# Extended-State-Observer-Based Terminal Sliding Mode Tracking Control for Synchronous Fly-Around with Space Tumbling Target

Authors: Zhijun Chen; Yong Zhao; Yuzhu Bai; Dechao Ran; Liang He
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper presents a robust controller with an extended state observer to solve the Synchronous Fly-Around problem of a chaser spacecraft approaching a tumbling target in the presence of unknown uncertainty and bounded external disturbance. The rotational motion and time-varying docking trajectory of tumbling target are given in advance and referred as the desired tracking objective. Based on dual quaternion framework, a six-degree-of-freedom coupled relative motion between two spacecrafts is modeled, in which the coupling effect, model uncertainties, and external disturbances are considered. More specially, a novel nonsingular terminal sliding mode is designed to ensure the convergence to the desired trajectory in finite time. Based on the second-order sliding mode, an extended state observer is employed to the controller to compensate the closed-loop system. By theoretical analysis, it is proved that the modified extended-state-observer-based controller guarantees the finite-time stabilization. Numerical simulations are taken to show the effectiveness and superiority of the proposed control scheme. Finally, Synchronous Fly-Around maneuvers can be accomplished with fast response and high accuracy.


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## Body

## 1. Introduction

Synchronous Fly-Around (SFA) technique represents the concept of proximity operation for driving a chaser spacecraft to fly around a space tumbling target with attitude synchronized. With the number of failed spacecraft and space debris increasing, on-orbit service operations are urgently required to extend life of the failed spacecrafts, such as assembly, repairing, module replacement, detumbling, refueling, and orbital debris removal [1, 2]. Under the influence of terrestrial gravitational perturbation, most of the failed spacecrafts have evolved into space tumbling targets, developing the noncooperative feature [3]. The SFA approach (see Figure 1) can be identified as a significant technology for on-orbit service missions since it performs high accuracy in close-range proximity to noncooperative tumbling target. Some studies have been performed with the intention to solve the SFA problem [4–7].Figure 1
Synchronous fly-around for a chaser approaching a tumbling target.As pointed out in [8], it also introduces a series of problems and challenges in SFA process. Different from traditional cooperative space missions, rendezvous and docking (RVD), and spacecraft formation flying (SFF), the noncooperative characteristics of SFA will result in deficiency of shape structure and quality information, which causes low accuracy of navigation in close proximity. Another problem is complicated modeling. It is obvious that SFA is a six-degree-of-freedom (6-DOF) relative motion, in which both the translational motion and rotational one are included. Mutual coupling between the two motions will also lead to complexity in modeling. Therefore, high precise modeling is an essential step to overcome the highly coupled drawback during SFA mission. Ma et al. [9] used C-W dynamics equation to establish the relative translational motion and adopted modified Rodrigues parameters to describe the relative rotational dynamics. Similar modeling method has also been performed in [10]. More recently, Xu et al. [11] considered the dynamical coupling inside the 6-DOF model and presented a nonlinear suboptimal control law for SFA problem. However, all the abovementioned studies gave the two-part relative motion and ignored the nonlinear coupling effect [12], such that it would cause the problem that rotational and translational control will not actuate simultaneously. Therefore, it is significant to establish integrated 6-DOF relative motion considering coupled term. Dual quaternion [13–15] gives us inspiration to solve SFA problem since it has been an effective tool for motion description in physical problems and mathematical calculation. It combines traditional Euler quaternions with dual numbers and inherits elegant properties of both. Based on dual quaternion, the coupled 6-DOF dynamics motion between two spacecrafts can be derived successfully. Wang et al. [12] made attempt to use dual quaternion to describe the coupled dynamics of rigid spacecraft and investigated the coordinated control problem for SFF problem. Wu et al. [16, 17] proposed a nonlinear suboptimal control based on dual quaternion for synchronized attitude-position control problem. Under dual-quaternion framework, Filipe and Tsiotras [18] proposed an adaptive tracking controller for spacecraft formation problem and ensured global asymptotical stability in the presence of unknown