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Optimal commands based multi-stage drag de-orbit design for a tethered system during large space debris removal
Acta Astronautica 163 (2019) 238–249
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Optimal commands based multi-stage drag de-orbit design for a tethered system during large space debris removal
T
Zhongyi Chua,∗, Tao Weia, Tao Shena, Jingnan Dia, Jing Cuib, Jun Sunc a
School of Instrumental Science and Opto-electronics Engineering, Beihang University, 100191, China School of Mechanical Engineering and Applied Electronics, Beijing University of Technology, 101100, China c Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai Aerospace Control Technology Institute, Shanghai, 201109, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Space tethered system Large space debris removal De-orbit strategy Gauss pseudospectral method
There is a serious challenge to the safe operation of orbiting satellites as the number of space debris increases for its high risks and the possible crippling effects of collisions. Consequently, the active removal scenario of space debris has drawn wide attention in recent years, and a tethered system is considered to be a promising method for its large operating distance and low power consumption. However, the flexible tether of the system will bring the coupling of the orbit motion, the sway motion and the variation of large debris attitude, which brings about great danger in de-orbiting phase and a huge challenge in later control. Hence, aiming to de-orbit large space debris safely with a tethered system, a multi-stage horizontal drag de-orbit strategy that consists of two stages is designed. At the first stage, the orbit altitude is rose with a tug thrust whose direction is consistent with the tether, which does not provoke the in-plane oscillation of the tethered system. At the second stage, the orbit is circularized by changing the size and direction of the tug thrust. Especially, optimal commands based on the minimum of tangling risks are planned using Gauss pseudospectral method to avoid target tangling and achieve decoupling of the orbit motion, the sway motion and the variation of the target attitude. Then, the stable attitude control of the tethered system is achieved by designing a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) in the de-orbit process. Finally, numerical simulation is implemented to verify the effectiveness of the proposed de-orbit design.
1. Introduction There is a serious challenge to the safe operation of orbiting satellites as the number of space debris on the GEO (geostationary orbit) increases. Over the past 40 years, GEO orbital debris has continued to accumulate. Large debris (spent rocket stages, defunct satellites, etc.) is more prone to collisions, which can produce tens of thousands of pieces of new debris [1], and it makes the orbit danger further strengthened. Therefore, dragging the GEO debris, especially the large debris out of the orbit safely and economically is one of the most important issues to solve, which ensures the safety of the spacecraft operating in orbit [2]. Some studies have shown that the tethered/tethered-net system is a promising technology to capture and remove space debris for its large operating distance and low energy consumption [3]. Therefore, the tethered system has become a study hotspot in the past two decades. The formation missions and debris removal missions have all been considered as applications of tethered systems [4], some methodologies for deployment/retrieval optimization of tethered satellite systems have
∗
been presented in Refs. [5,6], and many enabling space debris capturing and removal methods have been proposed and several methods have been tested on ground or in parabolic flight experiments [7]. High-risk debris (such as large massive debris) has been proposed to be safely removed by the concept of active debris removal (ADR) [8–10], which achieves de-orbit of the debris captured by a tethered spacecraft via active thrust. The 7# satellites of JAXA (Japan Aerospace Exploration Agency) has been conducted, which is considered as a space robot verification experiment [11]. The project ROGER (Robotic Geostationary Orbit Restorer) developed by the ESA (European Space Agency) [12,13] considered to use a tethered net or mechanical claw to de-orbit a redundant GEO (geostationary orbit) satellite. And ESA also conducted a pre-assessment study for capturing a large space debris to produce preliminary system design in 2012 [14]. In addition, some influencing factors and stability control problems have also been considered by some scholars. For example, the chaos behavior of tethered systems was considered by Aslanov V S in their studies, and they found the conditions for the existence of the chaos and illustrated its impact
Corresponding author. E-mail addresses: [email protected] (Z. Chu), [email protected] (J. Cui), [email protected] (J. Sun).
https://doi.org/10.1016/j.actaastro.2018.12.038 Received 31 July 2018; Received in revised form 7 December 2018; Accepted 28 December 2018 Available online 03 January 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
Acta Astronautica 163 (2019) 238–249
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Fig. 1. Schematic diagram of the space tethered system.
system. For this strategy, the thruster installed on the space tug keeps its direction consistent with the tether to avoid the in-plane oscillation of the tethered system at the first stage, and its size and direction are changed to achieve the orbital rounding at the second stage. Meanwhile, optimal commands are planned using Gauss pseudospectral method to avoid target tangling and achieve the decoupling of the orbit motion, the sway motion and the variation of the target attitude. Then, stable attitude control of the tethered system is achieved by designing a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) in the de-orbit process. Finally, numerical simulation is implemented to verify the effectiveness of the proposed de-orbit design. This paper includes five main sections. In Section 2, a multi-stage horizontal drag de-orbit strategy which upgrades the orbit altitude at the first stage and lets orbit round at the second stage with the tug thrust by a tethered system is designed. Based on that, in Section 3, the optimal commands is designed using Gauss pseudospectral method to avoid target tangling. Then, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) is designed to achieve stability control of the tethered system. In Section 4, the numerical simulation is described. Finally, conclusions are summarized in Section 5.
on the behavior of the two-body tethered system on a circular orbit [15]. Some scholars investigated the dynamical stability and the eigenfrequencies of the tethered system under the action of the thrust [16]. Furthermore, a tension control law to stabilize the motions of a tethered space-tug (TST) system during its deorbiting process by regulating the tension in the tether is presented in Ref. [17]. Specifically, a tethered satellite system (TSS) is formed when debris is captured by a manipulator (such as a harpoon, a mechanical hand or a net) to implement de-orbiting of a passive, non-cooperative, possibly spinning debris. However, due to the altitude of the orbit, the removal method of debris from high orbit is different from low orbit. It takes a long time to drag debris from high orbit to the surface of the Earth; and because of the flexibility of the tether, it takes great challenges to stabilize the TSS and makes it more difficult for the control design. Therefore, it is generally considered to drag the debris to the graveyard orbit that is several hundred kilometers above the GEO [18,19], which involves the technology and theory of satellite orbit transfer. In the existing satellite orbit transfer mode, the generalized Hohman orbit transfer principle which realizes orbit transfer with two speed pulses is widely used [20], and some scholars extended the timescale separation to Hohmann orbital transfer to achieve optimal control of TST systems [21]. However, there are two problems with this principle for a flexible tethered system. On the one hand, the pulse of the start-stop phase will cause the tether to become loose and tight when the pulse thrust is larger, which makes it easy to tangle for the tethered system. On the other hand, it changes the altitude of the orbit by means of free flight of spacecraft during the time outside the pulse, and it is very easy to cause collisions between satellites. Some scholars have proposed using gravity gradient to stabilize the tethered system during de-orbit process [22]. However, the gravitational gradient in the high-orbital regions has only a slight effect and only plays a part in the long-tether system of a few kilometers long, yet the obvious flexible characteristics of the long tether bring more complicated problems for system modeling and control. Moreover, it does not consider the coupling problem of the orbit motion, the sway motion and the variation of large debris attitude of the tethered system in the process of transferring orbit, which might cause tangling of the system. In recent years, some scholars have studied the decoupling regarding the tethered system [23,24]. But they mainly considered the decoupling of the in-plane angle and out-plane angle of the tethered system, and did not consider the tangling risks. Regarding the issues above, and to meet the stable demand of drag de-orbit and achieve mission objective, the novelty of the work is as follows: First, the dynamic equations of a tethered system are formulated and the impact of the sway motion on tether tangling is analyzed. In addition, a multi-stage horizontal drag de-orbit strategy which rises the orbit altitude at the first stage and realizes the orbital rounding at the second stage is designed to de-orbit large space debris safely with a tethered
2. Dynamic equations and mission analysis of tethered satellite system during large space debris removal 2.1. Assumption and reference frames As shown in Fig. 1, the tethered system will be formed after the mission satellite completes the capture of the target using the space tether. It consists of the mission satellite, the target, and the flexible tether. After stability control, the system achieves de-orbit by the thrust applied by the thruster installed on the mission satellite. In this paper, large space debris is regarded as a rigid body, which refers to be the target. The space tug is considered to be a particle, which refers to be the mission satellite. A viscoelastic tether is used to connect the space tug and the debris, its mass is ignored because it is too small compared with the mass of the tug and the debris. We assume that the space tug has completed the capture of the debris, so the capture phase is not part of our work and therefore not addressed in this paper. The thruster installed on the space tug can be instantaneously oriented on the desired thrust direction and change its size to rise the orbit altitude and achieve the orbital rounding. Based on the assumptions mentioned above, the reference frames (shown in Fig. 1) used in this paper are introduced as follows: (1) R1 is the Earth-centred inertial reference frame, the origin of the ⎯⎯⎯⎯⎯⇀ frame coincides with the centroid of the earth Oe , Oe x e points to the 239
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⎯⎯⎯⎯⎯⇀ vernal equinox, Oe z e is perpendicular to the earth equatorial plane, ⎯⎯⎯⎯⇀ and Oe ye is determined afterwards using the right-hand rule. (2) R o is the local orbital coordinate frame, the origin of the frame is ⎯⎯⎯⎯⎯⇀ attached to centroid of the system O , Oe z 0 is consistent with the ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ vector OOe , Ox 0 is perpendicular to Oz 0 in orbital plane and lies ⎯⎯⎯⇀ behind the debris, and Oy0 is determined afterwards using the righthand rule. (3) RT is the body fixed frame of the debris, the origin of the frame ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ coincides with the debris centroid Ot ; Ot x t , Ot yt and Ot z t coincide with the three principal inertia axes respectively and conform to the right-hand rule.
