Adaptive fuzzy finite-time control for spacecraft formation with communication delays and changing topologies

Accepted Manuscript

Adaptive Fuzzy Finite-time Control for Spacecraft Formation with Communication Delays and Changing Topologies Jianqiao Zhang, Dong Ye, Ming Liu, Zhaowei Sun PII: DOI: Reference:

S0016-0032(17)30207-7 10.1016/j.jfranklin.2017.04.018 FI 2971

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

23 October 2016 3 March 2017 16 April 2017

Please cite this article as: Jianqiao Zhang, Dong Ye, Ming Liu, Zhaowei Sun, Adaptive Fuzzy Finitetime Control for Spacecraft Formation with Communication Delays and Changing Topologies, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.04.018

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Adaptive Fuzzy Finite-time Control for Spacecraft Formation with Communication Delays and Changing Topologies Jianqiao Zhang, Dong Ye*, Ming Liu, Zhaowei Sun

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Research Center of Satellite Technology, Harbin Institute of Technology, Harbin, 150001, P.R. China.

Abstract

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This paper aims to solve the finite time consensus control problem for spacecraft formation flying (SFF) while accounting for multiple time varying communication delays and changing topologies among SFF members. First, in the presence of model uncertainties and external disturbances, the coupled dynamics of relative position and attitude are derived based on the Lie group SE(3), in which the position and attitude tracking errors with respect to the virtual leader whose trajectory is computed offline are described by exponential coordinates. Then, a nonsingular fast terminal sliding mode (NFTSM) constructed by the exponential coordinates and velocity tracking errors is developed, based on which adaptive fuzzy NFTSM control schemes are proposed to guarantee that the ideal configurations of the SFF members with respect to the virtual leader can be achieved in finite time with high accuracy and all the aforementioned drawbacks can be overcome. The convergence and stability of the closed-loop system are proved theoretically by Lyapunov methods. Finally, numerical simulations are presented to validate the effectiveness and feasibility of the proposed controllers.

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Keywords: Consensus control, Communication delays, Lie group SE(3), Adaptive fuzzy NFTSM control schemes, High accuracy.

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1. Introduction

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Spacecraft formation flying (SFF) is a technology realizing the functionality of a large complex spacecraft with a set of smaller, less-expensive, and cooperative ones, with several benefits including (1) system robustness by eliminating the single-point failures; (2) flexibility because different mission goals can be satisfied by selecting different configurations. SFF provides several applications such as scientific experiments, monitoring of the Earth, deep space exploration, on orbit servicing and maintenance of spacecraft, stereo imaging, and spacecraft rendezvous and docking[1–4]. These applications bring forward high synchronization accuracy requirements for the SFF, which cannot be satisfied by traditional control methods. The reason is that the control of attitude and orbit for spacecraft is described independently which makes meeting the accuracy requirements of orbit and attitude simultaneously be impossible for the coupling characteristics of spacecraft motion [5]. To solve these questions, modeling of SFF is the first significant design basis. In recent years, six degree-of-freedom (6-DOF) motion coordinated control, considering both three-DOF relative translational motion and three-DOF rotational motion, has drawn increasing attention from experts. In [3], the coupled relative translational and attitude motion for the leader-follower spacecraft formation was modeled using the vectrix formalism, and in this ✩ Corresponding author Email addresses: [email protected](Jianqiao Zhang); [email protected](Dong Ye); [email protected](Ming Liu).

Preprint submitted to Nuclear Physics B

April 21, 2017

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framework the mutual couplings of the translational motion and the attitude one induced by their gravitational interaction were considered. In [6], the translational and attitude motion were modeled without taking the coupled terms into consideration. In [7], a Hamiltonian approach was developed to model spacecraft motion relative to a circular reference orbit, and the Hamiltonian formulation facilitates the modeling of high-order terms and orbital perturbations. In [8], a six-DOF follow-up control of both relative orbit and chaser attitude using osculating reference orbit was designed. More recently, in [9], the 6-DOF relativemotion was modeled by combining the relative translational motion and the attitude one represented by quaternion, with taking the dynamical coupling inside the model. Though the 6-DOF relative-motion models in these studies are considered in a united framework, the separate describing of position and attitude makes the controller design and calculation be complicated. In order to describe the 6-DOF relative-motion in a united framework and facilitate the controller synthesis work, constructing coupled dynamics by the relative position and attitude described on the Lie group SE(3) has been an effective tool and received increasing attention in [2], [10–15]. In [2], a sliding mode control scheme on the coupled dynamics on SE(3) was presented to achieve the desired formation with respect to the virtual leader whose trajectory was computed offline. Meanwhile, special attention was paid for the actuator saturation such that the control thrust was restricted to a bounded magnitude. In [14], a Lyapunov based feedback control law on SE(3) was designed to make each follower track a desired relative configuration with respect to the virtual leader and a collision avoidance scheme was implemented to make sure each two followers be collision-free. Although the aforementioned control schemes were feasible on a SFF configuration control, they can only guarantee asymptotical convergence of the system states, which means that the control objective is obtained as time tends to infinity. Apart from this, the effects of uncertain parameters and unknown external disturbances which may seriously affect the control performance and increase the control effort haven’t been considered neither. Moreover, they didn’t take the consensus problem among the followers into consideration to make the system perform as a whole, which can’t be ignored for SFF systems. Because spacecraft formation control including formation establishment and maintenance should be satisfied to achieve the mission goals, in which the consensus control among the followers plays an important role for it can not only improve the control precision of formation maintenance but also enhance the system’s robustness in case of some of the followers losing the information of the leader. All the issues mentioned above show that there exists few literature solving the finite-time-highaccuracy control problem of SFF system modeled by Lie group SE(3) in the presence of structured and unstructured uncertainties and considering interconnection among SFF members to promise high accuracy of formation maintenance. This problem is absolutely a challenging problem, not only because the total uncertainties are unknown in practical situation but also time delays always exist in the communication links when information is exchanged among SFF members, which motivates us for the current work. Terminal sliding mode control (TSMC), which was proposed in [16], can render finite-time stability for control systems. But the slower convergence and the singularity problem are two disadvantages of conventional TSMC. Therefore, a fast TSM in [17] and a nonsingular TSM in [18] were proposed to overcome these two disadvantages, respectively. Then a continuous finite-time control scheme using a new form of TSM for rigid robotic manipulators was proposed in [19], which has faster and high-precision in both the reaching and sliding phase compared with the controllers in [16–18] . Motivated by the work of [19], many modified TSMs have been applied successfully in spacecraft control [20–25]. In comparison with the modified methods in [22–25] to avoid singularity, the recently developed nonsingular fast terminal sliding mode control (NFTSMC) method in [26], promises fast convergence speed. For on-orbit spacecraft, it suffers the external disturbances and parameter uncertainties due to fuel consuming and environmental influence. Considering the lumped disturbances of the system in the developed model, the dynamics of the SFF system will be strongly nonlinear because these disturbances are inevitable and can’t be linearly parameterized. If we want to guarantee the fine performance of the proposed controller, approximating the total disturbances with high accuracy is needed. There are several methods to realize this objective, in which fuzzy logic systems (FLSs) is an effective way. Moreover, among various fuzzy control methods, in particular, the approach based on Takagi-Sugeno (T-S) model has drawn rapidly growing attention in recent years for its capability of approximating any smooth nonlinear functions over a compact set to arbitrary accuracy [27, 28]. Recently, a number of significant results of engineering applications have 2

