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Nutation damping and spin orientation control of tethered space debris
Acta Astronautica 160 (2019) 683–693
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Nutation damping and spin orientation control of tethered space debris Xin Sun, Rui Zhong
T
∗
School of Astronautics, Beihang University, Beijing, 100191, China
ABSTRACT
As a preliminary of an active debris removal process, the attitude motion control of the debris need to be carried out to ensure security of following operations. The tethered space system is one of the most attractive methods for active debris removal missions. However, the tether tension becomes the only input to manipulate the attitude motion of the debris, causing a highly underactuated problem. Fortunately, the fully attitude stabilization of the debris is not a prerequisite for a secure deorbit operation. As long as the nutation of the debris is not so violent that makes the tether winds, the towing process can be applied continually, which is a significant advantage of the tether method. In this paper, a switched tension control law is presented to damp the nutation of the debris. Only two constant tension values are needed in the control method. Under the assumption that the tether direction is fixed in the inertia frame, the bounded stability of the control algorithm is proved based on the Barbalat's Lemma, and an asymptotic stability can be achieved for an asymmetrical debris symmetrically captured. Numerical simulations with tether libration considered are carried out to show the capability of the control method.
1. Introduction The number of artificial space objects are growing rapidly, occupying the limited orbit resources [1]. After a satellite finishes its tasks, runs out its fuel or encounters fatal errors, it will lose efficacy and turn into space debris. Unless applying some pertinent measures, those expired satellites will not drop into the atmosphere in a very long period of time. Meanwhile, longtime perturbations, such as the Earth's oblateness and atmospheric drag, will affect their orbits, increasing the risk of collisions with other operating or non-operating satellites and generation of more debris. The situation will become even worse if the chain reaction of collision occurs. Therefore, to maintain a relatively secure space environment, dependable scenarios for active debris removal (ADR) need to be presented and demonstrated [2]. The tethered system is considered to be one of the most feasible ways to accomplish space debris removal missions, which benefits from its remote manipulation and higher security [3]. For the capture of the debris, a popular method is to use a flexible net to wrap it up, forming a tethered space net (TSN) system. In the deorbit process of such a TSN system, there are two main concerns that need to be solved, one is the libration control of the tether, and the other one is the attitude motion control of the debris. For the libration control of the tether, when the tether is relatively long and the size of the debris and the net is negligible, the TSN system is simplified to a dumbbell model, and the dynamics and tether libration control of such a system is sufficiently studied by researchers [4–11]. For the attitude dynamics of the debris, Aslanov derived the equations of the motion based on the Lagrange formalism and checked the critical modes of tether entanglement by
∗
numerical simulations [12]. Chu studied the effect of the attachment point bias caused by capture error during its removal process [13]. Hovell validated the postcapture dynamics of the subtether configuration experimentally [14]. Qi studied the attitude dynamics of doubletethered debris [15]. Generally, in the forepart of an ADR process, it is necessary to manipulate the attitude of the debris to avoid tether entanglement. Yudintsev proposed a modified yo-yo mechanism to despin the debris [16]; Huang adopted a tethered space robot (TSR) with a gripper and use the thrusters on the TSR to realize stabilization of a tumbling robot-target combination [17]. Wang also used a TSR system and employed a specified tethered manipulator to change the position of the tether attachment point [18]. Zhang used the offset of tether attachment point to suppress the spin of debris [19]. All those detumbling methods need additional mechanisms to achieve the attitude control of the debris, for the reason that the tether can only provide a sole positive tension along the tether direction. However, for space debris removal missions, the complete three-axis despun of the debris is not always necessary, and the aim is to avoid the winding and alleviate the effect on the tether libration caused by the irregular attitude motion of the debris. Thus the key problem is to restrain the motion of the tether attachment point perpendicular to the tether direction. This paper focuses on the circumstance that a noncooperative space debris on the geosynchronous orbit (GEO) captured by a space tether system with uncertain attitude motion and unknown inertia information. There are no additional mechanisms assisting with the attitude control of the debris, and the tether tension is the only input to manipulate. The transverse motion of the tether attachment point is restrained by the nutation damping of the debris. The control
Corresponding author. E-mail addresses: [email protected] (X. Sun), [email protected] (R. Zhong).
https://doi.org/10.1016/j.actaastro.2019.03.019 Received 17 January 2019; Received in revised form 2 March 2019; Accepted 5 March 2019 Available online 12 March 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
Acta Astronautica 160 (2019) 683–693
X. Sun and R. Zhong
¨+ =
o
+ 2( +
o)
m1 (m2 + m t /2) L mm L
tan
1.5µR
3
sin 2
Q m L2 cos2
(2)
¨ + 2 m1 (m2 + mt /2) L + 1 [3µR mm L 2
3
sin2
+(
o
+ )2 ]sin 2
=
Q m L2
(3)
where o is the orbital angular velocity of the space tether system, µ is the geocentric gravitational constant, R is the distance from the center of the Earth to the COM of the space tether system, T is the tension of the tether and L is the length of the tether. The masses of the main satellite, the debris and the tether are m1, m2 and mt respectively, and m = m1 + m2 + mt is the total mass of the space tether system. The mass of the tether mt equals cL , where is the density and c is the m, equaling to cross sectional area. The terms (m1 + 0.5mt )(m2 + 0.5mt ) 1 (m2 + mt ) m1 m m ¯ , and , equaling to , are defined to t m 6 m simplify the expressions of equations. QL , Q and Q in the right hand terms are the corresponding generalized forces of the thrust on the main satellite and can be derived according to the principle of virtual work
QL = Fig. 1. Illustration of different coordinate systems.
Q =
+ µR 3 (3 sin2
cos2
+ )2 cos2 +
1)] =
T + QL m ¯
2m2 + mt LPby 2m
d b2x dt d b2y B dt d C dtb2z
A
A typical space tether system consists of a main satellite with propulsion devices, a relatively long tether and an expired spacecraft called debris captured by a net or a gripper, as illustrated in Fig. 1. To describe the libration motion of the tether, it is first assumed that the main satellite and the debris are both point masses, and the attitude motion of the main satellite and the debris have little effect on the tether libration. This assumption is reasonable due to the sizable tether length compared with the size of the main satellite and the debris. Thus the libration motion of the tether can be decoupled from the attitude motion of the two end bodies. Another assumption is that the tether is light-weight and the tension is uniform along the tether. The origin of the orbit frame ox o yo z o is located at the center of mass (COM) of the space tether system, with its z o axis pointing from the center of the Earth to the COM of the space tether system and yo axis perpendicular to the orbit plane. The body frame of the space tether system oxb yb z b with the axis along the tether, is obtained by the sequential y-z rotations from the orbit frame and the corresponding in-plane and out-of-plane libration angles are and respectively. Then the dynamic equations of the tether libration can be derived using Lagrange's Equation and are given as in Ref. [20]. o
(5) (6)
where Pbx , Pby and Pbz are the three components of the thrust vector P in the body frame of the space tether system oxb yb z b . For the attitude motion of the debris, firstly, the body frame ob2 xb2 yb2 z b2 is defined with its origin at the COM of the debris as in Fig. 1. The xb2 , yb2 and z b2 axes align the principal axes of inertia, and the corresponding moment of inertia are A , B and C respectively. Then the three Euler angles , and are defined by the sequential z-y-x rotations from the orbit frame to the debris body frame. For the attitude dynamic equation, the Euler equation for rigid bodies is adopted as
2. Dynamic formulation of the tether libration and debris attitude motion
m L [( m ¯
(4)
2m2 + mt LPbz 2m
Q =
methodology presented in this paper does not need any inertia information or the measure of its attitude motion. Another advantage of this control methodology is the application of the switched system, only two constant tension values are needed and the switching criterion is simple. The bounded stability of the control law is proved based on the Barbalat's Lemma, with an asymptotic stable case studied based on the attitude dynamics analysis of a tethered rigid body. Moreover, the ability that only use the tether tension to achieve three-axis spin counteraction is explored. Some numerical simulations are carried out to demonstrate the effectiveness of this control method.
