Attitude tracking control for a space moving target with high dynamic performance using hybrid actuator

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Attitude tracking control for a space moving target with high dynamic performance using hybrid actuator

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a, b

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, Mo-Hong Zheng , Feng Wang , Bing Hua , Zhi-Ming Chen ,

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a r t i c l e

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Micro-Satellite Research Center, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao street, Nanjing, 210016, China b Research Center of Satellite Technology, Harbin Institute of Technology, No. 2 Yikuang street, Harbin, 150001, China

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Yun-Hua Wu , Feng Han Yue-Hua Cheng a

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Article history: Received 28 December 2017 Received in revised form 23 March 2018 Accepted 24 March 2018 Available online xxxx

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Attitude tracking control for a space moving target (such as debris or malfunctioning satellite) is investigated in this paper, which is different from the traditional agile attitude maneuvering and tracking control, and is a challenging problem for attitude control system, requiring agility, large control torque output, and high dynamic accuracy, etc. The rapidly moving target and spacecraft pose several tough issues such as agile attitude tracking control and actuator configuration design. A novel attitude tracking strategy is proposed to tackle the dynamic imaging process, including three phases, earth observation, attitude adjustment and dynamic tracking phase. With the accomplishment of attitude adjustment, the spacecraft will point toward the target to start the imaging task. For the maneuvers in the attitude adjustment and tracking phases, a combined control strategy consisting of saturation controller and backstepping controller is proposed. The former one constrains the attitude angular velocity as well as the required momentum on the actuators during the initial phase, while the backstepping controller guarantees the control accuracy with high dynamic performance in the imaging phase. A hybrid momentum exchanging actuator consisting of Control Moment Gyro (CMG) and Reaction Wheel (RW) is introduced to satisfy the great control torque demand. Null motion strategy is derived for the hybrid actuator to deal with CMG singularity and RW saturation simultaneously. Numerical simulations have demonstrated the advantages of the hybrid actuator and the proposed attitude control strategy, which not only enables the spacecraft to maneuver rapidly but also guarantees the tracking accuracy. © 2018 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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The growing demands of continuous observation or imaging rather than a static snap shot of specific earth targets, such as earthquake, flood and forest fire, etc., with High Resolution (HR) from space have forced the earth staring imaging technology, which enables the dynamic and continuous information form. This new information style is more objective and efficient for the users to benefit from. A large amount of commercial earth observation spacecraft corporations, which are competent to provide the HR images even the videos, consequently bloom and thrive throughout the world. For example, the Digital Globe’s satellite constellation can capture images with 0.3 m resolution and is quite advanced for civil use. There is no doubt that the video imaging mode of spacecraft would be more sophisticated, which, compared to the conventional earth imaging of a stationary target, is commanded to point at the

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E-mail address: [email protected] (Y.-H. Wu). https://doi.org/10.1016/j.ast.2018.03.041 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

target during the specific time segment. Due to the high orbital velocity and high altitude, the spacecraft attitude maintaining and adjusting becomes of great significance, and the required attitude angular velocity is around 1 deg/s. To accomplish these kinds of attitude tracking control missions with high pointing precision requirement, different models and control architectures have been proposed. Generally, two major problems should be tackled in this imaging process, attitude determination and tracking. In absence of angular velocity measurement, the attitude determination has been studied in Refs. [1] and [2]. Ref. [3] proposed an effective method to determine the desired spacecraft attitude command through the proper choice of the reference frame for the small staring satellites, and Ref. [4] covered the topic of fast prediction algorithms with attitude control. The attitude tracking problems during the imaging process also have been investigated in various papers. A quaternion dynamic output feedback for attitude tracking problem without velocity measurement was studied in Ref. [5], and high precision attitude tracking problem was discussed in Ref. [6] with an iterative learning control method to reject the effects of

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Nomenclature

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A ag as at Cs d En J hact I Q Qd

CMGRW system matrix CMG gimbal axis direction vector CMG flywheel spin axis direction vector CMG output torque axis direction vector RW spin axis direction vector null motion vector identity matrix of dimension n Jacobian matrix Momentum of the actuator moment of inertia of spacecraft [q0 , q1 , q2 , q3 ]T = [q0 , q]T , attitude quaternion desired attitude quaternion

Qe R AB S CMG S RW S atc W 1 , W2 Y

δ Ω RW

ω ωd ωe

attitude error quaternion direction cosine matrix from frame A to B CMG performance index RW performance index CMGRW performance index weighted matrix the state of the CMGRW system CMG gimbal angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . deg RW angular speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad/s spacecraft attitude angular velocity . . . . . . . . . . . . . . deg/s desired spacecraft attitude angular velocity . . . . . deg/s attitude angular velocity error . . . . . . . . . . . . . . . . . . . deg/s

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the repeating disturbances on satellite. All these comprehensive works have enabled the video imaging spacecraft very advanced at present and in the future. For the moving space targets, such as debris or malfunctioning satellites, how to observe these objects effectively to minimize the corresponding harm on spacecraft in the complex space environment? Also, for other circumstances like the space station maintenance and servicing mission, periodical supervision would serve as a greatly valued means to improve the mission efficiency. A candidate solution is to monitor these targets from a space based-platform, putting forward a new on-orbit work mode which although shares something in common with the earth staring imaging mode. Ref. [7] has investigated the orbital determination of the space-based optical space surveillance system. And, aiming to demonstrate the capability to conduct space surveillance, the Space-Based Space Surveillance and Space-Based Visible programs have been launched in America, which were expected to collect data on man-made resident space objects, from a spacebased platform [8–10]. Certain missions also have been designed and planned for these kinds of spacecraft [10]. This special mission and work mode would lead to some new attitude tracking problems, which requires the spacecraft three axes to maneuver quickly rather than a single axis attitude maneuver in the earth target staring mode, and puts forward new attitude control demands, including command attitude determination, attitude tracking with high dynamic performance, and actuator configuration and steering. The rapid attitude tracking, as considered above, can only be guaranteed by certain particular actuator, which are expected to afford large attitude control torque or to absorb considerable momentum. From the perspective of efficiency and long operating life time, Control Moment Gyroscope (CMG) transcends other actuators such as RW for its torque amplification capability and momentum storage capacity. CMG has played a significant role in the restto-rest attitude control of agile spacecraft and large spacecraft, like the World View series satellites and the International Space Station. Therefore, it will be adopted in this paper to afford the required torque. The major challenge in using CMG is the inherent geometric singularity, where all the units torque are coplanar, meaning no output along its normal direction [11–13]. There is no existing steering logic or strategy that can handle the elliptic and hyperbolic singularity completely. The frustrating situation in this work will be treated by introducing a kind of new hybrid momentum exchanging device, CMG and Reaction Wheel (CMGRW), which is efficient to overcome the CMG singularity and RW saturation problems by exploiting the null motion among these actuators. Large amount of works have concentrated on the attitude tracking problem in the Earth staring work mode, which involved adaptive fault tolerant control [14], iterative learning control [6], and

robust adaptive control with unknown actuator nonlinearity [15], etc. While this work would divide the entire control process into 3 different phases, earth observation, attitude adjustment, and dynamic tracking to reach the accomplishment of video image of a space target. For different stage, particular attitude control algorithms are introduced, consisting of a PID saturation controller and backsteepping controller to satisfy the requirements. The remainder of this work is outlined as followings. Section 2 first describes the dynamic video imaging mission and establishes the orbit and attitude dynamic model of the spacecraft. Section 3 will focus on the desired attitude and angular velocity of the imaging spacecraft. Hybrid actuator configuration about CMG and RW as well as the CMG singularity problem will be discussed in section 4. Section 5 represents a novel strategy for the accomplishment of the dynamic imaging mission. The spacecraft attitude tracking control method and hybrid actuator steering strategy are designed in section 6. Numerical simulations are carried out in section 7 to demonstrate the efficiency of the proposed control method as well as the strategy. Finally, section 8 concludes the entire work of this research.

