Decentralized adaptive sliding mode control for beam synchronization of tethered InSAR system

Acta Astronautica 127 (2016) 57–66

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Decentralized adaptive sliding mode control for beam synchronization of tethered InSAR system Jinxiu Zhang, Zhigang Zhang, Baolin Wu n Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150080, China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 June 2015 Received in revised form 13 February 2016 Accepted 20 May 2016 Available online 24 May 2016

Beam synchronization problem of tethered interferometric synthetic aperture radar (InSAR) is addressed in this paper. Two antennas of the system are carried by separate satellites connected through a tether to obtain a preferable baseline. A Total Zero Doppler Steering (TZDS) is implemented to mother-satellite to cancel the residual Doppler. Subsequently attitude reference trajectories for the two satellites are generated to achieve the beam synchronization and TZDS. Thereafter, a decentralized adaptive sliding mode control law is proposed to track these reference trajectories in the presence of model uncertainties and external disturbances. Finally, the stability of closed-loop system is proved by the corollary of Barbalat's Lemma. Simulation results show the proposed control law is effective to achieve beam synchronization of the system. & 2016 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Tethered InSAR system Beam synchronization Total zero Doppler steering Decentralized adaptive control

1. Introduction Recent two decade years, more and more latest technology had been introduced into Earth observation, especially on the basis of formation flying and multi-base SAR technology [1–3]. Single-pass SAR interferometry has been put forward and is still under development to enhance temporal correlation and decrease phase errors. In 2009, an innovative spaceborne radar interferometer, TanDEM-X comprising two TerraSAR-X radar satellites flying in close formation, was launched to obtain a consistent global digital elevation model with an unprecedented accuracy [4,5]. To meet with coverage requirement and stable effective baseline, the second satellite needs orbit maneuvers via the cold gas to adjust the relative configuration and suit different latitude observation mission. As a contrast, the tethered InSAR [6], proposed in 1990 s, could provide a relative stable configuration. It could provide a relative stable long and variable baseline with less fuel consumption because of the properties of tethered satellite system (TSS) [7–9], and its baseline is adequate to obtain interferometric image in one entire orbit cycle and all latitude. At the same time, it only need several hours for tethered InSAR to build up interfermetric system after launch and on-orbit. Besides, the tether also could be used as communication cable, which is beneficial to the system design. Since these potential merits of tethered InSAR system, it was applied to Earth observations or space interferometry by researchers [10–13]. n Correspondence to: Research Center of Satellite Technology, Harbin Institute of Technology, Block B3, No. 2 Yikuang Street, Nangang District, Harbin 150001, China. E-mail address: [email protected] (B. Wu).

http://dx.doi.org/10.1016/j.actaastro.2016.05.024 0094-5765/& 2016 IAA. Published by Elsevier Ltd. All rights reserved.

High and varying Doppler centroid are adverse to the calibration of SAR images. Yaw steering and yaw-then-pitch steering [14,15] were developed to compensate the Doppler shift of a single SAR satellite moving in circular orbits and elliptical orbits, respectively. When applying Doppler steering to multi-SAR system, it should not affect other important system parameters, for instance, swath overlap ratio. However, a high swath overlap could be realized by well performed beam synchronization of SARs, including two strategies: satellite attitude maneuver and antenna beam electronic steering [16]. For both strategies, satellites should track reference trajectories to achieve accurate beam synchronization, which is similar to the attitude coordination of satellite formation. Thus, it is helpful to refer to literature about attitude coordination of satellite formation [17–21]. Decentralized control laws were proposed to guarantee that all spacecraft would track a given attitude trajectory in Ref. [17–19]. However, in practice, the satellites of tethered InSAR need track two different reference trajectories, respectively, to cancel the Doppler shift of mothersatellite and achieve high beam synchronization performance. The proposed control law in Ref. [20,21] could ensure that each spacecraft attains its desired time-varying trajectory and maintains attitude synchronization of formation. However, the combined dynamics of tethered InSAR is more complex than the distributed spacecraft formation since the influence of tether. Thus, the effects of tether need to be taken into account to achieve accurate beam synchronization. The beam synchronization is a realistic problem for tethered InSAR system and a solution to this problem is proposed in this paper. Since the combined dynamics of tethered InSAR has significant effects on the position and attitude of satellites, a lumped

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J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

mass model of tether is derived to describe the system accurately at first. Then a total zero Doppler steering is applied to mothersatellite to cancel the residual Doppler. Thereafter, the desired attitude of sub-satellite to achieve beam synchronization is generated by using of footprint of mother-satellite and its position. A decentralized adaptive sliding mode control law is then proposed for the two satellites to track the different time-varying desired attitude in the presence of model uncertainties and external disturbances. To obtain a high swath overlap, both relative attitude errors and relative angular velocity errors between the two satellites are used in the proposed control law, which is effective to promote the tracking precision. For the applications requiring tracking errors of antenna phase center are less than 5%, the precision can improve from 91.32% to 96.63% under this control law. Finally, the stability of the closed-loop system is analyzed with Barbalat's Lemma. This paper is organized as follows. In Section 2, the combined dynamics of tethered InSAR system considering the tether characteristic is presented. Subsequently, a yaw-then-pitch steering strategy is applied to cancel the residual Doppler of active radar, which also leads the desired attitude and angular velocity of mother-satellite, in Section 3. Thereafter, a decentralized adaptive control law is proposed to achieve the beam synchronization of tethered InSAR system in Section 4. Finally, simulation results are shown to validate the beam synchronization control method in Section 5.