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been reported by using T-S fuzzy-model-based technique such as, the adaptive fuzzy controllers proposed in [29–31], which can be applied to industrial processes, real-time observer-based fault detection problems were solved in [32, 33] with the help of fuzzy control, reliable and robust control for nonlinear stochastic systems with actuator faults in [34, 35], controller design for network-based systems with communication constraints including time delays, packet dropouts, and signal quantization in [36, 37], controller design for nonlinear systems with time-delays in [38, 39], and also spacecraft control in [40–43]. More specifically, the fuzzy control schemes have been applied successfully to approximate the disturbance of spacecraft in [40] and [41], and the adaptive fuzzy controllers combined with NFTSMC to reject system uncertainties in [42] and [43] were effective. However, to the authors’ best knowledge, few results based on adaptive fuzzy finitetime control laws have been reported for the 6-DOF SFF system, not to mention considering the consensus control problem among the followers with signal transmission time delays. Inspired by the facts mentioned above, this paper makes the first attempt to investigate the finite time high accuracy consensus control problem for SFF system described by exponential coordinates on the Lie group SE(3) with model uncertainties, external disturbances, and data transmission time delay inside the SFF network addressed simultaneously. The novelty of this paper is summarized as follows: (1) In comparison with the 6-DOF model in [2], the developed model in this paper takes the structured and unstructured uncertainties into consideration. Based on the developed model, a novel nonsingular fast terminal sliding manifold is defined by the exponential coordinates and velocity tracking errors, and then adaptive fuzzy nonsingular fast terminal sliding mode control schemes are proposed, in which the adaptive fuzzy control technique is applied to reject the system disturbances and uncertainties, and the NTFSMC is applied to guarantee the finite time stability of the SFF system. As a result, the control performance obtained in this paper is superior to the results in [2, 14], because the controllers in these two papers can only guarantee the control objective come true as time tends to infinity, and lose efficacy in poor environment. (2) Moreover, motivated by [44] and [45], inter-coordination time-delay terms are added to the controllers, which guarantee that the controllers are effective to solve the consensus control problem among SFF members with arbitrary communication time-varying delays and dynamically changing topologies. The remainder of this paper is organized as follows: Preliminaries are given in the following section. In third section, a NFTSM is first proposed, then adaptive fuzzy nonsingular fast terminal sliding mode control schemes are designed and the finite time convergence analysis of the resulting closed-loop system is also provided. Finally, numerical simulation results are presented, followed by conclusions in fifth section. 2. Preliminaries

2.1. Notations

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The following notations, relative coupled dynamics and lemmas are presented to finish the theoretical analysis in this paper. 

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 0 −x3 x2 0 −x1 . Let (x)× represent the skew-symmetric matrix of vector x ∈ R3 , where (x)× =  x3 −x2 x1 0 Im ∈ Rm×m represents an identity matrix, 0 denotes the zero vector or matrix, and both of them have appropriate dimensions. λmax (·) and λmin (·) denote the maximum and minimum eigenvalues of a maT trix, respectively. For a vector y = [y1 , y2 , · · · , ym ] ∈ Rm , yi (or yj ) denotes the system states of spacecraft i (or spacecraft j) (i/j = 1, 2, · · · , n), where n is the number of the followers and yik γ is its kth component (k = 1, 2, · · · , 6), kyk denotes the Euclidean norm or its induced norm, [y] =   T γ γ |y1 | sgn(y1 ) · · · |ym | sgn(ym ) , where γ is a scalar and sgn(·) is the sign function. For a matrix D, det(D) is its determinant, Di is the parameter of spacecraft i and Dik is the k th element on the main diagonal of Di . 2.2. Relative coupled dynamics based on Lie group SE(3) In this paper, a group of n rigid spacecraft is considered. It is desired to control their position and attitude to a specified consensus state consisting of synchronized attitudes and predetermined position configuration. 3

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Here, we assume that formation keeping is carried out with respect to a virtual leader, whose trajectory is computed offline and is known to all the formation members. Based on the kinematics and dynamics of a single rigid spacecraft, the relative coupled dynamics of the leader and the followers based on Lie group SE(3) are derived.

Fig. 1. Definition of the coordinate reference frames

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Before developing the models, we introduce three kinds of coordinate frames which will be used in this paper as shown in Fig. 1 : the standard Earth centered inertial reference frame {I} with its origin at the center of the Earth, and the coordinates denoted by (xI , yI , zI ); the body-fixed frame {B}, with its origin in the mass center of the spacecraft and axes coincide with the principal axes of inertia, and the coordinates denoted by (xB , yB , zB ); the orbital-reference frame {O}, which is a right-handed orthogonal system with its origin located in the mass center of the spacecraft, x axis points to the opposite direction of the center of the Earth, y axis is in the flight direction and z axis is found by using the right-hand rule, and the coordinates denoted by (xO , yO , zO ). Let C ∈ SO(3) be the rotation matrix from {B} to {I}, and R ∈ R3 be the position of the mass center of the spacecraft in {I}, where SO(3) is the set of rotation matrices of a rigid body with C included, and it is a Lie group denoted by SO(3) = {C ∈ R3×3 : C T C = I3 , det(C) = 1}. R3 is the three-dimensional real Euclidean space of positions of the mass center of the spacecraft. Then the kinematics of a rigid spacecraft are given by ( C˙ = C(ω)× (1) R˙ = Cv

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where v and ω are translational velocity and angular velocity expressed in {B}. The special Euclidean group SE(3) is the set of all translational and rotational motions of a rigid body moving in three-dimensional Euclidean space, which is a Lie group expressed by the semidirect product SE(3) = R3 n SO(3), an element χ of SE(3) with the form   C R χ= ∈ SE(3) (2) 01×3 1

can be used to represent the configuration of a spacecraft, by which the kinematics can be expressed as χ˙ = χ(V )v

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(3)

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where V =



ωT

vT

T

, and the vector space isomorphism R6 → ςe(3) is defined as   ω× v v V = ∈ ςe(3) 01×3 0

(4)

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ςe(3) is the Lie algebra (tangent space at the identity element) associated with Lie group SE(3). Let the mass of the spacecraft be m and its moment of inertia matrix be J . Then the dynamics of a rigid spacecraft expressed in {B} as given in [9] are ( J ω˙ + ω × J ω = Mg + τc + τd (5) mv˙ + mω × v = fg + uc + ud where fg and Mg are gravity force and gravity gradient moment, uc and τc are the control force and control torque, ud and τd are external force and external torque on the spacecraft, respectively. Taking the Earth’s oblateness and the coupled relationship between position and attitude into consideration, the expression of fg and Mg are given by

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fg = G + fJ2 µ

µ(RbT J Rb )

15 G = −( 3 )Rb − 3( 5 )J1 Rb + 2 kRk kRk mµ

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(1 −  − 5 2kRk   3J µR2 R 2  e y (1 − = mC T  − 5  2kRk   3J2 µRe2 Rz − (3 − 5 2kRk

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fJ2

3J2 µRe2 Rx

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Mg = 3(

µ kRk

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kRk

5Rz2

2)

× 5 )(Rb J Rb )

Rb

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

    2)  kRk   5Rz2  ) 2 kRk kRk 5Rz2

!