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2. Mission scenario and spacecraft dynamics

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The proposed dynamic video imaging mission has great potential applications for future space debris and malfunctioning satellite monitoring from space, which can provide more specific information of the target compared with the ground based observing system. The mission scenario is illustrated in the followings together with dynamic model of the spacecraft.

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2.1. Mission scenario

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Considering an earth observing spacecraft and a space moving target with orbit altitude difference around tens of kilometers, as illustrated in Fig. 1, the observing spacecraft is expected to perform dynamic imaging of the target during a time period when running overpass the target. The mission consequently becomes more sophisticated because of the orbit motion of the spacecraft and the target. High attitude tracking dynamic performance, what’s more, must be guaranteed in order to capture clear image due to the narrow field of view of the on board camera, such as 0.1 deg. More specific illustration is presented in Fig. 2 based on STK. The demonstration process consists of the following main steps. The spacecraft in Fig. 2(a) is performing earth observation mission; when target approaching observation window (Fig. 2(b)), the spacecraft would adjust its attitude and point at the target. During the dynamic imaging phase, the spacecraft is to track the target at any time node accurately in order to guarantee the video quality (Fig. 2(c) and (d)), where the attitude angular ve-

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2.2. Spacecraft dynamic model

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Fig. 1. Mission scenario of space target tracking.

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locity would approach a peak value about 4–5 deg/s, indicating the considerable agility requirement. Finally, the spacecraft would go back to the earth observation mode when the target runs out of the observation window (Fig. 2(e) and (f)). The major problems in the above processes include initial target acquisition, spacecraft operation mode switching, and precise attitude tracking, etc.

The relevant coordinate frames and quantities are defined as followings. Inertial Frame O X I Y I Z I : Its origin is located at the center of mass of the earth, X I axis points to the equinox and Z I axis is along the earth rotation axis. The Y I axis consequently is formed by the right hand principle. Perifocal Frame O X F Y F Z F : Its origin is located at the center of mass of the earth (one of the focus points of the elliptic orbit), and the O X F Y F plane is the orbit plane, where X F axis points to the perigee and Z F axis is normal to the orbit plane. Y F axis can be decided according to the right hand principle. Orbit Reference Frame O X O Y O Z O : Its origin is located at the center of mass of the spacecraft or the target, Z O axis points to the earth center, Y O is along the anti-symmetric direction of the orbit momentum and Z O axis is decided by the right hand principle. The imaging spacecraft and target orbit frames are denoted as O X S Y S Z S and O X T Y T Z T respectively. Spacecraft Body Frame O X B Y B Z B : The origin of O X B Y B Z B is located at the center of mass of the spacecraft, and its three axes lie along the corresponding principal axes of the spacecraft. Desired Imaging Frame O X D Y D Z D : This frame is used to calculate the desired attitude command of the imaging spacecraft, whose Z D axis points to the target, and X D axis is determined by Z D × −Y S , where Y S is the Y axis of the spacecraft orbit frame. Thus the Y D axis can be decided by the right hand principle. Considering a rigid spacecraft equipped with momentum exchanging devices as attitude control actuator, it’s easy to figure out the dynamic model according to the momentum conservation law,

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Fig. 2. Different steps of the dynamic imaging task, (a) earth imaging 1, (b) attitude adjustment 1, (c) dynamic imaging 1, (d) dynamic imaging 2, (e) attitude adjustment 2, (f) earth imaging 2.

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˙ + h˙ act + ω × ( I ω + hact ) T ext = H˙ + ω × H = I ω

(1)

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where H = I ω + h act is the momentum of the whole spacecraft, composed by the momentum of the spacecraft body I ω and the momentum of the actuator h act , in which I is the moment of inertia of the spacecraft and ω is the attitude angular velocity. T ext is the external torque. The spacecraft attitude kinematic model represented by quaternion ( Q = [q0 q1 q2 q3 ]T = [q0 q]T , where q0 is the scalar part, and q is the vector part) is,

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Q˙ =

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Ξ ( Q )ω −q1 −q2 −q3 q0 −q3 q2 ⎥ ⎥ q3 q0 −q1 ⎦ −q2 q1 q0

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(3)

⎪ ⎩ q˙ 0 = − 1 ωT q

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ς = [ς1 ς2 ς3 ]

for any vector defined as,

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Assuming the desired or command attitude and attitude angular velocity are Q d and ωd , then the error attitude are defined as, ∗ e= Q d⊗ Q ωe = ω − ωd

Q

(5)

where Q e = [qe0 qe1 qe2 qe3 ]T is the error quaternion, and Q ∗ is conjugate quaternion of Q . Quaternion multiplication is defined as,

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p0 ⎢ p1 P ⊗ Q =⎢ ⎣ p2 p3

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− p1 − p2 − p3 q0 ⎢ q1 ⎥ p0 − p3 p2 ⎥ ⎥⎢ ⎥ p3 p 0 − p 1 ⎦ ⎣ q2 ⎦ − p2 p1 p0 q3

(6)

Substituting Eq. (5) into Eq. (1), we can figure out the tracking dynamic model as,

 ⎧ ˙ e = − I ω˙ d − h˙ act − (ωd + ωe )× I (ωd + ωe ) + hact + T ext Iω ⎪ ⎪ ⎪ ⎨  1 1 q˙ e = qe0 E 3 + q× P ωe e ωe = 2 2 ⎪ ⎪ ⎪ ⎩ q˙ = − 1 qT ω e0 e e 2

(7)

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where E 3 ∈ R 3×3 denotes an identity matrix. Choosing vector X e = [qe , ωe ]T as the state vector, then we have

ω˙ e

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Fig. 3. Orbit frame and perifocal frame.

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=

+ q× )ωe − 1 ˙ d − (ωd + ωe )× [ I (ωd + ωe ) + hact ]] I [− I ω   0 + −1 ( T ext − h˙ act ) 1 (qe0 E 3 2

I

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3.1. Orbit mechanics

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q˙ e

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⎧ 1 ⎪ ⎨ q˙ = (q0 ω − ω× q)



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Considering the general situation that the boresight of the camera is mounted along one of the principal axes, such as Z B axis, thus the spacecraft is expected to rotate its attitude to point the camera to the target throughout the entire process. This section will present the procedure for desired attitude calculation in the dynamic imaging phase.