disturbance. For the safety problem of RVD, Dong et al. [19] proposed a dual-quaternion-based artificial potential function (APF) control to guarantee the arrival at the docking port of the target with desired attitude. The above research results illustrate the effectiveness of 6-DOF modeling using dual-quaternion framework and also provide the motivation of this paper.The guidance, navigation, and control (GNC) systems for space proximity operations should be taken into account during on-orbit SFA missions [20]. Due to the nonlinear and highly coupled dynamics, designing a controller for the accurate SFA operation is still an open problem. In addition to the preceding interests in 6-DOF modeling, uncertainties and disturbances are another key issue that should be addressed in the control system, which will severely reduce the performance of the controller or lead to instability of the closed-loop system. Therefore, it is urgent to realize accurate and fast SFA tracking control, under strong coupling effects and multisource interferences in practical process. In view of this problem, various nonlinear control methods have been carried out to estimate the disturbance and compensate the system, such as state-dependent Riccati equation (SDRE) control [21], adaptive control [22] and suboptimal control [11]. Among these methods, terminal sliding mode (TSM) control is an effective technique due to its robustness to system uncertainties and can provide finite-time convergence. Wang and Sun [23] carried out an adaptive TSM control method for spacecraft formation flying problem within dual-quaternion framework. However, the major disadvantage of initial TSM is the singularity problem. In order to eliminate this problem, Zou et al. [24] proposed a nonsingular TSM control (NTSMC) to realize the finite-time stability of satellite attitude control. Extended-state-observer (ESO) methods have recently been introduced to compensate for unmodeled dynamics and system with external disturbance and uncertainty. Conventional ESO has shown fine performance in disturbance rejection. Based on a new technique known as the second-order sliding mode (SOSM) [25], a novel ESO can guarantee high accuracy and robustness of the estimation with finite-time convergence. Some studies have recently combined conventional TSM and ESO for spacecraft control problem [26, 27]. Moreover, the combination of NTSMC and ESO is also widely used, especially in aerospace engineering, such as attitude tracking control [28, 29], underactuated spacecraft hovering [30], and reusable launch vehicle [31]. In Reference [28], an ESO-based third-order NTSMC was proposed for the attitude tracking control problem. Ran et al. [32] proposed an adaptive SOSM-based ESO with fault-tolerant NTSMC for the model uncertainty, external disturbance, and limited actuator control during spacecraft attitude control. Zhang et al. [33] presented an adaptive fast NTSMC using estimated information by SOSM-based ESO to solve the problem of spacecraft 6-DOF coupled tracking maneuver in the presence of model uncertainties and actuator misalignment. Moreno and Osori [25] gave a Lyapunov finite-time stability proof for a closed-loop system combining SOSM controller and ESO. To the best of the authors’ knowledge, ESO-based NTSMC is still an open problem for SFA mission within dual-quaternion framework.This paper aims to investigate the Synchronous Fly-Around maneuver problem in the presence of model uncertainties and external disturbances. Compared to existing results, the major contributions and differences of this study are summarized as follows:(1)
To date, it is the first time to use dual quaternion to describe 6-DOF relative dynamics for tumbling target-capturing problem, in which the nonlinear coupled effect, model uncertainties, and external disturbances are taken into account. Moreover, the error kinematics and dynamics are given to improve modeling accuracy significantly.(2)
Motivated by the concept of SOSM, a novel ESO is proposed to reconstruct the model uncertainties and external disturbances with finite-time convergence. Based on the estimated information of ESO, a fast NTSMC is carried out to eliminate the total disturbances and ensure the finite-time stable of the closed-loop system. The proposed method can gain faster response and higher accuracy than existing methods.This paper is organized as follows. Some necessary mathematical preliminaries of dual quaternion and useful lemmas are given and coupled kinematics and dynamics are derived for the relative motion problem in Section2. In Section 3, NTSMC and ESO strategies are elaborated, and the finite-time stability analysis of the system is performed in Section 4. Following that, numerical simulations are illustrated and discussed in Section 5 to show the effectiveness of the proposed method. Finally, some appropriate conclusions are presented in Section 6.