(6)

(8)

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where Rb = C T R, µ = 398600.47km3 /s2 is the gravitational constant of the Earth, J1 = 0.5tr(J )I3 + J , tr(·)is used to calculate the trace of a matrix, J2 = 1.08263 × 10−3 , Re = 6378.14km is the equatorial radius of the Earth, and Rx , Ry , Rz are R’s components. The dynamics can be expressed in a compact form as follows: Ξ V˙ = ad∗V ΞV + f (Ξ) + Γc + Γd

(10)

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 T  T  T where Ξ = diag(J , mI3 ), f (Ξ) = MgT , fgT , Γc = τcT , uT , Γd = τdT , uT . ad∗V is the c d  T  T vT co-adjoint operator of V = ω defined as ad∗V

=

adT V

=





ω× v×

−ω × 03×3

−v × −ω ×



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and the adjoint operator adV is expressed as adV =

5

03×3 ω×



(12)

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Eqs. (3) and (10) are the kinematics and dynamics of a single rigid spacecraft on SE(3), respectively. Next, based on these two equations, the relative coupled dynamics will be given. Let χi and χo be the actual configuration of spacecraft i and the virtual leader, respectively. Then the configuration error between spacecraft i and the virtual leader is χi/o = χ−1 o χi

(13)

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when the desired configuration of spacecraft i with respect to the virtual leader is   Cid Rid χdi = 01×3 1

(14)

Then the configuration tracking error of spacecraft i with respect to the virtual leader is χ1 = (χdi )−1 χi/o

can be calculated by χ1

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Then the configuration tracking error expressed by exponential coordinates   ei Φ ηei = ∈ R6 ei ϕ logSE(3) χ1

−1

(15)

(16)

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v where (·)−1 is the inverse mapping  of (·) , log(·)  SE(3) is the logarithm map, i.e., the inverse of the exponential C1 R1 map, when χ1 is expressed as , logSE(3) can be calculated by 01×3 1

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logSE(3) χ1 =



e× Φ i 01×3

ei ϕ 0



(18)

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ei is the principal rotation vector and ϕ ei is the position tracking error, which are expressed as where Φ  03×3 θ=0 e× = (19) Φ θ i  (C1 − C1T ) θ ∈ (−π, π), θ 6= 0 2 sin θ ei )R1 , S(Φ e i ) = I3 + ei = S −1 (Φ ϕ

1 − cos θ e× θ − sin θ e× 2 Φi + (Φi ) θ2 θ3

(20)

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ei and corresponds to the principal rotation angle. where θ = arccos(0.5(tr(C1 ) − 1)), which is the norm of Φ The relative velocity of spacecraft i with respect to the virtual leader is Vei = Vi − Adχ−1 Vo

where the adjoint action map defined for χ is  C Adχ = R× C

i/o

03×3 C

6



∈ R6×6

(21)

(22)

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Then the kinematics in exponential coordinates as given in [2] is ηe˙i = G(ηei )Vei

where

(24)

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where G(ηei ) is expressed as a block-triangular matrix # " ei ) A(Φ 03×3 G(ηei ) = ei , ϕ ei ) ei ) A(Φ T (Φ

(23)

e i ) = I3 + 1 ( Φ ei )× + ( 1 − 1 + cos θ )(Φ e × )2 A(Φ i 2 θ2 2θ sin θ

1 ei )ϕ ei ) + ( 1 − 1 + cos θ )(Φ ei ϕ eT ei A(Φ ei )) ei )× A(Φ eT (S(Φ i + Φi ϕ 2 θ2 2θ sin θ (1 + cos θ)(θ − sin θ) e eT ϕ ei Φ eT eT + ( (1 + cos θ)(θ + sin θ) − 2 )Φ ei Φ ei Φ − S(Φi )ϕ i i 2 θ4 i 2θ sin θ 2θ3 sin2 θ

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ei , ϕ ei ) = T (Φ

According to (21) and d(Adχ−1 (t) )/dt = −adVei Adχ−1 (t) as proved in [2], the relative acceleration of spacei/o

i/o

craft i can be expressed as

Taking (10) into (25) yields

˙ −1 ˙ Vei = V˙ i + adVei Ad−1 χi/o Vo − Adχi/o Vo

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˙ −1 ˙ Ξ Vei = ad∗Vi Ξi Vi + f (Ξi ) + Γci + Γdi + Ξi (adVei Ad−1 χi/o Vo − Adχi/o Vo )

(25)

(26)

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Then the relative coupled dynamics between spacecraft i and the virtual leader based on Lie group SE(3) can be modeled by Eqs. (23) and (26). However, when taking actual system into consideration, the structured and unstructured uncertainties can’t be ignored. We assume that the actual matrix with moment of inertia matrix and mass is Ξi1 = ei , where ∆Ξ ei Ξi + ∆Ξi , where ∆Ξi is the uncertainty part. (Ξi + ∆Ξi )−1 can be expressed as Ξi−1 + ∆Ξ is also an uncertainty part. Then (26) can be rewritten as ˙ Vei = Hi + ∆dei + Ξi−1 Γci + Ξi−1 f (Ξi )

(27)

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∗ −1 ˙ e e where Hi = Ξi−1 ad∗Vi Ξi Vi + adVei Ad−1 χi/o Vo − Adχi/o Vo , and ∆di = ∆Ξi (adVi Ξi1 Vi + Γci + f (Ξi )) + −1 Ξi−1 ad∗Vi ∆Ξi Vi + Ξi1 (Γdi + f (Ξi1 ) − f (Ξi )) is the total disturbance of the system.

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Remark 1. Eqs. (23) and (27) form the relative coupled dynamics between spacecraft i and the virtual leader, in which the 6-DOF relative-motion is modeled in a united framework with configuration tracking errors described by exponential coordinates on the Lie group SE(3) in a compact form. In (23), the existing of G(ηei ) guarantees that the kinematics is decoupled, and in (27) the dynamical couplings is involved in Hi , which guarantees that the couplings can be handled by using feedback technique. From (27), we find that by designing a suitable control law Γci our control objective can be realized unlike the traditional control schemes where control laws are designed separately for orbit and attitude maneuvers, which facilitates the controller design and reduces the computational complexity of the SFF system. Remark 2. The trajectory of the virtual leader is computed offline in an ideal environment without the external force and gravity moment and is known to all following members without communication delays. In order to keep the SFF perform as a whole, information exchange among the SFF members is required. 7

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In this case, communication delays can’t be ignored, whose constraint will be discussed in the controller synthesis part. When the relative configuration approaches the desired configuration, the relative configuration tracking error ηei represented by the exponential coordinates goes to zero, and the desired formation is achieved. And for (19), the logarithm map SE(3) → ςe(3) is bijective when θ ∈ (−π, π), which means that a designed tracking control scheme on the 6-DOF coupled model is almost global in its convergence over the state space. 2.3. Lemmas

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In order to facilitate the stability analysis of the controller, the following lemmas are provided.