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3. Desired attitude calculation

Eq. (2) can also be written as,

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(2)

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q0 ⎢ q1 Ξ( Q ) = ⎢ ⎣ q2 q3



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where

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Assuming that the orbit elements of both the spacecraft and the target are known through ground station measuring. Without loss of generality, an elliptic orbit is assumed with the orbit elements OE = ( A , B , e , i , Ω, ζ, f ), where A and B are the semi-major axis and semi-minor axis, respectively, e is the eccentricity ratio, i is the orbit inclination, Ω is the right ascension of the ascending node, ζ is the argument of perigee, and f is the true anomaly. For a two-body problem, the orbit mechanism (see Fig. 3) can be represented in polar equation as

r=

h

2

1

=a

1−e

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2

(9)

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where r is the magnitude of the spacecraft position vector, h is the orbital angular momentum and μ is the earth gravity coefficient. Considering the orbit position as a function of time, then we have

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μ e cos f

1 + e cos f

⎧ 3 μ2 ⎪ ⎪ M = (1 − e 2 ) 2 t ⎪ e ⎪ ⎨ h3 M e = E − e sin E  ⎪ ⎪ ⎪ ⎪ ⎩ tan E = 1 − e tan f 2 1+e 2

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(10)

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where M e is the mean anomaly, and E is eccentric anomaly. At any given time, the position can be derived using Eqs. (9) and (10). And consequently the direction cosine matrix from the orbit frame to the inertial frame can be formed by

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R OI = R FI R OF

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(11)

R OF = R x (0.5π ) R y (−0.5π − f ) R FI = R z (Ω) R x (i ) R z (ω)

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where



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(12)

where R x (0.5π ) for example denotes the rotation along the x axis about 0.5π rad.

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3.2. Desired tracking attitude

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(8)

Therefore, the control target of the dynamic imaging phase is to ensure X e →0 to acquire high quality images/videos.

The desired attitude of the imaging spacecraft will be represented in this section. During the attitude tracking phase, only the roll and pitch axes will maneuver, while the yaw axis or the boresight axis will keep steady. The orbits relationship of both the

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Table 1 Orbit elements of spacecraft and target.

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OE

Imaging spacecraft

Space target

A, km B, km i, deg Ω , deg

, deg f 0 , deg

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6900 6670 96 120 30 131

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Let R DO = R, thus the corresponding Euler angles are

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Fig. 4. Dynamic imaging attitude determination.

= arctan

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spacecraft and the space target are illustrated in Fig. 4, where X S Y S Z S and X T Y T Z T denote the orbit frames of the two bodies respectively, R S and R T are the position vectors, R ST is the error position vector and X D Y D Z D depicts the desired body frame. At time node t, the position vectors of the two bodies are determined in the inertial frame, consequently the error vector is

R ST = R S − R T

ZD =

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(13)

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(18)

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where x denotes the two norm of vector x. Consequently the other two axes can be determined by

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(19)

≡0

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Due to the proper choice of the desired imaging frame, the feasible situation emerges where the yaw angle is guaranteed as zero during the whole dynamic imaging process, meaning no rotation along the boresight axis. This non-rotating boresight offers a stable condition for tracking and imaging observation and thus the quality of the videos or images could be guaranteed.

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XD · Y S YD · YS ZD · Y S

According to the definition of attitude and the rotation sequence, the desired attitude angular velocity can be figured out using Eq. (20)

⎡

ϕ˙

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⎡ ⎤

⎤

XD · ZS Y D · ZS ⎦ ZD · ZS

(16)

⎤⎡

1 0 0 cos θ sin ϕ ⎦ ⎣ 0 R = ⎣ 0 cos ϕ sin θ 0 − sin ϕ cos ϕ ⎤ ⎡ cos ψ sin ψ 0 × ⎣ − sin ψ cos ψ 0 ⎦ 0 0 1  cos ψ cos θ cos ψ sin θ sin ϕ − sin ψ cos ϕ cos ψ sin θ cos ϕ + sin ψ sin ϕ

⎤

0 − sin θ ⎦ 1 0 0 cos θ

sin ψ cos θ sin ψ sin θ sin ϕ + cos ψ cos ϕ sin ψ sin θ cos ϕ − cos ψ sin ϕ



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(20)

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ωd = ⎣ 0 ⎦ + R x (ϕ ) ⎣ θ˙ ⎦

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ϕ˙ 0

ϕ˙

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0 0

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1 0 = ⎣ 0 ⎦ + ⎣ 0 cos ϕ 0 0 − sin ϕ

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⎤⎡ ⎤

⎡

⎤

ϕ˙ 0 0 sin ϕ ⎦ ⎣ θ˙ ⎦ = ⎣ θ˙ cos ϕ ⎦ (21) cos ϕ 0 −θ˙ sin ϕ

Moreover, considering kinematics Eq. (2) or Eq. (3), the desired attitude angular velocity can also be calculated by

ωd = 2 q0 E 3 + q

− sin θ cos θ sin ϕ cos θ cos ϕ

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⎡ ⎤

⎡

⎤

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0

Substituting Eq. (19) into Eq. (20) and applying appropriate simplification, we have



(17)

0

⎤

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+ R x (ϕ ) R y (θ) R z (ψ) ⎣ 0 ⎦ ψ˙

⎡

Given the assumption that the Euler angle from the orbit frame to the body frame is (ψ, θ, ϕ ) with rotation sequence of Z → Y → X (or 3 → 2 → 1). The rotation matrix can be determined as

⎡

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0

0

(15)

Note that the above basis vectors are represented in the inertial frame, and they can be projected into the spacecraft orbit frame X S Y S Z S to calculate the desired attitude quaternion. Thus the corresponding cosine matrix is

=

 Z D × (−Y S )( X D · X S )

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ωd = R x (ϕ ) ⎣ 0 ⎦ + R x (ϕ ) R y (θ) ⎣ θ˙ ⎦

⎧ ⎨ X = Z D × (−Y S ) D  Z D × (−Y S ) ⎩ Y D = ZD × XD

⎡

0

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3.3. Tracking attitude angular velocity

(14)

 R ST 

XD · XS R DO = ⎣ Y D · X S ZD · XS

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R ST

⎡

⎪ ⎪ ⎪ ⎪ ⎩ ˙ ψ ≡ 0

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according to the definition of the desired imaging frame,

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⎧ XD · Y S Z D × (−Y S ) · Y S ⎪ ⎪ = arctan ψ = arctan ⎪ ⎪ X · X  Z ⎨ D S D × (− Y S )( X D · X S )

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Substituting Eq. (15) into Eq. (18) for angle ψ ,

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⎧ Y D · ZS ⎪ ⎪ ϕ = arctan ⎪ ⎪ ZD · ZS ⎨ θ = arcsin(− X D · Z S ) ⎪ ⎪ X ·Y ⎪ ⎪ ⎩ ψ = arctan D S XD · XS

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 × −1 ˙

q

(22)

Note that Eq. (22) is the same as Eq. (21) since the quaternion is derived from the direction cosine matrix in Eq. (16). In this work we assume that the orbit elements of both the imaging spacecraft and the space target are known and are listed in Table 1.