(2m1 m) mt L2 L¨ + m1 (m2 + mt ) L
2m2 + mt Pbx 2m
+ (C
B)
b2y
b2z
= Mb2x
+ (A
C)
b2z
b2x
= Mb2y
+ (B
A)
b2x
b2y
= Mb2z
(7)
For simplicity, the capture is assumed rigid. The right hand terms are the moments generated by the tether tension along the three axes of the debris body frame. Furthermore, the capture is named symmetric along xb2 axis, when the node linking the tether and the net or gripper (tether attachment point) is on the xb2 axis, otherwise asymmetric. The term Mb2x becomes zero when the debris is captured symmetrically because the tension does not generate any moment along the xb2 axis. 3. Nutation damping and spin orientation control methodology 3.1. Nutation damping methodology design To simplify the analysis of the attitude motion of the debris, we first check the case that the orientation of the tether is fixed in the inertial frame. The angular rate of the orbit motion of the system is relatively small and can be neglected in GEOs. Thus the fixed tether orientation is a reasonable assumption when the tether is relatively long and the libration angles of the tether is under effective control. Then we set up another two reference frames os xs ys z s and ob2 xb2 yb2 z b2 , of which origins coincide with the COM of the debris, as is shown in Fig. 2. The directions of xs , ys and z s axes coincide with the , yb and z b axes of the space tether system body frame. The direction of the xb2 axis points from the COM of the debris to the tether attachment point and the ob2 xb2 yb2 z b2 frame can be obtained by sequential z-y
2
(1) 684
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Fig. 3. Illustration of another observation method.
tension is that T¯ < 2 mbx max , where Pbx max is the maximum thrust that the main satellite can provide along the direction. In real applications, the switching criterion based on the observation of in Eq. (11) can be equivalently replaced by r , the time derivative of the distance from the tether attachment point to the line connecting the tether connection point on the main satellite and the COM of debris, as is illustrated in Fig. 3. The observation of r is considered as applicable because only the sign is concerned. The measurement of r can be achieved by setting a mark on the tether attachment point in advance, which can be observed through the camera on the main satellite. Meanwhile, the position of the COM of the debris can be approximated by the centroid of the debris in the field of view, thus the sign of r can be directly measured in the field of view. m P
Fig. 2. Illustration of different coordinate systems of which origins are on the COM of the debris.
rotations from the debris body frame ob2 xb2 yb2 z b2 . The spatial angle between the xb2 and xb2 axes is defined as , a constant angle as the capture is assumed rigid. To describe the relative attitude from the reference frame os xs ys z s to ob2 xb2 yb2 z b2 , the sequential x-y-x rotations and the corresponding Euler angles , and are used. In the reference frame os xs ys z s , the tension of the tether always aligns xs axis. To avoid tether entanglement, the tether should be manipulated to a proper direction beforehand to limit in the range 0, 2 . Although the os xs ys z s frame is non-inertial, it is not rotational due to the assumption of fixed tether orientation. Therefore, the inertia force always acts on the COM of the debris, having no effect on its attitude motion. According to the Theorem of Translation of A Force, the action point of the tension can be moved to the COM of the debris as well, cancelling out the effect of the inertia force, with an extra moment left. The value of the moment generated by the tension can be derived as
3.2. Stability analysis of the nutation damping methodology As is described in Eq. (10), for a constant tension, the total mechanical energy is constant, that is
E=
V = =
T cos
+
1 2
b2 I2
b2
= const
T¯ ,
d
(
b2 I2
T cos ¯
b2
=0
(12)
+
1 b2 I2 b2 2
)
dt
+ T¯ cos
T cos ¯
+
(
T¯ cos
+
1 b2 I2 b2 2
)
dt d ( T cos ¯
+ T¯ cos dt
)
(T¯
T ) sin (13) ¯ It is found in Eq. (13) that V < 0 given > 0 . Therefore, the time derivative of V is always negative semidefinite using the control law presented in Eq. (11). Noticing that is continuous, the switching of the index function from zero to Eq. (13) following switching criterion only happens at = 0 . Thus the uniformly continuity of V is guaranteed. T all the time. Thus by Moreover, V has its lower bound because V ¯ applying the Barbalat's Lemma [21], we can conclude that =
(10)
where Ep equals T cos . In fact, as both the direction and value of the tension are fixed in the frame os xs ys z s , the tether tension plays a same role as a conservative force. Thus Ep and E can be interpreted as the potential energy of the tension and the total mechanical energy respectively. Based on the discussions above, the nutation damping control methodology is presented as
T=
d
=
The minus in the right term appears because the directions of Mt and d are opposite. Furthermore, if the value of the tension is constant, integrate both sides we can obtain that
E = Ep + Er =
1 2
dt
(9)
Mt d
+
Assume an index function V = T cos + b2 I2 b2 , thus V = 0 ¯ when T = T according to Eq. (12). If the tether tension is switched to a ¯ larger value T¯ at any instant,
where is the distance from the COM of the debris to the tether attachment point, and the moment always aligns xb2 × xs direction. Thus the change of the rotational energy Er of the debris can be obtained according to the work-energy principle
dEr =
T cos
1 2
(8)
Mt = T sin
d
lim V = 0
t
(14)
V = 0 corresponds to the case of T = T , thus 0 according to Eq. ¯ (11). As the range of is 0 , which indicates that the lower 2
bound of is zero, and is also uniformly continuous in time. Again, according to the Barbalat's Lemma, we can conclude that
>0
T, 0 (11) ¯ where T¯ and T are two different positive designed values of tension and ¯ satisfy T¯ > T > 0 . It should be noted that the designed value of the ¯ tension cannot be very large, because the propulsion force might be insufficient to separate the main satellite from the debris and they will collide with each other. A conservative estimate of the maximum
lim = 0
t
(15)
Therefore, will reduce gradually and approach a constant value finally. Moreover, as the spatial angle between xb2 and xb2 is a constant , the spatial angle between xs and xb2 is bounded, which indicates the nutation angle is bounded as well. Thus the bounded stability of the 685
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X. Sun and R. Zhong
control method is proved. Let us check a special case that the rigid debris is symmetrically captured by the space tether system along its xb2 axis, and A is the 0, 2 . maximum moment of inertia, with For the symmetrically-captured case, the tether attachment point is on the xb2 axis, thus the ob2 xb2 yb2 z b2 frame coincides with the ob2 xb2 yb2 z b2 frame. The rotation matrix from the frame os xs ys z s to the debris body frame ob2 xb2 yb2 z b2 can be derived as
cos
sin
sin
sin
cos
sin
cos
sin
cos
cos
Ab2s =
cos
cos
sin
sin
sin
cos
cos
cos
cos
=
b2
=
b2y b2z
cos
+
sin
cos
sin
cos
sin sin
[(A
[(A
cos cos
sin
sin
sin b2x
sin
B ) b2x b2y + T sin C
sin
( +
cos
C cos
sin
g
]
BC
( cos cos
2
) + B ( cos
+ C ( sin
cos
+
sin
)
sin
+
sin
sin
sin
)
)
sin
)
sin2 + C
T sin
f
=
sin
(20)
A
C B
+1
Hxsf = A cos 2Erf = A
f
b2x
2
b2x
+ (sin2
+ (sin2
)
cos2
+
2
A
B C
sin2
sin2 f
f
sin2
+ cos2
+ A cos2
f
f
) Erf
f
=
(g ) sin
f
2 Ag 2 sin2 f
(g sin2
f
f
f)
+ A2 g cos2
+ cos2
f
Erf
2 Ag Hxsf
f)
2 Hxsf
(28)
f
(g ) cos
h f
BC
)
+1
b 2x
(g )
(30)
cos2
sin
(T + (A
C)
2
cos
f)
sin2
B sin f [T + (A
B)
2
cos
f]
f
C
(31)
As A is the maximum moment of inertia, the right hand in Eq. (31) is always negative semidefinite and ¨ = 0 only if f = 0 . For the case 0 and it contradicts with the condition f = const . 0, that f 0 when Using this reduction to absurdity, we can conclude that 0 . Thus T varies with continuously, which is against our f = const 0 cannot become an equilicontrol methodology, indicating that f brium point. Then we can conclude that f = 0 is the only equilibrium point when B C . Therefore, the nutation elimination of the debris can be achieved using the control methodology presented in Eq. (11).