is determined by
⎧Tp = εl0[kt (l − l 0) + ct l˙] ⎨ T tp = Tp e trope ⎩
where l 0 is the original length of the tether; e trope represents tether tension unit vector described in RT ; T tp represents the tether tension vector described in RT ; kt and ct are the stiffness and damping of the tether, respectively, and
1 l > l0 εl 0 = ⎧ ⎨ ⎩0 l ≤ l0
2.2.1. Orbital motion As shown in Fig. 1, The equations of the orbital motion of the tethered system described in R1 based on the Newtonian method are as follows:
v v
μ r2
⎨ v˙s = − rr s + ⎪ ⎪ m˙ 1 = − I Fthg sp 0 ⎩
2.3. Analysis of the impact of the sway motion on tether winding for the system's nonlinear and under-actuated characteristics In practice, because of the under-actuated characteristic, high nonlinearity and strong coupling of the tethered system, the tether tangling should be included. The angle α (shown in Fig. 1) is used to evaluate the tangling risks of the tethered system, and describe the relative attitude between the tether and the debris based on our previous work [26]. Therefore, the angle α is controlled to reduce the risks of tangling, and ensure the safety of the tethered system. It is defined as follows:
Fth cos γ m Fth sin γ m
+
(1)
where r is the orbital radius of the centroid of the system; vr is the ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ velocity along − Oz 0 , and − Oz 0 is the negative vector of Oz 0 ; while vs ⎯⎯⎯⎯⇀ is the velocity along Ox 0 ; Fth is the value of the thrust supplied by the thruster on the space tug; and γ represents the direction angle of the de⎯⎯⎯⎯⇀ orbit force referring to − Oz 0 ; m1 is the mass of the tug containing the fuel, and m = m1 + m2 is the total mass of the system where m2 represents the mass of the debris. In addition, Isp is the specific impulse of the thruster; and μ and g0 are the geocentric gravitational constant and normal acceleration of gravity respectively.
T
t
⎛ p ⋅e rope ⎞ α = arccos ⎜ t pt ⎟ ⎠ ⎝
2.4. Mission analysis of the drag de-orbit strategy In order to meet the stable demand of drag de-orbit and achieve mission objective, a multi-stage horizontal drag de-orbit strategy is designed to de-orbit large space debris safely with a tethered system. The mission objective is to release the orbit resources by dragging the debris in GEO to graveyard orbit that is 350 km higher than GEO. The whole process of de-orbit consists of two stages (shown in Fig. 2 and Table 1). At the first stage, a tug thrust is used to rise the orbit
μl
⎧ l¨ = l (θ˙ + vs / r ) + 3 (3 sin2 θ − 1) r ⎪ ⎪ + Fth (sin γ cos θ − cos γ sin θ) − Tp + d1 ⎪ ¯ m1 m ⎨ θ¨ = 2 vs vr − 2 l˙ (θ˙ + v / r ) + 3μ cos θ sin θ s l ⎪ r2 r3 ⎪ Fth (sin γ sin θ + cos γ cos θ ) Fth sin γ − mr + d2 ⎪ − m1 l ⎩
(2)
where l represents the distance between the space tug and the debris; m ¯ = m1 m2 / m is the reduced mass of the system; T is tether tension; d1 and d2 are interferences of debris attitude; θ represents the in-plane angle of the system and its definition is as follows:
θ = arctan( −roz / rox )
(6)
where vector pt (shown in Fig. 1) is the vector pointing from the debris centroid to the tether attachment point described in RT , and it represents the attachment point bias of the tether and the debris. The attachment point is marked red in Fig. 1
2.2.2. Equations of relative position and attitude of tethered system Dynamic equations of the sway angle and the distance between the space tug and the debris are obtained as Equation (2) using the Lagrange method according to our previous work [25]. The initial state of the system has been stable through [25], besides, the system is in GEO, where the effect of the ambient force on the debris motion outside the orbital plane is relatively small. So the debris motion outside the orbital plane is of small magnitude all the time, we do not consider its influence on the system. The tether length of the system is short in this mission and the gravity in GEO is small, so the influence of gravity gradient torque is not considered, either. 2
(5)
In fact, in order to avoid the sudden loose and tightness of the tether, the expected value of εl0 is 1 in our research, which means that the tether is always tensioned.
2.2. Dynamic modeling of the tethered system
⎧ r˙ = vr vs2 ⎪ ⎪ v˙r = r −
(4)
(3)
where r0 is the vector pointing from the debris centroid to the tug centroid described in R o , roz and rox are its components in the z and x axes, respectively. Subsequently, in this paper, the tether is regarded as a visco-elastic tether, and its elasticity and damping are considered. The tension force
Fig. 2. The drag de-orbit process. 240
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Table 1 The multi-stage dragging de-orbit strategy design. Stage
Technical approach
Objective
The first stage The second stage
Using a tug thrust whose direction is consistent with the tether Changing the size and direction of the tug thrust
Rising the orbit altitude Achieving the orbital rounding
this stage is to design the size and direction of the required thrust, and based on that to achieve stable attitude control and avoid the tangling of the tethered system. 3. Optimal commands and controller design for the tethered system during large space debris removal Fig. 3. The first stage of the drag de-orbit strategy.
3.1. Optimal commands design for the multi-stage drag de-orbit strategy
altitude, and its direction is consistent with the tether so it does not provoke the in-plane oscillation of the tethered system. At the second stage, the orbit is circularized by changing the size and direction of the tug thrust. The detailed description of the drag de-orbit strategy is as follows:
In order to enable the tethered system to reach the graveyard orbit safely, and based on the introduction in Section 2.4, the orbital maneuvering position at the first stage needs to be designed. Moreover, the size and direction of required thrust at the second stage also needs to be planned.
2.4.1. The first stage of the drag de-orbit strategy As shown in Fig. 3, a constant tug thrust Fth is used to rise the orbit altitude at the first stage, and it is provided by the thruster installed on the space tug. The initial state of the system has been stable through our previous work in Ref. [25]. The literature [25] proposed a hybrid tension control method to stabilize tethered systems during large space debris removal, and stabled the oscillations caused by the attachment point bias. The initial direction of the tether is tangent to the circular orbit in the orbital plane. In the whole process of rising the orbit altitude, the direction of Fth is controlled to consist with the tether, so it does not provoke the in-plane oscillation of the tethered system, and the oscillations of the debris will be limited enough for the system to be stable. However, the use of the tug thrust makes the entire orbit not meet the Keplerian orbits' condition, and the orbital altitude of the tethered system still maintains a rise trend when it reaches the desired altitude. Regarding this issue, the orbit circulization is designed at the second stage, and the key design parameter of the first stage is to select the orbital maneuvering position.
3.1.1. The cost function of the two stages In this paper, the optimal control problem is to minimize the tangling risks and fuel consumption of the tethered system during large space debris removal, so as to avoid debris tangling and achieve decoupling of the orbit motion, the sway motion and the variation of the debris attitude. Therefore, the cost function of the multi-stage drag deorbit strategy is designed as:
∫t
Jdeo = Δm2f + 0.05
t2f
α⋅dt
1f
(7)
where Δm2f = m1 − 0 − m2f is the fuel consumption mass of the entire deorbit phase; m1 − 0 is the initial mass of the space tug at the first stage; m2f is the ultimate mass of the space tug at the second stage; t1f , t2f are the termination times of the two stages, respectively. 3.1.2. The boundary conditions and constraint conditions at the first stage As shown in section 2.4, the constant tug thrust Fth is controlled to consist with the tether at the first stage of the de-orbit strategy, and it does not provoke the in-plane oscillation of the tethered system at this stage. Thus the command design of the first stage only needs to consider the orbital dynamics of the tethered system. Therefore, the boundary conditions and constraint conditions of the state variables of the tethered system at the first stage are designed as:
2.4.2. The second stage of the drag de-orbit strategy As shown in Fig. 4, the final orbital maneuvering moment of the first stage is the initial moment of the second stage, and the orbit is circularized by changing the size and direction of Fth at the second stage. The coupling problem of the orbit motion, the sway motion and the variation of large debris attitude of the tethered system needs to be considered, where the orbit motion is the movement of the tethered system around the earth in space; the sway motion is the rotation of the tethered system around its centroid; the variation of the debris attitude is the rotation of the debris around its centroid. Therefore, the focus of
T ⎧ x oh − 0 = [r vr vs ] x oh − f = [r1f vr1f vs1f ]T ⎨ ⎩ x oh − min ≤ x oh(t )≤ x oh − max
(8)
where x oh − 0 and x oh − f are the initial and final values of the orbital parameters at the first stage, respectively; x oh − min and x oh − max are the
Fig. 4. The second stage of the drag de-orbit strategy. 241
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3.1.4. Optimal commands of the two stages Based on the boundary conditions and variable constraints of the two-stage of tethered system from Equation (7) to Equation (12) and the optimization objective in Equation (7). The optimal commands of the time corresponding to the maneuvering position t1f ; the size and direction of Fth at the second stage can be obtained based on the Gauss pseudospectral method. Meanwhile, the first stage orbital state parameter x oh − d ; the second stage orbital state parameter x o − d and the inplane attitude control commands x s − d of the tethered system can also be obtained based on the Gauss pseudospectral method. and x oh − d , x o − d , x s − d mean the time histories of the parameters during the de-orbit phase. The optimization process achieves the minimization of the cost function in Equation (7), which makes the angle α minimum throughout the de-orbit process so that the orbit motion and the sway motion have the least impact on the variation of the debris attitude. Thus, the decoupling of the orbit motion, the sway motion and the variation of the debris attitude is achieved.