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Fig. 5. Desired attitude (a) Euler angle, (b) angular velocity.

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The attitude and attitude angular velocity command indicate the great demands of momentum storage and torque output. Among all the actuators for spacecraft attitude control, only CMG can afford such large torque and absorb the momentum without consumption any propellant. CMG is a kind of momentum exchanging device, which contrary to RW, generates control torque by rotating a flywheel, that’s to say, changing the direction of the momentum rather than changes the magnitude of momentum. Therefore, CMG will be used as the major device for control torque output in this work also. 4.1. Hybrid actuator

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4. Hybrid actuator configuration

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Using the parameters in Table 1, Fig. 5 illustrates the desired attitude and the corresponding angular velocity. As expected there’s no attitude motion along the yaw axis, which is steady throughout the process. About 140 deg attitude maneuver is required along the roll and pitch axes, the rotation in the pitch axis changes dramatically around 180 s corresponding to the situation when the imaging spacecraft flying over the target. The pitch angle passes rapidly across zero attitude, while the roll angle reaches the maximum value, resulting into the extreme attitude angular velocity on the Z axis (which can be explained using Eq. (21)). The considerable angular velocity definitely poses a great momentum exchanging requirement beyond the capacity of RWs and magnetic rods.

CMG, owing to the great torque amplification and amazing momentum storage capacity, has been widely used in large spacecraft or agile spacecraft control, like International Space Station. The major problem in CMG application is the inherent geometric singularity and despite the substantial steering logics, none of them is capable of guaranteeing singularity escape in all situations. In order to fully exploit the advantages of CMG, a kind of hybrid actuator consisting of CMG and RW is adopted in this work. Both the CMG singularity and RW saturation will be considered simultaneously through the cooperation of the two kinds of devices. The most studied CMG configuration is a pyramid CMG cluster, where four CMGs are mounted on a platform with skew angle β as illustrated in Fig. 6. Considering the possibility of singularity in all direction in three dimensional space, three orthogonal RWs are needed at least. The configuration of the hybrid actuator is given in Fig. 6. Projecting the momentum of all CMG and RW units into the spacecraft body frame without considering the influence of the momentum generated by gimbal movement, we have

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Fig. 6. Hybrid actuator configuration.

105 106

H act = H CMG + H RW = A s I s Ω CMG + C s I RW Ω RW

107

(23)

108 109

where A s = [as1 , . . . , as4 ] is the CMG flywheel momentum direction, C s = [c s1 , . . . , c s3 ] is the RW momentum direction, I s = diag ( I si ) and I RW = diag( I RWi ) are moment of inertia matrixes, and Ω CMG = [ΩCMG1 , . . . , ΩCMG4 ]T and Ω RW = [ΩRW1 , . . . , ΩRW3 ]T are the angular speeds of the CMG flywheel and the RW. Assuming that all the CMGs are identical and the flywheels share the same constant angular speed, and we define h0 = I si ΩCMGi = 1 Nms. Therefore,

110 111 112 113 114 115 116 117 118 119

H act = h0 A s + I RW Ω RW ⎡ − cos β sin δ1

120

− cos δ2 cos β sin δ3 cos δ1 − cos β sin δ2 − cos δ3 sin β sin δ1 sin β sin δ2 sin β sin δ3 ⎤ cos δ4 I RW1 ΩRW1 ⎦ cos β sin δ4 I RW2 ΩRW2 sin β sin δ4 I RW3 ΩRW3

= ⎣

121 122 123 124 125 126 127 128

(24)

129 130

where δi is the ith gimbal angle. Applying time differential to Eq. (24), we have,

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⎡

21 22 23

H˙ act

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

58 59 60 61 62 63 64 65 66

− cos β cos δ1 sin δ2 cos β cos δ3 − cos β cos δ2 sin δ3 = ⎣ − sin δ1 sin β cos δ1 sin β cos δ2 sin β cos δ3 ⎤ ⎡ δ˙1 ⎢ ˙ ⎥ ⎤ ⎢ δ2 ⎥ ⎢ δ˙3 ⎥ − sin δ4 I RW1 ⎥ ⎢ ⎦ ⎢ δ˙4 ⎥ cos β cos δ4 I RW2 ⎥ ⎢ ⎥ ˙ sin β cos δ4 I RW3 ⎢ ⎢ ΩRW1 ⎥ ⎣ Ω˙ RW2 ⎦ Ω˙ RW3   

δ˙ = J R Ω˙ RW = A Y˙

(25)

4.2. CMG singularity A CMG cluster would trap into a singularity state, only when the output torque of each CMG unit lies in a plane, meaning there is no output along its normal vector direction (singular direction). The corresponding Jacobian matrix satisfies,

rank( J ) = 2or det J J



=0

(26)

which means the row vectors of the Jacobian matrix are linear dependent, consequently a vector η ∈ R 3 and η = 0 exists and satisfies,

⎡

η

T⎣

⎤

J1 J2 ⎦ = 0 J3

(27)

(28)

96 97 98 99 100 101

(30)

102 103 104 105 106 107

(31)

108 109 110 111 112 113 114 115 117

±ati × agi

i =1

4 

agi × u s

εi

95

116

i =1

i =1

where J i is the ith row vector of matrix J . Eq. (27) can be represented by,

⎧ −η1 c β c 1 − η2 s1 = sβ c 1 ⎪ ⎪ ⎪ ⎨ η s − η c β c = sβ c 1 2 2 2 2 ⎪ η 1 c β c 3 + η2 s 3 = s β c 3 ⎪ ⎪ ⎩ −η1 s4 + η2 c β c 4 = sβ c 4

=

asi =

92 94

Wie also found the above singular angle sets using Binet– Cauchy Identity [13]. Moreover, using the CMG singularity mechanism and structure character, the singular momentum (H s ) along the given direction (u s ) can be determined as,

Hs =

89

93

  ⎧ −2c 2 β + c β t 1 − t 1t 2 ⎪ ⎪ ⎨ δ3 = arctan cβ + t2  2  ⎪ 2c β + c β t 2 + t 1 t 2 ⎪ ⎩ δ4 = arctan cβ − t1

4 

88

91

(29)

and

4 

87

90

⎧ sβ(c β − t 1 ) ⎪ ⎪ ⎨ η1 = − 2 c β + t1t2 sβ(c β + t 2 ) ⎪ ⎪ ⎩ η2 = − 2 c β + t1t2

are the control variables, J is Jacobian Matrix. The steering logic is to solve the projection from H˙ act ∈ R 3 to Y˙ ∈ R 7 and to guarantee singularity avoidance and RW desaturation at the same time, which will be discussed in section 6 in detail.