2. Dynamics of tethered InSAR system The tethered InSAR system moving in a Keplerian orbit could be seen in Fig. 1. To describe dynamics of the system, some coordinate frames are used: the orbital frame Ocxoyo zo fixing its origin at the mass center of system. The axis yo points along the local vertical, zo axis is normal to the orbital plane, and xo axis completes the right-hand triad. For brevity, it is also denoted as frame C. The inertial coordinate fixed on Earth center is represented as frame I. The coordinates fixed on two satellites are omx mbymb zmb and osxsbysb zsb , and their desired attitude are presented by omx mdymd zmd and osxsdysd zsd coordinates. For simplicity, the instantaneous body frame and the desired coordinate are denoted by frame B and frame D, respectively.

Fig. 1. Tethered InSAR system moving in orbit.

2.1. Description of the tether characteristic The combined dynamics of satellites could strongly affect the phase accuracy of the interferometric images. Therefore, the dynamics of TSS should be modelled accurately by three different oscillations to describe the effect of the tether tension on the satellites’ attitude. For the up to now, there are three mathematical models of TSS, i.e. the dumbbell model, the lumped mass model and the flexible model. As described in [22], the dumbbell models, which are also called bar models, are often used for estimating calculations for simplicity; the flexible models could describe the dynamics of a tether most precisely while the numerical integration requires great computing expenditure. As a compromise, the lumped mass models have a relative accurate description and less computation cost. Hence the lumped mass model is chosen to describe the dynamics of tether. Naturally, more parts the tether is divided, more accurate dynamics of tether is obtained. In YES2 project, a 30 km tether was divided into 20 particles and had an accuracy of one meter [22]. Thus, the 1 km tether used in this paper divided into 10 parts could describe the dynamic of the tether with adequate precision. Satellites of the system are seen as point mass and the atmospheric drag is neglected, which is reasonable when the tether is

short. The lumped mass model could be obtained using the method in [23]. The three dimensional equations of motion for the ith element of tether can be expressed in the frame C as:

⎡ r¨i ⎤ ⎢ ⎥ −1 −1 ⎢ α¨ i ⎥ = B (A D − C) ⎢ β¨ ⎥ ⎣ ⎦ i

(1)

⎡ sin α cos β − cos α sin α sin β ⎤ ⎡ −1 0 0⎤ i i i i i ⎥ ⎢ ⎥ ⎢ r 0 cos 0⎥ β ⎥ ⎢ A = − cos αi cos βi − sin αi − cos αi sin βi B = ⎢ i i ⎥ ⎢ ⎢⎣ 0 ri ⎥⎦ 0 − sin βi 0 cos βi ⎦ ⎣ ⎤ ⎡ 2 ⎡f⎤ ri(αi̇ + n)2cos2βi + riβi̇ ⎥ ⎢ ⎢ 1⎥ ⎥ ⎢ ̇ ⎢ f2 ⎥ ̇ ̇ C = rn D = r n r n cos 2 sin 2 cos β − ( α + ) β β + ( α + ) β ̇ ̇ i i i i i i i i⎥ ⎢ i ⎢ ⎥ 2 ⎥ ⎢ ⎢⎣ f3⎥⎦ 2riβ̇ i̇ + ri(αi̇ + n) sin βi cos βi ⎦ ⎣

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

f1 =

μri sin αi cos βi Rc3 +

f2 = 2

f3 = −

Ti sin αi cos βi mi

μri cos αi cos βi

−

Ti sin αi cos βi

+

Rc3

mi + 1

Rc3

mi + 1

− −

Ti cos αi cos βi

+

Ti − 1 cos αi − 1 cos βi − 1

mi −

Ti + 1 sin αi + 1 cos βi + 1

Ti − 1 sin αi − 1 cos βi − 1

Ti cos αi cos βi μri sin βi

−

mi mi + 1

Ti sin βi mi + 1

+

Ti + 1 sin βi + 1 mi + 1

qiė =

;

1 (q I3 + qie × )ωie 2 ie0

Ti + 1 cos αi + 1 cos βi + 1 mi + 1

mi +

−

qiė 0 = −

;