(22) (23)
For simplicity, let
g = sin2
cos
f
A cos
T sin
¨ =
(21)
)
+ cos2
2
f
b2x
+ cos2
+ cos2
cos
f
f
which indicates that T is a function of g and h . (1) For the case that B = C , g and h are constants and no longer a function of , which indicates that for specific Hxsf and Erf , there exists a constant T that satisfies Eq. (29). In fact, according to the Euler equation for rigid bodies, Eq. (7), b2x is constant when B = C , which means that the angular momentum Hb2x along its xb2 axis is constant as well. If the xb2 axis finally coincides with the xs axis, Hb2xf = Hb2x = Hxs = Hxsf . This is not a common case because Hb2x Hxs for most debris after capture. Thus, the value of will approach a positive value instead of zero. (2) For the case that B C , g and h are continuous functions of , and the tension T = T ( ) is also a continuous function of , which means T is a constant only if is a constant, that is, = 0 . Assuming that = 0 and substituting T = const > 0 , = 0 , = f = const and 0, into the expression of ¨ , Eq. (18) becomes
2
2
sin
f
=
T = T (g , h
According to Eq. (15), as the time t approaches infinity, will approach a constant value, which indicates that the tension no longer does work and there is no energy exchange between Ep and Er . Thus Er will approach a constant value as well. Generally speaking, the system under designed control tends to the moment, when Er = Erf = const , = f = const , = 0 and ¨ = 0 . Substitute them Hxs = Hxsf = const , into Eqs. (18)-(20), we get
cos2 B
(g sin2
2 Ag Hxsf
Erf
f
When sin f = 0 , that is, f = 0 , Eq. (29) becomes an identity and is independent of the tension T . It indicates that f = 0 is a possible equilibrium point of the control law. 0 , the Assume that there exists a non-zero equilibrium point, f expression of T can be written as
Because the direction of the tension is supposed to be fixed and always aligns the xs axis, the tension generates no moment in the xs direction. Thus the angular momentum in the xs direction is constant. Moreover, the expression of Er , the rotational energy of the debris, can be derived as
cos
+ A2 g cos2
(29)
(19)
2Er = A ( +
f
f
From the expressions of and b2x , it can be found that only the term g is a function of Euler angle and other terms are constant. For simplicity, Eq. (26) and Eq. (27) can be written as = (g ) and b2x = b2x (g ) . Substitute them into Eq. (21), we can obtain that
(18)
( sin
g sin2
2 (g sin2
(17)
]
) + B sin sin
2 Ag 2 sin2
f
According to Eqs. (26) and (27), the existence of the solutions to and b2x requires that
The expression of the debris angular momentum in the xs direction Hxs can be derived as
Hxs = A cos
can be derived as
(27)
sin
sin
cos
Hxsf A cos2
cos
sin
C ) b2x b2z + T cos B
f
b2x
(26)
According to Euler equation for rigid body, Eq. (7), we can obtain the second time derivative of
¨=
± sin
and
f
=
sin
Then the relationship between the angular velocity of the debris in its body frame and the three Euler angular rates can be written as
+
Hxsf g
sin2
(25)
C )cos2
+ C (A
Using Eq. (22) and Eq. (23),
(16)
b2x
B )sin2
h = B (A
sin
+ cos
cos
and
(24) 686
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Above analysis proves that the debris' nutation angle results in the equilibrium f = 0 for most symmetrical capture cases (B C ), and the xb2 axis of the debris will finally coincide with the tether direction. Although a symmetrical capture is difficult in application, it can be safely concluded that the debris’ nutation will reduce using the proposed control methodology and the nutation angle will be small if a near symmetrical capture is achieved. This will be further validated in the numerical simulation part below. Moreover, spin orientation control of the debris is possible. By firstly changing the direction of the tether through an orbital maneuver of the main satellite, and then applying the control law in Eq. (11), the debris will finally spin around the tether, reaching the desired spin orientation.
4. Results and discussion To check the effectiveness of the control method presented in Section III, numerical simulations are carried out in this section. Both symmetrical and asymmetrical capture cases are taken into consideration. It should be pointed out that the tether libration equations described by Eqs. (1)–(3) are oversimplified for the controller design. In order to test the efficiency of the controller, the dynamic model used in the simulation no longer treats the debris as a point mass and takes the effects of the attitude motion of the debris on the tether length and libration into consideration, which is more accurate. The dynamic formulation can be derived as in Ref. [12]. The main concern using the simplified dynamic model in the controller design is that the moment of inertia and attitude information of the debris are included in the accurate model. A control law based on the accurate model will introduce those hardly measurable terms. Whereas one of the key advantages of the nutation damping method presented in this paper is the avoidance of those terms. The mass and geometric parameters of the space tether system are listed in Table 1. The initial states of the system including the tether libration and the attitude motion of the debris are listed in Table 2.
3.3. Tether libration control For the tether libration control, a simple PD controller is applied, according to Eqs. (1)–(3), and the expressions for Ql , Q and Q are (2m1 m ) mm ¯ t L2 m1 (m2 + mt ) L
m L [(
+ µR 3 (3 sin2
cos2
QL =
+T
Q =
om
m ¯ [KDL (L
L2 cos2
1.5µR 3m L2 cos2
+ )2 cos2 +
o
2
1)]
Lr ) + KPL (L
(32)
Lr )] m1 (m2 + mt / 2) L mm L
L2 cos2
+ 2( +
o) m
sin 2
m L2 cos2
[KD (
r)
4.1. Symmetrical capture case
tan
+ KP (
For the symmetrical capture case, the position of the tether attachment point in the debris body frame is (5 m, 0 m, 0 m). In the first simulation, the higher tension T¯ is chosen to be 30 N and the lower tension T is chosen to be 20 N. The desired tether length Lr is 100 m, ¯ and the desired tether in-plane and out-of-plane libration angles r and r are both zero. Meanwhile, the parameters for the PD controller in Eqs. (32)-(34) are all chosen to be 0.005 in order to limit the required propulsion force. The simulation duration is chosen to be 3000 s and the results are shown in Fig. 4. As is illustrated in Fig. 4, the tether length and libration angles are well stabilized in about 2000 s. That is because the attitude motion of the debris indeed has insignificant effects on the tether motion and the PD controller has its inherent anti-interference ability. Moreover, the
r )]
(33)
Q = 2 m LL
m1 (m2 + mt / 2) mm
1
3
m L2 [KD (
r)
+ 2 [3µR
sin2
+(
+ KP (
o
+ ) 2 ] m L2 sin 2 r )]
(34) where KDL > 0 , KPL > 0 , KD > 0 , KP > 0 , KD > 0 and KP > 0 are adjustable coefficients for the PD controller, and Lr , Lr , r , r , r and r are the desired states. Stated thus, the tension control law in Eq. (11) focuses on the nutation damping of the debris and makes the xb2 axis of the debris approach the tether direction, while the thrust provided by the main satellite is used to change the tether direction and stabilize the libration of the tether according to Eqs. (32)-(34).
