- Email: [email protected]
Anti-sway control of tethered satellite systems using attitude control of the main satellite
Author's Accepted Manuscript
Anti-sway control of tethered satellite Systems using attitude control of the main satellite Peyman Yousefian, Hassan Salarieh
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(15)00080-6 http://dx.doi.org/10.1016/j.actaastro.2015.02.027 AA5365
To appear in:
Acta Astronautica
Received date: 19 October 2014 Revised date: 14 February 2015 Accepted date: 23 February 2015 Cite this article as: Peyman Yousefian, Hassan Salarieh, Anti-sway control of tethered satellite Systems using attitude control of the main satellite, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.02.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anti-Sway Control of Tethered Satellite Systems Using Attitude Control of the Main Satellite Peyman Yousefiana, Hassan Salariehb a
Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567,Azadi Street, Tehran, Iran, E-mail: [email protected] b
Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567,Azadi Street, Tehran, Iran, E-mail: [email protected]
Corresponding Author: Hassan Salarieh Department of Mechanical Engineering Sharif University of Technology Azadi Street, Tehran, Iran P.O. Box 11155-9567 E-mail: [email protected] Tel: +98 21 6616 5538 Fax: +98 21 6600 0021
1
Abstract: In this study a new method is introduced to suppress libration of a tethered satellite system (TSS). It benefits from coupling between satellites and tether libration dynamics. The control concept uses the main satellite attitude maneuvers to suppress librational motion of the tether, and the main satellite's actuators for attitude control are used as the only actuation in the system. The study considers planar motion of a two body TSS system in a circular orbit and it is assumed that the tether's motion will not change it. Governing dynamic equations of motion are derived using the extended Lagrange method. Controllability of the system around the equilibrium state is studied and a linear LQG controller is designed to regulate libration of the system. Tether tension and satellite attitude are assumed as only measurable outputs of the system. The Extended Kalman Filter (EKF) is used to estimate states of the system to be used as feedback to the controller. The designed controller and observer are implemented to the nonlinear plant and simulations demonstrate that the controller lead to reduction of the tether libration propoerly. By the way, because the controller is linear, it is applicable only at low amplitudes in the vicinity of equilibrium point. To reach global stability, a nonlinear controller is demanded. Keywords: TSS, Lagrange, Satellite, Dynamics, EKF, LQG, Control
1 Introduction The concept of Tethered satellite systems (TSS) has been introduced by Tsiollovsky in 1895 for the first time. In recent decades the idea of connecting satellites with tether has attracted attentions of many researchers. Numerous missions have been conducted to examine concepts of these systems [1]. Possible applications of these systems include: scientific experiments like gravity gradient, provision of desired microgravity environment, generation of electricity, deployment of payloads into new orbits and retrieval of satellites for reusing and etc. [2, 3]. In general form, TSS dynamics is very complicated which is governed by a set of ordinary and partial nonlinear, non-autonomous and coupled equations. A complete review of recent works on dynamics and control of these systems is presented in [1, 4-9]. A typical TSS mission involves deployment, station-keeping and retrieval phases. There is a broad literature in controlling the motion of TSS in these phases [2, 10-16]. Deployment phase is the first and indispensable phase of the mission which is inherently stable provided that certain deployment speeds are not exceeded. In [14-17] various controlling methods in deployment phase are discussed. The next phase is station keeping. Controlling the tether in the direction of nadir during this phase, is one of the most interested missions in TSS applications. Many studies have been conducted to reach this goal by using various actuator and control methods. Thrusters on satellites [18-21], tether tension control [22], tether rate control, and attachment point position control [4] are among the methods that are used to reduce the librational motion of tether. Sometimes the last phase is retrieval which is known to be unstable. Some controlling schemes have been developed to retrieve tethered satellites [16, 23, 24]. From the dynamical view point, tether and satellites dynamics are coupled. To study this, it is necessary to model the satellite as a rigid body. In [25] abstract form of the equations of motion for multibody TSS in a three dimensional Keplerian orbit is derived. In this model, satellites are free to undergo three dimensional attitude motions. In [26] oscillation of the main satellite due to tether is studied, but it is assumed that the mass of sub-satellite, i.e. the satellite which is attached to the main satellite through a tether, is negligible and center of gravity of the whole system is approximated with the center of gravity of the main satellite. In [27] the effects of the tether on satellite have been modeled with a perturbation force. In all methods that have been implemented to control the libration of TSS in station keeping phase, it is required to utilize special set of actuation system on the satellite to change the length, rate, and angle or tension of the tether. In this study, a new control approach is presented to reduce tether libration just by using the attitude control actuators of the main satellite. In this method, attitude maneuver of the main satellite and the dynamic coupling between the main satellite attitude and tether libration are used to suppress libration of the tether. To implement such controlling idea, it is necessary to consider the coupling between rigid body motion of the satellite and tether libration. In this study, station keeping phase of a pico-satellite deployed from a microsatellite is considered. 2D dynamic equations of the TSS are extracted in a circular orbit. It is assumed that the center of mass (CM) of the system is on the circular orbit and motion of the tether does not affect the orbital motion of the system and it would remain on the same orbit. The main satellite is modeled as a rigid body and the other one as a point mass. Tether is assumed to be inelastic with constant length which has not any longitudinal or transverse vibration. Based on the new control approach, a linear feedback controller is designed to reduce libration using the main satellite attitude control. The only actuator of the system is a reaction wheel and there are not any other actuators like thruster and magnetic torquers. the designed controller is full state feedback. It means that all states of the system are required to be measured and fed to the controller, but some of them are not feasible to be measured. Therefore, it is necessary to design an observer to estimate
2
them. To do this, an Extended Kalman Filter (EKF) is designed which uses tether tension force and the main satellite attitude to
Nomenclatures A
State matrix
qi
a ij
R ( t ) Measurement noise covariance matrix
d1
Coefficients of Lagrange multipliers Input matrix Output matrix Distance of satellites center of mass Distance of main satellite from origin
d2 F f g H h I J (u )
Distance of sub satellite from origin State transition matrix State function of state space representation Input function of state space representation Observation matrix Input function of state space representation Main satellite moment of inertia Cost Function
B C
d
K Kinetic energy of satellite K ( t ) Kalman gain KC L m1
Linear Controller gain Lagrangian of the system Length of tether Main satellite mass
m2
Sub-satellite mass
l
Generalized coordinates
ROrbit Orbital radius r
Distance of tether junction point from COM
TOrbit Period of orbit UGravity Potential energy of the system
u
Control signal V Weight matrix at LQG method v ( t ) Measurement noise
v1
Velocity of main satellite
v2 Velocity of sub satellite W Wight matrix at LQG method w ( t ) Modeling noise x
y η θ
λi τ ϕ ωSat ω
P ( t ) Covariance matrix of EKF Q ( t ) Modeling noise covariance matrix
State vector of the system Output vector of the system Angle of system with Nadir Satellite absolute rotation angle Lagrange multiplier Torque acting on main satellite Angle of satellite relative to tether Absolute angular velocity of satellite Orbital angular velocity
estimate all of the states of the system and fed them back to the controller.