1 T q ωie 2 ie 3×3

where Ji ∈ 

Ti sin βi mi

+

Ti − 1 sin βi − 1 mi

;

where ri is the length of the ith tether element, αi and βi are the inplane and out-of-plane angle in the frame C of the ith tether, Ti is the tether tension, μ is the gravitational constant of the Earth, Rc is the radius vector of the mass center of the system, n is the rate of change of the true anomaly, i.e. the rotation velocity of frame C with respect to I. The tether tension must be positive and could be obtained by:

⎧ ri ≤ r0 0 ⎪ Ti = ⎨ EA (ri − r0) ri > r0 ⎪ ⎩ r0

(2)

2.2. Attitude dynamics of satellite with tether The satellites are seen as rigid body with three perpendicular control torques along their body fixed coordinate axes. The unit quaternion q¯ is used to represent the motion of satellites. Meanwhile q0 and q denote the scalar part and vector part of the quaternion, respectively. Let q¯i(i = m , s ) represents the instantaneous orientation of mother-satellite or sub-satellite with respect to the reference frame C, q¯id(i = m , s ) denotes the desired orientation D with respect to frame C, and q¯ie(i = m , s ) expresses the error quaternion between the instantaneous body frame B and its desired orientation D. The meaning of subscript of angular velocities is similar, such as ωi , ωid , ωie (i = m , s ). Define the error quaternion of satellite as:

q¯ie = q¯id−1 ⊗ q¯i (i = m, s )

(3)

(4)

where R(q¯ ) ∈ SO(3) is the rotation matrix corresponding to the quaternion q¯ , and given by:

)

R(q¯ ) = q02 − qT q I3 + 2qqT − 2q0q×

(8) is a constant, positive definite inertia matrix of

satellite expressed in frame B, τi ∈ 3 is the control torque, d i ∈ 3 is the torque of external disturbances, I3 is a 3 order identity matrix, Mi ∈ 3 is the torque caused by tether tension, which could be represented as:

Mi = Δi × Ti

(9)

where Δi(i = m , s ) denote the offset from the center of mass of satellite to tether connecting point for mother-satellite and subsatellite, respectively.

The Doppler centroid is the center frequency of the azimuth spectrum of SAR, and plays an important role during azimuth compression, range migration correction and geolocalization processes. However, residual Doppler introduced by the Earth rotation increases the focusing and processing difficulties. So it is sensible to find a way to cancel the residual Doppler. But performing Doppler steering to all satellites would be detrimental to high swath overlap. Consequently, only mother-satellite who carries the active radar performs Doppler compensation after a tradeoff. An advanced method, total zero Doppler steering (TZDS), was published to cancel the residual Doppler of SAR in elliptical orbits [15]. It consists of both yaw and pitch maneuvers. Similar to Ref. [14], a coordinate system free TZDS of mother-satellite could be derived as follow:

fD = −

2 ^ ρ ̇⋅ρ λ

(5)

The notation q× is a skew-symmetric matrix.

⎡ 0 −q q ⎤ 3 2 ⎥ ⎢ 0 −q1⎥ q × = ⎢ q3 ⎥ ⎢ ⎣ −q2 q1 0 ⎦

m f dc =−2

RA⋅VA λRA

(11)

where RA means the vector from the radar's apparent phase origin of mother-satellite to the target (as seen in Fig. 1), VA = RȦ , describing the relative motion between the radar antenna phase center of mother-satellite and the target. Using operational laws of vector and geometrical relationship of the system, the expression becomes as follow: m

f dc = =

There are some properties of the rotation matrix:

(10)

where λ is the wavelength of the radar, ρ and ρ^ are the vector from the radar to the target and its unit vector, respectively. ρ̇ means the relative motion between the target and the radar. According to the formula in Ref. [14], the centroid Doppler of the mother-satellite could be written:

The angular velocity error could be expressed as:

ωie = ωi − R(q¯ie )ωid

(7)

3. Total zero Doppler steering of mother-satellite

where EA denotes the stiffness of the tether, and r0 is the length of unstrained tether, equaling 100 m here.

(

59

−2R A −2R A −2R A ⋅(Vm − Vt ) = ⋅(Vm − ωe × R t ) = ⋅[Vm − ωe × (R A + R m)] λR A λR A λR A −2R A −2 ⋅(R m × ωe + Vm) = R A⋅Q A λR A λR A

(12)

where Q A = R m × ωe + Vm , is equivalent with VA , called equivalent velocity. It only depends on the position and velocity of mothersatellite, as well as the rotation of the Earth. The position vector of mother-satellite could be express as:

R−1(q¯ ) = RT (q¯ ), ‖R(q¯ )‖ = 1 Ṙ (q¯ ) = − ω × R(q¯ ) = R(q¯ )ω× Then the error dynamics of rigid satellites could be written as:

Ji ω̇ie = − ωi × Ji ωi + τi + di + Mi + Ji [ωie × R(q¯ie )ωid − R(q¯ie )ω̇id ]

(6)

R m = R c + R cm

(13)

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J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

where R m and Rc denote the vectors from the earth center to center mass of the mother-satellite and the system, respectively. R ij represents the vector from the ith element to the jth element. The ratio of the tether mass to satellites mass is negligible for the length is short. Hence the position vector Rcm could be expressed as:

R cm = −

ms R ms mm + ms

(14)

Obviously, the vector from mother-satellite to sub-satellite could be calculated by: n

R ms =

4.1. Attitude reference trajectories of satellites

∑ ri

(15)

i=1

By differentiating both sides of Eq. (13), the velocity of mothersatellite could be represented as:

Vm = Vc + Vcm

(16)

The position vector and velocity vector of sub-satellite could be obtained similarly, and would be used later.