Table 1 Inertia and geometric parameters of the system.
3.4. Further discussions on the nutation damping methodology A further application of this control methodology can be explored by checking the angular momentum of the debris. As is discussed above, both the directions of its angular momentum and xb2 axis will finally coincide with the tether direction, and the magnitude of the angular momentum will be reduced to the same value of Hxs0 when asymmetrical debris is captured symmetrically. Thus after the nutation of the debris is eliminated, we can manipulate the tether to another orientation using the thrust provided by the main satellite and repeat the process of the nutation damping to achieve further reduction of its angular momentum. Assume that the acute angle between the current tether direction and the next desired direction is 1, then the final angular momentum of the debris after the second nutation damping process can be derived as
Hxs1 = Hxs0 cos
As 0 < cos 1 < 1, the magnitude of the debris’ angular momentum decreases. Moreover, if we proceed with the procedure mentioned above, the angular momentum will decrease unceasingly, that is
Hxsn = Hxs0 cos
1
cos
2
cos
n
Values
Mass of the main satellite (m1 ) Mass of the debris (m2 ) Density of the tether ( ) Cross sectional area of the tether (c ) The x b2 axis moment of inertia ( A ) The yb2 axis moment of inertia (B ) The z b2 axis moment of inertia (C )
1000 kg 2000 kg 1440 kg/m3 5 mm2 4000 kg m2 3000 kg m2 2000 kg m2
Table 2 Initial states of the system.
(35)
1
Parameters
(36)
Then the magnitude of the debris angular momentum will approach zero, which indicates that a three-axis spin counteraction can be achieved using this method after infinite times of tether direction alteration. 687
States
Initial values
Semi-major axis Eccentricity True anomaly Length of the tether Tether in-plane angle Tether out-of-plane angle Angular velocity of the debris around x b2 axis Angular velocity of the debris around yb2 axis Angular velocity of the debris around z b2 axis Euler angle Euler angle Euler angle
42314 km 1 × 10−3 0 deg 80 m −10 deg −10 deg 25 deg/s 10 deg/s 10 deg/s 0 deg 0 deg 0 deg
Acta Astronautica 160 (2019) 683–693
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Fig. 4. Illustrations of the controlled libration motion, propulsion and attitude motion of the debris.
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Fig. 5. Illustrations of the controlled motion with changing the desired tether direction repeatedly.
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Fig. 6. Illustrations of the controlled libration motion, propulsion and attitude motion of the debris.
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Fig. 7. Illustrations of the controlled motion with changing the desired tether direction repeatedly.
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X. Sun and R. Zhong
chosen parameters for the PD controller can make the three components of the propulsion force in the body frame of the space tether system all below 150 N. Meanwhile, the semimajor axis and eccentricity of the system fluctuate in the beginning and then increase steadily because of the incipient fluctuant propulsion force for the tether libration control. The results in Fig. 4 also reveals the effects of the nutation damping method presented in Eq. (11). As the coefficients in the PD controller are chosen very small to limit the propulsion force, relatively long time is needed to stabilize the tether libration compared with the nutation damping process. The effect of the nutation damping law is to make the debris spin around the tether direction, thus unless the libration of the tether is stabilized, the debris attitude angles and will not stop fluctuating. Therefore, the transverse angular velocities b2y and b2z are well eliminated in 2000 s as well, in the wake of the libration stabilization process, and the Euler angles and approach zero, which indicates the xb2 axis of the debris finally coincides with the tether direction , verifying the asymptotic stability of the nutation damping method from the aspect of numerical results. The following simulation is carried out to test the ability of the control method in the application of three-axis spin counteraction mentioned in the end of Section III. The parameters of the space tether system and the initial states are the same with the former simulation, which are shown in Tables 1–2 The desired tether length L is still 100 m, and the desired tether out-of-plane libration angle is zero as well, whereas the desired tether in-plane libration angle changes every 2000 s between 0° and 20° to dissipate the angular momentum of the debris including component along the xb2 direction, and the simulation duration is chosen to be 10000 s. Fig. 5 shows the results of the simulation. Fig. 5 shows that the PD controller performs well in the frequent tether attitude maneuver procedure. Comparing the results of with , and with , it can be observed that the spin axis of the debris tracks the tether direction throughout the simulation. Therefore, to achieve the control of the spin orientation of the debris, we only need to manipulate the direction of the tether and apply the nutation damping method in Eq. (11). Besides, as the tether attitude maneuver procedure proceeds, the spin velocity b2x drops stepwise, following Eq. (36) in the end of Section III. This demonstrated that the proposed method of decreasing the debris’ angular momentum by alternating the tether directions successively with nutation damping is feasible.
nutation causing no tether entanglement is acceptable. It should be noted that the tether tension tend to flutter as the debris approaches the steady state. A feasible way to avoid tension flutter is to keep a constant tension when the tension variation frequency exceeds a threshold value. The next simulation is homologous with the second simulation in the symmetrical capture case to test the capability of the control method to retard the spin rate in the asymmetrical case. The parameters are all the same with the former simulation, and the desired tether inplane libration angle changes every 2000 s between 0° and 20°. Fig. 7 shows that although the transverse angular velocities cannot be completely eliminated, it is still effective to diminish the axial angular momentum. 5. Conclusion In this paper, the attitude dynamic characteristics of a space debris under a constant tether tension is analyzed, and a switching control method using only two constant tensions is presented, which can achieve both the nutation damping and the spin orientation control of a non-cooperative space debris. The bounded stability of the control law is proved based on the work-energy principle and the Barbalat's Lemma under a fixed tension direction assumption. For an asymmetrical debris which is symmetrically captured, the debris under proposed controller tends to spin around the tether direction with constant rate. As the transverse motion of the tether attachment point is well restrained, the risk of entanglement of the tether can be alleviated. The simulation results show the effectiveness of this control method in real applications when the tether libration is taken into consideration. Moreover, the results indicate that the nutation damping and the tether libration control can be carried out simultaneously when the libration is not violent. The ability to achieve three-axis despun of the debris by changing the direction of the tether repeatedly is revealed by both theoretical analysis and simulation results. This control method does not require knowing the moment of inertia, neither accurate attitude measurement of the debris, making it highly applicable in real missions. Acknowledgments This study is supported by the National Natural Science Foundation of China (Grant no. 11772023) and the Academic Excellence Foundation of BUAA for PhD students.
4.2. Asymmetrical capture case
References
For a more common case, that is, the asymmetrical capture case, the position of the tether attachment point in the debris body frame is chosen to be (5 m, 1 m, 1 m), which is no longer on the xb2 axis, but other than that the rest parameters are the same as those in the symmetrical capture case. Fig. 6 shows the results of the control methodology. The tether length is pretty well stabilized at 100 m in 2000 s, which appears to be the same with the symmetrical capture case in Fig. 4. However, due to the fact that a rigid body cannot spin around a nonprincipal axis of inertia, the nutation of the debris cannot be completely eliminated. The attitude motion of the debris tends to a combination of nutation and procession caused by the tether tension. Thus the tether attachment point swings continually, and the tether libration angles can no longer be well stabilized at zero asymptotically using the PD controller presented in Eqs. (33) and (34) consequently. Instead, both the in-plane and out-of-plane angles of the tether is restricted in a small region of zero. The debris attitude motion reaches a steady state after 1500 s, with the transverse angular velocities b2y and b2z approach a positive constant instead of zero as in the former case. The amplitudes of attitude angles and are finally restricted under 10°, demonstrating the stability of the control methodology from the aspect of numerical simulation. In real ADR applications by means of space tethers, a complete nutation elimination is not always required, and a bounded
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Nutation damping and spin orientation control of tethered space debris Xin Sun, Rui Zhong
T
∗
School of Astronautics, Beihang University, Beijing, 100191, China
ABSTRACT
As a preliminary of an active debris removal process, the attitude motion control of the debris need to be carried out to ensure security of following operations. The tethered space system is one of the most attractive methods for active debris removal missions. However, the tether tension becomes the only input to manipulate the attitude motion of the debris, causing a highly underactuated problem. Fortunately, the fully attitude stabilization of the debris is not a prerequisite for a secure deorbit operation. As long as the nutation of the debris is not so violent that makes the tether winds, the towing process can be applied continually, which is a significant advantage of the tether method. In this paper, a switched tension control law is presented to damp the nutation of the debris. Only two constant tension values are needed in the control method. Under the assumption that the tether direction is fixed in the inertia frame, the bounded stability of the control algorithm is proved based on the Barbalat's Lemma, and an asymptotic stability can be achieved for an asymmetrical debris symmetrically captured. Numerical simulations with tether libration considered are carried out to show the capability of the control method.