2 Modeling To design a controller, it is required to model the dynamics of the system. Various methods for the dynamics modeling of a tethered system exist, where the tether is considered either elastic or inelastic, satellites are modeled as either rigid body or point mass, orbit is assumed circular or non-circular and etc. [1]. To maintain the advantage of simplicity required for analysis and controller design, the following assumptions are made: • Because of the disturbances in elliptic orbits which affect the libration of tether [10], the system is assumed to be in a circular orbit to study just the effect of satellite maneuver on libration. • To perform tether libration controlling maneuvers, the main satellite is modeled as a rigid body. The sub-satellite is modeled by a point mass, since its small dimensions do not affect on the tether libration. • Because the typical tether mass for this type of application amount to 0.3 kg km-1 [15], it is idealized to be massless. • In these systems tether has longitudinal and transverse vibration with low amplitude and high frequency [4]. Because of the low frequency nature of the orbital motions and satellite maneuvers, vibrations of tether will not affect the system. Therefor the tether could be assumed inelastic.
3
Geometry of the system is shown in Figure 1. The origin of the orbit coordinate system is located at the center of mass of the whole system which moves on a circular orbit. l Is the tether length and d is the distance between CM of satellites. η is angle of d with nadir and θ denotes angle of rotation of satellite. The CM of whole system is on a circular orbit with radius of ROrbit and angular velocity of ω . η l d
r
Figure 1. Geometry of system.
2-1 Dynamics Modeling Satellite velocities are described by Eq. (1). d 1 and d 2 are distances of satellites from origin in Eq. (2). Angular velocity of the main satellite is given by Eq. (3). Having velocities, kinetic energy of the system in the orbital coordinate, can be derived by Eq. (4).
v1 = d 1 (η + ω ) ( − cos (η ) i + sin (η ) k ) v 2 = d 2 (η + ω ) ( cos (η ) i − sin (η ) k ) d1 =
m2 d , m1 + m 2
d2 =
m1 d m1 + m 2
ωSat = θ + ω K =
Eq. (1)
Eq. (2) Eq. (3)
1 1 1 2 m 1 v 12 + m 2 v 22 + I ωSat 2 2 2
Eq. (4)
Potential energy caused by the gravity gradient, is described in Eq. (5). It should be noted that potential energy term due to gravity gradient of the rigid satellite is small comparing to the term caused by the tether, so it is not considered in the potential energy [28].
U Gravity M =
µM l2 3 C
2R
(1 − 3cos (η ) ) 2
m1 m 2 m1 + m 2
Eq. (5)
Eq. (6)
Equations of motion would be derived using extended Lagrange method. Generalized coordinates of system are defined as below: q1 = η , q 2 = θ , q 3 = d
Eq. (7)
Length of the tether is derived from below equation: l 2 = d 2 + r 2 − 2 r d cos (η − θ )
Eq. (8)
4
Assuming rigid tether with constant length, time derivative of the above equation would be Eq. (9). It presents the Holonomic constraint equation of the system: a 11η + a 12θ + a 13 d = 0
Eq. (9)
a 11 = r d sin (η − θ ) a 12 = −r d sin (η − θ )
Eq. (10)
a 13 = d − r cos (η − θ ) Defining Lagrangian of the system as Eq. (11) and the constraint, Lagrange equations of the system are given by: Eq. (11)
L = K − UGravity d ∂L ∂L = a 11λ1 − dt ∂η ∂η d ∂L ∂L − = τ + a 12 λ1 dt ∂θ ∂θ
Eq. (12)
d ∂L ∂L = a 13λ 1 − dt ∂d ∂d
λ1 is the Lagrange multiplier and refers to the tether tension in this system. τ is external torque acting on the main satellite and would be used as controlling force. It can be generated using reaction wheel, magnetic torquer, thruster and any other typical instruments which are used in satellites. Finally dynamic equations of the system are derived as below: d η +
1 d d ω 2 sin ( 2η ) + r sin (η − θ ) η 2 − d 2
(
+η 2r ω sin (η − θ ) + 2 d
)
(
)
2 + ω 3 r ω cos (η ) sin (η − θ ) + 2 d = 0
Eq. (13)
I θ − r sin (η − θ )( λ1 ) − τ = 0
(
2
)
2
d
λ1 + M d ω 2 1 − 3cos (η ) − M d (ω + η ) − = 0 d
Because r l , using Eq. (8), d can be approximated as below:
d l + r cos (η − θ ) or d l
Eq. (14)
d r 2 2 2 2 Using Eq. (8), it can be shown that d is in the order of O r η , r θ , r η, r θ . Therefore we have O = O O η ,θ ,η,θ . d d 2 r r d Because 1 , In the first and the third equations, the term is very small relative to (ω + η ) and η 2 . So this term can be d l d neglected for the sake of simplicity. By defining state variables as below, equations of motion in state space are written as Eq. (16).
(
x1 = η , x2 = η , x3 = θ , x4 = θ , u = τ
)
Eq. (15)
5
(
)
x1 = f1 ( x, u ) = x2 x 2 = f 2 ( x, u ) Eq. (16)
x3 = f 3 ( x, u ) = x3 x 4 = f 4 ( x, u )
2-2 Observer Design As mentioned previously, the satellite attitude and the tether tension are two measurable states of the system. Attitude of the satellite could be measured using conventional attitude determination systems which are very common in most satellites. Tension of the tether is measurable by using a force sensor at the tether releasing mechanism. In this part, an observer would be designed to estimate all of the states of the system using the measured outputs. The simplified form of tether tension force can be evaluated by Eq. (13) as below: 2 2 x2 λ 1 ( x ) M l ω − 1 − 3cos ( x 1 ) + 1 + ω
(
2
)
Eq. (17)
The first output is defined as satellite attitude and the second one is normalized tension of tether which is defined by Eq. (18):
y2 =
λ1 ( x ) Eq. (18)
M l ω2
Therefore, the output vector of the system is given by: x3 2 y = h (x) = 2 x2 − 1 − 3cos ( x 1 ) + 1 + ω
(
)
Eq. (19)
From [29], for a plant with model and measurement noises of w ( t ) and v ( t ) given by Eq. (20), equations of the EKF are obtained as Eq. (21). x (t ) = f ( x (t ) ,u (t ) ) + w (t ) z (t ) = h ( x (t ) ) + v (t )
Eq. (20)
w (t ) ∼ N ( 0, Q (t ) ) v (t ) ∼ N ( 0, R (t ) )
(
xˆ (t ) = f ( xˆ (t ) ,u (t ) ) + K (t ) y (t ) − h ( xˆ (t ) )
)
T
P (t ) = F (t ) P (t ) + P (t ) F (t ) − K (t ) H (t ) P (t ) + Q (t ) Eq. (21) T
K (t ) = P (t ) H (t ) R (t )
−1
In these equations, H and F are calculated as below:
6
F=
H=
∂f ( x , t ) ∂x
x = xˆ (t )
Eq. (22)
∂h ( x , t ) ∂x
x = xˆ (t )
Q and R are respectively covariances of the noise of the model and measurement. To study local stability of EKF, it is required to show observability of the linearized system about equilibrium point. The equilibrium point is defined when the tether is in stationary position at radial direction [12]. The linearized system is derived as below:
x = A x + B u Eq. (23)
y = Cx 0 2 3(l + r ) ω − ∂f l A= = 0 ∂ x ( X = 0,u = 0 ) 2 3l M r ω I ∂f B= = 0 0 0 ∂ u ( X = 0,u = 0 )
0 0 C= 0 1 ω 2
1 0 0 0 −
0 3rω l 0
2
3M l rω I
2
0 0 1 0
Eq. (24)
T
1 I
Eq. (25)
1 0 0 0
Eq. (26)
Eq. (27) describes the observability matrix of the system in which the first four rows are linearly independent. Therefore the matrix is full rank and the linear system is observable. In addition, A is a full rank matrix, hence it can be concluded that the designed EKF is locally asymptotically stable. 0 0 0 C −3ω 2 l + r CA l CA2 = 2 3l M r ω CA3 I 0 ... ...