T

⎧ −η1 c β − η2 t 1 = sβ ⎪ ⎪ ⎪ ⎨ η t − η c β = sβ 1 2 2 ⎪ η c β + η 1 2 t 3 = sβ ⎪ ⎪ ⎩ −η1 t 4 + η2 c β = sβ

where t i ≡ tan δi . Thus it’s easy to figure out all the singular gimbal angle sets because only two of four above equations are independent. For example, for all δ1 and δ2 , using the former two equations in Eq. (29), we have,

T

Ω˙ RW = Ω˙ RW1 Ω˙ RW2 Ω˙ RW3 ∈ R 3



86

where c i ≡ cos δi and si ≡ sin δi , and c β ≡ cos β and sβ ≡ sin β . Without loss of the generality, supposing c i = 0, and then we have,

T where δ˙ = δ˙1 δ˙2 δ˙3 δ˙4 ∈ R 4 ,

56 57

85

Fig. 7. Singular momentum surface, (a) {εi } = {−1, 1, 1, 1}, (b) {εi } = {1, 1, 1, 1}.

20

agi × u s 

118 119

× agi , u s = agi

(32)

where asi is the ith flywheel spin axis, agi is the ith gimbal axis, ati is the ith unit output axis and εi = ±1. The most significant cases are that εi = 1 and only one is −1, the corresponding singular momentum surface is shown in Fig. 7. The momentum envelop is formed by combining Fig. 7(a) and Fig. 7(b), every point on which corresponds to a singular state. Although the singular surface structure and momentum envelop would not change greatly, because of the small RWs adopted in the work, the cooperation will help to escape the singularity effectively between CMG and RW.

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Fig. 8. Different phases of dynamic imaging tracking mission.

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5. Attitude tracking strategy

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

To accomplish the dynamic imaging tracking task, the spacecraft should switch from one work mode to another. However, as illustrated in Fig. 5(a), the initial state in the tracking mode is quite different form that in the earth observation phases and consequently a transmission phase is required for the spacecraft to adjust its attitude. Considering the general scenario, before capturing the target, the imaging spacecraft is executing other tasks (sun as earth imaging) and then the spacecraft is commanded to monitor the target in appropriate time range. Since the target’s orbit parameters could be determined, and also in order to fully utilize the short imaging period, the spacecraft is designed to keep the previous attitude at the beginning of the imaging window. Thus, prior to the imaging phase, it is advised that the spacecraft should maneuver its attitude rapidly in advance and then keep that until the beginning of the imaging phase. Under this philosophy, the spacecraft is always being in the “ready state”, and it can execute the imaging task immediately rather than rotating its axes during the imaging phase. The proposed strategy in this work is represented in Fig. 8, where the capital letters denote the time nodes and lines in different types indicate different phases in the dynamic imaging process. In order to fully utilize the relatively short acquisition period, the spacecraft would be commanded to maneuver in advance at a kind of preparation state, and hold this attitude till the tracking phase, rather than switch the work mode from one to other immediately. In Fig. 8, the whole process is divided into three phases, earth imaging, attitude adjustment and holding, and dynamic imaging, where the spacecraft should accomplish different attitude maneuver tasks. Before time node A, the spacecraft is observing the earth or executing other missions, which is considered as the earth imaging mode. Once the target is approaching the observation window of the spacecraft, the control system would observe and calculate the orbit of the target. Simultaneously, the spacecraft would predict the beginning of the dynamic imaging phase. Hence, the spacecraft maneuvers rapidly during the time range A and B to adjust its attitude and hold the attitude till time node C. The imaging process would last from C to D. Note that because the spacecraft has reached the desired attitude at C, it will track the target smoothly until the target runs out of the observation window at time node D. Finally, the spacecraft will return to the Earth observing mode. Table 2 summarizes the above different phases. Comment 1: During the mission process, the imaging spacecraft would experience different operation modes, which correspond to particular object and control requirements. The adjustment phase

Table 2 Working schedule of the spacecraft.

86 87

Mode

Time node or period

Object

Requirement

88

Earth imaging Attitude adjustment Attitude holding Dynamic imaging

A A–B B–C C–D

Earth Target at C Target at C Target

/ Rapid maneuver Attitude holding Tracking

89 90 91 92 93

would sever as a preparation or initial condition for the spacecraft to capture the target. Thus the adjustment should be accomplished rapidly without any oscillation or overshoot, while preciseness is required and is necessary during the dynamic imaging stage. The spacecraft attitude controller should satisfy all the above demands.

94 95 96 97 98 99 100

6. Attitude controller and hybrid actuator steering logic

101

The whole control architecture will be built in this section with the consideration of different control targets in different phases. A combined controller, consisting of saturation control method and backsteepping control method, is proposed to satisfy the control requirement. What’s more, null motion among the actuators will be exploited, which is feasible for CMG singularity escape as well as RW desaturation.

104 105 106 107 108 109 111

Section 5 has described different control requirements for different phases. Firstly, the spacecraft will perform large angle maneuver from three-axis stabilization mode without tracking the attitude angular velocity during the adjustment phase, as shown in Fig. 5(a). To restrict the attitude angular velocity and acceleration due to the capability of the attitude control actuator, saturation control method is introduced



103

110

6.1. Attitude control architecture

u c = −satU kI sat L (qe ) + c I ω

102



112 113 114 115 116 117 118 119 120

(33)

121 122

where u c is control torque, U and L are the control constraints which can be determined according to the actuator output capacity, k and c are the control gains. More feasible, the control limit L can be modified as following to overcome the overshoot and chattering problem,

L=

c k



min

4a|qe |, |ω|max



123 124 125 126 127 128

(34)

where a is the spacecraft attitude maneuver acceleration. The saturation function in Eq. (33) for any vector x ∈ R n is defined as,

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 1 2

sat X (x) =

X xx

∞

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24



29 30 31 32



q˙ e q¨ e



=

35 36 37 38 39 40 41 42 43 44 45 46 47

q˙ e + 2 P −1 q¨ e





q˙ e

1 2

+

˙ d − (ωd + 2 P −1 q˙ e )× [ I (ωd + 2 P −1 q˙ e ) + hact ] − 2I P˙ P I −1 [− I ω

1 2

P I −1



0



−1

q˙ e ]

(37)

T ext − h˙ act

Here the new state is chosen as X = [x1 x2 x3 ]T ,

(38)

x2 = q e ⎪ ⎪ ⎩ x3 = q˙ e Then the system model can be represented as,

⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 = x3 −1 ˙

×

⎪ ˙ d − (ωd + 2 P qe ) x˙ 3 = P I [− I ω ⎪ ⎪ 2 ⎪ ⎩ −1 × [ I (ωd + 2 P −1 q˙ e ) + hact ] + ( T ext − h˙ act )] − P P˙ q˙ e (39)

49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

(43)

69

substituting Eq. (43) into Eq. (42), V˙ 1 = exp(−0.5κ xT1 x1 − 0.5xT2 x2 ) × xT2 ( z − k1 x2 ). Furthermore,

70

1

V2 = V1 +

2

71 72 73

z2

(44)

74 75

similarly

76





77

V˙ 2 = exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 (−k1 x2 + z ) + z z˙

  = −k1 exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 x2   + exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 z + z T z˙   = −k1 exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 x2

   + z T z˙ + exp −0.5κ xT1 x1 − 0.5xT2 x2 x2

78 79

(45)