minimum and maximum constraint conditions of the orbital state variables, respectively. The minimum constraint condition x oh − min ensures that the tethered system has enough kinetic energy to achieve deorbit; and the maximum constraint condition x oh − max avoids some bad situations (collisions or tangling of the tethered system, etc.) from occurring at the first stage of the drag de-orbit strategy. 3.1.3. The boundary conditions and constraint conditions at the second stage After finishing the first stage, the orbit circular will be made at the second stage by changing the size and direction of Fth . Because of the strong coupling of the orbit motion, the sway motion and the variation of the debris attitude, the size and direction of Fth and the in-plane relative motion state parameters need to be designed. Therefore, the boundary conditions and constraint conditions of state variables of the tethered system at the second stage are designed as:
⎧ ⎪ ⎪ ⎪ ⎪
3.2. Coordinated control design of the tethered system
x o−0 = [ r1f vr1f vs1f ]T T
x s−0 = [l 0 l˙ 0 θ0 θ˙0 ]
Based on the above optimal commands that achieve decoupling of the orbit motion, the sway motion and the variation of the debris attitude, the focus of the next step is to design the controller to track the obtained optimal commands. Because the full states of the passive debris are difficult to obtain and the system's nonlinearity is strong during the de-orbit process, in this paper, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) which can achieve effective control of the system like this is designed (shown in Fig. 5). Therefore, it is not necessary to know the full state with the proposed control strategy. The fuzzy adaptive PD controller is designed for the suppression of angle α by the variations in the in-plane angle and the distance between the space tug and the debris; and control tension is considered a time-varying parameter added in the HSMC to realize hybrid control of the overall tethered system. Regarding the requirement above, in this paper, an improved HSMC is designed based on [27], it adds the control tension of α as a timevarying parameter to achieve simultaneous control of the swing angle and the distance between the space tug and the debris using tension, meanwhile suppress the oscillation of the angle of the debris. The specific process is as follows. As shown in Section 3.1, the states of the tethered system are defined as
T
T T] x att − 0 = [ωt0 Qt0 = [ rgrave vr−grave vs−grave ]T
⎨ x o−f T ⎪ x = x = [l l˙ e e e θe θ˙e ] ⎪ s−f ⎪ x o−min ≤ x o (t ) ≤ x o−max ⎪ x s−min ≤ x s (t ) ≤ x s−max ⎩
(9)
where x o − 0 , x s − 0 and x att − 0 represent the initial values of the orbital state parameters, the in-plane state parameters and the debris attitude parameters at the second stage, respectively; x o − f and x s − f represent the final value of the orbital state parameters and final value of the inT T plane state parameters, respectively; ωt0 and Qt0 are the initial angular velocity value and the initial attitude quaternion of the debris, respectively; x o − min and x o − max are the minimum and maximum constraint conditions of the orbital state parameters, respectively; x s − min and x s − max are the minimum and maximum constraint conditions of the in-plane state parameters, respectively. The minimum boundary conditions x o − min and x s − min ensure that the tethered system has enough kinetic energy to achieve de-orbit; and the maximum boundary conditions x o − max and x s − max avoid some bad situations (collisions or tangling of the tethered system, etc.) from occurring at the second stage of the drag de-orbit strategy. In addition, in order to avoid the sudden disappear of the thrust that causes the satellite to collide or the tug thrust so high that causes the tether to break at the second stage, the size of Fth should be constrained as follows:
0 ≤ Fmin ≤ Fth ≤ Fmax
x = [ x1 x2 x3 x 4 ]T = [l l˙ θ θ˙ ]T Then, the dynamic model of the system is as follows:
⎧ x˙1 = x2 ⎪ x˙2 = f (x) − l ⎨ x˙ 3 = x 4 ⎪ ⎩ x˙ 4 = fθ (x)
(10)
where Fmin and Fmax are the minimum and maximum constraints of the tug thrust. Meantime, in order to avoid the sudden loose and tightness of the tether, the tether tension Tp should be constrained as follows:
0 < Tmin ≤ Tp ≤ Tmax
Tα ¯ m
+ bsat(Tc ) (14)
where
(11)
b=−
where Tmin and Tmax are the minimum and maximum constraints of the tether tension. In addition, in order to avoid the sudden relaxation of the tether tension caused by a large change of thrust direction at the second stage, the in-plane thrust direction needs to be constrained relative to tether direction as follows:
γ − θ ≤ Δang
(13)
1 m ¯ 2
fl (x) = l (θ˙ + vs / r ) +
F (sin γ cos θ − cos γ sin θ) μl (3 sin2 θ − 1) + th m1 r3
+ dˆ1
3μ cos θ sin θ vs vr l˙ − 2 (θ˙ + vs / r ) + r3 r2 l Fth (sin γ sin θ + cos γ cos θ) F sin γ − − th + dˆ2 m1 l mr
fθ (x) = 2
(12)
where Δang represents the maximum angular deviation between the inplane thrust direction and the tether direction.
In the controller, Tα is the control tension of the fuzzy adaptive PD 242
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Fig. 5. The coordinated control strategy of tethered system.
T l¨d + kl e˙ l − fl (x) + α − bsat(Teq) + aλ − gΔT + k l˙ e l˙ = 0 m ¯
controller, and the disturbance produced by Tα is suppressed by the robustness of the HSMC controller when it is added to the control system, then the control tension of the HSMC controller Tc is produced. The control tension of angle α is considered a time-varying component of the dynamic model (As shown in Eq. (14)). The tension constraint is expressed as
⎧Tmax Tc ≥ Tmax Tmin < Tc < Tmax sat(Tc ) = Tc ⎨ T Tc ≤ Tmin min ⎩
Then, set g = b , sat(Teq) + ΔT = Teq , thus
Teq = b−1(l¨d + kl e˙ l − fl (x) +
s = σl s l˙ + σθ s θ˙
(15)
(16)
(21)
1 2 s 2
(22)
Thus,
where a is a positive coefficient, and g is a gain in saturation error ΔT = Tc − Tp . The states of sway motion are divided into two parts to conduct the HSMC, namely ( l, l˙ ) and ( θ , θ˙ ) . Each part consists of a fast loop and a slow loop. The first layer of sliding surfaces is designed as
T V˙ = ss˙ = s (σl s˙ l˙ + σθ s˙ θ˙ ) = sσl ⎡l¨d + kl e˙ l − fl (x) + α + aλ + k l˙ e l˙ ⎤ m ¯ ⎣ ⎦ ¨ − sσl b [sat(Teq + Tsw ) + ΔT ] + sσθ [θd + kθ e˙ θ − fθ (x) + k θ˙ e θ˙ ]
(23)
where sat(Teq + Tsw ) + ΔT = Teq + Tsw . We can rewrite Eq. (23) as follows according to Eq. (20) when let it conform to the exponential approach law.
⎧ slow − loop: t ⎪ sl = el + kl ∫0 el dτ ⎪ ⎪ t sθ = eθ + kθ ∫0 eθ dτ ⎪
V˙ = s { −σl bTsw + σθ [θ¨d + kθ e˙ θ − fθ (x) + k θ˙ e θ˙ ]} = −ks 2 − εs⋅sats (s ) ≤ 0 (24) where k and ε are positive coefficients and (17)
⎧1 ⎪s sats (s ) = ⎨0 ⎪− 1 ⎩
where el = ld − l , eθ = θd − θ , e l˙ = vc − l˙ − λ , e θ˙ = ωc − θ˙ is the defined error of each part and each loop; vc = l˙d + kl el and ωc = θ˙d + kθ eθ are virtual control variables, and ld , l˙d , θd and θ˙d are components of the optimal command x s − d introduced in Section 3.1. kl , kθ , k l˙ , k θ˙ are the given positive coefficients. We can write the overall control tension of the sliding mode controller as
Tc = Teq + Tsw
(20)
where σl and σθ are weight coefficients. Therefore, the Lyapunov function can be defined as follows:
V=
⎨ fast − loop: ⎪ t s l˙ = el˙ + k l˙ ∫0 e l˙ dτ ⎪ ⎪ t ⎪ s θ˙ = eθ˙ + k θ˙ ∫0 e θ˙ dτ ⎩
Tα + aλ + k l˙ e l˙) m ¯
We can design and express the second layer of sliding surfaces as
The following anti-saturation module is added to the control system to solve the problem caused by tension saturation according to Ref. [27].