0 1
1 0
ω2 0
0
0
3 r ω2 l
0 −3
−
l+r l ... ...
3 M l r ω2 I 0 ... ...
0 0 1 0 0 3 r l ... ...
Eq. (27)
2-3 Controller Design In this part a linear controller would be designed using the LQG method. Then it would be implemented on the nonlinear equations of the system. Controllability matrix is calculated by Eq. (28) to check the controllability of system. As shown in the equation, the matrix is full rank and the linear system is controllable.
7
C = B |A B | A2 B | A3 B 0 0 = 0 1 I
0
0
0
3 r ω2 Il
1 I
0
0 −
3 lM r ω 2 I2
0 2 3 lM r ω − I2 0 3 r ω2 Il
Eq. (28)
At LQG method, the state feedback law, u = − KC xˆ , is designed to minimizes the following quadratic cost function:
∞ J ( x, u ) = E xT W x + uT V u dt 0 u = −K C xˆ
∫(
)
Eq. (29)
W and V are weight matrices, and E is the expectation operator. The requirement that J ≥ 0 , implies that all these matrices should be positive definite.
3 Numerical analysis and discussion Here, performance of the designed controller and observer is studied. The designed linear controller with estimated states as feedback is applied on the full nonlinear equations of the plant (Eq. (13)). Parameters of the system are assumed as below:
ω = 0.0010781 Rad/s
m1 = 50 kg
ROrbi = 700 Km
m 2 = 1 kg
TOrbit = 5828 Sec
I = 2 kg .m 2
l = 200 m
r = 2m
Eq. (30)
Initial conditions are assumed to be:
η (t = 0 ) = 6
θ (t = 0 ) = 6
η (t = 0 ) = 0
θ (t = 0 ) = 0
Eq. (31)
Covariance of the Noise of the sensors is assumed to be 0.5% of signals' maximum amplitude and Covariance of noise of modeling is assumed to be 1% of states' maximum amplitude. Therefore covariance matrices are:
10 0 Q = 10−4 × 0 0
0 0 0 0.005 R= 0 1 0 5 × 10−7 Eq. (32) 0 0 0 100
0 0 1 0
It should be noted that during control process, the satellite is not permitted to tumble. It means that the angle between the satellite and the tether should remain under 90 degrees as a constraint. Tumbling of the satellite may result in twisting or slackness of the tether. To avoid this important event, gain of the controller should be adjusted in a way that the constraint is satisfied. In the LQG method, gain of the controller is adjusted with weigh functions:
8
150 0 0 0 0 10 0 0 V = 100 W= 0 0 1 0 0 0 1 0
Eq. (33)
Using the above weight functions, gain of the controller is obtained as: T
KC = [ 0.06 706 0.1 0.64]
Eq. (34)
Before applying the controller, the observer needs to be converged. Therefore, the First 1000 seconds of simulation is allocated for convergence of the observer and the controller is off. After that, the controller is implemented and controlling torque acts on the system. The way that controller works is reduction of tether libration angle by controlling attitude of the main satellite. The concept in behind is that, controlling the attitude of the satellite relative to tether causes to change the distance of the two satellites ( d ). This results in a linear velocity along the tether direction. In the presence of angular velocity of tether and linear velocity, a couple of Coriolis forces perpendicular to the tether are produced. These forces result in a torque on the tether and reduce the amplitude of libration. Figure 2 and Figure 3 show estimated values of the tether angle and the angular velocity. Estimation error is under 1 and 0.0004 / sec respectively. According to the figures, the controller has reduced tether libration in 6 orbits. Figure 4 and Figure 5 show satellite attitude and angular velocity. Estimation error is under 0.5 / sec and 0.1 / sec respectively. Errors are result of sensors noise and would decrease with reduction of the noise. As shown in the figures, at initial orbits, the main satellite performs a special kind of attitude maneuvers to create Coriolis forces and suppress libration. After decreasing libration, attitude of the main satellite is also regulated at nadir direction. Figure 6 shows controlling torque acting on the main satellite. In simulations, it is assumed that the actuator saturates in 0.0008 Nm . As illustrated in the figure, it is saturated at the first orbit, when the amplitude of tether libration has the maximum value. Figure 7 and Figure 8 show measured outputs of the system which are fed into the EKF. As illustrated in figures, signals are noisy. The noise is intentionally added to the signals to see effect of noise in the performance of the system.
Figure 2. Estimated value of the tether angle, xˆ1 .
9
Figure 3. Estimated value of the tether angular velocity,
Figure 4. Estimated satellite attitude,
xˆ 3 .
Figure 5. Estimated satellite angular velocity,
10
xˆ 4 .
xˆ 2 .
Figure 6. Control torque,
u.
Figure 7. Measured value of the satellite attitude, y 1 .
Figure 8. Measured value of the tether tension,
y2 .