1. Introduction The number of artificial space objects are growing rapidly, occupying the limited orbit resources [1]. After a satellite finishes its tasks, runs out its fuel or encounters fatal errors, it will lose efficacy and turn into space debris. Unless applying some pertinent measures, those expired satellites will not drop into the atmosphere in a very long period of time. Meanwhile, longtime perturbations, such as the Earth's oblateness and atmospheric drag, will affect their orbits, increasing the risk of collisions with other operating or non-operating satellites and generation of more debris. The situation will become even worse if the chain reaction of collision occurs. Therefore, to maintain a relatively secure space environment, dependable scenarios for active debris removal (ADR) need to be presented and demonstrated [2]. The tethered system is considered to be one of the most feasible ways to accomplish space debris removal missions, which benefits from its remote manipulation and higher security [3]. For the capture of the debris, a popular method is to use a flexible net to wrap it up, forming a tethered space net (TSN) system. In the deorbit process of such a TSN system, there are two main concerns that need to be solved, one is the libration control of the tether, and the other one is the attitude motion control of the debris. For the libration control of the tether, when the tether is relatively long and the size of the debris and the net is negligible, the TSN system is simplified to a dumbbell model, and the dynamics and tether libration control of such a system is sufficiently studied by researchers [4–11]. For the attitude dynamics of the debris, Aslanov derived the equations of the motion based on the Lagrange formalism and checked the critical modes of tether entanglement by
∗
numerical simulations [12]. Chu studied the effect of the attachment point bias caused by capture error during its removal process [13]. Hovell validated the postcapture dynamics of the subtether configuration experimentally [14]. Qi studied the attitude dynamics of doubletethered debris [15]. Generally, in the forepart of an ADR process, it is necessary to manipulate the attitude of the debris to avoid tether entanglement. Yudintsev proposed a modified yo-yo mechanism to despin the debris [16]; Huang adopted a tethered space robot (TSR) with a gripper and use the thrusters on the TSR to realize stabilization of a tumbling robot-target combination [17]. Wang also used a TSR system and employed a specified tethered manipulator to change the position of the tether attachment point [18]. Zhang used the offset of tether attachment point to suppress the spin of debris [19]. All those detumbling methods need additional mechanisms to achieve the attitude control of the debris, for the reason that the tether can only provide a sole positive tension along the tether direction. However, for space debris removal missions, the complete three-axis despun of the debris is not always necessary, and the aim is to avoid the winding and alleviate the effect on the tether libration caused by the irregular attitude motion of the debris. Thus the key problem is to restrain the motion of the tether attachment point perpendicular to the tether direction. This paper focuses on the circumstance that a noncooperative space debris on the geosynchronous orbit (GEO) captured by a space tether system with uncertain attitude motion and unknown inertia information. There are no additional mechanisms assisting with the attitude control of the debris, and the tether tension is the only input to manipulate. The transverse motion of the tether attachment point is restrained by the nutation damping of the debris. The control
Corresponding author. E-mail addresses: [email protected] (X. Sun), [email protected] (R. Zhong).
https://doi.org/10.1016/j.actaastro.2019.03.019 Received 17 January 2019; Received in revised form 2 March 2019; Accepted 5 March 2019 Available online 12 March 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
Acta Astronautica 160 (2019) 683–693
X. Sun and R. Zhong
¨+ =
o
+ 2( +
o)
m1 (m2 + m t /2) L mm L
tan
1.5µR
3
sin 2
Q m L2 cos2
(2)
¨ + 2 m1 (m2 + mt /2) L + 1 [3µR mm L 2
3
sin2
+(
o
+ )2 ]sin 2
=
Q m L2
(3)
where o is the orbital angular velocity of the space tether system, µ is the geocentric gravitational constant, R is the distance from the center of the Earth to the COM of the space tether system, T is the tension of the tether and L is the length of the tether. The masses of the main satellite, the debris and the tether are m1, m2 and mt respectively, and m = m1 + m2 + mt is the total mass of the space tether system. The mass of the tether mt equals cL , where is the density and c is the m, equaling to cross sectional area. The terms (m1 + 0.5mt )(m2 + 0.5mt ) 1 (m2 + mt ) m1 m m ¯ , and , equaling to , are defined to t m 6 m simplify the expressions of equations. QL , Q and Q in the right hand terms are the corresponding generalized forces of the thrust on the main satellite and can be derived according to the principle of virtual work
QL = Fig. 1. Illustration of different coordinate systems.
Q =
+ µR 3 (3 sin2
cos2
+ )2 cos2 +
1)] =
T + QL m ¯
2m2 + mt LPby 2m
d b2x dt d b2y B dt d C dtb2z
A
A typical space tether system consists of a main satellite with propulsion devices, a relatively long tether and an expired spacecraft called debris captured by a net or a gripper, as illustrated in Fig. 1. To describe the libration motion of the tether, it is first assumed that the main satellite and the debris are both point masses, and the attitude motion of the main satellite and the debris have little effect on the tether libration. This assumption is reasonable due to the sizable tether length compared with the size of the main satellite and the debris. Thus the libration motion of the tether can be decoupled from the attitude motion of the two end bodies. Another assumption is that the tether is light-weight and the tension is uniform along the tether. The origin of the orbit frame ox o yo z o is located at the center of mass (COM) of the space tether system, with its z o axis pointing from the center of the Earth to the COM of the space tether system and yo axis perpendicular to the orbit plane. The body frame of the space tether system oxb yb z b with the axis along the tether, is obtained by the sequential y-z rotations from the orbit frame and the corresponding in-plane and out-of-plane libration angles are and respectively. Then the dynamic equations of the tether libration can be derived using Lagrange's Equation and are given as in Ref. [20]. o
(5) (6)
where Pbx , Pby and Pbz are the three components of the thrust vector P in the body frame of the space tether system oxb yb z b . For the attitude motion of the debris, firstly, the body frame ob2 xb2 yb2 z b2 is defined with its origin at the COM of the debris as in Fig. 1. The xb2 , yb2 and z b2 axes align the principal axes of inertia, and the corresponding moment of inertia are A , B and C respectively. Then the three Euler angles , and are defined by the sequential z-y-x rotations from the orbit frame to the debris body frame. For the attitude dynamic equation, the Euler equation for rigid bodies is adopted as
2. Dynamic formulation of the tether libration and debris attitude motion
m L [( m ¯
(4)
2m2 + mt LPbz 2m
Q =
methodology presented in this paper does not need any inertia information or the measure of its attitude motion. Another advantage of this control methodology is the application of the switched system, only two constant tension values are needed and the switching criterion is simple. The bounded stability of the control law is proved based on the Barbalat's Lemma, with an asymptotic stable case studied based on the attitude dynamics analysis of a tethered rigid body. Moreover, the ability that only use the tether tension to achieve three-axis spin counteraction is explored. Some numerical simulations are carried out to demonstrate the effectiveness of this control method.