λ˙ = −aλ + gΔT
(19)
s > Δ1 others − Δ2 < s < Δ2 s < − Δ1
(25)
Hence, it can infer the switching law Tsw according to Eq. (24) as
Tsw =
1 {ks + ε⋅sats (s ) + σθ [θ¨d + kθ e˙ θ − fθ (x) + k θ˙ eθ˙ ]} σl b
(26)
(18) Then, the overall control tension Tc is obtained using Eq. (18), Eq. (20) and Eq. (26). Based on Lyapunov's stability theory, it is observed that the asymptotical stability of the sliding system is ensured according to Eq. (24).
where Teq is the equivalent law to ensure the states of the system stay at surface level, and Tsw is the switching law which prompts the states to the desired surfaces. In order to obtain the equation of Teq , make s˙ l˙ = 0 ; then, 243
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and Qt0 in Table 2 are in stable ranges. The initial direction of tether is tangent to the circular orbit as also introduced in Section 2.4. The boundary conditions and constraint conditions of the state variables of the tethered system are shown in Table 3:
Table 2 The parameters of the tethered system. Parameter
Value
Parameter
Value
m1 − 0 m2
989.12 kg 3000 kg
ct pt
[2 0.8 − 0.8]T m
Ixx
3000 kg m2
ωt0
[4.94 1.00 − 1.20]T deg/s
Iyy
1500 kg m
2
Qt0
Izz
2000 kg m2
[0.51 0.85 0.02 − 0.15]T 300 s
kt
33 GPa
10 N/m
Isp
4.2. Simulation results Hence, the orbit parameters change of the tethered system at the two stages is obtained as follows. The orbit altitude, radial speed, tangential speed and the variation of the mass of the space tug caused by fuel consumption at the first stage are shown in Fig. 6, Fig. 7, Fig. 8, Fig. 9, respectively. According to the simulation results of the first stage, it can be seen that the orbit altitude of the tethered system increases steadily, and the maneuver node of the tethered system can be obtained as t1f = 13771.77s . Based on the first stage, the orbit altitude, radial velocity, tangential velocity, the variation of the mass of the space tug caused by fuel consumption, the variation of the thrust and the variation of γ at the second stage can be obtained and shown in Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15 respectively, and the maneuver node t1f is viewed as the initial state of the second stage. According to the simulation results of the second stage, it can be seen that the final orbit of the tethered system reaches the graveyard orbit, and the final radial velocity becomes zero. Thus, the orbital rounding is achieved finally. The entire de-orbit mission consumes about 110.12 kg of fuel. As described in Section 2.1, the thruster can be instantaneously oriented on the desired thrust direction and size. In order to further study the tracking effect of the controller on the optimization commands, we study the tracking results of the distance between the space tug and the debris and the in-plane angle next. The control commands of the in-plane angle and the distance between the space tug and the debris are obtained as follows. Fig. 16 represents the control commands of the distance between satellites, and Fig. 17 represents the control commands of in-plane angle. (The initial moment of the second stage is taken as the moment of t = 0 below). Based on the above commands, the stable attitude control of the tethered system is achieved by the hybrid tension control designed in Section3.2. Fig. 18 shows the tracking error of the control commands of the distance between the space tug and the debris, and it is in the range of −0.15 m to 0.15 m. Fig. 19 shows the tracking error of the control commands of the in-plane angle, and it is in the range of −0.04deg to 0.1deg. Thus, it achieves an excellent tracking accuracy. Fig. 20 shows the change of the attitude angle α , and it is in the range of 0–45° in the de-orbit process. Moreover, it is under 15° for a long time. The safety threshold under this bias is 60° [26]. Thus, the tethered system does not tangle during the entire de-orbit process.
Table 3 Boundary conditions and constraint conditions. Parameter
Value
[Tmin , Tmax ] [Fmin , Fmax ] [γmin , γmax ] x oh − 0
[2, 90] N [0, 60] N [-180, 270] deg
[42167102 m 2.32 m/s 3082.45 m/s]T
x o−f
[42514000 m 0 m/s 3061.98 m/s]T
x oh − min
[42167102 m 0 m/s 3050 m/s]T
x oh − max
[42514000 m 50 m/s 3100 m/s]T
x o − min
[42167102 m 0 m/s 3050 m/s]T
x o − max
[42514000 m 50 m/s 3100 m/s]T
xs − 0
[200 m 0 m/s 0 deg 0 deg/s]T
xs − f
[200m 0m/s 180deg 0deg/s]T
xs − min
[100 m − 1 m/s − 90 deg − 1.14 deg/s]
xs − max
[1500 m 2 m/s 180 deg 0.3 deg/s]T 90 deg
Δang
T
4. Numerical simulation 4.1. Simulation parameters Based on Section 2 and Section 3, a simulation is implemented to verify the effectiveness of the proposed de-orbit strategy. The objective of the simulation is to drag the debris in GEO to the graveyard orbit that is 350 km above the GEO. It is assumed that the space tug is a particle and its energy consumption is considered. The specific parameters of the tethered red system are shown in Table 2. First of all, there are some instructions for the initial conditions of the tethered system. In fact, the parameters in Table 2 are inherited from Ref. [25]. As introduced in Section 2.3, pt represents the attachment point bias of the tether and the debris. Moreover, as introduced in Section 2.4, the oscillations caused by pt has been stabled by the control method proposed in Ref. [25], so the initial state of the system has been stable. Therefore, ωt0
Fig. 6. The orbit altitude at the first stage. 244
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Fig. 7. The radial speed at the first stage.
Fig. 8. The tangential speed at the first stage.
Fig. 9. The variation of the mass of the space tug at the first stage.
safe de-orbit of the tethered system, and rise the orbit altitude of the tethered system to reach the graveyard orbit finally.
Fig. 21 shows the change of tether tension, and it is under 90 N in the de-orbit process. Such a tether strength requirement can be easily met by selecting suitable materials. In addition, the tether remains tensioned all the time during de-orbit process. However, the commands and tracking results of the distance between the space tug and the debris show that the distance is larger than 200 m. Therefore, the system will not collide, so it meets the safety requirements of the deorbit process. From the above simulation results, it can be seen that the designed horizontal de-orbit strategy and hybrid tension control can achieve the
5. Conclusion It will take a long time to drag space debris in GEO to the surface of the Earth, because its orbit altitude is very high. Therefore, for the debris in GEO, dragging them to the graveyard orbit is researched by many scholars. In this paper, a multi-stage horizontal drag de-orbit strategy is designed to achieve de-orbit of the debris in GEO. For this 245
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Fig. 10. The orbit altitude at the second stage.
Fig. 11. The radial speed at the second stage.
Fig. 12. The tangential speed at the second stage.
Fig. 13. The variation of the mass of the space tug at the second stage.
motion, the sway motion and the variation of the debris attitude is also considered in this paper, and the optimal commands based on the minimum of tangling risks are planned using Gauss pseudospectral method to solve this problem effectively. Then, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) is designed to achieve stable control of the tethered
strategy, the thruster that can be instantaneously oriented on the desired thrust direction is used at the two stages of de-orbit, and its direction is consistent with the tether at the first stage to avoid provoking the in-plane oscillation of the tethered system; yet it changes its size and direction at the second stage to achieve the orbital rounding at the second stage. Furthermore, the problem of decoupling of the orbit 246
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Fig. 14. The variation of the thrust at the second stage. Fig. 15. The variation of γ at the second stage.
Fig. 16. The control commands of the distance between the space tug and the debris.
Fig. 17. The control commands of the in-plane angle.
247
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Fig. 18. The tracking error of the control commands of the distance between the space tug and the debris.
Fig. 19. The tracking error of the control commands of the in-plane angle.
Fig. 20. The variation of the angle α at the second stage.
Acknowledgments
system in the de-orbit process. Finally, the numerical simulation is implemented and verifies that the proposed de-orbit design is efficacious. It can be seen that the deployment and the tension control are the key points of the tethered system. Thus, for the focus of the future work, the development mechanism and its control technology will be studied, and the corresponding ground experiments will be carried out to lay the foundation for future space application.
This research is supported by National Natural Science Foundation of China (61773028, 51375034) and Natural Science Foundation of Beijing (4172008).