Changing parameters of the system, would affect the performance of the controller. Increasing the length of the tether ( l ), would increase the effect of the gravity gradient and tension force. Therefore it would require more control effort and more time to settle the libration. Additionally, increasing mass of the satellites would have the same effect. Likely, decreasing the r would lower the effect of Coriolis force and subsequently more control effort would be required. To demonstrate these effects, behavior of the system is
11
considered when the length of the tether is increased to l = 300 m and the results are compared with the previous case with l = 200 m . As illustrated in Figure 9, effects of gravity gradient and tether tension would be increased by increasing the length of the tether. Therefore it would require more controlling maneuver of the main satellite to surpass the libration. Figure 10 depicts increase of the satellite's maneuver with increase of tether length. As the maneuver of satellite increases, it needs more controlling torque. According to Figure 11, in the case with long tether, the actuator is saturated 4 periods more than the case with short tether. Finally Figure 12 illustrates that with the same saturation limit of the actuator, it would take more time to overcome libration, when the length of the tether is increased.
Figure 9. Comparison of the tether tension for different tether lengths
Figure 10. Comparison of satellite maneuvers to damp libration for different tether lengths
Figure 11. Comparison of required controlling torque on satellite for different tether lengths
12
Figure 12. Comparison of suppression of libration for different tether lengths
4 Conclusion In this study, dynamic equations of a TSS system in orbital plane are derived. The main satellite is modeled as a rigid body and the sub-satellite as a point mass. An EKF is used to successfully estimate states of the system by using satellite attitude and tether tension force as the system outputs which are measurable. It is shown that the linearized system is controllable and a full state feedback linear controller is designed by the LQG method to reduce tether libration using the attitude maneuvers of the main satellite. Estimated states are used as feedback to the controller. Finally the linear controller is applied to the nonlinear system and it is successful to suppress low amplitude tether libration. Effect of some parameters like mass of the satellites and length of the tether are also investigated on the performance of the controller. It illustrates that change of parameters which result in increase of the gravity gradient force, would require more control effort. The controller diverges at large amplitudes of libration and a nonlinear controller would be required to suppress them.
13
References [1] K.D. Kumar, Review on Dynamics and Control of Nonelectrodynamic Tethered Satellite Systems, J AEROSPACE ENG, 43 (2006) 705-720. [2] V.J. Modi, G. Gilardi, A.K. Misra, Attitude Control of Space Platform Based Tethered Satellite System, J AEROSPACE ENG, 11 (1998) 24-31. [3] G. Liu, C. Li, G. Ma, J. Huang, Stabilization control of the tethered satellite system using nonlinear model predictive control, in: Control Conference (CCC), 2014 33rd Chinese, 2014, pp. 7679-7684. [4] M.P. Cartmell, D.J. McKenzie, A review of space tether research, PROG AEROSP SCI, 44 (2008) 1-21. [5] H. Wen, D.P. Jin, H.Y. Hu, Advances in dynamics and control of tethered satellite systems, ACTA MECH SINICA, 24 (2008) 229-241. [6] K. Ishimura, K. Higuchi, Coupling among Pitch Motion, Axial Vibration, and Orbital Motion of Large Space Structures, J AEROSPACE ENG, 21 (2008) 61-71. [7] W. Zhang, F.B. Gao, M.H. Yao, Periodic solutions and stability of a tethered satellite system, MECH RES COMMUN, 44 (2012) 24-29. [8] W. Jung, A. Mazzoleni, J. Chung, Dynamic analysis of a tethered satellite system with a moving mass, Nonlinear Dyn, 75 (2014) 267-281. [9] G. Avanzini, M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations, ACTA ASTRONAUT, 94 (2014) 338-350. [10] M. Krupa, W. Poth, M. Schagerl, A. Steindl, W. Steiner, H. Troger, G. Wiedermann, Modelling, Dynamics and Control of Tethered Satellite Systems, Nonlinear Dyn, 43 (2006) 73-96. [11] Godard, K.D. Kumar, B. Tan, Nonlinear optimal control of tethered satellite systems using tether offset in the presence of tether failure, ACTA ASTRONAUT, 66 (2009) 1434-1448. [12] A.K. Misra, Dynamics and control of tethered satellite systems, ACTA ASTRONAUT, 63 (2008) 1169-1177. [13] H. Kojima, Y. Sugimoto, Y. Furukawa, Experimental study on dynamics and control of tethered satellite systems with climber, ACTA ASTRONAUT, In Press, Corrected Proof (2011) 96–108. [14] H. Wen, D. Jin, H. Hu, Optimal feedback control of the deployment of a tethered subsatellite subject to perturbations, Nonlinear Dyn, 51 (2008) 501-514. [15] H. Gläßel, F. Zimmermann, S. Brückner, U.M. Schöttle, S. Rudolph, Adaptive neural control of the deployment procedure for tether-assisted re-entry, AEROSP SCI TECHNOL, 8 (2004) 73-81. [16] P. Williams, Deployment/retrieval optimization for flexible tethered satellite systems, Nonlinear Dyn, 52 (2008) 159-179. [17] A. Steindl, Optimal control of the deployment (and retrieval) of a tethered satellite under small initial disturbances, Meccanica, 49 (2014) 1879-1885. [18] Pradham, Planar Dynamics and Control of Tethered Satellite Systems, in: Mechanical Engineering, The University of British Columbia, 1994. [19] G. Zhao, L. Sun, S. Tan, H. Huang, Librational characteristics of a dumbbell modeled tethered satellite under small, continuous, constant thrust, P I MECH ENG G-J AER, 227 (2013) 857-872. [20] G. Zhao, L. Sun, H. Huang, Thrust control of tethered satellite with a short constant tether in orbital maneuvering, P I MECH ENG G-J AER, (2014). [21] J. Huang, G. Ma, G. Liu, Nonlinear dynamics and reconfiguration control of two-satellite Coulomb tether formation at libration points, AEROSP SCI TECHNOL, (2014). [22] K. Kumar, K.D. Kumar, Tethered dual spacecraft configuration: a solution to attitude control problems, AEROSP SCI TECHNOL, 4 (2000) 495-505.
14
[23] G. Sun, Z.H. Zhu, Fractional order tension control for stable and fast tethered satellite retrieval, ACTA ASTRONAUT, 104 (2014) 304-312. [24] H. Wen, D. Jin, H. Hu, Feedback Control for Retrieving an Electro-Dynamic Tethered Sub-Satellite, Tsinghua Science & Technology, 14 (2009) 79-83. [25] S. Kalantzis, V.J. Modi, S. Pradhan, A.K. Misra, Dynamics and control of multibody tethered systems, ACTA ASTRONAUT, 42 (1998) 503-517. [26] Aslanov, Oscillations of a spacecraft with a vertical tether, Proc World Congr Enging, (2009) 1827–1831. [27] S. Cho, Analysis of The Motion of a Tether-Perturbed Satellite, J.Astron, (2003). [28] V.J. Modi, G. Chang-Fu, A.K. Misra, D.M. Xu, On the control of the space shuttle based tethered systems, ACTA ASTRONAUT, 9 (1982) 437-443. [29] K.J. Astrom, Introduction to Stochastic Control Theory, Academic Press, New York and Londen, 1970.
