ N −1 > 0, the follow condition must be satisfied,

6 − 12 A + 6 A 2 − 2 A 3 − 6e + 15 Ae − 12 A 2 e + 4 A 3 e

− 3 Ae 2 + 6 A 2 e 2 − 2 A 3 e 2 > 0 4. Stability of control 4.1. Sliding mode control law design In this work, the tension control law is realized by regulating the tethers’ reeling in/out velocity. The reel velocity control in Eq. (10) is an open-loop control and is vulnerable to external perturbations. The detailed relationship between the reel speed and the dynamics of libration has been shown in [30]. Different from existing tension control laws, the velocity control can directly control the Coriolis force at each subsatellite to balance the Coriolis force acting on the climber due to orbital transfer movement. The latter is the main source of disturbance for the libration. Thus, using reeling in/out velocity as the control input can suppress the libration directly compared to the common tension control inputs. It should be noted that, since the velocity control inputs can be transformed into tension by Eq. (6b), the velocity control is equivalent to the tension control essentially. To suppress the impact of perturbations, the sliding mode control is adopted due to its simplicity and robustness in dealing with perturbations. Defining a sliding mode manifold that drives the libration angle θi (i = 1, . . . , N − 1) to the desired value, such that,



 si = c i (θi − θid ) + θi − θid



(12)

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G. Shi et al. / Aerospace Science and Technology 91 (2019) 41–48

Fig. 7. Libration angles with control modification.

where c i is a positive constant that defines the bandwidth of error  are the desired libration angle and its dynamics of si , θid and θid angular velocity follow Eqs. (8) and (9), respectively. The error dynamics can be derived by taking derivative with respect to τ at each side of Eq. (12)







 + θi − θid si = c i θi − θid



(13)

The reel velocity input is li . Recast Eq. (6a) in the form θi = f i + b i u i , where f i is the nonlinear function of θi and b i is a gain function depending on θ  , l , and u = l . i

i

i

i

Substituting Eqs. (6a), (8) and (9) into Eq. (13) yields,





 − θid + f i + bi u i si = c i θi − θid

(14)

Equation (14) denotes the derivation of the i-th sliding mode manifold. For the entire system, Eq. (14) can be extended as





s = C θ  − θ d − θ d + f + Bu

(15)

where f = [ f 1 , . . . , f N −1 ] T , u = [u 1 · · · u N −1 ] T , C is a diagonal matrix with c i = positive constant, which defines the bandwidth of the error dynamics. Thus, the sliding mode control law can be derived from Eq. (18) as,

⎧ ⎨ u = u eq + u sw u eq = − B −1 [ f + C (θ  − θ d ) − θ d ] ⎩ u sw = − B −1 K sgn(s)

(16)

where sgn(s) = [sign(s1 ), . . . , sign(s N −1 )] T and K is a diagonal matrix of control gain with ki = positive constant. The reel velocity input u represents reeling velocities of the (1 − N)-th subsatellite, respectively. To avoid chattering in the sign function of Eq. (16), sign(s) is replaced by the saturation function



sat(si ) = where

si s i /ε i

if si  ≥ εi if si  < εi

(17)

εi ∈ R + is a small constant.

4.2. Stability and robust of the control law Consider a candidate Lyapunov function as

V = s T s /2

(18)

Taking the derivative of V with respect to the true anomaly yields





V  = s T s = − k1 sign(s1 )s1 + · · · k N sign(s N )s N ≤ 0

(19)

Therefore, the system is stable. Under the proposed control law, the states can be kept following the desired trajectory.

Furthermore, if there is a bonded disturbance d acting on the system, Eqs. (12) and (13) can be written as,

fd = f +d

(20)

Submitting Eq. (20) into Eq. (19) yields





V  = s T s = −sgn(s) T d sgn(s) + K s

(21)

If the disturbance is bonded, ki ≥ |di |, the matrix [d sgn(s) + K ] is positive definite and V  ≤ 0. Thus, the system is robust when subjected to a bonded disturbance. Equation (16) defines the reel velocity inputs of the system. The reel velocity inputs are carried out by tensions in each piece of tether. Thus, the tensions T = [ T 1 · · · T N −1 ] T in tethers, which can also be used as the control inputs, can be derived by substituting Eq. (16) into Eq. (6b). 4.3. Case study The system parameters are given in Table 1, the initial condition of the system is given as follows: θ1 (0) = 0.1, θ2 (0) = 0, θ1 (0) = θ2 (0) = A ω cos(ωϑ), where A = 0.2 and ω = 1. As the control input (tension produced by the tether) is limited in spacecraft, we assume the acceleration along each tether is less than 0.01 m/s2 , such that