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Fig. 21. The variation of the tether tension at the second stage.
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Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Optimal commands based multi-stage drag de-orbit design for a tethered system during large space debris removal
T
Zhongyi Chua,∗, Tao Weia, Tao Shena, Jingnan Dia, Jing Cuib, Jun Sunc a
School of Instrumental Science and Opto-electronics Engineering, Beihang University, 100191, China School of Mechanical Engineering and Applied Electronics, Beijing University of Technology, 101100, China c Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai Aerospace Control Technology Institute, Shanghai, 201109, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Space tethered system Large space debris removal De-orbit strategy Gauss pseudospectral method
There is a serious challenge to the safe operation of orbiting satellites as the number of space debris increases for its high risks and the possible crippling effects of collisions. Consequently, the active removal scenario of space debris has drawn wide attention in recent years, and a tethered system is considered to be a promising method for its large operating distance and low power consumption. However, the flexible tether of the system will bring the coupling of the orbit motion, the sway motion and the variation of large debris attitude, which brings about great danger in de-orbiting phase and a huge challenge in later control. Hence, aiming to de-orbit large space debris safely with a tethered system, a multi-stage horizontal drag de-orbit strategy that consists of two stages is designed. At the first stage, the orbit altitude is rose with a tug thrust whose direction is consistent with the tether, which does not provoke the in-plane oscillation of the tethered system. At the second stage, the orbit is circularized by changing the size and direction of the tug thrust. Especially, optimal commands based on the minimum of tangling risks are planned using Gauss pseudospectral method to avoid target tangling and achieve decoupling of the orbit motion, the sway motion and the variation of the target attitude. Then, the stable attitude control of the tethered system is achieved by designing a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) in the de-orbit process. Finally, numerical simulation is implemented to verify the effectiveness of the proposed de-orbit design.
1. Introduction There is a serious challenge to the safe operation of orbiting satellites as the number of space debris on the GEO (geostationary orbit) increases. Over the past 40 years, GEO orbital debris has continued to accumulate. Large debris (spent rocket stages, defunct satellites, etc.) is more prone to collisions, which can produce tens of thousands of pieces of new debris [1], and it makes the orbit danger further strengthened. Therefore, dragging the GEO debris, especially the large debris out of the orbit safely and economically is one of the most important issues to solve, which ensures the safety of the spacecraft operating in orbit [2]. Some studies have shown that the tethered/tethered-net system is a promising technology to capture and remove space debris for its large operating distance and low energy consumption [3]. Therefore, the tethered system has become a study hotspot in the past two decades. The formation missions and debris removal missions have all been considered as applications of tethered systems [4], some methodologies for deployment/retrieval optimization of tethered satellite systems have
∗
been presented in Refs. [5,6], and many enabling space debris capturing and removal methods have been proposed and several methods have been tested on ground or in parabolic flight experiments [7]. High-risk debris (such as large massive debris) has been proposed to be safely removed by the concept of active debris removal (ADR) [8–10], which achieves de-orbit of the debris captured by a tethered spacecraft via active thrust. The 7# satellites of JAXA (Japan Aerospace Exploration Agency) has been conducted, which is considered as a space robot verification experiment [11]. The project ROGER (Robotic Geostationary Orbit Restorer) developed by the ESA (European Space Agency) [12,13] considered to use a tethered net or mechanical claw to de-orbit a redundant GEO (geostationary orbit) satellite. And ESA also conducted a pre-assessment study for capturing a large space debris to produce preliminary system design in 2012 [14]. In addition, some influencing factors and stability control problems have also been considered by some scholars. For example, the chaos behavior of tethered systems was considered by Aslanov V S in their studies, and they found the conditions for the existence of the chaos and illustrated its impact
Corresponding author. E-mail addresses: [email protected] (Z. Chu), [email protected] (J. Cui), [email protected] (J. Sun).
https://doi.org/10.1016/j.actaastro.2018.12.038 Received 31 July 2018; Received in revised form 7 December 2018; Accepted 28 December 2018 Available online 03 January 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic diagram of the space tethered system.
system. For this strategy, the thruster installed on the space tug keeps its direction consistent with the tether to avoid the in-plane oscillation of the tethered system at the first stage, and its size and direction are changed to achieve the orbital rounding at the second stage. Meanwhile, optimal commands are planned using Gauss pseudospectral method to avoid target tangling and achieve the decoupling of the orbit motion, the sway motion and the variation of the target attitude. Then, stable attitude control of the tethered system is achieved by designing a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) in the de-orbit process. Finally, numerical simulation is implemented to verify the effectiveness of the proposed de-orbit design. This paper includes five main sections. In Section 2, a multi-stage horizontal drag de-orbit strategy which upgrades the orbit altitude at the first stage and lets orbit round at the second stage with the tug thrust by a tethered system is designed. Based on that, in Section 3, the optimal commands is designed using Gauss pseudospectral method to avoid target tangling. Then, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) is designed to achieve stability control of the tethered system. In Section 4, the numerical simulation is described. Finally, conclusions are summarized in Section 5.
on the behavior of the two-body tethered system on a circular orbit [15]. Some scholars investigated the dynamical stability and the eigenfrequencies of the tethered system under the action of the thrust [16]. Furthermore, a tension control law to stabilize the motions of a tethered space-tug (TST) system during its deorbiting process by regulating the tension in the tether is presented in Ref. [17]. Specifically, a tethered satellite system (TSS) is formed when debris is captured by a manipulator (such as a harpoon, a mechanical hand or a net) to implement de-orbiting of a passive, non-cooperative, possibly spinning debris. However, due to the altitude of the orbit, the removal method of debris from high orbit is different from low orbit. It takes a long time to drag debris from high orbit to the surface of the Earth; and because of the flexibility of the tether, it takes great challenges to stabilize the TSS and makes it more difficult for the control design. Therefore, it is generally considered to drag the debris to the graveyard orbit that is several hundred kilometers above the GEO [18,19], which involves the technology and theory of satellite orbit transfer. In the existing satellite orbit transfer mode, the generalized Hohman orbit transfer principle which realizes orbit transfer with two speed pulses is widely used [20], and some scholars extended the timescale separation to Hohmann orbital transfer to achieve optimal control of TST systems [21]. However, there are two problems with this principle for a flexible tethered system. On the one hand, the pulse of the start-stop phase will cause the tether to become loose and tight when the pulse thrust is larger, which makes it easy to tangle for the tethered system. On the other hand, it changes the altitude of the orbit by means of free flight of spacecraft during the time outside the pulse, and it is very easy to cause collisions between satellites. Some scholars have proposed using gravity gradient to stabilize the tethered system during de-orbit process [22]. However, the gravitational gradient in the high-orbital regions has only a slight effect and only plays a part in the long-tether system of a few kilometers long, yet the obvious flexible characteristics of the long tether bring more complicated problems for system modeling and control. Moreover, it does not consider the coupling problem of the orbit motion, the sway motion and the variation of large debris attitude of the tethered system in the process of transferring orbit, which might cause tangling of the system. In recent years, some scholars have studied the decoupling regarding the tethered system [23,24]. But they mainly considered the decoupling of the in-plane angle and out-plane angle of the tethered system, and did not consider the tangling risks. Regarding the issues above, and to meet the stable demand of drag de-orbit and achieve mission objective, the novelty of the work is as follows: First, the dynamic equations of a tethered system are formulated and the impact of the sway motion on tether tangling is analyzed. In addition, a multi-stage horizontal drag de-orbit strategy which rises the orbit altitude at the first stage and realizes the orbital rounding at the second stage is designed to de-orbit large space debris safely with a tethered
2. Dynamic equations and mission analysis of tethered satellite system during large space debris removal 2.1. Assumption and reference frames As shown in Fig. 1, the tethered system will be formed after the mission satellite completes the capture of the target using the space tether. It consists of the mission satellite, the target, and the flexible tether. After stability control, the system achieves de-orbit by the thrust applied by the thruster installed on the mission satellite. In this paper, large space debris is regarded as a rigid body, which refers to be the target. The space tug is considered to be a particle, which refers to be the mission satellite. A viscoelastic tether is used to connect the space tug and the debris, its mass is ignored because it is too small compared with the mass of the tug and the debris. We assume that the space tug has completed the capture of the debris, so the capture phase is not part of our work and therefore not addressed in this paper. The thruster installed on the space tug can be instantaneously oriented on the desired thrust direction and change its size to rise the orbit altitude and achieve the orbital rounding. Based on the assumptions mentioned above, the reference frames (shown in Fig. 1) used in this paper are introduced as follows: (1) R1 is the Earth-centred inertial reference frame, the origin of the ⎯⎯⎯⎯⎯⇀ frame coincides with the centroid of the earth Oe , Oe x e points to the 239
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⎯⎯⎯⎯⎯⇀ vernal equinox, Oe z e is perpendicular to the earth equatorial plane, ⎯⎯⎯⎯⇀ and Oe ye is determined afterwards using the right-hand rule. (2) R o is the local orbital coordinate frame, the origin of the frame is ⎯⎯⎯⎯⎯⇀ attached to centroid of the system O , Oe z 0 is consistent with the ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ vector OOe , Ox 0 is perpendicular to Oz 0 in orbital plane and lies ⎯⎯⎯⇀ behind the debris, and Oy0 is determined afterwards using the righthand rule. (3) RT is the body fixed frame of the debris, the origin of the frame ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ coincides with the debris centroid Ot ; Ot x t , Ot yt and Ot z t coincide with the three principal inertia axes respectively and conform to the right-hand rule.