15
Anti-sway control of tethered satellite Systems using attitude control of the main satellite Peyman Yousefian, Hassan Salarieh
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(15)00080-6 http://dx.doi.org/10.1016/j.actaastro.2015.02.027 AA5365
To appear in:
Acta Astronautica
Received date: 19 October 2014 Revised date: 14 February 2015 Accepted date: 23 February 2015 Cite this article as: Peyman Yousefian, Hassan Salarieh, Anti-sway control of tethered satellite Systems using attitude control of the main satellite, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.02.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anti-Sway Control of Tethered Satellite Systems Using Attitude Control of the Main Satellite Peyman Yousefiana, Hassan Salariehb a
Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567,Azadi Street, Tehran, Iran, E-mail: [email protected] b
Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567,Azadi Street, Tehran, Iran, E-mail: [email protected]
Corresponding Author: Hassan Salarieh Department of Mechanical Engineering Sharif University of Technology Azadi Street, Tehran, Iran P.O. Box 11155-9567 E-mail: [email protected] Tel: +98 21 6616 5538 Fax: +98 21 6600 0021
1
Abstract: In this study a new method is introduced to suppress libration of a tethered satellite system (TSS). It benefits from coupling between satellites and tether libration dynamics. The control concept uses the main satellite attitude maneuvers to suppress librational motion of the tether, and the main satellite's actuators for attitude control are used as the only actuation in the system. The study considers planar motion of a two body TSS system in a circular orbit and it is assumed that the tether's motion will not change it. Governing dynamic equations of motion are derived using the extended Lagrange method. Controllability of the system around the equilibrium state is studied and a linear LQG controller is designed to regulate libration of the system. Tether tension and satellite attitude are assumed as only measurable outputs of the system. The Extended Kalman Filter (EKF) is used to estimate states of the system to be used as feedback to the controller. The designed controller and observer are implemented to the nonlinear plant and simulations demonstrate that the controller lead to reduction of the tether libration propoerly. By the way, because the controller is linear, it is applicable only at low amplitudes in the vicinity of equilibrium point. To reach global stability, a nonlinear controller is demanded. Keywords: TSS, Lagrange, Satellite, Dynamics, EKF, LQG, Control
1 Introduction The concept of Tethered satellite systems (TSS) has been introduced by Tsiollovsky in 1895 for the first time. In recent decades the idea of connecting satellites with tether has attracted attentions of many researchers. Numerous missions have been conducted to examine concepts of these systems [1]. Possible applications of these systems include: scientific experiments like gravity gradient, provision of desired microgravity environment, generation of electricity, deployment of payloads into new orbits and retrieval of satellites for reusing and etc. [2, 3]. In general form, TSS dynamics is very complicated which is governed by a set of ordinary and partial nonlinear, non-autonomous and coupled equations. A complete review of recent works on dynamics and control of these systems is presented in [1, 4-9]. A typical TSS mission involves deployment, station-keeping and retrieval phases. There is a broad literature in controlling the motion of TSS in these phases [2, 10-16]. Deployment phase is the first and indispensable phase of the mission which is inherently stable provided that certain deployment speeds are not exceeded. In [14-17] various controlling methods in deployment phase are discussed. The next phase is station keeping. Controlling the tether in the direction of nadir during this phase, is one of the most interested missions in TSS applications. Many studies have been conducted to reach this goal by using various actuator and control methods. Thrusters on satellites [18-21], tether tension control [22], tether rate control, and attachment point position control [4] are among the methods that are used to reduce the librational motion of tether. Sometimes the last phase is retrieval which is known to be unstable. Some controlling schemes have been developed to retrieve tethered satellites [16, 23, 24]. From the dynamical view point, tether and satellites dynamics are coupled. To study this, it is necessary to model the satellite as a rigid body. In [25] abstract form of the equations of motion for multibody TSS in a three dimensional Keplerian orbit is derived. In this model, satellites are free to undergo three dimensional attitude motions. In [26] oscillation of the main satellite due to tether is studied, but it is assumed that the mass of sub-satellite, i.e. the satellite which is attached to the main satellite through a tether, is negligible and center of gravity of the whole system is approximated with the center of gravity of the main satellite. In [27] the effects of the tether on satellite have been modeled with a perturbation force. In all methods that have been implemented to control the libration of TSS in station keeping phase, it is required to utilize special set of actuation system on the satellite to change the length, rate, and angle or tension of the tether. In this study, a new control approach is presented to reduce tether libration just by using the attitude control actuators of the main satellite. In this method, attitude maneuver of the main satellite and the dynamic coupling between the main satellite attitude and tether libration are used to suppress libration of the tether. To implement such controlling idea, it is necessary to consider the coupling between rigid body motion of the satellite and tether libration. In this study, station keeping phase of a pico-satellite deployed from a microsatellite is considered. 2D dynamic equations of the TSS are extracted in a circular orbit. It is assumed that the center of mass (CM) of the system is on the circular orbit and motion of the tether does not affect the orbital motion of the system and it would remain on the same orbit. The main satellite is modeled as a rigid body and the other one as a point mass. Tether is assumed to be inelastic with constant length which has not any longitudinal or transverse vibration. Based on the new control approach, a linear feedback controller is designed to reduce libration using the main satellite attitude control. The only actuator of the system is a reaction wheel and there are not any other actuators like thruster and magnetic torquers. the designed controller is full state feedback. It means that all states of the system are required to be measured and fed to the controller, but some of them are not feasible to be measured. Therefore, it is necessary to design an observer to estimate
2
them. To do this, an Extended Kalman Filter (EKF) is designed which uses tether tension force and the main satellite attitude to
Nomenclatures A
State matrix
qi
a ij
R ( t ) Measurement noise covariance matrix
d1
Coefficients of Lagrange multipliers Input matrix Output matrix Distance of satellites center of mass Distance of main satellite from origin
d2 F f g H h I J (u )
Distance of sub satellite from origin State transition matrix State function of state space representation Input function of state space representation Observation matrix Input function of state space representation Main satellite moment of inertia Cost Function
B C
d
K Kinetic energy of satellite K ( t ) Kalman gain KC L m1
Linear Controller gain Lagrangian of the system Length of tether Main satellite mass
m2
Sub-satellite mass
l
Generalized coordinates
ROrbit Orbital radius r
Distance of tether junction point from COM
TOrbit Period of orbit UGravity Potential energy of the system
u
Control signal V Weight matrix at LQG method v ( t ) Measurement noise
v1
Velocity of main satellite
v2 Velocity of sub satellite W Wight matrix at LQG method w ( t ) Modeling noise x
y η θ
λi τ ϕ ωSat ω
P ( t ) Covariance matrix of EKF Q ( t ) Modeling noise covariance matrix
State vector of the system Output vector of the system Angle of system with Nadir Satellite absolute rotation angle Lagrange multiplier Torque acting on main satellite Angle of satellite relative to tether Absolute angular velocity of satellite Orbital angular velocity
estimate all of the states of the system and fed them back to the controller.