## 2. Mathematical Preliminaries

This study considers the leader-follower spacecraft formation, including a chaser and a tumbling target (Figure2). In order to facilitate the description of the model, the following coordinate system is used: OI−XIYIZI is the Earth-centered inertial coordinate system, with the Cartesian right-hand reference frame; Ol−xlylzl is LVLH (local-vertical, local-horizontal) reference frame with the origin at the center of mass of target, xl is pointing to the spacecraft radially outward; zl is the axis normal to the target orbital plane, and yl is the axis established by right-hand rule; Of−xfyfzf (respectively, the tumbling target, Ol−xlylzl) is the body-fixed coordinate system of the chaser, with its origin in the center of mass of the chaser and axes pointing to the axis of inertial; Pl−xlylzl is the desired tracking reference frame, with the capturing point Pl of the tumbling target. The control objective is to control the chaser (Of−xfyfzf) synchronize with desired state (Pl−xlylzl) so that the on-orbit service can be guaranteed in the next step.Figure 2
Coordinate system definition.


### 2.1. Dual Quaternion

Dual quaternion can be regarded as a quaternion whose element is dual numbers and can be defined as [14, 15, 23].(1)q^=η^,ξ^=q+εq′,where η^ is the dual scalar part; ξ^=ξ^1,ξ^2,ξ^3 is the dual vector part; q and q′ are the traditional Euler quaternions, representing the real part and dual part of dual quaternion, respectively; and ε is the called dual unit with ε2=0 and ε≠0. In this paper, ℕ is employed to denote the set of dual quaternions and ℝ denotes the set of dual numbers. Therefore, it is supposed that q^∈ℕ4, ξ^∈ℝ3, and η^∈ℝ.Similar to the traditional quaternion and dual vector (see Appendix), the basic operations for dual quaternions are given as follows:(2)q^1+q^2=η^1+η^2,ξ^1+ξ^2,λq^=λη^,λξ^,q^∗=η^,−ξ^,q^1⊗q^2=η^1η^2−ξ^1ξ^2,η^1ξ^2+η^2ξ^1+ξ^1×ξ^2,where q^1=η^1,ξ^1 and q^2=η^2,ξ^2, λ is a scalar, and q^∗ represents the conjugate of q^.Using dual quaternion, the frameOf−xfyfzf with respect to frame Ol−xlylzl can be defined as(3)q^fl=q^l∗⊗q^f=qfl+ε12pfll⊗qfl=qfl+ε12qfl⊗pflf,where q^l and q^f represent the dual quaternion of frame of the target and the chaser, respectively; qfl is the relative quaternion; pfll and pflf are the relative position vectors of the chaser relative to the target, expressed in frames Ol−xlylzl and Of−xfyfzf, respectively.


### 2.2. Relative Kinetics and Coupled Dynamics

Referring to [12, 18, 23], the kinematics and coupled dynamics equation of Ol−xlylzl with respect to Of−xfyfzf using dual quaternion representation can be described as follows:(4)2q^˙fl=q^fl⊗ω^flf,ω^˙flf=−M^f−1ω^flf+q^fl∗⊗ω^ll⊗q^fl×M^fω^flf+q^fl∗⊗ω^ll⊗q^fl+M^f−1F^ff−q^fl∗⊗ω^˙ll⊗q^fl+ω^flf×q^fl∗⊗ω^ll⊗q^fl,where ω^flf=ωflf+εp˙flf+ωflf×pflf∈ℕ3 is the relative velocity-expressed frame Of−xfyfzf; ωflf is the relative angular velocity; and p˙flf is the relative linear velocity between two frames. As can be seen from the definition, ω^flf can also be written as ω^flf=ω^ff−q^fl∗⊗ω^ll⊗q^fl. Besides, the force F^ff=F^uf+F^gf+F^df is the total dual force; F^gf=fgf+ετgf is the dual gravity; F^uf=fuf+ετuf is the dual control force; F^df=fdf+ετdf is the dual disturbance; and fgf is the dual gravity force and τgf is the dual gravity torque referred to the chaser (F^uf=fuf+ετuf and F^df=fdf+ετdf, respectively). M^f=mfd/dεI+εJf is the dual inertia matrix of chaser, in which mf and Jf are the mass and moment of inertia of chaser.