Lemma 1. ([19]) If a continuous positive definite function V (x) : Rn → R satisfies the following differential inequality, where ρ1 > 0, ρ2 > 0, and 0 < ρ < 1: V˙ (x) ≤ −ρ1 V (x) − ρ2 V ρ (x), ∨t > t0

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then V (x) can reach the equilibrium in finite time, and the convergence time satisfies ρ1 V 1−ρ (x(t0 )) + ρ2 1 ln ρ1 (1 − ρ) ρ2

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tf ≤ t0 + where x(t0 ) is the initial value of x.

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Lemma 2. ([19]) Suppose that there exists positive constants a1 , a2 , · · · , aN and 0 < ρ < 1, and then the following inequality holds: 2ρ 2ρ 2 (a21 + a22 + · · · + a2N )2ρ ≤ (a2ρ 1 + a2 + · · · + aN )

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Lemma 3. ([20]) For a continuous positive definite function V (x) : Rn → R, which satisfies the following differential inequality:

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V˙ (x) + $1 V $2 (x) ≤ 0, (x ∈ Ω, x 6= 0)

(31)

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where $1 > 0, $2 ∈ (0, 1), and Ω ∈ Rn is a neighborhood of origin, then V (x) can reach the equilibrium in finite time t, where t=

V 1−$2 (x(t0 )) $1 (1 − $2 )

(32)

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e are all positive. Lemma 4. The eigenvalues of G(η)

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e in [2] Proof. The following equation holds for G(η)

e ηe = ηe G(η)

(33)

e ηe = ηeT (G(η) e η) e = ηeT ηe ηeT G(η)

(34)

Multiplying both sides of (33) by ηeT yields

e is a positive definite matrix, and then its eigenvalues are all positive; When ηe 6= 0, ηeT ηe > 0, we have G(η) e is an identity matrix, which means its when ηe is equal to zero, by substituting it into (24), we have G(η) eigenvalues are all equal to one. The above analysis concludes the proof.

Lemma 5. ([46]) For a positive definite function V (x) : Rn → R, which is homogeneous of degree σ with respect to the dilation (r1 , · · · , rN ). If for a continuous function V¯ (x) satisfies V¯ (ρr1 x1 , · · · , ρr1 xN ), as ρ → 0 with any fixed xκ 6= 0, κ = 1, 2, · · · , N , then (V + V¯ ) is locally positive definite. 8

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3. Adaptive fuzzy NFTSM controller design and stability analysis

3.1. Design of the NFTSM

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In this section, a SFF system with n-rigid-spacecraft tracking a dynamic desired reference state is considered. Our purpose is to design control schemes on the relative coupled dynamics so that the configuration of each spacecraft can converge to the desired state in the presence of model uncertainties and external disturbances in finite time. In order to make the spacecraft within a formation perform as a whole rather than several individuals, interconnection between spacecraft in the formation is required. When taking actual system into consideration, communication delays exist due to information exchange, which may affect the stability of the system. Fortunately, the controllers can not only achieve the control objective in the presence of such an adverse factor but also be useful for changing topologies.

To obtain finite time convergence, TSM and NTSM are first developed. However, their convergence time is longer than that of the conventional SMC when far away from the equilibrium point. To enhance the convergence speed, FTSM is designed with the following form: S = e˙ + α[e]γ1 + β[e]γ2

(35)

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where α, β ∈ R6×6 are both positive definite diagonal matrices, γ1 > 1, and 0 < γ2 < 1. If γ1 = 1, (35) has the same form of the FTSM in [19], and if at the same time γ2 = h/g, it will have the form of the FTSM in [17]. Though as the analysis in [17] and [19], the surface has a high convergence rate either near ˙ which may cause singularity due to the equilibrium or far away from it, since its derivative contains eγ2 −1 e, e˙ 6= 0, e = 0. In order to solve this problem, numbers of methods were developed in [23–25]. They all find a region Ω, where the singularity problem will be avoided. But when the sliding surface is in the region, the expression is not proper for the 6-DOF model proposed in this paper because of the existence of G(ηei ). Thus based on the properties of the NFTSM in [26], we design a novel nonsingular fast terminal sliding surface to improve the system transient state performance, which is given by Si = ηei + αi [ηei ]g/h + βi [Vei ]p/q

(36)

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where αi , βi ∈ R6×6 are both positive definite diagonal matrices, 1 < p/q < g/h < 2, and they are all odd integers.

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Remark 3. In the proposed NFTSM manifold (36), the sliding surface is defined by configuration tracking errors expressed by exponential coordinates ηei and velocity tracking errors Vei , which will converge to zero in finite time without explicit reference states if a suitable controller is designed. However, when Si reaches origin, the existence of G(ηei ) makes the finite time stability proof of the two components of Si be difficult. By proposing Lemma 4, the problem can be solved in Theorem 1.

Theorem 1. When the sliding manifold Si reaches the equilibrium, which is Si = Λi ηei + αi [ηei ]g/h + βi [Vei ]p/q = 0, and Λi , αi , βi ∈ R6×6 are all positive definite diagonal matrices, then ηei , Vei will reach the equilibrium in finite time.

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Proof. When Si = Λi ηei + αi [ηei ]g/h + βi [Vei ]p/q = 0, we have

Vei = [−(Λi ηei + αi [ηei ]g/h )/βi ]γ

(37)

where 0 < γ = q/p < 1. Then we consider the following candidate Lyapunov function V =

1 T ηe ηei 2 i

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(38)

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calculating the derivative of V with respect to time and substituting (37) into the expression yields V˙ = ηeiT ηe˙i = ηeiT G(ηei )[−(Λi ηei + αi [ηei ]g/h )/βi ]γ ≤ −λmin (G(ηei ))

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[ηeik ]γ+1 [(Λki + αki [ηeik ]g/h−1 )/βik ]γ ≤ −ΘV

1+γ 2

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k=1

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where Θ = λmin (G(ηei ))2(1+γ)/2 min([(Λki + αki [ηeik ]g/h−1 )/βik ]γ ). Since λmin (G(ηei )) is positive by using Lemma 4, thus we have Θ > 0. Then the finite time stability of ηei is guaranteed by using Lemma 3 for 0.5 < (1 + γ)/2 < 1, and it is easy to obtain that Vei converges to zero in finite time by taking ηei into (37). The above analysis concludes the proof. 3.2. Controller synthesis

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Based on the NFTSM designed in subsection 3.1, the relative coupled dynamics can be rewritten as the following form: ( x˙ 1 = ηe˙i = G(ηei )Vei = G(x1 )x2 (40) ˙ x˙ 2 = Vei = Hi + ∆dei + Ξi−1 Γci + Ξi−1 f (Ξi )