is determined by
⎧Tp = εl0[kt (l − l 0) + ct l˙] ⎨ T tp = Tp e trope ⎩
where l 0 is the original length of the tether; e trope represents tether tension unit vector described in RT ; T tp represents the tether tension vector described in RT ; kt and ct are the stiffness and damping of the tether, respectively, and
1 l > l0 εl 0 = ⎧ ⎨ ⎩0 l ≤ l0
2.2.1. Orbital motion As shown in Fig. 1, The equations of the orbital motion of the tethered system described in R1 based on the Newtonian method are as follows:
v v
μ r2
⎨ v˙s = − rr s + ⎪ ⎪ m˙ 1 = − I Fthg sp 0 ⎩
2.3. Analysis of the impact of the sway motion on tether winding for the system's nonlinear and under-actuated characteristics In practice, because of the under-actuated characteristic, high nonlinearity and strong coupling of the tethered system, the tether tangling should be included. The angle α (shown in Fig. 1) is used to evaluate the tangling risks of the tethered system, and describe the relative attitude between the tether and the debris based on our previous work [26]. Therefore, the angle α is controlled to reduce the risks of tangling, and ensure the safety of the tethered system. It is defined as follows:
Fth cos γ m Fth sin γ m
+
(1)
where r is the orbital radius of the centroid of the system; vr is the ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ ⎯⎯⎯⎯⇀ velocity along − Oz 0 , and − Oz 0 is the negative vector of Oz 0 ; while vs ⎯⎯⎯⎯⇀ is the velocity along Ox 0 ; Fth is the value of the thrust supplied by the thruster on the space tug; and γ represents the direction angle of the de⎯⎯⎯⎯⇀ orbit force referring to − Oz 0 ; m1 is the mass of the tug containing the fuel, and m = m1 + m2 is the total mass of the system where m2 represents the mass of the debris. In addition, Isp is the specific impulse of the thruster; and μ and g0 are the geocentric gravitational constant and normal acceleration of gravity respectively.
T
t
⎛ p ⋅e rope ⎞ α = arccos ⎜ t pt ⎟ ⎠ ⎝
2.4. Mission analysis of the drag de-orbit strategy In order to meet the stable demand of drag de-orbit and achieve mission objective, a multi-stage horizontal drag de-orbit strategy is designed to de-orbit large space debris safely with a tethered system. The mission objective is to release the orbit resources by dragging the debris in GEO to graveyard orbit that is 350 km higher than GEO. The whole process of de-orbit consists of two stages (shown in Fig. 2 and Table 1). At the first stage, a tug thrust is used to rise the orbit
μl
⎧ l¨ = l (θ˙ + vs / r ) + 3 (3 sin2 θ − 1) r ⎪ ⎪ + Fth (sin γ cos θ − cos γ sin θ) − Tp + d1 ⎪ ¯ m1 m ⎨ θ¨ = 2 vs vr − 2 l˙ (θ˙ + v / r ) + 3μ cos θ sin θ s l ⎪ r2 r3 ⎪ Fth (sin γ sin θ + cos γ cos θ ) Fth sin γ − mr + d2 ⎪ − m1 l ⎩
(2)
where l represents the distance between the space tug and the debris; m ¯ = m1 m2 / m is the reduced mass of the system; T is tether tension; d1 and d2 are interferences of debris attitude; θ represents the in-plane angle of the system and its definition is as follows:
θ = arctan( −roz / rox )
(6)
where vector pt (shown in Fig. 1) is the vector pointing from the debris centroid to the tether attachment point described in RT , and it represents the attachment point bias of the tether and the debris. The attachment point is marked red in Fig. 1
2.2.2. Equations of relative position and attitude of tethered system Dynamic equations of the sway angle and the distance between the space tug and the debris are obtained as Equation (2) using the Lagrange method according to our previous work [25]. The initial state of the system has been stable through [25], besides, the system is in GEO, where the effect of the ambient force on the debris motion outside the orbital plane is relatively small. So the debris motion outside the orbital plane is of small magnitude all the time, we do not consider its influence on the system. The tether length of the system is short in this mission and the gravity in GEO is small, so the influence of gravity gradient torque is not considered, either. 2
(5)
In fact, in order to avoid the sudden loose and tightness of the tether, the expected value of εl0 is 1 in our research, which means that the tether is always tensioned.
2.2. Dynamic modeling of the tethered system
⎧ r˙ = vr vs2 ⎪ ⎪ v˙r = r −
(4)
(3)
where r0 is the vector pointing from the debris centroid to the tug centroid described in R o , roz and rox are its components in the z and x axes, respectively. Subsequently, in this paper, the tether is regarded as a visco-elastic tether, and its elasticity and damping are considered. The tension force
Fig. 2. The drag de-orbit process. 240
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Table 1 The multi-stage dragging de-orbit strategy design. Stage
Technical approach
Objective
The first stage The second stage
Using a tug thrust whose direction is consistent with the tether Changing the size and direction of the tug thrust
Rising the orbit altitude Achieving the orbital rounding
this stage is to design the size and direction of the required thrust, and based on that to achieve stable attitude control and avoid the tangling of the tethered system. 3. Optimal commands and controller design for the tethered system during large space debris removal Fig. 3. The first stage of the drag de-orbit strategy.
3.1. Optimal commands design for the multi-stage drag de-orbit strategy
altitude, and its direction is consistent with the tether so it does not provoke the in-plane oscillation of the tethered system. At the second stage, the orbit is circularized by changing the size and direction of the tug thrust. The detailed description of the drag de-orbit strategy is as follows:
In order to enable the tethered system to reach the graveyard orbit safely, and based on the introduction in Section 2.4, the orbital maneuvering position at the first stage needs to be designed. Moreover, the size and direction of required thrust at the second stage also needs to be planned.
2.4.1. The first stage of the drag de-orbit strategy As shown in Fig. 3, a constant tug thrust Fth is used to rise the orbit altitude at the first stage, and it is provided by the thruster installed on the space tug. The initial state of the system has been stable through our previous work in Ref. [25]. The literature [25] proposed a hybrid tension control method to stabilize tethered systems during large space debris removal, and stabled the oscillations caused by the attachment point bias. The initial direction of the tether is tangent to the circular orbit in the orbital plane. In the whole process of rising the orbit altitude, the direction of Fth is controlled to consist with the tether, so it does not provoke the in-plane oscillation of the tethered system, and the oscillations of the debris will be limited enough for the system to be stable. However, the use of the tug thrust makes the entire orbit not meet the Keplerian orbits' condition, and the orbital altitude of the tethered system still maintains a rise trend when it reaches the desired altitude. Regarding this issue, the orbit circulization is designed at the second stage, and the key design parameter of the first stage is to select the orbital maneuvering position.
3.1.1. The cost function of the two stages In this paper, the optimal control problem is to minimize the tangling risks and fuel consumption of the tethered system during large space debris removal, so as to avoid debris tangling and achieve decoupling of the orbit motion, the sway motion and the variation of the debris attitude. Therefore, the cost function of the multi-stage drag deorbit strategy is designed as:
∫t
Jdeo = Δm2f + 0.05
t2f
α⋅dt
1f
(7)
where Δm2f = m1 − 0 − m2f is the fuel consumption mass of the entire deorbit phase; m1 − 0 is the initial mass of the space tug at the first stage; m2f is the ultimate mass of the space tug at the second stage; t1f , t2f are the termination times of the two stages, respectively. 3.1.2. The boundary conditions and constraint conditions at the first stage As shown in section 2.4, the constant tug thrust Fth is controlled to consist with the tether at the first stage of the de-orbit strategy, and it does not provoke the in-plane oscillation of the tethered system at this stage. Thus the command design of the first stage only needs to consider the orbital dynamics of the tethered system. Therefore, the boundary conditions and constraint conditions of the state variables of the tethered system at the first stage are designed as:
2.4.2. The second stage of the drag de-orbit strategy As shown in Fig. 4, the final orbital maneuvering moment of the first stage is the initial moment of the second stage, and the orbit is circularized by changing the size and direction of Fth at the second stage. The coupling problem of the orbit motion, the sway motion and the variation of large debris attitude of the tethered system needs to be considered, where the orbit motion is the movement of the tethered system around the earth in space; the sway motion is the rotation of the tethered system around its centroid; the variation of the debris attitude is the rotation of the debris around its centroid. Therefore, the focus of
T ⎧ x oh − 0 = [r vr vs ] x oh − f = [r1f vr1f vs1f ]T ⎨ ⎩ x oh − min ≤ x oh(t )≤ x oh − max
(8)
where x oh − 0 and x oh − f are the initial and final values of the orbital parameters at the first stage, respectively; x oh − min and x oh − max are the
Fig. 4. The second stage of the drag de-orbit strategy. 241
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3.1.4. Optimal commands of the two stages Based on the boundary conditions and variable constraints of the two-stage of tethered system from Equation (7) to Equation (12) and the optimization objective in Equation (7). The optimal commands of the time corresponding to the maneuvering position t1f ; the size and direction of Fth at the second stage can be obtained based on the Gauss pseudospectral method. Meanwhile, the first stage orbital state parameter x oh − d ; the second stage orbital state parameter x o − d and the inplane attitude control commands x s − d of the tethered system can also be obtained based on the Gauss pseudospectral method. and x oh − d , x o − d , x s − d mean the time histories of the parameters during the de-orbit phase. The optimization process achieves the minimization of the cost function in Equation (7), which makes the angle α minimum throughout the de-orbit process so that the orbit motion and the sway motion have the least impact on the variation of the debris attitude. Thus, the decoupling of the orbit motion, the sway motion and the variation of the debris attitude is achieved.