(2m1 m) mt L2 L¨ + m1 (m2 + mt ) L
2m2 + mt Pbx 2m
+ (C
B)
b2y
b2z
= Mb2x
+ (A
C)
b2z
b2x
= Mb2y
+ (B
A)
b2x
b2y
= Mb2z
(7)
For simplicity, the capture is assumed rigid. The right hand terms are the moments generated by the tether tension along the three axes of the debris body frame. Furthermore, the capture is named symmetric along xb2 axis, when the node linking the tether and the net or gripper (tether attachment point) is on the xb2 axis, otherwise asymmetric. The term Mb2x becomes zero when the debris is captured symmetrically because the tension does not generate any moment along the xb2 axis. 3. Nutation damping and spin orientation control methodology 3.1. Nutation damping methodology design To simplify the analysis of the attitude motion of the debris, we first check the case that the orientation of the tether is fixed in the inertial frame. The angular rate of the orbit motion of the system is relatively small and can be neglected in GEOs. Thus the fixed tether orientation is a reasonable assumption when the tether is relatively long and the libration angles of the tether is under effective control. Then we set up another two reference frames os xs ys z s and ob2 xb2 yb2 z b2 , of which origins coincide with the COM of the debris, as is shown in Fig. 2. The directions of xs , ys and z s axes coincide with the , yb and z b axes of the space tether system body frame. The direction of the xb2 axis points from the COM of the debris to the tether attachment point and the ob2 xb2 yb2 z b2 frame can be obtained by sequential z-y
2
(1) 684
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X. Sun and R. Zhong
Fig. 3. Illustration of another observation method.
tension is that T¯ < 2 mbx max , where Pbx max is the maximum thrust that the main satellite can provide along the direction. In real applications, the switching criterion based on the observation of in Eq. (11) can be equivalently replaced by r , the time derivative of the distance from the tether attachment point to the line connecting the tether connection point on the main satellite and the COM of debris, as is illustrated in Fig. 3. The observation of r is considered as applicable because only the sign is concerned. The measurement of r can be achieved by setting a mark on the tether attachment point in advance, which can be observed through the camera on the main satellite. Meanwhile, the position of the COM of the debris can be approximated by the centroid of the debris in the field of view, thus the sign of r can be directly measured in the field of view. m P
Fig. 2. Illustration of different coordinate systems of which origins are on the COM of the debris.
rotations from the debris body frame ob2 xb2 yb2 z b2 . The spatial angle between the xb2 and xb2 axes is defined as , a constant angle as the capture is assumed rigid. To describe the relative attitude from the reference frame os xs ys z s to ob2 xb2 yb2 z b2 , the sequential x-y-x rotations and the corresponding Euler angles , and are used. In the reference frame os xs ys z s , the tension of the tether always aligns xs axis. To avoid tether entanglement, the tether should be manipulated to a proper direction beforehand to limit in the range 0, 2 . Although the os xs ys z s frame is non-inertial, it is not rotational due to the assumption of fixed tether orientation. Therefore, the inertia force always acts on the COM of the debris, having no effect on its attitude motion. According to the Theorem of Translation of A Force, the action point of the tension can be moved to the COM of the debris as well, cancelling out the effect of the inertia force, with an extra moment left. The value of the moment generated by the tension can be derived as
3.2. Stability analysis of the nutation damping methodology As is described in Eq. (10), for a constant tension, the total mechanical energy is constant, that is
E=
V = =
T cos
+
1 2
b2 I2
b2
= const
T¯ ,
d
(
b2 I2
T cos ¯
b2
=0
(12)
+
1 b2 I2 b2 2
)
dt
+ T¯ cos
T cos ¯
+
(
T¯ cos
+
1 b2 I2 b2 2
)
dt d ( T cos ¯
+ T¯ cos dt
)
(T¯
T ) sin (13) ¯ It is found in Eq. (13) that V < 0 given > 0 . Therefore, the time derivative of V is always negative semidefinite using the control law presented in Eq. (11). Noticing that is continuous, the switching of the index function from zero to Eq. (13) following switching criterion only happens at = 0 . Thus the uniformly continuity of V is guaranteed. T all the time. Thus by Moreover, V has its lower bound because V ¯ applying the Barbalat's Lemma [21], we can conclude that =
(10)
where Ep equals T cos . In fact, as both the direction and value of the tension are fixed in the frame os xs ys z s , the tether tension plays a same role as a conservative force. Thus Ep and E can be interpreted as the potential energy of the tension and the total mechanical energy respectively. Based on the discussions above, the nutation damping control methodology is presented as
T=
d
=
The minus in the right term appears because the directions of Mt and d are opposite. Furthermore, if the value of the tension is constant, integrate both sides we can obtain that
E = Ep + Er =
1 2
dt
(9)
Mt d
+
Assume an index function V = T cos + b2 I2 b2 , thus V = 0 ¯ when T = T according to Eq. (12). If the tether tension is switched to a ¯ larger value T¯ at any instant,
where is the distance from the COM of the debris to the tether attachment point, and the moment always aligns xb2 × xs direction. Thus the change of the rotational energy Er of the debris can be obtained according to the work-energy principle
dEr =
T cos
1 2
(8)
Mt = T sin
d
lim V = 0
t
(14)
V = 0 corresponds to the case of T = T , thus 0 according to Eq. ¯ (11). As the range of is 0 , which indicates that the lower 2
bound of is zero, and is also uniformly continuous in time. Again, according to the Barbalat's Lemma, we can conclude that
>0
T, 0 (11) ¯ where T¯ and T are two different positive designed values of tension and ¯ satisfy T¯ > T > 0 . It should be noted that the designed value of the ¯ tension cannot be very large, because the propulsion force might be insufficient to separate the main satellite from the debris and they will collide with each other. A conservative estimate of the maximum
lim = 0
t
(15)
Therefore, will reduce gradually and approach a constant value finally. Moreover, as the spatial angle between xb2 and xb2 is a constant , the spatial angle between xs and xb2 is bounded, which indicates the nutation angle is bounded as well. Thus the bounded stability of the 685
Acta Astronautica 160 (2019) 683–693
X. Sun and R. Zhong
control method is proved. Let us check a special case that the rigid debris is symmetrically captured by the space tether system along its xb2 axis, and A is the 0, 2 . maximum moment of inertia, with For the symmetrically-captured case, the tether attachment point is on the xb2 axis, thus the ob2 xb2 yb2 z b2 frame coincides with the ob2 xb2 yb2 z b2 frame. The rotation matrix from the frame os xs ys z s to the debris body frame ob2 xb2 yb2 z b2 can be derived as
cos
sin
sin
sin
cos
sin
cos
sin
cos
cos
Ab2s =
cos
cos
sin
sin
sin
cos
cos
cos
cos
=
b2
=
b2y b2z
cos
+
sin
cos
sin
cos
sin sin
[(A
[(A
cos cos
sin
sin
sin b2x
sin
B ) b2x b2y + T sin C
sin
( +
cos
C cos
sin
g
]
BC
( cos cos
2
) + B ( cos
+ C ( sin
cos
+
sin
)
sin
+
sin
sin
sin
)
)
sin
)
sin2 + C
T sin
f
=
sin
(20)
A
C B
+1
Hxsf = A cos 2Erf = A
f
b2x
2
b2x
+ (sin2
+ (sin2
)
cos2
+
2
A
B C
sin2
sin2 f
f
sin2
+ cos2
+ A cos2
f
f
) Erf
f
=
(g ) sin
f
2 Ag 2 sin2 f
(g sin2
f
f
f)
+ A2 g cos2
+ cos2
f
Erf
2 Ag Hxsf
f)
2 Hxsf
(28)
f
(g ) cos
h f
BC
)
+1
b 2x
(g )
(30)
cos2
sin
(T + (A
C)
2
cos
f)
sin2
B sin f [T + (A
B)
2
cos
f]
f
C
(31)
As A is the maximum moment of inertia, the right hand in Eq. (31) is always negative semidefinite and ¨ = 0 only if f = 0 . For the case 0 and it contradicts with the condition f = const . 0, that f 0 when Using this reduction to absurdity, we can conclude that 0 . Thus T varies with continuously, which is against our f = const 0 cannot become an equilicontrol methodology, indicating that f brium point. Then we can conclude that f = 0 is the only equilibrium point when B C . Therefore, the nutation elimination of the debris can be achieved using the control methodology presented in Eq. (11).