R s = R c + R cs

To obtain a preferable measurement in one entire orbit cycle, the passive radar need track the area illuminated by active radar rapidly and accurately. Two attitude reference trajectories are derived firstly. One is for mother-satellite, along which the active radar could cancel the residual Doppler. The other is for sub-satellite, and sub-satellite could track the footprint of mother-satellite precisely along it. Then the two satellites should track their reference to finish the mission in the presence of system uncertainties, external disturbances, and effects of tether tension as well.

Vs = Vc + Vcs

(17)

Assuming the antenna is fixed on the satellite, and there is no offset from the radar phase center to the center mass of satellite for simplicity. Moreover, the center line of radar beam is in the omymb zmb plane. Then driving omx mb axis to coincide with Q A would cancel the residual Doppler. The angles between equivalent velocity and omx mbymb zmb could be denoted by yaw angle θy and pitch angle θp :

⎧ Q θy = arctan Az ⎪ Q Ax ⎪ ⎨ Q Ay ⎪ θ = arctan ⎪ p 2 2 + Q Az Q Ax ⎩

As mentioned before, both satellites are assumed to have allangles attitude maneuvering capability, and all the strategies are fulfilled by satellite attitude maneuverings. Since the offset between antenna phase center and mass center of satellite could be compensated, the two centers are assumed to coincide for simplicity here. In addition, the look angel of the antennas are both assumed to be γ . The desired attitude of mother-satellite is obtained in Section 3, and as shown in Fig. 2, the area illuminated by active radar could be determined by the beam pointing of mother-satellite. To achieve swath overlap, the beam of sub-satellite should point to the area as well. The unit vectors of the beam pointing of mother-satellite and sub-satellite are denoted by eAm and eBs in their body fixed coordinates, respectively. Then, the components along the coordinate axes are:

(18)

where Q Ax , Q Ay and Q Az are the projection of Q A along the axes of omxmbymb zmb . Drive mother-satellite to rotate around omyb about θp clockwise and then around omzb about θy . As a result, the omxb axis would coincide with equivalent velocity. Thus, the residual Doppler of mother-satellite is canceled. The temporal body fixed coordinate could represent the desired attitude of mother-satellite. Assuming omxmbymb zmb coincides with frame C before maneuvers, then the desired attitude of mothersatellite could be described by a rotation matrix:

R c2md = R z(θy)R y(θp)

(19)

⎡ cos θ 0 sin θ ⎤ ⎡ cos θ sin θy 0⎤ y p p ⎥ ⎢ ⎥ ⎢ R z(θy) = ⎢ − sin θy cos θy 0⎥ R y(θp) = ⎢ 0 1 0 ⎥ ⎢ − sin θ 0 cos θ ⎥ ⎥ ⎢ ⎣ ⎣ p p⎦ 0 0 1⎦ The desired attitude of mother-satellite is represented by direction cosine matrix here, and it would be transferred to unit quaternion to avoid the singularity problem [24] when used later.

4. Beam synchronization of tethered InSAR system The synchronization between antennas plays a key role in the interferometric measurement. Here, only beam synchronization is considered to ensure the two antenna's footprint overlap on the surface of the earth. As a result of Ref. [12], the antennas direction need to change to finish a global measurement mission, and the antennas pointing should be synchronized to achieve the adequate overlapping ratio.

Fig. 2. Antenna directing of tethered InSAR.

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

Assumption 1. There is a constant but unknown bound of the uncertainty part ΔJi ≤ cΔJ .

T eAm = ⎡⎣ 0 cos γ sin γ ⎤⎦ T eBs = ⎡⎣ 0 cos γ sin γ ⎤⎦

(20)

The beam pointing of mother-satellite in frame C could be obtained:

eAc = R Tc2meAm

(21)

where R c2m is the rotation matrix from frame C to omxmbymb zmb . As shown in Fig. 2, beam synchronization would approach if the antenna pointing of sub-satellite is steered along the vector RB , which denotes the vector from sub-satellite to the target point. osxsbysb zsb is assumed to coincide with frame C before maneuvers. Then just steer the beam pointing eBs to the direction of RB , and beam synchronization could achieve. The rotation angle will be determined if RB is known. The unit vector of RB is denoted as:

RB Rt − Rs = eBc = ‖RB‖ ‖R t − R s‖

(22)