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In order to make the controller have high-accuracy, the most important thing is to deal with the total disturbance of the system, which has large uncertainties and strong nonlinearities. Here we employ the fuzzylogic-based estimation scheme, which has been shown to be capable to approximate any smooth function over a compact set to any degree of accuracy, to compensate the total disturbance of the system. Before the controller design, the structure of the FLS used in our controller will be illustrated first. Suppose that there are N input variables X = (x1 , · · · , xN )T ∈ RN , and then the FLS is constructed by considering the case where the fuzzy rule base consists of M rules in the following form: IF x1 is Al1 and · · · and xN is AlN , T HEN z is Bl

(41)

CE

PT

ED

where l = 1, · · · , M is the number of fuzzy rules for every input variable xκ , z is the output of the fuzzy system, Alκ and Bl are fuzzy sets. If the fuzzy system with singleton fuzzifier, center-average defuzzifier, and product inference engine, then the output of the FLS can be expressed as [27, 40] Q  N M z¯l X κ=1 µAlκ (xκ )  z= (42) PM QN l=1 l=1 κ=1 µAlκ (xκ )

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Usually, µAlκ (xκ ) is given by Gaussian membership function of the form   ¯lκ 2 1 xκ − x ) µAlκ (xκ ) = alκ exp − ( 2 ζκl

(43)

where alκ , x ¯lκ , ζκl are all positive real parameters with 0 < alκ ≤ 1. x ¯lκ is the point where µAlκ (xκ ) gets its maximum value. Let W = (¯ z 1 , z¯2 , · · · , z¯M ), and then (42) can be rewritten as z = Wξ

10

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where ξ(X) =

Q

N κ=1

M X

µAlκ (xκ )

PM QN

l=1



κ=1 µAlκ (xκ )

l=1



(45)

is regarded as a basis function. Then the total disturbance of the system can be approximated by (46)

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F = Ξ∆de = W ∗ ξ + ε

where W ∗ is the optimal weight matrix and ε is the approximation error of FLS. Let W be the estimation of W ∗ , and then the output of FLS can be expressed by F¯ = W ξ + ε

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where Si is taken as the input variables of ξ. Second, the following assumptions are stated for the stability analysis of the overall closed-loop system before controller design. Assumption 1. The output of FLS is bounded, such that k F¯ k≤ FM where FM is a positive constant.

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Assumption 2. The approximation error of FLS is bounded such that

where εM is a positive constant.

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k ε k≤ εM

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Assumption 3. The optimal weight matrix W ∗ is bounded such that tr(W ∗T W ) ≤ WM

(50)

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where WM is a positive constant.

Γci = Γeqi + Γsci + Γrci

 g/h−1 fi ]2−p/q − f (Ξi ) − Ξi Hi  )G(ηei )[V Γeqi = −Ξi Kβi (I6 + Kαi [ηei ] Γsci = −K1i Si − K2i [Si ]γ − λi (t)(Wi ξi + ψi ) − (1 − λi (t))Ψi  Pn  1/2 Γrci = j=1 (−kij Si + oij kj1 Sj (t − Tij )− oij kj2 kSj (t − Tij )k sgn(Si ))

AC

where

CE

To this end, based on the basic mind of behavior-based control, we can design adaptive fuzzy nonsingular fast terminal sliding mode states feedback control schemes directly, given as follows: (51)

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1 6 where Kβi = diag(q/(pβi1 ), · · · , q/(pβi6 )), Kαi = diag(q/(pα1i ), · · · , q/(pα6i )), K1i = diag(k1i , · · · , k1i ), 1 6 K2i = diag(k2i , · · · , k2i ), they are all positive definite constant matrices. ψi and Ψi are the robust control terms, which will be determined later. λi (t) is the switch function given by ( 0, kWi ξi k > FM (53) λi (t) = 1, kWi ξi k ≤ FM

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and the adaptive law for Wi is given by ˙ i = λi (t)σ1 (K −1 [Vei ]p/q−1 Si ξ T − σ2 Wi ) W i βi

(54)

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kij , kj1 , kj2 are all positive definite constant matrices, and when i = j, kij = kj1 = kj2 = 0. oij is a binary number, if there is interconnection between spacecraft i and j, oij = 1, otherwise oij = 0. Tij represents the communication delay from the jth spacecraft to the ith spacecraft which is assumed to be a time-varying quantity satisfying 0 < T˙ij < 1. Sj (t − Tij ) is the delayed information that the ith spacecraft receives from the jth spacecraft. ψi and Ψi are given by   k k  ψik = κk1i tanh 6ku κ1i Si τk k  (55)   Ψ k = κk tanh 6ku κ2i Si i 2i τ

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where κk1i > εM , κk2i > FM , ku = 0.2785, τ is a positive scalar. Since for a hyperbolic tangent function 0 ≤ |δ| − δ tanh(δ/ϑ) ≤ 0.2785ϑ holds, then it is easy to verify that  6 P   (|Sik |||εi || − Sik ψik ) ≤ τ 0 ≤ SiT εi − SiT ψi ≤ k=1 (56) 6 P  T ¯ T k k k  ¯ 0 ≤ Si Fi − Si Ψi ≤ (|Si |||Fi || − Si Ψi ) ≤ τ k=1

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Remark 4. The proposed controller (51) consists of three parts: Γeqi is used to satisfy S˙ i = 0, Γsci is the station-keeping control action used to drive the ith spacecraft to the desired states, Γrci is the formation control action used to maintain the desired relative configuration within the formation. Since the controller is designed on (40), which means that the initial attitude tracking error is required to satisfy the constraint θ ∈ (−π, π). If the initial error is not in this region, we can first use an alternative controller to drive the attitude tracking error into this region, and then switch to the proposed controller to guarantee our control objective finished in finite time.

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Remark 5. Assumption 1 is reasonable because the mass, the mass moment of inertia tensor, the input variables Si , and the external disturbance are all bounded. And Assumption 2 and 3 are based on the fact that any continuous function can be approximated by a FLS to any degree of accuracy.

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Remark 6. The output of FLS, Wi ξi , and the robust controller ψi consist of the adaptive fuzzy controller, where the FLS is used to approximate the total disturbance of the system and the robust controller is used to compensate the approximation error of FLS. In general, the FLS has poor approximation at the beginning and the output may exceed the bound. In order to solve this problem, a switching function λi (t) is used to make a switching between the adaptive FLS and the robust controller Ψi . The output of the adaptive FLS works only when kWi ξi k ≤ FM , which guarantees the feasibility of the controller. 3.3. Stability analysis With control law (51), we have the following theorems.