(22) (23)
For simplicity, let
g = sin2
cos
f
A cos
T sin
¨ =
(21)
)
+ cos2
2
f
b2x
+ cos2
+ cos2
cos
f
f
which indicates that T is a function of g and h . (1) For the case that B = C , g and h are constants and no longer a function of , which indicates that for specific Hxsf and Erf , there exists a constant T that satisfies Eq. (29). In fact, according to the Euler equation for rigid bodies, Eq. (7), b2x is constant when B = C , which means that the angular momentum Hb2x along its xb2 axis is constant as well. If the xb2 axis finally coincides with the xs axis, Hb2xf = Hb2x = Hxs = Hxsf . This is not a common case because Hb2x Hxs for most debris after capture. Thus, the value of will approach a positive value instead of zero. (2) For the case that B C , g and h are continuous functions of , and the tension T = T ( ) is also a continuous function of , which means T is a constant only if is a constant, that is, = 0 . Assuming that = 0 and substituting T = const > 0 , = 0 , = f = const and 0, into the expression of ¨ , Eq. (18) becomes
2
2
sin
f
=
T = T (g , h
According to Eq. (15), as the time t approaches infinity, will approach a constant value, which indicates that the tension no longer does work and there is no energy exchange between Ep and Er . Thus Er will approach a constant value as well. Generally speaking, the system under designed control tends to the moment, when Er = Erf = const , = f = const , = 0 and ¨ = 0 . Substitute them Hxs = Hxsf = const , into Eqs. (18)-(20), we get
cos2 B
(g sin2
2 Ag Hxsf
Erf
f
When sin f = 0 , that is, f = 0 , Eq. (29) becomes an identity and is independent of the tension T . It indicates that f = 0 is a possible equilibrium point of the control law. 0 , the Assume that there exists a non-zero equilibrium point, f expression of T can be written as
Because the direction of the tension is supposed to be fixed and always aligns the xs axis, the tension generates no moment in the xs direction. Thus the angular momentum in the xs direction is constant. Moreover, the expression of Er , the rotational energy of the debris, can be derived as
cos
+ A2 g cos2
(29)
(19)
2Er = A ( +
f
f
From the expressions of and b2x , it can be found that only the term g is a function of Euler angle and other terms are constant. For simplicity, Eq. (26) and Eq. (27) can be written as = (g ) and b2x = b2x (g ) . Substitute them into Eq. (21), we can obtain that
(18)
( sin
g sin2
2 (g sin2
(17)
]
) + B sin sin
2 Ag 2 sin2
f
According to Eqs. (26) and (27), the existence of the solutions to and b2x requires that
The expression of the debris angular momentum in the xs direction Hxs can be derived as
Hxs = A cos
can be derived as
(27)
sin
sin
cos
Hxsf A cos2
cos
sin
C ) b2x b2z + T cos B
f
b2x
(26)
According to Euler equation for rigid body, Eq. (7), we can obtain the second time derivative of
¨=
± sin
and
f
=
sin
Then the relationship between the angular velocity of the debris in its body frame and the three Euler angular rates can be written as
+
Hxsf g
sin2
(25)
C )cos2
+ C (A
Using Eq. (22) and Eq. (23),
(16)
b2x
B )sin2
h = B (A
sin
+ cos
cos
and
(24) 686
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Above analysis proves that the debris' nutation angle results in the equilibrium f = 0 for most symmetrical capture cases (B C ), and the xb2 axis of the debris will finally coincide with the tether direction. Although a symmetrical capture is difficult in application, it can be safely concluded that the debris’ nutation will reduce using the proposed control methodology and the nutation angle will be small if a near symmetrical capture is achieved. This will be further validated in the numerical simulation part below. Moreover, spin orientation control of the debris is possible. By firstly changing the direction of the tether through an orbital maneuver of the main satellite, and then applying the control law in Eq. (11), the debris will finally spin around the tether, reaching the desired spin orientation.
4. Results and discussion To check the effectiveness of the control method presented in Section III, numerical simulations are carried out in this section. Both symmetrical and asymmetrical capture cases are taken into consideration. It should be pointed out that the tether libration equations described by Eqs. (1)–(3) are oversimplified for the controller design. In order to test the efficiency of the controller, the dynamic model used in the simulation no longer treats the debris as a point mass and takes the effects of the attitude motion of the debris on the tether length and libration into consideration, which is more accurate. The dynamic formulation can be derived as in Ref. [12]. The main concern using the simplified dynamic model in the controller design is that the moment of inertia and attitude information of the debris are included in the accurate model. A control law based on the accurate model will introduce those hardly measurable terms. Whereas one of the key advantages of the nutation damping method presented in this paper is the avoidance of those terms. The mass and geometric parameters of the space tether system are listed in Table 1. The initial states of the system including the tether libration and the attitude motion of the debris are listed in Table 2.
3.3. Tether libration control For the tether libration control, a simple PD controller is applied, according to Eqs. (1)–(3), and the expressions for Ql , Q and Q are (2m1 m ) mm ¯ t L2 m1 (m2 + mt ) L
m L [(
+ µR 3 (3 sin2
cos2
QL =
+T
Q =
om
m ¯ [KDL (L
L2 cos2
1.5µR 3m L2 cos2
+ )2 cos2 +
o
2
1)]
Lr ) + KPL (L
(32)
Lr )] m1 (m2 + mt / 2) L mm L
L2 cos2
+ 2( +
o) m
sin 2
m L2 cos2
[KD (
r)
4.1. Symmetrical capture case
tan
+ KP (
For the symmetrical capture case, the position of the tether attachment point in the debris body frame is (5 m, 0 m, 0 m). In the first simulation, the higher tension T¯ is chosen to be 30 N and the lower tension T is chosen to be 20 N. The desired tether length Lr is 100 m, ¯ and the desired tether in-plane and out-of-plane libration angles r and r are both zero. Meanwhile, the parameters for the PD controller in Eqs. (32)-(34) are all chosen to be 0.005 in order to limit the required propulsion force. The simulation duration is chosen to be 3000 s and the results are shown in Fig. 4. As is illustrated in Fig. 4, the tether length and libration angles are well stabilized in about 2000 s. That is because the attitude motion of the debris indeed has insignificant effects on the tether motion and the PD controller has its inherent anti-interference ability. Moreover, the
r )]
(33)
Q = 2 m LL
m1 (m2 + mt / 2) mm
1
3
m L2 [KD (
r)
+ 2 [3µR
sin2
+(
+ KP (
o
+ ) 2 ] m L2 sin 2 r )]
(34) where KDL > 0 , KPL > 0 , KD > 0 , KP > 0 , KD > 0 and KP > 0 are adjustable coefficients for the PD controller, and Lr , Lr , r , r , r and r are the desired states. Stated thus, the tension control law in Eq. (11) focuses on the nutation damping of the debris and makes the xb2 axis of the debris approach the tether direction, while the thrust provided by the main satellite is used to change the tether direction and stabilize the libration of the tether according to Eqs. (32)-(34).