The rotation axis could be represented as:

e=

eBs × eBc ‖eBs × eBc‖

(23)

The rotate angle is:

⎛ e ⋅e δ = arccos⎜⎜ Bs′ Bc ⎝ eBs′ eBc

⎞ ⎟⎟ = arccos(eBs′⋅eBc ) ⎠

(24)

‖R t‖ = Re

(25)

R t − R m = ‖R t − R m‖eAc

(26)

With Eqs. (25) and (26), the target's position could be determined. Subsequently, rotation axis and angle could be obtained. Then the two attitude reference trajectories are obtained. The desired quaternions could be obtained through the transformation from rotation matrix represented by Eq. (19) and Euler rotation method described by Eqs. (23) and (24). Moreover, the desired angular velocity could be achieved by:

( )

( )

E q¯ d

Assumption 2. Assuming the tether tension torque uncertainty part satisfies ΔMi ≤ cΔM . Assumption 3. The external disturbances satisfy ‖d i‖ ≤ cd . cΔM and cd are an unknown non-negative constant. Then error dynamics described by Eq. (6) could be rewritten as:

¯J ω̇ = − ω × ¯J ω + τ + ρ + M ¯ + ¯J [ω × R(q¯ )ω − R(q¯ )ω̇ ] (28) i i i ie i ie i i i i ie id ie id where ρi contains the uncertainty terms and external disturbance, and its expression is:

ρi = − ΔJi ωie − ωi × ΔJi ωi + di + ΔMi + ΔJi ⎡⎣ ωie × R(q¯ie )ωid − R(q¯ie )ωid ⎤⎦

(27)

⎡ −qd1 qd0 qd3 −qd2 ⎤ ⎢ ⎥ = ⎢ −qd2 −qd3 qd0 qd1 ⎥ ⎢⎣ −qd3 qd2 −qd1 qd0 ⎥⎦

The derivative of desired quaternion q¯ḋ could be calculated by finite difference method.

(29)

The desired angular velocity of satellites and their derivatives should be bounded to assure the attitude could be tracked under finite control torque. With the Assumptions 1–3, the uncertainty term expressed by Eq. (29) is bounded:

‖ρi ‖ ≤ ¯Ji bi

As long as the target point is determined, the detail of the maneuvers could be known. For simplicity, assume the earth is sphere with the radius Re , and the target point is the intersection between mother-satellite's beam center and earth surface. The following equations are satisfied according to this.

ωd = 2E q¯ d q¯ ḋ

61

(30)

where bi are unknown non-negative constant. Then the problem of beam synchronization turns into the problem of tracking desired attitudes. Hence, when the error attitude and angular velocity converge to zeros as time increases, i.e., lim qie → 0, lim ωie → 0, the beam synchronization would t →∞

t →∞

realize. A decentralized adaptive sliding model control law is developed to realize beam synchronization of tethered InSAR system when system uncertainties and external disturbances are taken into account. The sliding model vectors of satellites are defined as:

si = ωie + βqie + C[(ωie − ωje ) + β(qie − qje )](i = m, s ; j = s , m)

(31)

3×3

where β ∈  is a positive definite constant matrix; C is the control weight parameter for the interacting relative attitude synchronization between two satellites and is a positive constant. Theorem 1. : Consider the satellites attitude tracking dynamics are governed by Eqs. (6)–(8) with the decentralized adaptive sliding model control law as follow:

¯ − ¯J [ω × R(q¯ )ω − R(q¯ )ω̇ + βq̇ − K s τi = ωi × ¯Ji ωi − M i ie i i i ie id ie id ie ^ − bi sgn(si )]

(32)

4.2. Decentralized adaptive sliding model control design

^̇ bi = σi‖si‖

The fuel consumptions or payload motion would have impacts on the inertial parameter of satellites. But assume inertial parameter Ji will keep positive definite and bounded. Let Ji = ¯Ji + ΔJi , ¯Ji and ΔJi are the nominal part and uncertainty part of the inertial parameter, respectively. In addition, because of the installation and mathematical model errors, the connecting point of the tether and the direction of tether tension cannot be determined exactly. As¯ is the nominal part of tether ¯ + ΔM , where M suming Mi = M i i i tension torque, calculated by measured values. The following assumptions are made:

If the assumptions are satisfied, and K i > 0, σi > 0, which are chosen by designer, then the sliding mode surface si = 0 described by Eq. (31) will asymptotically stable. The sign function sgn (
) used in Eq. (32) is defined as:

⎧ 1 x>0 ⎪ sgn( x) = ⎨ 0 x = 0 ⎪ ⎩ −1 x < 0 Proof: Consider the candidate Lyapunov function as:

(33)

62

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

where sat (s ) is a saturation function defined as:

^ 2 ^ 2 1 T −1 (b − b m ) −1 (bs − bs ) S (K ⊗ I3)−1S + ¯Jm m + ¯J s 2 2σm 2σs ⎤ ⎡ 1 C I C I + − ⎡ sm ⎤ ( )3 3 ⎥ S=⎢ ⎥ K=⎢ ⎣ ss ⎦ ⎢⎣ −C I3 ( C + 1)I3⎥⎦ V=