2 Modeling To design a controller, it is required to model the dynamics of the system. Various methods for the dynamics modeling of a tethered system exist, where the tether is considered either elastic or inelastic, satellites are modeled as either rigid body or point mass, orbit is assumed circular or non-circular and etc. [1]. To maintain the advantage of simplicity required for analysis and controller design, the following assumptions are made: • Because of the disturbances in elliptic orbits which affect the libration of tether [10], the system is assumed to be in a circular orbit to study just the effect of satellite maneuver on libration. • To perform tether libration controlling maneuvers, the main satellite is modeled as a rigid body. The sub-satellite is modeled by a point mass, since its small dimensions do not affect on the tether libration. • Because the typical tether mass for this type of application amount to 0.3 kg km-1 [15], it is idealized to be massless. • In these systems tether has longitudinal and transverse vibration with low amplitude and high frequency [4]. Because of the low frequency nature of the orbital motions and satellite maneuvers, vibrations of tether will not affect the system. Therefor the tether could be assumed inelastic.
3
Geometry of the system is shown in Figure 1. The origin of the orbit coordinate system is located at the center of mass of the whole system which moves on a circular orbit. l Is the tether length and d is the distance between CM of satellites. η is angle of d with nadir and θ denotes angle of rotation of satellite. The CM of whole system is on a circular orbit with radius of ROrbit and angular velocity of ω . η l d
r
Figure 1. Geometry of system.
2-1 Dynamics Modeling Satellite velocities are described by Eq. (1). d 1 and d 2 are distances of satellites from origin in Eq. (2). Angular velocity of the main satellite is given by Eq. (3). Having velocities, kinetic energy of the system in the orbital coordinate, can be derived by Eq. (4).
v1 = d 1 (η + ω ) ( − cos (η ) i + sin (η ) k ) v 2 = d 2 (η + ω ) ( cos (η ) i − sin (η ) k ) d1 =
m2 d , m1 + m 2
d2 =
m1 d m1 + m 2
ωSat = θ + ω K =
Eq. (1)
Eq. (2) Eq. (3)
1 1 1 2 m 1 v 12 + m 2 v 22 + I ωSat 2 2 2
Eq. (4)
Potential energy caused by the gravity gradient, is described in Eq. (5). It should be noted that potential energy term due to gravity gradient of the rigid satellite is small comparing to the term caused by the tether, so it is not considered in the potential energy [28].
U Gravity M =
µM l2 3 C
2R
(1 − 3cos (η ) ) 2
m1 m 2 m1 + m 2
Eq. (5)
Eq. (6)
Equations of motion would be derived using extended Lagrange method. Generalized coordinates of system are defined as below: q1 = η , q 2 = θ , q 3 = d
Eq. (7)
Length of the tether is derived from below equation: l 2 = d 2 + r 2 − 2 r d cos (η − θ )
Eq. (8)
4
Assuming rigid tether with constant length, time derivative of the above equation would be Eq. (9). It presents the Holonomic constraint equation of the system: a 11η + a 12θ + a 13 d = 0
Eq. (9)
a 11 = r d sin (η − θ ) a 12 = −r d sin (η − θ )
Eq. (10)
a 13 = d − r cos (η − θ ) Defining Lagrangian of the system as Eq. (11) and the constraint, Lagrange equations of the system are given by: Eq. (11)
L = K − UGravity d ∂L ∂L = a 11λ1 − dt ∂η ∂η d ∂L ∂L − = τ + a 12 λ1 dt ∂θ ∂θ
Eq. (12)
d ∂L ∂L = a 13λ 1 − dt ∂d ∂d
λ1 is the Lagrange multiplier and refers to the tether tension in this system. τ is external torque acting on the main satellite and would be used as controlling force. It can be generated using reaction wheel, magnetic torquer, thruster and any other typical instruments which are used in satellites. Finally dynamic equations of the system are derived as below: d η +
1 d d ω 2 sin ( 2η ) + r sin (η − θ ) η 2 − d 2
(
+η 2r ω sin (η − θ ) + 2 d
)
(
)
2 + ω 3 r ω cos (η ) sin (η − θ ) + 2 d = 0
Eq. (13)
I θ − r sin (η − θ )( λ1 ) − τ = 0
(
2
)
2
d
λ1 + M d ω 2 1 − 3cos (η ) − M d (ω + η ) − = 0 d
Because r l , using Eq. (8), d can be approximated as below:
d l + r cos (η − θ ) or d l
Eq. (14)
d r 2 2 2 2 Using Eq. (8), it can be shown that d is in the order of O r η , r θ , r η, r θ . Therefore we have O = O O η ,θ ,η,θ . d d 2 r r d Because 1 , In the first and the third equations, the term is very small relative to (ω + η ) and η 2 . So this term can be d l d neglected for the sake of simplicity. By defining state variables as below, equations of motion in state space are written as Eq. (16).
(
x1 = η , x2 = η , x3 = θ , x4 = θ , u = τ
)
Eq. (15)
5
(
)
x1 = f1 ( x, u ) = x2 x 2 = f 2 ( x, u ) Eq. (16)
x3 = f 3 ( x, u ) = x3 x 4 = f 4 ( x, u )
2-2 Observer Design As mentioned previously, the satellite attitude and the tether tension are two measurable states of the system. Attitude of the satellite could be measured using conventional attitude determination systems which are very common in most satellites. Tension of the tether is measurable by using a force sensor at the tether releasing mechanism. In this part, an observer would be designed to estimate all of the states of the system using the measured outputs. The simplified form of tether tension force can be evaluated by Eq. (13) as below: 2 2 x2 λ 1 ( x ) M l ω − 1 − 3cos ( x 1 ) + 1 + ω
(
2
)
Eq. (17)
The first output is defined as satellite attitude and the second one is normalized tension of tether which is defined by Eq. (18):
y2 =
λ1 ( x ) Eq. (18)
M l ω2
Therefore, the output vector of the system is given by: x3 2 y = h (x) = 2 x2 − 1 − 3cos ( x 1 ) + 1 + ω
(
)
Eq. (19)
From [29], for a plant with model and measurement noises of w ( t ) and v ( t ) given by Eq. (20), equations of the EKF are obtained as Eq. (21). x (t ) = f ( x (t ) ,u (t ) ) + w (t ) z (t ) = h ( x (t ) ) + v (t )
Eq. (20)
w (t ) ∼ N ( 0, Q (t ) ) v (t ) ∼ N ( 0, R (t ) )
(
xˆ (t ) = f ( xˆ (t ) ,u (t ) ) + K (t ) y (t ) − h ( xˆ (t ) )
)
T
P (t ) = F (t ) P (t ) + P (t ) F (t ) − K (t ) H (t ) P (t ) + Q (t ) Eq. (21) T
K (t ) = P (t ) H (t ) R (t )
−1
In these equations, H and F are calculated as below:
6
F=
H=
∂f ( x , t ) ∂x
x = xˆ (t )
Eq. (22)
∂h ( x , t ) ∂x
x = xˆ (t )
Q and R are respectively covariances of the noise of the model and measurement. To study local stability of EKF, it is required to show observability of the linearized system about equilibrium point. The equilibrium point is defined when the tether is in stationary position at radial direction [12]. The linearized system is derived as below:
x = A x + B u Eq. (23)
y = Cx 0 2 3(l + r ) ω − ∂f l A= = 0 ∂ x ( X = 0,u = 0 ) 2 3l M r ω I ∂f B= = 0 0 0 ∂ u ( X = 0,u = 0 )
0 0 C= 0 1 ω 2
1 0 0 0 −
0 3rω l 0
2
3M l rω I
2
0 0 1 0
Eq. (24)
T
1 I
Eq. (25)
1 0 0 0
Eq. (26)
Eq. (27) describes the observability matrix of the system in which the first four rows are linearly independent. Therefore the matrix is full rank and the linear system is observable. In addition, A is a full rank matrix, hence it can be concluded that the designed EKF is locally asymptotically stable. 0 0 0 C −3ω 2 l + r CA l CA2 = 2 3l M r ω CA3 I 0 ... ...