li (ϑ) =

li (ϑ) − li (ϑ − h) h

where h denotes the simulation step. All simulations were carried by MATLAB with RK-4 integrator. The simulation step is 0.001 rad. The simulation results of e = 0.2 are shown in Figs. 7–9. The associated parameters for the control law are designed as k1 = k2 = 1, c 1 = c 2 = 1, ε1 = ε2 = 0.1. Because of the initial perturbation, the libration angles are greater than the desired state in the first half orbit. By the end of the first orbit, the libration motion of the subsatellites converges to the desired one, see Fig. 7. The reel velocity inputs are shown in Fig. 8. An obvious fluctuation appears in the first half orbit corresponding to the libration in Fig. 7. After first orbit, the reel velocity inputs follow a periodic motion that indicates the motion of the system has converged to the desired periodic motion. The reel velocity inputs in Fig. 8 also denote the real changes of tether lengths. By the end of the first orbit, the total dimensionless tether length of the system (l1 + l2 ) changes around 0.9 smoothly and periodically. This indicates, to keep the tethered three-satellites inline and liberating periodically, slightly periodic changes of the tethers are need, which is very good. Fig. 9 shows the required tensions in tethers, which are the real control inputs applied by the reeling devices. After the dramatic changes

G. Shi et al. / Aerospace Science and Technology 91 (2019) 41–48

47

Fig. 8. Tether lengths with control modification.

Fig. 9. Control tensions.

in the first orbit, the dimensionless tensions converge to periodic motions. Although charting is still existing in each period, it is very small and bounded. It should be noted that in this work, the sliding mode control is mainly used to test the control strategy based on the libration angles and tether length rate. To improve the control performance, more advanced control methods should be studied which will be our work in the future. 5. Conclusions This work studies the dynamics and stability control of the tethered multi-satellites in elliptic orbits. A new tension control strategy is proposed to keep the tethered multi-satellite in a stable periodic motion with collinear configuration. Based on the control scheme, a tether length function is obtained to ensure the desired libration motion. When the orbital eccentricity is small, the desired magnitude of the stable libration can be derived by A = e, which minimizes the length of reeled in/out tether. Furthermore, a sliding mode control is designed to suppress the impact of initial perturbations. Numerical results show that the tethered multi-satellites can be controlled to follow a periodic motion in elliptic orbits with tension control only. Declaration of Competing Interest There is no Competing Interest.

Acknowledgements This work is funded by the National Natural Science Foundation of China, Grant No. 11472213, China Scholarship Council Scholarship No. 201606290135 and the Discovery Grant (RGPIN2018-05991) of the Natural Sciences and Engineering Research Council of Canada. References [1] A.K. Mirsa, Z. Amier, V.J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronaut. 17 (1988) 1059–1068. [2] M.L. Cosmo, E.C. Lorenzini (Eds.), Tethers in Space Handbook, NASA, 1997. [3] K. Kumar, K.D. Kumar, Tethered dual spacecraft configuration a solution to attitude control problems, Aerosp. Sci. Technol. 4 (2000) 495–505. [4] R.L. Abel, Dynamics and Control of Tethered Satellite System, Rochester Institute of Technology, 2006. [5] M.P. Cartmell, D.J. McKenzie, A review of space tether research, Prog. Aerosp. Sci. 44 (2008) 1–21, https://doi.org/10.1016/j.paerosci.2007.08.002. [6] V.S. Aslanov, A.S. Ledkov, Dynamics of Tethered Satellite Systems, Woodhead Publishing, 2012. [7] H. Fujii, S. Ishijima, Mission function control for deployment and retrieval of a subsatellite, J. Guid. Control Dyn. 12 (1989) 243–247, https://doi.org/10.2514/3. 20397. [8] A.K. Misra, V.J. Modi, Three-dimensional dynamics and control of tetherconnected n-body systems, Acta Astronaut. 26 (1992) 77–84. [9] H.A. Fujii, T. Watanabe, T. Kusagaya, D. Sato, M. Ohta, Dynamics of a flexible space tether equipped with a crawler mass, J. Guid. Control Dyn. 31 (2008) 436–440, https://doi.org/10.2514/1.26240. [10] P. Huang, F. Zhang, L. Chen, Z. Meng, Y. Zhang, Z. Liu, Y. Hu, A review of space tether in new applications, Nonlinear Dyn. 94 (2018) 1–19, https://doi.org/10. 1007/s11071-018-4389-5.

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