⎧ sgn(s ) s ≥ ς sat (s ) = ⎨ s <ς ⎩ s /ς (34)

Let

V1 =

−1 (b m

(35)

(36)

It is obvious that K and ¯Ji are positive definite constant matrixes and σi is a positive definite constant. Hence V is positive definite. Take the first derivative of Eq. (35):

⎡ sm ̇ ⎤ V1̇ = ST K −1S ̇ = ST K −1⎢ ⎥ ⎣ sṡ ⎦ ⎡ ( C + 1)I 3 = ST K −1⎢ ⎢⎣ −C I3

⎤⎡ ω̇ + βq̇ ⎤ me ⎥⎢ me ⎥ ( C + 1)I3⎥⎦⎢⎣ ω̇ se + βqsė ⎥⎦ −C I3

(37)

Using the Eqs. (28) and (32), Eq. (37) could be rewritten as:

⎡ ^ ¯ −1 ⎤ ⎢ −K msm − bmsgn(sm) + Jm ρm ⎥ V1̇ = ⎡⎣ smT ssT ⎤⎦⎢ ⎥ −1 ^ ⎢⎣ −K sss − bssgn(ss ) + ¯J s ρs ⎥⎦ −1 −1 ^ ^ = − K m‖sm‖2 + smT ¯Jm ρm − smT bm − K s‖ss‖2 + ssT ¯J s ρs − ssT bs ^ ^ ≤ − K m‖sm‖2 − K s‖ss‖2 + (bm − bm)‖sm‖ + (bs − bs )‖ss‖

(38)

(44)

Under the modified control law Eq. (41) and adaption law Eq. (44), the system is ultimately converging into the boundary layer. The proof process is similar to Theorem 1 and it is omitted here for brevity.

To verify the control laws in Eq. (41), numerical simulations are performed. As mentioned earlier, the use of relative attitude errors and relative angular velocity errors in the controller can reduce the tracking errors, and increase the swath overlap. Therefore, two cases are studied to demonstrate this in the simulations: Case A: The relative attitude errors and relative angular velocity errors are used in the control laws, i.e., C ¼1 in Eq. (31). Case B: Contrary to case A, these information are not utilized in the control laws, i.e., C ¼0 in Eq. (31). Then the sliding mode vector is similar with that used in Ref. [26]. The parameters of tethered InSAR system are shown in Table 1. The antennas mounted on the satellites both have a 3 dB elevation aperture of 3.8° and a 3 dB azimuth aperture of 0.38°. There are inertia uncertainties and disturbances existing in the system. The actual inertia matrices of satellites are assumed as:

⎡ 30 2 1.9⎤ ⎡ 22 2 1.9⎤ ⎢ ⎥ ⎢ ⎥ Jm = ⎢ 2 27 1.5⎥(kg m2) Js = ⎢ 2 19 1.5⎥(kg m2) ⎣ 1.9 1.5 25 ⎦ ⎣ 1.9 1.5 15 ⎦

The derivative of Eq. (36) is:

^ ^ (b − bm) (b − bs ) ^̇ ^̇ V2̇ = m ( − bm) + s ( − bs ) σm σs ^ ^ = − (bm − bm)‖sm‖ − (bs − bs )‖ss‖

(39)

The nominal inertia matrices are assumed as:

¯J = diag ⎡⎣ 30, 30, 30⎤⎦ kg m2 ¯J = diag ⎡⎣ 20, 20, 20⎤⎦ kg m2 m s

(

Then the first derivative of V is: −1

V̇ = V1̇ + V2̇ ≤ − ¯Jm K m‖sm‖2 − ¯J s K s‖ss‖2

(40)

^ From Eq. (40), it can be concluded si ∈ L∞ and bi − bi ∈ L∞. ^ Consequently, it could be obtained that bi ∈ L∞. With Eq. (32) the conclusion will be obtained τi ∈ L∞. Hence the error parameters ω̇ ie , qiė are all bounded with error dynamics constraints, which will lead si̇ ∈ L∞. Using the corollary of Barbalat's Lemma, it is obtained that lim si(t ) = 0, (i = m , s ). This leads to lim ‖ωie ‖ = lim ‖qie‖ = 0, t →∞

)

(

)