Theorem 2. Consider the system described by (40) and Assumptions 1, 2, 3 are satisfied. The controller is given by (51), the adaptive law is given by (54), and 4`ij (λmin (K3i kij ) − `ij )(1 − T˙ij ) > λ2max (K3i kj1 ) and λmin (K3i kij ) > `ij > 0 hold. For a sufficiently large positive constant cmax , if the initial condition satisfies   f T (0)W fi (0) ≤ cmax SiT (0)Si (0) + tr W (57) i 12

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fi = W ∗ − Wi is the FLS weight matrix approximation error, then Si and W fi are uniformly where W i ultimately bounded. Proof. We consider the following candidate Lyapunov function V =

n  X 1

2

i=1

SiT Ξi Si

 X n X n Z t 1 Tf f + tr(Wi Wi ) + `ij SiT Si dx 2σ1 t−T ij i=1 j=1

(58)

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calculating the derivative of V with respect to time and substituting (40) into it, yields  X n X n h i 1 T T f TW ˙ i) + ˙ tr(W ` S S − (1 − T )S (t − T )S (t − T ) ij i ij ij j ij i i j σ1 i=1 j=1 i=1  n  X βi p e p/q−1 e˙ 1 αi g g/h−1 T ˙ T f e e [ηei ] [Vi ] G(ηei )Vi + Vi ) − tr(W Wi ) = Si Ξi (G(ηei )Vi + h q σ1 i=1 n  X

+

SiT Ξi S˙ i −

n X n X i=1 j=1

h i `ij SiT Si − (1 − T˙ij )SjT (t − Tij )Sj (t − Tij )

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V˙ =

Then substituting controller (51) and the adaptive law (54) into (59) and the inequalities (56) is applied, we have n    X fi ξi + εi − ψi ) + (1 − λi (t))(Fi − Ψi ) − K1i Si − K2i [Si ]γ SiT K3i λi (t)(W V˙ = i=1

n X n   X h  f T λi (t)(K3i Si ξ T − σ2 Wi ) SiT K3i − kij Si + oij kj1 Sj (t − Tij ) + − tr W i

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i=1 j=1

i h i 1/2 T T ˙ − oij kj2 kSj (t − Tij )k sgn(Si ) + `ij Si Si − (1 − Tij )Sj (t − Tij )Sj (t − Tij ) i=1

 X n n X   1/2 − SiT K3i K1i Si − K2i [Si ]γ + ci − oij λmin (K3i kj2 ) kSj (t − Tij )k kSi k

n n X X i=1 j=1

T νij νij −

n  n X X i=1 j=1

i=1 j=1

`ij (1 − T˙ij ) −

oij λ2max (K3i kj1 )

4λmin (K3i kij ) − `ij



SjT (t − Tij )Sj (t − Tij )

 p where K3i = diag pβik (Veik )p/q−1 /q , ci = λmax (K3i )τ + σ2 WM /2, and νij = λmin (K3i kij ) − `ij Si −  

o λ (K k ) p ij max 3i j1 Sj (t − Tij ). When 4`ij λmin (K3i kij ) − `ij 1 − T˙ij > λ2max K3i kj1 , and with the 2 λmin (K3i kij ) − `ij following definitions   n  P  1 T 1 f TW fi )  S Ξ S + tr( W V = i i 1  i i 2 2σ 1   i=1  n P n R  P t V2 = ` S T Si dx (61) t−Tij ij i  i=1 j=1   n n P P  1/2   oij λmin (K3i kj2 ) kSj (t − Tij )k kSi k V3 =

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−

n  X

PT

≤

(60)

i=1 j=1

Then (60) can be changed into V˙ ≤

n  X i=1

−

SiT K3i



K1i Si − K2i [Si ] 13

γ



+ ci



− V3

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and the following equation holds V˙ 1 = V˙ − V˙ 2 ≤

n  X i=1

−

SiT K3i



K1i Si − K2i [Si ]

γ



+ ci



− V3 − V˙ 2

(63)

From (61), we know that V˙ 2 is homogeneous of degree σ = 2 with respect to the dilation (r1 = 1, r2 = 1), since V˙ 2 (ρSi , ρSi (t − Tij )) = ρ2 V˙ 2 (Si , Si (t − Tij )); V3 is homogeneous of degree σ = 3/2 with respect to the dilation (r1 = 1, r2 = 1), since V3 (ρSi , ρSi (t − Tij )) = ρ3/2 V3 (Si , Si (t − Tij )). Then we have

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ρ2 V˙ 2 (Si , Si (t − Tij )) V˙ 2 (ρSi , ρSi (t − Tij )) = lim = lim ρ1/2 V˙ 2 (Si , Si (t − Tij )) = 0 3/2 ρ→0 ρ→0 ρ→0 ρ ρ3/2 lim

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Then according to Lemma 5, we have that V3 + V˙ 2 is locally positive definite. Then from (63), one can obtain  X n  n     X V˙ 1 ≤ − SiT K3i K1i Si − K2i [Si ]γ + ci ≤ − λmin (K3i K1i )SiT Si + ci (65) i=1

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i=1

Thus V˙ 1 is strictly negative outside the region  r ΩSi = Si kSi k ≤

ci λmin (K3i K1i )



(66)

fi are uniformly ultimately bounded. The above analysis concludes the proof. which means that Si and W

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Remark 7. The controller proposed in this paper is effective for any type of the communication topology. The weight parameters kj1 and kj2 are used to express the information transmission from the jth spacecraft to the ith spacecraft. The binary number oij switching between 0 and 1 is used to describe the time-varying communication topology, which means that whether a communication failure or a new member jointing the system, the controller will always be useful.

PT

In Theorem 2, we have shown the stability of the closed-loop system, which means that the proposed control schemes can solve the consensus problem in the presence of model uncertainties, external disturbances, communication delays and changing topologies. However, it doesn’t provide the finite-time stability. Therefore, the following theorem is proposed to guarantee the finite-time stability of the sliding surface in both the reaching and sliding phase.

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Theorem 3. Consider the system described by (40) and Assumptions 1, 2, 3 are satisfied. The controller is given by (51), the adaptive law is given by (54), and 4`ij (λmin (K3i kij ) − `ij )(1 − T˙ij ) > λ2max (K3i kj1 ) and λmin (K3i kij ) > `ij > 0 hold. For a sufficiently large positive constant cmax , if the initial condition satisfies   f T (0)W fi (0) ≤ cmax SiT (0)Si (0) + tr W (67) i

f and the parameter κk1i is chosen such that κk1i > εM + W i ξi , then Si will converge to the region ∆i in finite time, where q o np 2 ∆i = min c1i /(a1i a3i ), (c1i /a2i ) γ+1 /a3i ) (68)

Furthermore, the configuration tracking error ηei will converge to the region n o k ηei ≤ min ∆i , (∆i /αki )h/g 14

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in finite time. Proof. We consider the following candidate Lyapunov function V4 =

n X 1

2

i=1

SiT Ξi Si

+

n X n Z X i=1 j=1

t

t−Tij

`ij SiT Si dx

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The proof process is the same as Throrem 2, the only difference is the selection of V5 . n X

V5i =

i=1

n X 1 i=1

2

SiT Ξi Si

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V5 = After calculating, we have V˙ 5 = V˙ 4 − V˙ 2 ≤

n  X

−

i=1

SiT K3i



K1i Si − K2i [Si ]