minimum and maximum constraint conditions of the orbital state variables, respectively. The minimum constraint condition x oh − min ensures that the tethered system has enough kinetic energy to achieve deorbit; and the maximum constraint condition x oh − max avoids some bad situations (collisions or tangling of the tethered system, etc.) from occurring at the first stage of the drag de-orbit strategy. 3.1.3. The boundary conditions and constraint conditions at the second stage After finishing the first stage, the orbit circular will be made at the second stage by changing the size and direction of Fth . Because of the strong coupling of the orbit motion, the sway motion and the variation of the debris attitude, the size and direction of Fth and the in-plane relative motion state parameters need to be designed. Therefore, the boundary conditions and constraint conditions of state variables of the tethered system at the second stage are designed as:
⎧ ⎪ ⎪ ⎪ ⎪
3.2. Coordinated control design of the tethered system
x o−0 = [ r1f vr1f vs1f ]T T
x s−0 = [l 0 l˙ 0 θ0 θ˙0 ]
Based on the above optimal commands that achieve decoupling of the orbit motion, the sway motion and the variation of the debris attitude, the focus of the next step is to design the controller to track the obtained optimal commands. Because the full states of the passive debris are difficult to obtain and the system's nonlinearity is strong during the de-orbit process, in this paper, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) which can achieve effective control of the system like this is designed (shown in Fig. 5). Therefore, it is not necessary to know the full state with the proposed control strategy. The fuzzy adaptive PD controller is designed for the suppression of angle α by the variations in the in-plane angle and the distance between the space tug and the debris; and control tension is considered a time-varying parameter added in the HSMC to realize hybrid control of the overall tethered system. Regarding the requirement above, in this paper, an improved HSMC is designed based on [27], it adds the control tension of α as a timevarying parameter to achieve simultaneous control of the swing angle and the distance between the space tug and the debris using tension, meanwhile suppress the oscillation of the angle of the debris. The specific process is as follows. As shown in Section 3.1, the states of the tethered system are defined as
T
T T] x att − 0 = [ωt0 Qt0 = [ rgrave vr−grave vs−grave ]T
⎨ x o−f T ⎪ x = x = [l l˙ e e e θe θ˙e ] ⎪ s−f ⎪ x o−min ≤ x o (t ) ≤ x o−max ⎪ x s−min ≤ x s (t ) ≤ x s−max ⎩
(9)
where x o − 0 , x s − 0 and x att − 0 represent the initial values of the orbital state parameters, the in-plane state parameters and the debris attitude parameters at the second stage, respectively; x o − f and x s − f represent the final value of the orbital state parameters and final value of the inT T plane state parameters, respectively; ωt0 and Qt0 are the initial angular velocity value and the initial attitude quaternion of the debris, respectively; x o − min and x o − max are the minimum and maximum constraint conditions of the orbital state parameters, respectively; x s − min and x s − max are the minimum and maximum constraint conditions of the in-plane state parameters, respectively. The minimum boundary conditions x o − min and x s − min ensure that the tethered system has enough kinetic energy to achieve de-orbit; and the maximum boundary conditions x o − max and x s − max avoid some bad situations (collisions or tangling of the tethered system, etc.) from occurring at the second stage of the drag de-orbit strategy. In addition, in order to avoid the sudden disappear of the thrust that causes the satellite to collide or the tug thrust so high that causes the tether to break at the second stage, the size of Fth should be constrained as follows:
0 ≤ Fmin ≤ Fth ≤ Fmax
x = [ x1 x2 x3 x 4 ]T = [l l˙ θ θ˙ ]T Then, the dynamic model of the system is as follows:
⎧ x˙1 = x2 ⎪ x˙2 = f (x) − l ⎨ x˙ 3 = x 4 ⎪ ⎩ x˙ 4 = fθ (x)
(10)
where Fmin and Fmax are the minimum and maximum constraints of the tug thrust. Meantime, in order to avoid the sudden loose and tightness of the tether, the tether tension Tp should be constrained as follows:
0 < Tmin ≤ Tp ≤ Tmax
Tα ¯ m
+ bsat(Tc ) (14)
where
(11)
b=−
where Tmin and Tmax are the minimum and maximum constraints of the tether tension. In addition, in order to avoid the sudden relaxation of the tether tension caused by a large change of thrust direction at the second stage, the in-plane thrust direction needs to be constrained relative to tether direction as follows:
γ − θ ≤ Δang
(13)
1 m ¯ 2
fl (x) = l (θ˙ + vs / r ) +
F (sin γ cos θ − cos γ sin θ) μl (3 sin2 θ − 1) + th m1 r3
+ dˆ1
3μ cos θ sin θ vs vr l˙ − 2 (θ˙ + vs / r ) + r3 r2 l Fth (sin γ sin θ + cos γ cos θ) F sin γ − − th + dˆ2 m1 l mr
fθ (x) = 2
(12)
where Δang represents the maximum angular deviation between the inplane thrust direction and the tether direction.
In the controller, Tα is the control tension of the fuzzy adaptive PD 242
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Fig. 5. The coordinated control strategy of tethered system.
T l¨d + kl e˙ l − fl (x) + α − bsat(Teq) + aλ − gΔT + k l˙ e l˙ = 0 m ¯
controller, and the disturbance produced by Tα is suppressed by the robustness of the HSMC controller when it is added to the control system, then the control tension of the HSMC controller Tc is produced. The control tension of angle α is considered a time-varying component of the dynamic model (As shown in Eq. (14)). The tension constraint is expressed as
⎧Tmax Tc ≥ Tmax Tmin < Tc < Tmax sat(Tc ) = Tc ⎨ T Tc ≤ Tmin min ⎩
Then, set g = b , sat(Teq) + ΔT = Teq , thus
Teq = b−1(l¨d + kl e˙ l − fl (x) +
s = σl s l˙ + σθ s θ˙
(15)
(16)
(21)
1 2 s 2
(22)
Thus,
where a is a positive coefficient, and g is a gain in saturation error ΔT = Tc − Tp . The states of sway motion are divided into two parts to conduct the HSMC, namely ( l, l˙ ) and ( θ , θ˙ ) . Each part consists of a fast loop and a slow loop. The first layer of sliding surfaces is designed as
T V˙ = ss˙ = s (σl s˙ l˙ + σθ s˙ θ˙ ) = sσl ⎡l¨d + kl e˙ l − fl (x) + α + aλ + k l˙ e l˙ ⎤ m ¯ ⎣ ⎦ ¨ − sσl b [sat(Teq + Tsw ) + ΔT ] + sσθ [θd + kθ e˙ θ − fθ (x) + k θ˙ e θ˙ ]
(23)
where sat(Teq + Tsw ) + ΔT = Teq + Tsw . We can rewrite Eq. (23) as follows according to Eq. (20) when let it conform to the exponential approach law.
⎧ slow − loop: t ⎪ sl = el + kl ∫0 el dτ ⎪ ⎪ t sθ = eθ + kθ ∫0 eθ dτ ⎪
V˙ = s { −σl bTsw + σθ [θ¨d + kθ e˙ θ − fθ (x) + k θ˙ e θ˙ ]} = −ks 2 − εs⋅sats (s ) ≤ 0 (24) where k and ε are positive coefficients and (17)
⎧1 ⎪s sats (s ) = ⎨0 ⎪− 1 ⎩
where el = ld − l , eθ = θd − θ , e l˙ = vc − l˙ − λ , e θ˙ = ωc − θ˙ is the defined error of each part and each loop; vc = l˙d + kl el and ωc = θ˙d + kθ eθ are virtual control variables, and ld , l˙d , θd and θ˙d are components of the optimal command x s − d introduced in Section 3.1. kl , kθ , k l˙ , k θ˙ are the given positive coefficients. We can write the overall control tension of the sliding mode controller as
Tc = Teq + Tsw
(20)
where σl and σθ are weight coefficients. Therefore, the Lyapunov function can be defined as follows:
V=
⎨ fast − loop: ⎪ t s l˙ = el˙ + k l˙ ∫0 e l˙ dτ ⎪ ⎪ t ⎪ s θ˙ = eθ˙ + k θ˙ ∫0 e θ˙ dτ ⎩
Tα + aλ + k l˙ e l˙) m ¯
We can design and express the second layer of sliding surfaces as
The following anti-saturation module is added to the control system to solve the problem caused by tension saturation according to Ref. [27].