The attitude of satellite is affected by gravity gradient, atmosphere drag, solar radiation pressure and so on. The total external disturbances are assumed worse than actual situation according to Ref. [27,28]: ⎤ ⎡ ⎤ ⎡ 1 × 10−3 sin(0.11t ) 1 × 10−3 cos(0.11t ) ⎥ ⎢ ⎥ ⎢ − − 3 3 ⎢ ⎥ ⎢ d m = 2 × 10 cos(0.11t + π /6) Nm d s = 2 × 10 sin(0.11t + π /6) ⎥Nm ⎥ ⎢ ⎥ ⎢ ⎢⎣ 1 × 10−3 cos(0.11t + π /3)⎥⎦ ⎢⎣ 1 × 10−3 cos(0.11t + π /3) ⎥⎦

t →∞

which means the satellites would track their desired trajectories well. As a result, the beam synchronization of tethered InSAR system achieves. When sliding mode control is applied in practical system, the system would experience undesirable phenomenon known as chattering [25]. The discontinuous control leads to oscillations in the vicinity of surface. A method to mitigate chattering is smoothing out the control input within a boundary layer near the surface. Furthermore, the control law could be modified as:

¯ − ¯J [ω × R(q¯ )ω − R(q¯ )ω̇ + β¯J q̇ + K s ′ τi = ωi × ¯Ji ωi − M i ie i i i ie id ie id i ie ^ + bisat (si )]

The adaption law depicted by Eq. (33) is modified as:

5. Numerical simulation

⎡ ω̇ me + βq̇ ⎤ me ⎥ = ST ⎢ ̇ ⎥⎦ ⎢⎣ ω̇ se + βqse

−1

(43)

^̇ bi = σi‖si′‖

^ ^ − bm)2 ¯ −1 (bs − bs )2 + Js 2σm 2σs

t →∞

where ς is the thickness of this boundary layer. Another parameter s′ in Eq. (41) is represented as:

s′ = s − ςsat (s )

1 T −1 S K S 2

V2 = ¯Jm

(42)

(41)

Table 1 The parameters of tethered InSAR system. Parameter

Value

Orbit eccentricity ecc Orbit semi-major axis a Orbit inclination inc Right ascending node Ω Argument of perigee w Look angle of antenna γ Mother-satellite mass Sub-satellite mass Tether length

0.005 6828 km 95° 100° 0 30° 300 kg 200 kg 1 km

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

For simplicity, it is assumed the connecting points between tether and satellites lie on the inertial axes of satellites, and they are expressed in body-fixed frame as:

14 pitch yaw roll

12

T rs = ⎡⎣ 0 −0.5 0⎤⎦ m

10

The initial attitude and angular velocity of the satellites are assumed in the frame C as: T q¯m(0) = ⎡⎣ 0.9887 0.0865 0.0865 0.0865⎤⎦

ωm (0)

T = ⎡⎣ 0 0 0⎤⎦ rad/s T q¯s (0) = ⎡⎣ 0.9887 0.0865 0.0865 0.0865⎤⎦

ωs (0)

attitude error(deg)

T rm = ⎡⎣ 0 0.5 0⎤⎦ m

63

8

0.02

6

0

4

-0.02 50

100

150

200

250

300

2

T

= ⎡⎣ 0 0 0⎤⎦ rad/s

0

To acquire a satisfied performance of the satellite, the controller parameters are chosen as:

-2

β = I3, C = 1, K = I3, σ = 0.01, ς = 0.01.

0

50

100

150

200

250

300

time(s) Fig. 4. Attitude tracking error of mother-satellite in case A.

0.5 0 -0.5

control torque τm(Nm)

It's worth noting that the choice of these values should be elaborate, since they have an important effect on the transitional period and stable phase of the system. Case A: with coordianation between two satellites in the controller (C ¼1) The residual Doppler of mother-satellite, attitude tracking errors, control torque of satellites and torques caused by tether tension are shown in Figs. 3–9. It is shown that both the satellites can track the desired attitudes in the presence of system uncertainties and external disturbances. The residual Doppler of mother-satellite after entering steady state is shown in Fig. 3. The Doppler centroid ranges between 7124.3 Hz, which is related to attitude tracking accuracy of mother-satellite. For comparison, the Doppler centroid of TerraSAR-X reaches 7100 Hz as the steering accuracy of attitude is 70.01°. As depicted in Fig. 4, the tracking errors undergo a rapidly decrease and the steady attitude tracking error of the mother-satellite remains under 0.02°. As a consequence, it needs considerable control torques to compensate the initial attitude errors, which corresponds to the Fig. 5. During the steady state, the control torques driving the satellite to track its attitude reference, as well as counteracting the external disturbances and tether tension torque, are less than 0.002 Nm. The tether tension torques of the mother-satellite fluctuates rapidly in Fig. 6, and the magnitudes of tether tension torques are less than control torques. Moreover, comparing the control torques

-3

x 10 -1

2 0

-1.5

-2

-2

50

100

150

200

250

300

-2.5 τmx

-3

τ my

-3.5

τ

mz

-4

0

50

100

150

200

250

300

time(s) Fig. 5. Control torque of mother-satellite in case A.

-3

1

150

x 10

0

100

Mm(Nm)

residual Doppler(Hz)

-4

x 10

-1

50

0

6 4 2 0

-2

50

-3

-50

100

150

200

250

300 M

mx

-4

-100

Mmy Mmz

-150

-5

50

100

150 time(s)

200

Fig. 3. Doppler centroid of mother-satellite.