Table 1 Inertia and geometric parameters of the system.
3.4. Further discussions on the nutation damping methodology A further application of this control methodology can be explored by checking the angular momentum of the debris. As is discussed above, both the directions of its angular momentum and xb2 axis will finally coincide with the tether direction, and the magnitude of the angular momentum will be reduced to the same value of Hxs0 when asymmetrical debris is captured symmetrically. Thus after the nutation of the debris is eliminated, we can manipulate the tether to another orientation using the thrust provided by the main satellite and repeat the process of the nutation damping to achieve further reduction of its angular momentum. Assume that the acute angle between the current tether direction and the next desired direction is 1, then the final angular momentum of the debris after the second nutation damping process can be derived as
Hxs1 = Hxs0 cos
As 0 < cos 1 < 1, the magnitude of the debris’ angular momentum decreases. Moreover, if we proceed with the procedure mentioned above, the angular momentum will decrease unceasingly, that is
Hxsn = Hxs0 cos
1
cos
2
cos
n
Values
Mass of the main satellite (m1 ) Mass of the debris (m2 ) Density of the tether ( ) Cross sectional area of the tether (c ) The x b2 axis moment of inertia ( A ) The yb2 axis moment of inertia (B ) The z b2 axis moment of inertia (C )
1000 kg 2000 kg 1440 kg/m3 5 mm2 4000 kg m2 3000 kg m2 2000 kg m2
Table 2 Initial states of the system.
(35)
1
Parameters
(36)
Then the magnitude of the debris angular momentum will approach zero, which indicates that a three-axis spin counteraction can be achieved using this method after infinite times of tether direction alteration. 687
States
Initial values
Semi-major axis Eccentricity True anomaly Length of the tether Tether in-plane angle Tether out-of-plane angle Angular velocity of the debris around x b2 axis Angular velocity of the debris around yb2 axis Angular velocity of the debris around z b2 axis Euler angle Euler angle Euler angle
42314 km 1 × 10−3 0 deg 80 m −10 deg −10 deg 25 deg/s 10 deg/s 10 deg/s 0 deg 0 deg 0 deg
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Fig. 4. Illustrations of the controlled libration motion, propulsion and attitude motion of the debris.
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Fig. 5. Illustrations of the controlled motion with changing the desired tether direction repeatedly.
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Fig. 6. Illustrations of the controlled libration motion, propulsion and attitude motion of the debris.
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Fig. 7. Illustrations of the controlled motion with changing the desired tether direction repeatedly.
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chosen parameters for the PD controller can make the three components of the propulsion force in the body frame of the space tether system all below 150 N. Meanwhile, the semimajor axis and eccentricity of the system fluctuate in the beginning and then increase steadily because of the incipient fluctuant propulsion force for the tether libration control. The results in Fig. 4 also reveals the effects of the nutation damping method presented in Eq. (11). As the coefficients in the PD controller are chosen very small to limit the propulsion force, relatively long time is needed to stabilize the tether libration compared with the nutation damping process. The effect of the nutation damping law is to make the debris spin around the tether direction, thus unless the libration of the tether is stabilized, the debris attitude angles and will not stop fluctuating. Therefore, the transverse angular velocities b2y and b2z are well eliminated in 2000 s as well, in the wake of the libration stabilization process, and the Euler angles and approach zero, which indicates the xb2 axis of the debris finally coincides with the tether direction , verifying the asymptotic stability of the nutation damping method from the aspect of numerical results. The following simulation is carried out to test the ability of the control method in the application of three-axis spin counteraction mentioned in the end of Section III. The parameters of the space tether system and the initial states are the same with the former simulation, which are shown in Tables 1–2 The desired tether length L is still 100 m, and the desired tether out-of-plane libration angle is zero as well, whereas the desired tether in-plane libration angle changes every 2000 s between 0° and 20° to dissipate the angular momentum of the debris including component along the xb2 direction, and the simulation duration is chosen to be 10000 s. Fig. 5 shows the results of the simulation. Fig. 5 shows that the PD controller performs well in the frequent tether attitude maneuver procedure. Comparing the results of with , and with , it can be observed that the spin axis of the debris tracks the tether direction throughout the simulation. Therefore, to achieve the control of the spin orientation of the debris, we only need to manipulate the direction of the tether and apply the nutation damping method in Eq. (11). Besides, as the tether attitude maneuver procedure proceeds, the spin velocity b2x drops stepwise, following Eq. (36) in the end of Section III. This demonstrated that the proposed method of decreasing the debris’ angular momentum by alternating the tether directions successively with nutation damping is feasible.
nutation causing no tether entanglement is acceptable. It should be noted that the tether tension tend to flutter as the debris approaches the steady state. A feasible way to avoid tension flutter is to keep a constant tension when the tension variation frequency exceeds a threshold value. The next simulation is homologous with the second simulation in the symmetrical capture case to test the capability of the control method to retard the spin rate in the asymmetrical case. The parameters are all the same with the former simulation, and the desired tether inplane libration angle changes every 2000 s between 0° and 20°. Fig. 7 shows that although the transverse angular velocities cannot be completely eliminated, it is still effective to diminish the axial angular momentum. 5. Conclusion In this paper, the attitude dynamic characteristics of a space debris under a constant tether tension is analyzed, and a switching control method using only two constant tensions is presented, which can achieve both the nutation damping and the spin orientation control of a non-cooperative space debris. The bounded stability of the control law is proved based on the work-energy principle and the Barbalat's Lemma under a fixed tension direction assumption. For an asymmetrical debris which is symmetrically captured, the debris under proposed controller tends to spin around the tether direction with constant rate. As the transverse motion of the tether attachment point is well restrained, the risk of entanglement of the tether can be alleviated. The simulation results show the effectiveness of this control method in real applications when the tether libration is taken into consideration. Moreover, the results indicate that the nutation damping and the tether libration control can be carried out simultaneously when the libration is not violent. The ability to achieve three-axis despun of the debris by changing the direction of the tether repeatedly is revealed by both theoretical analysis and simulation results. This control method does not require knowing the moment of inertia, neither accurate attitude measurement of the debris, making it highly applicable in real missions. Acknowledgments This study is supported by the National Natural Science Foundation of China (Grant no. 11772023) and the Academic Excellence Foundation of BUAA for PhD students.
4.2. Asymmetrical capture case
References
For a more common case, that is, the asymmetrical capture case, the position of the tether attachment point in the debris body frame is chosen to be (5 m, 1 m, 1 m), which is no longer on the xb2 axis, but other than that the rest parameters are the same as those in the symmetrical capture case. Fig. 6 shows the results of the control methodology. The tether length is pretty well stabilized at 100 m in 2000 s, which appears to be the same with the symmetrical capture case in Fig. 4. However, due to the fact that a rigid body cannot spin around a nonprincipal axis of inertia, the nutation of the debris cannot be completely eliminated. The attitude motion of the debris tends to a combination of nutation and procession caused by the tether tension. Thus the tether attachment point swings continually, and the tether libration angles can no longer be well stabilized at zero asymptotically using the PD controller presented in Eqs. (33) and (34) consequently. Instead, both the in-plane and out-of-plane angles of the tether is restricted in a small region of zero. The debris attitude motion reaches a steady state after 1500 s, with the transverse angular velocities b2y and b2z approach a positive constant instead of zero as in the former case. The amplitudes of attitude angles and are finally restricted under 10°, demonstrating the stability of the control methodology from the aspect of numerical simulation. In real ADR applications by means of space tethers, a complete nutation elimination is not always required, and a bounded
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