80 81 82 83 84

advisably



z˙ = − exp −0.5κ

xT1 x1

− 0.5xT2 x2



85

x2 − k 2 z

(46)

therefore V˙ 2 = −k1 exp(−0.5κ xT1 x1 − 0.5xT2 x2 )xT2 x2 − k2 zz T ≤ 0,

86 87 88 89

x˙ 3 = −k1 x˙ 2 − κ x˙ 1 + z˙

91

  = −k1 x˙ 2 − κ x˙ 1 − exp 0.5κ xT1 x1 + 0.5xT2 x2 x2 − k2 (x3 + k1 x2 + κ x1 )   = −k1 x˙ 2 − κ x˙ 1 − exp 0.5κ xT1 x1 + 0.5xT2 x2 x2 − k2 x3 − k1k2 x2 − k2 κ x1

= −k2 κ x1 − (k1 + k2 )x3 − κ + k1k2   + exp 0.5κ xT1 x1 + 0.5xT2 x2 x2

92 93 94

(47)

˙d + uc = I ω



−1 ˙

× 

−1 ˙



qe I ωd + 2 P qe + hact  −2I P −1 (k1 + k2 )q˙ e     + k1k2 + exp −0.5κ xT1 x1 − 0.5xT2 x2 + κ qe   −1 + k2 κ qe dt + 2I P˙ q˙ e

ωd + 2 P

95 96 97 98 99



100 101 102 103 104 105

(48)

106 107 108 109 110

−1

48 50

68

α = −k1 x2 − κ x1

consequently according to Eq. (39), using Eq. (47) the attitude controller is designed as,

 ⎧ ⎪ ⎪ ⎨ x1 = qe dt

1

67

90

(36)

Substituting Eq. (36) into Eq. (8) and we have,

33 34

−1

α is set as

thus the vector

meaning that the system is stable. And

ωe = 2 P −1 q˙ e ω˙ e = 2 P˙

26 28

(35)

where X is the saturation value and x∞ denotes the infinite norm of vector x. Obviously, Eq. (35) doesn’t change the direction for any vector x. Under the above saturation method, the spacecraft would experience three stages when accomplishing an attitude maneuver task, including acceleration motion, uniform motion and deceleration motion, during which the angular velocity is always under the constraints. Once the attitude adjustment is finished, the attitude would remain until the beginning of the imaging phase. In the imaging phase, both the spacecraft attitude and the attitude velocity should be exactly considered for precise control to guarantee the video quality. The above saturation control algorithm aiming at large angle maneuver is unsuitable for precise attitude tracking. The backstepping control method, which divides a dynamic system into several cascading subsystems [15], could take every state into consideration and has been used in attitude tracking control [16–19]. Therefore, we adopt the backsteepping control method to resolve the problem in this paper. For the error quaternion in Eq. (7), we have,

25 27

x∞ ≤ X x∞ > X

x

9

First we define the parameter as

(40)

where α is an intermediate parameter, and z can be considered as the control input for the subsystem x˙ 1 and x˙ 2 . Choosing the Lyapunov function as a Gauss function in terms of x1 and x2 ,

V 1 = 1 − exp −0.5κ xT1 x1 − 0.5xT2 x2

infinite when the system error state is approaching zero. So the above control law can be modified as,

˙d + uc = I ω

z = x3 − α ( x1 , x2 )



Note that u c = −h˙ act , which is generated by the hybrid momen−1 −1 would be tum exchanging devices. Because P˙ = (q˙ e0 E 3 + q˙ × e )



ωd + 2 P

−1 ˙

qe

×  I

ωd + 2 P

−1 ˙



qe + hact

− 2I P −1 (k1 + k2 )q˙ e   

+ k1k2 + exp −0.5κ xT1 x1 − 0.5xT2 x2 + κ qe 

 −1 + k2 κ qe dt + 2I (q˙ e0 + σ ) E 3 + q˙ × q˙ e e



112 113 114 115 116 117 118

(49)

119 120 121 122 123

(41)

where κ is a positive scalar. Applying time differential to Eq. (41) and considering Eqs. (39) and (40) meanwhile,

  T  κ x1 x˙ 1 + xT2 x˙ 2    = exp −0.5κ xT1 x1 − 0.5xT2 x2 κ xT1 x2 + xT2 x3   = exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 (κ x1 + x3 )   = exp −0.5κ xT1 x1 − 0.5xT2 x2 xT2 (κ x1 + z + α )



111

V˙ 1 = exp −0.5κ xT1 x1 − 0.5xT2 x2

(42)

where σ is a small positive scalar. When the system is reaching the stable state, qe and its differential would become zero, therefore the modification does not affect the performance of the spacecraft. Comment 2: Two unique controllers are designed for different control demands in different operation modes. The former controller guarantees the smooth maneuver and constrains the maneuver attitude angular velocity, while Eq. (49) cannot. However, the latter controller aims to track the target attitude with high accuracy.

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6.2. Null motion steering logic

67

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

68



min

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1 2

T

δ˙ P 1 δ˙

subject to J δ˙ = u c



min

1

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

71 72 73 74 75

(50)

76 77

2

78 79 80 81 82 83 84

T Y˙ P 2 Y˙

subject to A Y˙ = u c −1

(51)

7×7

where P 2 = W 2 ∈  is the corresponding weighted matrix. The corresponding weighted pseudo-inverse solution can be figured out as,

Y˙ = W 2 A

T



AW 2 A

 T −1

uc

(52)

Owing to the existence of the RW in the hybrid system, the system matrix is always full rank, rank( A ) = rank([ J R ]) = rank( R ) ≡ 3, which guarantees the solution of the above steering logic. In practice, the introduction of the matrix R however couldn’t influence the whole system dramatically, because it’s quite “small”. Thus the null motion among the hybrid system is discussed below. For the hybrid actuator system, the dynamic performance, which is the function of CMG singularity and RW saturation, is designed as,

37 38

70

where P 1 ∈ 4×4 ( P 1−1 = W 1 ) is a positive definite diagonal weighted matrix. The solution is a typical weighted pseudo-inverse solution or weighted pseudo-inverse steering logic, which cannot deal with the singularity problem. For the hybrid system CMGRW, the following problem is considered,

18 19

69

The above subsection has determined the attitude control law, while the actuator would not always guarantee the torque output especially accurately for CMG cluster. The most existing CMG steering logics fail to handle the singularity problem, but perform well in some situations. Considering the following optimal problem,



2 2 S act = 1 − r1 exp −λ1 S CMG − λ2 S RW −1

− r2 exp

2 −λ1 S CMG



− r3 exp

2 



2 −λ2 S RW

−1

2 

(53)

where r i and λi are positive parameters satisfying sum(r i ) = 1, S CMG and S RW are singularity and saturation indexes of CMG and RW, respectively,

⎧ ⎪ ⎪ ⎨ S CMG =

T

det( J J )

(54)

Y˙ N = ρ E 7 − W 2 A

T



AW 2 A

 T −1

d=

∂ S act ∂Y

T

 =

∂ S act ∂δ

∂ S act ∂Ω RW

86

T

87

(56)

88 89 90

And then

91

∂ S act ˙ S˙ act,N = YN ∂Y  T  − 1  ∂ S act ∂ S act =ρ E 7 − W 2 A T AW 2 A T A W2 ∂Y ∂Y

92 93 94

(57)

95 96

T −1

The matrix [ E 7 − W A ( F W F ) A ] W has been proven as positive semi-definite (see Appendix A), thus

97

S act ∈ [0, 1]& S˙ act,N ≥ 0

100

T

(58)

Above results indicate that the performance index would increase under null motion as expected, indicating proper actuator states. The null motion vector can be found in Ref. [18]. And consequently the steering logic is composed as,

Y˙ = Y uc + Y˙ N = W A T ( AW A T )−1 u c + ρ [ E 7 − W A T ( AW A T )−1 A ] W d (59)

98 99 101 102 103 104 105 106 107 108 109 110

which is always existed and feasible for singularity escape. In the future, consensus theory would be used to optimize the momentum management of the hybrid devices [20].