250

300

0

50

100

150

200

250

time(s) Fig. 6. Tether tension torque of mother-satellite in case A.

300

64

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

12 pitch yaw roll

attitude error(deg)

10

8

0.04 0.02

6

0 4

-0.02

2

-0.04 50

100

150

200

250

300

0

-2

0

50

100

150

200

250

300

time(s) Fig. 7. Attitude tracking error of sub-satellite in case A.

mainly working on the counterbalance of external disturbances, excepting the torque along omzmb axis. With the increase of tether tension torque along the axis, the control torque τmz also changes to compensate the variation. The attitude tracking errors, control torques and tether tension torques of sub-satellite are shown in Figs. 7–9. Obviously, the attitude tracking errors of sub-satellite are bigger than that of mother-satellite, as well as the torques acting on sub-satellite, especially the tether tension torques. Moreover, the tether tension torques also overweigh the external disturbances along the oszsb axis. Thus, the frequency of control torque τmz depends on the tether tension torques. It is easy to conclude that the tether tension has a significant impact on the sub-satellite. Case B: without coordianation between two satellites in the controller (C ¼ 0) In case B, only attitude tracking errors and angular velocity errors are used in the design of the sliding mode vector. Comparing the attitude tracking errors of two cases, it could be concluded that the control laws including the relative attitude errors and relative angular velocity errors drive the satellites to track reference trajectories more accurately under the equivalent control torques (Figs. 10–13). The maximum tracking errors of antennas phase center of two cases are compared in Fig. 14. Both the tracking errors of range direction and azimuth direction of case B are much bigger than that of case A. Hence, tethered InSAR system of case A could obtain a higher swath overlap ratio. For the antennas used here, the swath width of range direction and azimuth direction are approximately 30 km and 3 km, respectively. Thus, as depicted in Fig. 14, tracking accuracies of range direction and azimuth direction of case A both increase comparing to case B, especially the azimuth direction, from 91.32% to 96.63% specifically.

6. Conclusions The beam synchronization is a realistic problem for InSAR system requiring a high swath overlap ratio. A decentralized control law is proposed to tackle this problem in presence of system uncertainties, external disturbances and tether effects. The numerical results show that both the satellites can track the attitude reference trajectories. Moreover, the use of relative attitude errors and relative angular velocity errors in the attitude controller could improve the performance of the attitude tracking. The

Fig. 8. Control torque of sub-satellite in case A.

0.015 M

sx

Msy Msz

14

0.01

pitch yaw roll

Ms(Nm)

12

attitude error(deg)

10

0.005

0

0

50

100

150

200

250

300

time(s)

8

0.02

6

0

4

-0.02 50

100

150

200

250

300

2 0

Fig. 9. Tether tension torque of sub-satellite in case A.

-2

and external disturbances of mother-satellite, the magnitude and phase of control torques coincide with that of disturbances. Thus, during the steady state, the control torques of mother-satellite are

0

50

100

150

200

250

time(s) Fig. 10. Attitude tracking error of mother-satellite in case B.

300

J. Zhang et al. / Acta Astronautica 127 (2016) 57–66

65

300

0.5

case A case B

0

x 10

phase center error (m)

control torque τm(Nm)

250 -3

-0.5 2

-1

0 -1.5

-2 50

100

150

200

250

300

-2

200

150

100

τ

-2.5

mx

τ

50

my

-3

τ

mz

-3.5

0

50

100

150

200

250

300

time(s)

0

range direction

azimuth direction

Fig. 14. Maximum tracking errors of antenna phase center.

Fig. 11. Control torque of mother-satellite in case B.

12 pitch yaw roll

attitude error(deg)

10

8

6

0.05

0

4

2

-0.05 50

100

150

200

250

300

0

-2 0

steady attitude tracking errors of both satellites are much reduced with the proposed cooperative attitude controller, which is beneficial to the Doppler compensation of mother-satellite and obtaining high swath overlap. For instance, the tracking precision of antenna phase center increases from 91.32% to 96.63% along the azimuth direction. It is obvious that the tether tension has a significant impact on the control of attitude tracking, especially on the sub-satellite. The tether tension torque along oszsb direction is almost an order of magnitude higher than external disturbance, and it is compensated directly in this paper. Thus, it need much control torque to counteract the tether tension torque. If the tether tension torque could be used as control torque, it will decrease the fuel consumption. Meanwhile it would increase the difficulty of system design, and this would be further studied in the future.

Acknowledgments 50

100

150

200

250

time(s) Fig. 12. Attitude tracking error of sub-satellite in case B.

300

Funded under the National Natural Science Foundation of China (61503093, 91438202), Project Agreement no. AUGA5710053114 with Harbin Institute of Technology, and Open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology (Grant number HIT.KLOF.MST.201502).

References

Fig. 13. Control torque of sub-satellite in case B.

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