λ˙ = −aλ + gΔT
(19)
s > Δ1 others − Δ2 < s < Δ2 s < − Δ1
(25)
Hence, it can infer the switching law Tsw according to Eq. (24) as
Tsw =
1 {ks + ε⋅sats (s ) + σθ [θ¨d + kθ e˙ θ − fθ (x) + k θ˙ eθ˙ ]} σl b
(26)
(18) Then, the overall control tension Tc is obtained using Eq. (18), Eq. (20) and Eq. (26). Based on Lyapunov's stability theory, it is observed that the asymptotical stability of the sliding system is ensured according to Eq. (24).
where Teq is the equivalent law to ensure the states of the system stay at surface level, and Tsw is the switching law which prompts the states to the desired surfaces. In order to obtain the equation of Teq , make s˙ l˙ = 0 ; then, 243
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and Qt0 in Table 2 are in stable ranges. The initial direction of tether is tangent to the circular orbit as also introduced in Section 2.4. The boundary conditions and constraint conditions of the state variables of the tethered system are shown in Table 3:
Table 2 The parameters of the tethered system. Parameter
Value
Parameter
Value
m1 − 0 m2
989.12 kg 3000 kg
ct pt
[2 0.8 − 0.8]T m
Ixx
3000 kg m2
ωt0
[4.94 1.00 − 1.20]T deg/s
Iyy
1500 kg m
2
Qt0
Izz
2000 kg m2
[0.51 0.85 0.02 − 0.15]T 300 s
kt
33 GPa
10 N/m
Isp
4.2. Simulation results Hence, the orbit parameters change of the tethered system at the two stages is obtained as follows. The orbit altitude, radial speed, tangential speed and the variation of the mass of the space tug caused by fuel consumption at the first stage are shown in Fig. 6, Fig. 7, Fig. 8, Fig. 9, respectively. According to the simulation results of the first stage, it can be seen that the orbit altitude of the tethered system increases steadily, and the maneuver node of the tethered system can be obtained as t1f = 13771.77s . Based on the first stage, the orbit altitude, radial velocity, tangential velocity, the variation of the mass of the space tug caused by fuel consumption, the variation of the thrust and the variation of γ at the second stage can be obtained and shown in Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15 respectively, and the maneuver node t1f is viewed as the initial state of the second stage. According to the simulation results of the second stage, it can be seen that the final orbit of the tethered system reaches the graveyard orbit, and the final radial velocity becomes zero. Thus, the orbital rounding is achieved finally. The entire de-orbit mission consumes about 110.12 kg of fuel. As described in Section 2.1, the thruster can be instantaneously oriented on the desired thrust direction and size. In order to further study the tracking effect of the controller on the optimization commands, we study the tracking results of the distance between the space tug and the debris and the in-plane angle next. The control commands of the in-plane angle and the distance between the space tug and the debris are obtained as follows. Fig. 16 represents the control commands of the distance between satellites, and Fig. 17 represents the control commands of in-plane angle. (The initial moment of the second stage is taken as the moment of t = 0 below). Based on the above commands, the stable attitude control of the tethered system is achieved by the hybrid tension control designed in Section3.2. Fig. 18 shows the tracking error of the control commands of the distance between the space tug and the debris, and it is in the range of −0.15 m to 0.15 m. Fig. 19 shows the tracking error of the control commands of the in-plane angle, and it is in the range of −0.04deg to 0.1deg. Thus, it achieves an excellent tracking accuracy. Fig. 20 shows the change of the attitude angle α , and it is in the range of 0–45° in the de-orbit process. Moreover, it is under 15° for a long time. The safety threshold under this bias is 60° [26]. Thus, the tethered system does not tangle during the entire de-orbit process.
Table 3 Boundary conditions and constraint conditions. Parameter
Value
[Tmin , Tmax ] [Fmin , Fmax ] [γmin , γmax ] x oh − 0
[2, 90] N [0, 60] N [-180, 270] deg
[42167102 m 2.32 m/s 3082.45 m/s]T
x o−f
[42514000 m 0 m/s 3061.98 m/s]T
x oh − min
[42167102 m 0 m/s 3050 m/s]T
x oh − max
[42514000 m 50 m/s 3100 m/s]T
x o − min
[42167102 m 0 m/s 3050 m/s]T
x o − max
[42514000 m 50 m/s 3100 m/s]T
xs − 0
[200 m 0 m/s 0 deg 0 deg/s]T
xs − f
[200m 0m/s 180deg 0deg/s]T
xs − min
[100 m − 1 m/s − 90 deg − 1.14 deg/s]
xs − max
[1500 m 2 m/s 180 deg 0.3 deg/s]T 90 deg
Δang
T
4. Numerical simulation 4.1. Simulation parameters Based on Section 2 and Section 3, a simulation is implemented to verify the effectiveness of the proposed de-orbit strategy. The objective of the simulation is to drag the debris in GEO to the graveyard orbit that is 350 km above the GEO. It is assumed that the space tug is a particle and its energy consumption is considered. The specific parameters of the tethered red system are shown in Table 2. First of all, there are some instructions for the initial conditions of the tethered system. In fact, the parameters in Table 2 are inherited from Ref. [25]. As introduced in Section 2.3, pt represents the attachment point bias of the tether and the debris. Moreover, as introduced in Section 2.4, the oscillations caused by pt has been stabled by the control method proposed in Ref. [25], so the initial state of the system has been stable. Therefore, ωt0
Fig. 6. The orbit altitude at the first stage. 244
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Fig. 7. The radial speed at the first stage.
Fig. 8. The tangential speed at the first stage.
Fig. 9. The variation of the mass of the space tug at the first stage.
safe de-orbit of the tethered system, and rise the orbit altitude of the tethered system to reach the graveyard orbit finally.
Fig. 21 shows the change of tether tension, and it is under 90 N in the de-orbit process. Such a tether strength requirement can be easily met by selecting suitable materials. In addition, the tether remains tensioned all the time during de-orbit process. However, the commands and tracking results of the distance between the space tug and the debris show that the distance is larger than 200 m. Therefore, the system will not collide, so it meets the safety requirements of the deorbit process. From the above simulation results, it can be seen that the designed horizontal de-orbit strategy and hybrid tension control can achieve the
5. Conclusion It will take a long time to drag space debris in GEO to the surface of the Earth, because its orbit altitude is very high. Therefore, for the debris in GEO, dragging them to the graveyard orbit is researched by many scholars. In this paper, a multi-stage horizontal drag de-orbit strategy is designed to achieve de-orbit of the debris in GEO. For this 245
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Fig. 10. The orbit altitude at the second stage.
Fig. 11. The radial speed at the second stage.
Fig. 12. The tangential speed at the second stage.
Fig. 13. The variation of the mass of the space tug at the second stage.
motion, the sway motion and the variation of the debris attitude is also considered in this paper, and the optimal commands based on the minimum of tangling risks are planned using Gauss pseudospectral method to solve this problem effectively. Then, a hybrid fuzzy adaptive proportion differentiation (PD) with hierarchical sliding-mode controller (HSMC) is designed to achieve stable control of the tethered
strategy, the thruster that can be instantaneously oriented on the desired thrust direction is used at the two stages of de-orbit, and its direction is consistent with the tether at the first stage to avoid provoking the in-plane oscillation of the tethered system; yet it changes its size and direction at the second stage to achieve the orbital rounding at the second stage. Furthermore, the problem of decoupling of the orbit 246
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Fig. 14. The variation of the thrust at the second stage. Fig. 15. The variation of γ at the second stage.
Fig. 16. The control commands of the distance between the space tug and the debris.
Fig. 17. The control commands of the in-plane angle.
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Fig. 18. The tracking error of the control commands of the distance between the space tug and the debris.
Fig. 19. The tracking error of the control commands of the in-plane angle.
Fig. 20. The variation of the angle α at the second stage.
Acknowledgments
system in the de-orbit process. Finally, the numerical simulation is implemented and verifies that the proposed de-orbit design is efficacious. It can be seen that the deployment and the tension control are the key points of the tethered system. Thus, for the focus of the future work, the development mechanism and its control technology will be studied, and the corresponding ground experiments will be carried out to lay the foundation for future space application.
This research is supported by National Natural Science Foundation of China (61773028, 51375034) and Natural Science Foundation of Beijing (4172008).
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Fig. 21. The variation of the tether tension at the second stage.
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