### 2.3. Lemmas

The lemmas useful for this study are introduced as follows. Firstly, the following system is considered:(5)x˙=fx,u,t,f0,t=0,where x∈Rn is the system status and u∈Rm is the control input.Lemma 1.
(see [34]). Suppose Φ is a positive definite function and there exists scalar υ>0, 0<α<1, and satisfies(6)Φ˙x+υΦαx≤0,x∈U0,then there is an field U0⊂Rn so that the Φx can reach equilibrium point in finite time. And the convergence time is satisfied as(7)T1≤Φ01−αυ−υα.Lemma 2.
(see [35]). Suppose Θ is a positive definite function and there are scalars ρ1>0 and ρ2>0, 0<β<1, and satisfies(8)Θ˙x≤−ρ1Θx−ρ2Θβx,x∈U1,then there is a field U1⊂Rn so that the Θx can reach equilibrium point in finite time. And the convergence time is satisfied as(9)T2≤1ρ11−βlnρ1Θ01−β+ρ2ρ2.


### 2.4. Problem Formulation

The control objective of this paper is to design a controller so that the relative motion stateq^flt,ω^flft of the chaser converges to the desired state q^dt,ω^dt in finite time. In order to describe the relative motion between the chaser and the capturing point of tumbling target, the relative motion between the reference frame Pl−xlylzl and frame Of−xfyfzf is established. Firstly, the description of target’s attitude is given which is considered as the general Euler attitude motion. The kinematics and dynamic are shown as follows, respectively:(10)2q˙l=ql∘ωll,Jlω˙ll=−ωll×Jlωll+τd,where ql is the quaternion of the tumbling target; ωll is the angular velocity expressed in Ol−xlylzl; Jl is the moment of inertia of tumbling target; and τd is the total disturbance torque acting on target. The operation ∘ represents the multiplication of traditional quaternion.The position vector of reference pointPl with respect to the centroid Ol on the tumbling target in the frame Of−xfyfzf is represented as follows:(11)pflf=qfl∗∘Pl∘qfl,where qfl is the attitude quaternion of the chaser relative to the target and Pl is the position of the noncentroid point Pl with respect to the centroid Ol at the initial moment.Remark 1.
Referring to [36], some information of relative motion can be measured. The model of 3D reconstruction which is obtained by the mathematical alignment of the image plane and binocular stereo matching is established to determine the relative position and pose of the target’s capturing point. In such a way, relative rotational and translational motion between two spacecrafts can be obtained.
Considering the external disturbances and model uncertainty of the spacecraft, the error kinematics and dynamics equations based onq^e and ω^e can be obtained as(12)2q^˙e=q^e⊗ω^e,M^0ω^˙e=−ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e−M^0q^e∗⊗ω^˙d⊗q^e+M^0ω^e×q^e∗⊗ω^d⊗q^e+F^u+F^g0+F^d+ΔF^g+ΔΣ^,where ω^e=ω^flf−q^e∗⊗ω^d⊗q^e=ωe+ενe and q^e=q^d−1⊗q^fl=qe+εpe are the dual tracking error; qe and pe are the rotation error and translation error, respectively; ωe and νe are the angular velocity error and linear velocity error, respectively; M^0 and ΔM^ are the nominal and uncertain parts of the dual matrix inertia matrix; F^d is the bounded disturbance; F^g0 and ΔF^g are the dual gravity associated with M^0 and ΔM^; and ΔΣ^ is the uncertainty of the dual inertia matrix and the value of which is(13)ΔΣ^=−M^0ω^˙e−ω^e+q^e∗⊗ω^d⊗q^e×ΔM^ω^e+q^e∗⊗ω^d⊗q^e−ΔM^q^e∗⊗ω^˙d⊗q^e+ΔM^ω^e×q^e∗⊗ω^d⊗q^e.