0 1
1 0
ω2 0
0
0
3 r ω2 l
0 −3
−
l+r l ... ...
3 M l r ω2 I 0 ... ...
0 0 1 0 0 3 r l ... ...
Eq. (27)
2-3 Controller Design In this part a linear controller would be designed using the LQG method. Then it would be implemented on the nonlinear equations of the system. Controllability matrix is calculated by Eq. (28) to check the controllability of system. As shown in the equation, the matrix is full rank and the linear system is controllable.
7
C = B |A B | A2 B | A3 B 0 0 = 0 1 I
0
0
0
3 r ω2 Il
1 I
0
0 −
3 lM r ω 2 I2
0 2 3 lM r ω − I2 0 3 r ω2 Il
Eq. (28)
At LQG method, the state feedback law, u = − KC xˆ , is designed to minimizes the following quadratic cost function:
∞ J ( x, u ) = E xT W x + uT V u dt 0 u = −K C xˆ
∫(
)
Eq. (29)
W and V are weight matrices, and E is the expectation operator. The requirement that J ≥ 0 , implies that all these matrices should be positive definite.
3 Numerical analysis and discussion Here, performance of the designed controller and observer is studied. The designed linear controller with estimated states as feedback is applied on the full nonlinear equations of the plant (Eq. (13)). Parameters of the system are assumed as below:
ω = 0.0010781 Rad/s
m1 = 50 kg
ROrbi = 700 Km
m 2 = 1 kg
TOrbit = 5828 Sec
I = 2 kg .m 2
l = 200 m
r = 2m
Eq. (30)
Initial conditions are assumed to be:
η (t = 0 ) = 6
θ (t = 0 ) = 6
η (t = 0 ) = 0
θ (t = 0 ) = 0
Eq. (31)
Covariance of the Noise of the sensors is assumed to be 0.5% of signals' maximum amplitude and Covariance of noise of modeling is assumed to be 1% of states' maximum amplitude. Therefore covariance matrices are:
10 0 Q = 10−4 × 0 0
0 0 0 0.005 R= 0 1 0 5 × 10−7 Eq. (32) 0 0 0 100
0 0 1 0
It should be noted that during control process, the satellite is not permitted to tumble. It means that the angle between the satellite and the tether should remain under 90 degrees as a constraint. Tumbling of the satellite may result in twisting or slackness of the tether. To avoid this important event, gain of the controller should be adjusted in a way that the constraint is satisfied. In the LQG method, gain of the controller is adjusted with weigh functions:
8
150 0 0 0 0 10 0 0 V = 100 W= 0 0 1 0 0 0 1 0
Eq. (33)
Using the above weight functions, gain of the controller is obtained as: T
KC = [ 0.06 706 0.1 0.64]
Eq. (34)
Before applying the controller, the observer needs to be converged. Therefore, the First 1000 seconds of simulation is allocated for convergence of the observer and the controller is off. After that, the controller is implemented and controlling torque acts on the system. The way that controller works is reduction of tether libration angle by controlling attitude of the main satellite. The concept in behind is that, controlling the attitude of the satellite relative to tether causes to change the distance of the two satellites ( d ). This results in a linear velocity along the tether direction. In the presence of angular velocity of tether and linear velocity, a couple of Coriolis forces perpendicular to the tether are produced. These forces result in a torque on the tether and reduce the amplitude of libration. Figure 2 and Figure 3 show estimated values of the tether angle and the angular velocity. Estimation error is under 1 and 0.0004 / sec respectively. According to the figures, the controller has reduced tether libration in 6 orbits. Figure 4 and Figure 5 show satellite attitude and angular velocity. Estimation error is under 0.5 / sec and 0.1 / sec respectively. Errors are result of sensors noise and would decrease with reduction of the noise. As shown in the figures, at initial orbits, the main satellite performs a special kind of attitude maneuvers to create Coriolis forces and suppress libration. After decreasing libration, attitude of the main satellite is also regulated at nadir direction. Figure 6 shows controlling torque acting on the main satellite. In simulations, it is assumed that the actuator saturates in 0.0008 Nm . As illustrated in the figure, it is saturated at the first orbit, when the amplitude of tether libration has the maximum value. Figure 7 and Figure 8 show measured outputs of the system which are fed into the EKF. As illustrated in figures, signals are noisy. The noise is intentionally added to the signals to see effect of noise in the performance of the system.
Figure 2. Estimated value of the tether angle, xˆ1 .
9
Figure 3. Estimated value of the tether angular velocity,
Figure 4. Estimated satellite attitude,
xˆ 3 .
Figure 5. Estimated satellite angular velocity,
10
xˆ 4 .
xˆ 2 .
Figure 6. Control torque,
u.
Figure 7. Measured value of the satellite attitude, y 1 .
Figure 8. Measured value of the tether tension,
y2 .