112 113 114 115

which also have been normalized. Fig. 9 depicts the performance surface of Eq. (53) (the parameters are provided in section 7), where the minimum value point corresponds to the singular and saturated states of the system, and therefore S act = 0. Another two extreme value areas are dominated by the CMG singularity and RW saturation, respectively. For such a redundant CMGRW configuration, null motion, which generates zero net momentum change or torque on the spacecraft but changes the actuator state, is used to improve the efficiency of CMG. In the CMGRW system, null motion can be determined by



111

∈ [0, 1]

[det( J J )]max Ω RW 2 ⎪ ⎪ ⎩ S RW = ∈ [0, 1] Ω RW,max 2 T

85

Fig. 9. Performance surface of the hybrid actuator.



A W 2d

(55)

where ρ is the null motion strength and d is the null motion vector. Obviously, A Y˙ N ≡ 0. The null motion vector, which determines the direction of the actuator movement, is chosen using the gradient method,

7. Numerical simulation

116 117

The control architecture is presented in Fig. 10. According to the relative motion of the spacecraft and target, the system would switch to different controllers as discussed. The hybrid actuator would afford corresponding torque, where the singularity and saturation are both considered through the CMG and RW cooperation. Several simulations in this section will be carried out to demonstrate the dynamic imaging mission with particular initial conditions, including actuator singularity and saturation. The major difficulty in using CMG is the inherent geometric singularity of which the internal elliptic is the toughest problem. Therefore, the advised and well known strategy is to execute the simulation in the neighborhood of singularity directly, which is serious for CMG application. Both the hyperbolic and elliptic singularities will be considered to illustrate the efficiency of the hybrid system. Different simulations are summarized in Table 3.

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Fig. 10. Simulation diagram of the dynamic tracking control.

19 20 21 22 23 24

85 86

Table 3 Description of different simulation scenarios.

87

Scenario

Actuator

Description

Controller

Initial condition

1

CMGRW

Single controller test

Backsteepping controller

Ω RW = (0, 0, 0) rad/s δ = (0, 0, 0, 0) deg Euler = (0, 0, 0) deg

25 26 27

Combined controller

Ω RW = (10, 10, −15)π rad/s δ = (−105, 10, 95, 10) deg Euler = (0, 0, 0) deg

92

3

CMGRW

Test for CMG singularity escape at elliptic singularity

Combined controller

Ω RW = (10, 10, −15)π rad/s δ = (−105, 10, 95, 10) deg Euler = (0, 0, 0) deg

95

Test for CMG singularity escape at hyperbolic singularity and RW desaturation

Combined controller

Ω RW = (25, 25, −28)π rad/s δ = (−105, 10, 95, 170) deg Euler = (0, 0, 0) deg

98

4

CMGRW

33 34 35 37 38

43 44 45 46 47 48 49 50

Parameter

53

Spacecraft moment of inertia

I

RW moment of inertia CMG flywheel momentum Actuator limit (1) Actuator limit (2) Control gains (1) Torque limit Maximum attitude velocity Maximum attitude acceleration Weighted matrix Scalars Null motion strength

I RW h0

Ω RW, max δ˙max , Ω˙ RW,max c, k U

ωmax a W 1, W 2 κ, σ

ρ

Disturbance torque

Td

Sensor accuracy

/

54

Magnitude

⎡

59 60 61 62 63 64 65 66

99 100

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121

56 58

97

104

⎤

10 0.02 0.01 ⎣ 0.02 15 −0.01 ⎦ kg m2 0.01 −0.01 20 diag(0.005, 0.005, 0.005) kg m2 0.5 Nms (50, 50, 50) · 2π rad/s (1, 1, 1, 1) rad/s, (10, 10, 10) rad/s2 8, 4 0.5 Nm 2 deg/s 0.25U/ J x , 0.25U/ J y , 0.25U/ J z diag(5, 5, 5, 5), diag(5, 5, 5, 5, 500, 500, 500) 0.005, 10−5 200 exp(−10S CMG ) ⎡ ⎤ 3 cos(10ωt ) + 4 sin(3ωt ) − 10 ⎣ 1.5 sin(2ωt ) + cos(5ωt ) + 15 ⎦ · 10−5 Nm 3 sin(10ωt ) + 8 sin(4ωt ) + 10 attitude: 0.005 deg attitude rate: 0.001 deg/s

55 57

96

103

Symbol

51 52

94

102

41 42

93

101

Table 4 Numerical simulation parameters.

39 40

91

Pyramid CMG control test

31

36

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CMG

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The above four designed scenarios aim at different targets. The advantages of the combined controller, the tracking strategy proposed in section 5, as well as the singularity escape capability of the hybrid actuator compared with the pyramid CMG cluster will be demonstrated through these simulation as well as discussed above. During the attitude adjustment phase, the spacecraft is expected to maneuver to predetermined attitude in less than 50 seconds (corresponding to the attitude command (70.09, 64.57, 0) deg). The other simulation parameters are given in Table 4.

Scenario 1: The backstepping controller is adopted uniquely in this simulation to accomplish the mission throughout the whole process. In order to show the characteristics of this method in the attitude adjustment phase, the CMG singularity and RW saturation is not considered. Thus, the initial states of the actuator are set as Ω RW = (0, 0, 0) rad/s and δ = (0, 0, 0, 0) deg, which are feasible. The parameters of backstepping controller are k1 = 10 and k2 = 0.1. The simulation results are illustrated in Fig. 11. As shown in Fig. 11, the spacecraft maneuvers rapidly to accomplish the attitude adjustment task in about 50 s. The angular