γ



+ c1i



− V3 − V˙ 2

(71)

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V˙ 5 = V˙ 4 − V˙ 2 ≤

n  X i=1

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where c1i = λmax (K3i )τ . Since V2 and V3 are selected the same as (61), we have V3 + V˙ 2 is locally positive definite. Thus the following inequality holds    − SiT K3i K1i Si − K2i [Si ]γ + c1i

By using Lemma 2, we have

6 X

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− SiT K3i K2i [Si ]γ ≤ −λmin (K3i K2i )

≤

(γ+1)/2

6 X λ

max (Ξi )

2

(γ+1)/2 (Sik )

(74)

k=1 −λmin (K3i K2i )(2/λmax (Ξi ))(γ+1)/2 (SiT Ξi Si )(γ+1)/2 (γ+1)/2 −a2i V5i

PT

≤

[Sik ]γ+1

k=1

ED

≤ −λmin (K3i K2i )(2/λmax (Ξi ))

(73)


(γ+1)/2 where a2i = λmin (K3i K2i ) 2/λmax (Ξi ) > 0, and then (73) becomes

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where a1i

(γ+1)/2 V˙ 5i ≤ −a1i V5i − a2i V5i + c1i (75)
= λmin (K3i K1i ) 2/λmax (Ξi ) > 0, and then (75) can be rewritten as the following two forms ( (γ+1)/2 V˙ 5i ≤ −(a1i − c1i /V5i )V5i − a2i V5i (76) (γ+1)/2 (γ+1)/2 V˙ 5i ≤ −a1i V5i − (a2i − c1i /V5i )V5i

Let a3i = λmin (Ξi )/2. From (76), if a1i > c1i /V5i , then thep finite-time stability will be guaranteed by using Lemma 1, and Si will reach the region kSi k ≤ ∆1i = c1i /(a1i a3i ) in finite time. From (76), if (γ+1)/2 a2i > c1i /V5i , then the p finite-time stability will be guaranteed by using Lemma 1, and Si will reach the region kSi k ≤ ∆2i = (c1i /a2i )2/(γ+1) /a3i in finite time. Therefore, the sliding surface Si will reach the region ∆i in finite time, where kSi k ≤ ∆i = min {∆1i , ∆2i } 15

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k ¯ ≤ ∆i , then we have ¯ k and ∆ Once Si has been in the region ∆i , we have Sik = ∆ i i ¯ ki Sik = ηeik + αki [ηeik ]g/h + βik [Veik ]p/q = ∆

Then, (78) can be rewritten as the following two forms ( ¯ k /ηek )ηek + αk [ηek ]g/h + β k [Ve k ]p/q = 0 (1 − ∆ i i i i i i i ¯ k /[ηek ]g/h )[ηek ]g/h + β k [Ve k ]p/q = 0 ηek + (αk − ∆ i

i

i

i

i

(79)

i

if the following inequality holds

n
o k ¯ ki , ∆ ¯ ki /αki h/g ηei > min ∆

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i

(78)

(80)

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(79) will have the same structure as the sliding surface in Theorem 1, and then the configuration error ηeik will reach the region o n k (81) ηei ≤ min ∆i , (∆i /αki )h/g

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in finite time. The above analysis concludes the proof. Through Theorem 2 and 3, we conclude that by selecting approximate controller parameters, we can drive Si with its components ηei and Vei to a really small region in the presence of model uncertainties, external disturbances, communication delays and changing topologies. Then the high-accuracy-finite-time stability in both the reaching phase and the sliding phase can be guaranteed, which means that the ideal configurations of the spacecraft with respect to a dynamic reference state and the configurations between each two following members can be achieved in finite time with high accuracy, at the same time, their relative velocities will all be bounded in a small region with zero included. 4. Simulation results

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In this section, simulations of a formation composed of four spacecraft with respect to a virtual leader, which is specified to follow a circle orbit with a 400 km orbit altitude are presented to illustrate the effectiveness and robustness of the proposed control schemes. The nominal parts of the mass and inertia matrix are assumed to be identical and are chosen the same as [22]: m = 100 kg, and   of each spacecraft 22 0 0 J =  0 20 0  kg · m2 , which can be written as Ξ = diag(22, 20, 23, 100, 100, 100). 0 0 23 The trajectory of the virtual leader is generated offline, and the leader body fixed frame is always perfectly aligned with the leader orbit frame. The initial configuration and velocity of the virtual leader  −0.7660 −0.6428 0 −5268.96  0.4545 −0.5417 −0.7071 3126.25   and Vo = are expressed in {I} and{B}, respectively: χo =   0.4545 −0.5416 0.7071 3126.25  0 0 0 1  T 0, 0, 0.0011, 0, 7.6126, 0 , the displacements are in kilometers, the translational velocities are in kilometers per second and the angular velocities are in radians per second. Our formation flying mission is: the desired configuration of the four spacecraft are required to construct a tetrahedron-shaped formation in the body-fixed as illustrated in Fig. 2 with their p of the virtual leader √ √ frame coordinates are R1f = [0, 0, 0]T , R2f = [5, 5 3/6, 5 3/2]T m, R3f = [5, 5 3/2, 0]T m, R4f = [10, 0, 0]T m, while the attitude, the translational velocity and the angular velocity of each SFF members need to be the same as the virtual leader.

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S1

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S2

Fig. 2. The configuration of the four SFF members with respect to the virtual leader

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Initially, the four spacecraft are assumed to be performing different flying tasks, with various initial states which are given as follows: The coordinates in the body-fixed frame of the virtual leader are: R1 (0) = [2, 2, 2]T m, R2 (0) = [6, 3, 5]T m, R3 (0) = [6, 8, 0]T m, R4 (0) = [12, 0, 0]T m. The initial orientations of spacecraft 1-4 are assumed to be different from the initial orientation of the virtual leader through 1-3, 3-1-3, 3-1-3 and 3-1-3 with Euler angles [π/6, π/6], [π/6, π/6, π/6], [π/4, π/4, π/4], [π/12, π/12, π/12], respectively. Their translational velocities and angular velocities expressed in their body-fixed frames are:  T T  ω1 (0) = [0.000009, 0.0006, 0.00093] rad/s, v1 (0) = [3.4415, 5.69884, −3.692] km/s;  ω (0) = [0.00049, 0.0007, 0.0007]T rad/s, v2 (0) = [3.7436, 3.1795, −5.8162]T km/s; 2 (82) ω3 (0) = [0.00076, 0.00074, 0.00031]T rad/s, v3 (0) = [3.6254, −0.93764, −6.6279]T km/s;    ω4 (0) = [0.0002, 0.00044, 0.001]T rad/s, v4 (0) = [2.4639, 6.3969, −3.31067]T km/s.

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The real mass and inertia matrix cannot be obtained accurately because of fuel cost, measuring deviation, and other factors. Meanwhile, external disturbances caused by aerodynamic resistance exist ineluctably in practice. Considering all these factors, the uncertainty part of mass and inertia matrix and the external disturbances are chosen as follows: ( ∆Ξ1 = diag(0.3J , 0.1mI3 ); ∆Ξ2 = diag(0.4J , 0.15mI3 ); (83) ∆Ξ3 = diag(0.3J , 0.2mI3 ); ∆Ξ4 = diag(0.4J , 0.2mI3 ).