Changing parameters of the system, would affect the performance of the controller. Increasing the length of the tether ( l ), would increase the effect of the gravity gradient and tension force. Therefore it would require more control effort and more time to settle the libration. Additionally, increasing mass of the satellites would have the same effect. Likely, decreasing the r would lower the effect of Coriolis force and subsequently more control effort would be required. To demonstrate these effects, behavior of the system is
11
considered when the length of the tether is increased to l = 300 m and the results are compared with the previous case with l = 200 m . As illustrated in Figure 9, effects of gravity gradient and tether tension would be increased by increasing the length of the tether. Therefore it would require more controlling maneuver of the main satellite to surpass the libration. Figure 10 depicts increase of the satellite's maneuver with increase of tether length. As the maneuver of satellite increases, it needs more controlling torque. According to Figure 11, in the case with long tether, the actuator is saturated 4 periods more than the case with short tether. Finally Figure 12 illustrates that with the same saturation limit of the actuator, it would take more time to overcome libration, when the length of the tether is increased.
Figure 9. Comparison of the tether tension for different tether lengths
Figure 10. Comparison of satellite maneuvers to damp libration for different tether lengths
Figure 11. Comparison of required controlling torque on satellite for different tether lengths
12
Figure 12. Comparison of suppression of libration for different tether lengths
4 Conclusion In this study, dynamic equations of a TSS system in orbital plane are derived. The main satellite is modeled as a rigid body and the sub-satellite as a point mass. An EKF is used to successfully estimate states of the system by using satellite attitude and tether tension force as the system outputs which are measurable. It is shown that the linearized system is controllable and a full state feedback linear controller is designed by the LQG method to reduce tether libration using the attitude maneuvers of the main satellite. Estimated states are used as feedback to the controller. Finally the linear controller is applied to the nonlinear system and it is successful to suppress low amplitude tether libration. Effect of some parameters like mass of the satellites and length of the tether are also investigated on the performance of the controller. It illustrates that change of parameters which result in increase of the gravity gradient force, would require more control effort. The controller diverges at large amplitudes of libration and a nonlinear controller would be required to suppress them.
13
References [1] K.D. Kumar, Review on Dynamics and Control of Nonelectrodynamic Tethered Satellite Systems, J AEROSPACE ENG, 43 (2006) 705-720. [2] V.J. Modi, G. Gilardi, A.K. Misra, Attitude Control of Space Platform Based Tethered Satellite System, J AEROSPACE ENG, 11 (1998) 24-31. [3] G. Liu, C. Li, G. Ma, J. Huang, Stabilization control of the tethered satellite system using nonlinear model predictive control, in: Control Conference (CCC), 2014 33rd Chinese, 2014, pp. 7679-7684. [4] M.P. Cartmell, D.J. McKenzie, A review of space tether research, PROG AEROSP SCI, 44 (2008) 1-21. [5] H. Wen, D.P. Jin, H.Y. Hu, Advances in dynamics and control of tethered satellite systems, ACTA MECH SINICA, 24 (2008) 229-241. [6] K. Ishimura, K. Higuchi, Coupling among Pitch Motion, Axial Vibration, and Orbital Motion of Large Space Structures, J AEROSPACE ENG, 21 (2008) 61-71. [7] W. Zhang, F.B. Gao, M.H. Yao, Periodic solutions and stability of a tethered satellite system, MECH RES COMMUN, 44 (2012) 24-29. [8] W. Jung, A. Mazzoleni, J. Chung, Dynamic analysis of a tethered satellite system with a moving mass, Nonlinear Dyn, 75 (2014) 267-281. [9] G. Avanzini, M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations, ACTA ASTRONAUT, 94 (2014) 338-350. [10] M. Krupa, W. Poth, M. Schagerl, A. Steindl, W. Steiner, H. Troger, G. Wiedermann, Modelling, Dynamics and Control of Tethered Satellite Systems, Nonlinear Dyn, 43 (2006) 73-96. [11] Godard, K.D. Kumar, B. Tan, Nonlinear optimal control of tethered satellite systems using tether offset in the presence of tether failure, ACTA ASTRONAUT, 66 (2009) 1434-1448. [12] A.K. Misra, Dynamics and control of tethered satellite systems, ACTA ASTRONAUT, 63 (2008) 1169-1177. [13] H. Kojima, Y. Sugimoto, Y. Furukawa, Experimental study on dynamics and control of tethered satellite systems with climber, ACTA ASTRONAUT, In Press, Corrected Proof (2011) 96–108. [14] H. Wen, D. Jin, H. Hu, Optimal feedback control of the deployment of a tethered subsatellite subject to perturbations, Nonlinear Dyn, 51 (2008) 501-514. [15] H. Gläßel, F. Zimmermann, S. Brückner, U.M. Schöttle, S. Rudolph, Adaptive neural control of the deployment procedure for tether-assisted re-entry, AEROSP SCI TECHNOL, 8 (2004) 73-81. [16] P. Williams, Deployment/retrieval optimization for flexible tethered satellite systems, Nonlinear Dyn, 52 (2008) 159-179. [17] A. Steindl, Optimal control of the deployment (and retrieval) of a tethered satellite under small initial disturbances, Meccanica, 49 (2014) 1879-1885. [18] Pradham, Planar Dynamics and Control of Tethered Satellite Systems, in: Mechanical Engineering, The University of British Columbia, 1994. [19] G. Zhao, L. Sun, S. Tan, H. Huang, Librational characteristics of a dumbbell modeled tethered satellite under small, continuous, constant thrust, P I MECH ENG G-J AER, 227 (2013) 857-872. [20] G. Zhao, L. Sun, H. Huang, Thrust control of tethered satellite with a short constant tether in orbital maneuvering, P I MECH ENG G-J AER, (2014). [21] J. Huang, G. Ma, G. Liu, Nonlinear dynamics and reconfiguration control of two-satellite Coulomb tether formation at libration points, AEROSP SCI TECHNOL, (2014). [22] K. Kumar, K.D. Kumar, Tethered dual spacecraft configuration: a solution to attitude control problems, AEROSP SCI TECHNOL, 4 (2000) 495-505.
14
[23] G. Sun, Z.H. Zhu, Fractional order tension control for stable and fast tethered satellite retrieval, ACTA ASTRONAUT, 104 (2014) 304-312. [24] H. Wen, D. Jin, H. Hu, Feedback Control for Retrieving an Electro-Dynamic Tethered Sub-Satellite, Tsinghua Science & Technology, 14 (2009) 79-83. [25] S. Kalantzis, V.J. Modi, S. Pradhan, A.K. Misra, Dynamics and control of multibody tethered systems, ACTA ASTRONAUT, 42 (1998) 503-517. [26] Aslanov, Oscillations of a spacecraft with a vertical tether, Proc World Congr Enging, (2009) 1827–1831. [27] S. Cho, Analysis of The Motion of a Tether-Perturbed Satellite, J.Astron, (2003). [28] V.J. Modi, G. Chang-Fu, A.K. Misra, D.M. Xu, On the control of the space shuttle based tethered systems, ACTA ASTRONAUT, 9 (1982) 437-443. [29] K.J. Astrom, Introduction to Stochastic Control Theory, Academic Press, New York and Londen, 1970.
15