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velocity, unexpectedly, has reached 10 deg/s, beyond the agility of most spacecraft. The large angular velocity also poses a problem of momentum exchanging requirement for the actuator in the initial time period as illustrated in Fig. 11(b), which would lead to the actuator saturation. (The reason why the spacecraft could accomplish the task is that the CMG flywheel momentum is set as 2.5 Nm s. While in the following scenarios the magnitude is just 0.5 Nm s.) Therefore, the saturation controller is used to constraint the maximum slew angular velocity within a given value, reducing the requirements on the control system. Scenario 2: In this simulation, the proposed strategy as well as the combined control method are applied, while the spacecraft is actuated only by a pyramid CMG cluster together with pseudo-inverse logic. It aims to show the shortcomings of CMG system in torque output and momentum tracking. The initial states of the actuator are set as Ω RW = (10, 10, −15)π rad/s and δ = (−105, 10, 95, 10) deg, which is apparently located in the neighborhood of an elliptic singular point (−90, 0, 90, 0) deg. The parameters of backstepping controller are k1 = 20 and k2 = 2. The corresponding results are presented in Fig. 12. As shown in Fig. 12, the CMG encountered a singularity from the singularity neighborhood around 3.6 s and the spacecraft unexpectedly was out of control, meaning the space target cannot be captured. The gimbal angle set reached at (−83.93, 212.0, 180.8, −26.47) deg, and all the CMG output vectors are coplanar, corresponding to η = (1.47, 0.17) in Eq. (30). As there is no other strategy to escape this frustrating state, the CMG system would trap into the singularity forever, which is a serious problem in practical application. Scenario 3: The same initial conditions of the CMG gimbal angle and RW angular velocity as scenario 2 are considered in this scenario again, in which the hybrid actuator as well as the null motion steering logic is introduced. This simulation can be compared with scenario 2, which is demonstrated to be feasible for singularity escape. The parameters of backstepping controller are

k1 = 20 and k2 = 2. And the simulation results are shown in Fig. 13 and Fig. 14. Avoiding the singular situation, the hybrid actuator overcomes the issue in scenario 2, and the attitude error is less than 0.008 deg for the dynamic imaging phase. Under the saturated control method for the attitude adjustment phase, the spacecraft slew rate is constrained within 2 deg/s as shown in Fig. 13(b), which is quite small compared with that in scenario 1. Around at 180 seconds, the dramatically changing attitude of the imaging spacecraft (as illustrated in Fig. 5) causes an obvious attitude angular velocity error about 0.008 deg/s in Fig. 13(b). Also the large angular velocity has resulted into great momentum exchanging requirement, corresponding to the smaller performance index of the actuator in Fig. 14(b), because the actuator momentum would approached the momentum envelop. The cooperation of the CMG and RW can be found in Fig. 14(a). During the initial phase and around 180 s, the RW group accelerates apparently to help CMG escape the singularity. What is more significant is that the CMGRW system has represented particular work mode in the attitude adjustment and dynamic imaging phases, torque generation and momentum absorption. The considerable torque in initial stage helps the spacecraft maneuver rapidly, while in dynamic imaging stages the torque requirement is not so strong but the momentum approach the largest value in Fig. 14(b). Scenario 4: In this scenario, both the CMG singularity and the RW desaturation are considered. As shown in the above simulations, it is hard for the RW became saturated during the control process, which just provide more control degrees rather than afford the main control torque. Thus the strategy in this paper to cope with the RW saturation is to utilize the CMG to absorb the RW momentum, which in comparison with CMG is relatively small. Therefore, the initial conditions are set as Ω RW = (25, 25, −28)π rad/s, δ = (−105, 10, 95170) deg, which correspond to a near saturated RW state and a singular CMG hyperbolic singularity. The parameters of backstepping controller are k1 = 20

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and k2 = 2. The simulation results are illustrated in Fig. 15 and Fig. 16. The spacecraft has experienced acceleration motion, uniform motion, and deceleration motion to accomplish the attitude adjustment rapidly, as shown in Fig. 15. With the cooperation of RW, the CMG cluster escapes the singularity with a large performance

index throughout the entire dynamic tracking process (as shown in Fig. 16). At 400 s the desaturation program is enabled, and the RW is desaturated to a feasible state in less than 10 s. The performance index paths of scenarios 3 and scenario 4 are illustrated in Fig. 17, from which one can clearly find the cooperation relationship and performances of the CMG and RW. From the

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Fig. 15. Part 1 of simulation result of scenario 4, (a) attitude tricking error, (b) angular velocity error.

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unfeasible states, the actuator rapidly has escaped the singularity. Also, the paths in three dimensional space of scenario 4 depicts that the CMG cluster first escaped the singularity rapidly and then the RW desaturation was treated, as designed. Those feasible states suggests the spacecraft would maneuver and tack smoothly and accurately and also sever as a kind of practical initial state for next mission.

The simulations in this section have focused on different purposes. Comparing scenario 1 with 2, a single controller for all the phases has been demonstrated to be unfeasible, which would result into large torque and momentum demands. From the comparison of scenarios 2 and 3, it’s clear that the hybrid actuator as well as the null motion can deal with the CMG singularity, which is beyond the capability of the CMG cluster. The desaturation effect

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has demonstrated that the combined controller and hybrid actuator can guarantee the attitude tracking dynamic performance. The proposed method has great potential application for future space target such as debris and malfunction satellite monitoring activities.

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Acknowledgements

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The authors would like to express their acknowledgment for the support provided by the National Key Research and Development Plan (No. 2016YFB0500901), the Fundamental Research Funds for the Central Universities (No. 2016083) and the Natural Science Foundation of China (No. 61673206).

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Fig. 16. Performance index of CMG and RW of scenario 4.

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Fig. 17. Actuator performance paths of scenarios 3 and 4. The symbols “O” and “” represent the initial and terminal points respectively.

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has been shown in scenario 4. Above all, the spacecraft can track the command signals with high dynamic performance for various initial conditions considering the CMG singularity avoidance and RW desaturation.

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8. Conclusion

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E n − W A T AW A T 1 2

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AW W

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A novel hybrid actuator configuration and combined control strategy are proposed in this research for dynamic attitude tracking of a space moving target, which is different from the Earth staring imagining mission. The great agility and preciseness are guaranteed by a proposed tracking strategy, which divides the whole process into three different phases, including earth stabilization, attitude adjustment and holding, and dynamic imaging. On condition of the different requirements in corresponding control phases, a combined controller consisting of saturation controller and backstepping method is designed, actuated by mixed momentum exchanging devices, Control Moment Gyro (CMG) and Reaction Wheel (RW). Null motion is then derived among the CMG and RW to achieve the cooperation between the two actuators, resulting CMG singularity avoidance and RW desaturation at the same time. Numerical simulations based on different initial conditions

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which suggests that the matrix E n − W A T ( AW A T )−1 AW is non-negative definite. Consequently, we can effortlessly figure out that the matrix A [ E n − W A T ( AW A T )−1 A ] W is semi-positive definite.

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[ E n − W A ( AW A ) AW ] 1 1 × [ E n − W 2 A T ( AW A T )−1 AW 2 ] 1 1 = E n − 2W 2 A T ( AW A T )−1 AW 2 1 1 1 1 + W 2 A T ( AW A T )−1 AW 2 W 2 A T ( AW A T )−1 AW 2 1 1 = E n − W 2 A T ( AW A T )−1 AW 2

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Because of W is a diagonal matrix, then the matrix [ E n − W A T ( AW A T )−1 AW ] W can be rewritten as

=W

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Appendix A

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None declared.

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Conflict of interest statement

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