Γdi



   =   

0.005 sin(0.2t + i) N · m 0.005 cos(0.25t + i) N · m −0.005 sin(0.15t + i) N · m 0.05 sin(0.1t + i) N 0.05 cos(0.15t + i) N −0.05 sin(0.2t + i) N

       

(84)

Communication delays in the process of information collecting, computing, and transmitting cannot be ignored. To test the effectiveness of the proposed control schemes, we choose the communication delays as follows: T12 = T13 + 0.1 = T14 − 0.1 = T21 + 0.2 = T23 + 0.35 = T24 + 0.25 = T31 − 0.4 = T32 + 0.15 = T34 − 0.3

= T41 − 0.2 = T42 − 0.15 = T43 − 0.2 17

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with T12 = 0.8 + 0.5 sin(t/5) cos(t/5) in second. If t < Tij , the history function is selected to be constant and is equal to Sj (0). The time-varying communication topology is described as follows: o12 = o13 (t + 4.8) = o14 (t − 7.3) = o21 (t + 7.4) = o23 (t − 5.6) = o24 (t − 4.6) = o31 (t + 4.5) = o32 (t − 3.5) = o34 (t − 6.8)

mod (t, 7) ≤ 6 mod (t, 7) > 6

, where mod(t, 7) is used to calculate the remainder of dividing t by 7.

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( 1 with o12 (t) = 0

(86)

= o41 (t − 7.2) = o42 (t + 5.7) = o43 (t − 5.8)

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Sk In the simulations, let xκ = k i be the inputs of the fuzzy systems. Seven fuzzy membership Si + 0.0001 functions are defined as follows:  1 
µA1κ (xκ ) =    1 + exp 5(xκ + π/4)    xκ + 1
2    µA2κ (xκ ) = exp (−0.5 )   0.25  
x + 0.5 2  µA3 (xκ ) = exp (−0.5 κ )    κ 0.25 xκ
2 µA4κ (xκ ) = exp (−0.5 ) (87) 0.25  
 x − 0.5 2 κ   ) µA5κ (xκ ) = exp (−0.5   0.25  
x − 1  2 κ  µA6κ (xκ ) = exp (−0.5 )   0.25   1  
µA7κ (xκ ) = 1 + exp 5(xκ − π/4)

Parameter name

Value

αi = diag(0.2, 0.2, 0.2, 0.4, 0.4, 0.4), βi = diag(2, 2, 2, 4, 4, 4); p = 7, q = 5, g = 9, h = 5.

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Sliding surface

ED

Table 1: Control parameters for numerical simulation.

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Controller parameters

τ = 0.01, κ1i = diag(0.1, 0.1, 0.1, 1, 1, 1), κ2i = diag(2, 2, 2, 5, 5, 5), K1i = diag(10, 10, 10, 20, 20, 20),K2i = diag(2, 2, 2, 4, 4, 4), kj1 = diag(2, 2, 2, 5, 5, 5), kij = diag(4, 4, 4, 10, 10, 10), σ1 = 0.01, σ2 = 100, kj2 = diag(0.004, 0.004, 0.004, 0.01, 0.01, 0.01). Wi = 07×6

Initial weight matrix of FLS

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It should be pointed out that the control law (51) is designed using sliding mode control and adaptive fuzzy control theory, and it is first applied in SFF system control, in which the 6-DOF relative coupled dynamics are described by exponential coordinates on the Lie group SE(3). Therefore, there dose not exist a unified procedure to be followed for choosing the control parameters. Consequently, when implementing controller (51) in simulation, the guidelines of choosing the design parameters are by trial-and-error until a good consensus control performance is obtained. The simulation parameters are given in Table 1. Moreover, we assume that the follower spacecraft have available continuous control thrust in all body and pay axes, ui ≤ 10N and attention to the actuator saturation with a bounded magnitude of force and torque as τi ≤ 1N · m, respectively. Simulation results are presented in Figs. 3-9, which validate the stability analysis of the proposed control schemes. 18

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Fig. 3. Attitude tracking errors in terms of the exponential coordinates with respect to the virtual leader

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Fig. 3 and Fig. 4 show the time histories of the relative attitude and angular velocity tracking errors of the four SFF members with respect to the virtual leader. It can be seen that the tracking errors fall to the tolerance within 30s and high tracking accuracy with a bounded k is achieved magnitude of the relative e ≤ 2 × 10−3 (deg) and ∆ω k ≤ 4 × 10−4 (deg /s). attitude and angular velocity tracking errors as Φ i i

Fig. 4. Angular velocity tracking errors with respect to the virtual leader expressed in the body-fixed frame

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Fig. 5. Position tracking errors in terms of the exponential coordinates with respect to the ideal configuration

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Fig. 5 and Fig. 6 show the time histories of the relative translational and velocity tracking errors of the four SFF members with respect to the virtual leader, in which rapid transient behavior within 85s can be seen. The high tracking accuracy on spacecraft 1, 2, 3 is guaranteed. However, the tracking accuracy of spacecraft 4 is poor, because the initial translational errors of the fourth spacecraft is the largest, in order to make it fall to the tolerance with the same time as the previous three spacecraft, a little accuracy is sacrificed. By adjusting the control parameters, high control accuracy can be achieved but it needs long convergence time.

Fig. 6. Translational velocity tracking errors with respect to the virtual leader expressed in the body-fixed frame

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In this paper, the consensus control problem of spacecraft formation in the presence of model uncertainties, external disturbances, time varying communication delays and changing topologies has been solved. Each spacecraft was regarded as a rigid body, and by employing the Lie group SE(3), the coupled translational and rotational dynamics have been derived, in which the tracking errors of position and attitude were described by exponential coordinates. Based on the derived relative coupled dynamics, a NTFSM without explicit reference states was designed, and then a class of adaptive fuzzy finite-time control laws have been proposed to guarantee the following members to achieve the relative configurations with respect to the virtual leader and perform as a whole in the presence of the aforementioned drawbacks. It has been proved that the presented controllers can guarantee the high accuracy finite time stability in both the reaching phase and sliding phase. Meanwhile, they were effective under different topologies: disconnected or connected, directed or undirected, fixed or time-varying. Numerical simulations have been performed under poor conditions to illustrate the effectiveness of the proposed controllers. The simulation results demonstrated the effectiveness and robustness of the proposed control schemes. However, some practical problems such as, data quantization and packet dropouts, and actuator and sensor faults et al, are not considered in this paper. Developing the SFF system model via Markov jump systems as defined in [47, 48] to solve these problems will be the authors’ future work. 6. Acknowledgements This work is supported by the National Natural Science Foundation of China (61603115, 91438202, 61273096), the China Postdoctoral Science Foundation funded project(2015M81455), and the Heilongjiang 22

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Postdoctoral Fund (LBH-Z15085). References

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