## 2.1. Dual Quaternion

Dual quaternion can be regarded as a quaternion whose element is dual numbers and can be defined as [14, 15, 23].(1)q^=η^,ξ^=q+εq′,where η^ is the dual scalar part; ξ^=ξ^1,ξ^2,ξ^3 is the dual vector part; q and q′ are the traditional Euler quaternions, representing the real part and dual part of dual quaternion, respectively; and ε is the called dual unit with ε2=0 and ε≠0. In this paper, ℕ is employed to denote the set of dual quaternions and ℝ denotes the set of dual numbers. Therefore, it is supposed that q^∈ℕ4, ξ^∈ℝ3, and η^∈ℝ.Similar to the traditional quaternion and dual vector (see Appendix), the basic operations for dual quaternions are given as follows:(2)q^1+q^2=η^1+η^2,ξ^1+ξ^2,λq^=λη^,λξ^,q^∗=η^,−ξ^,q^1⊗q^2=η^1η^2−ξ^1ξ^2,η^1ξ^2+η^2ξ^1+ξ^1×ξ^2,where q^1=η^1,ξ^1 and q^2=η^2,ξ^2, λ is a scalar, and q^∗ represents the conjugate of q^.Using dual quaternion, the frameOf−xfyfzf with respect to frame Ol−xlylzl can be defined as(3)q^fl=q^l∗⊗q^f=qfl+ε12pfll⊗qfl=qfl+ε12qfl⊗pflf,where q^l and q^f represent the dual quaternion of frame of the target and the chaser, respectively; qfl is the relative quaternion; pfll and pflf are the relative position vectors of the chaser relative to the target, expressed in frames Ol−xlylzl and Of−xfyfzf, respectively.


## 2.2. Relative Kinetics and Coupled Dynamics

Referring to [12, 18, 23], the kinematics and coupled dynamics equation of Ol−xlylzl with respect to Of−xfyfzf using dual quaternion representation can be described as follows:(4)2q^˙fl=q^fl⊗ω^flf,ω^˙flf=−M^f−1ω^flf+q^fl∗⊗ω^ll⊗q^fl×M^fω^flf+q^fl∗⊗ω^ll⊗q^fl+M^f−1F^ff−q^fl∗⊗ω^˙ll⊗q^fl+ω^flf×q^fl∗⊗ω^ll⊗q^fl,where ω^flf=ωflf+εp˙flf+ωflf×pflf∈ℕ3 is the relative velocity-expressed frame Of−xfyfzf; ωflf is the relative angular velocity; and p˙flf is the relative linear velocity between two frames. As can be seen from the definition, ω^flf can also be written as ω^flf=ω^ff−q^fl∗⊗ω^ll⊗q^fl. Besides, the force F^ff=F^uf+F^gf+F^df is the total dual force; F^gf=fgf+ετgf is the dual gravity; F^uf=fuf+ετuf is the dual control force; F^df=fdf+ετdf is the dual disturbance; and fgf is the dual gravity force and τgf is the dual gravity torque referred to the chaser (F^uf=fuf+ετuf and F^df=fdf+ετdf, respectively). M^f=mfd/dεI+εJf is the dual inertia matrix of chaser, in which mf and Jf are the mass and moment of inertia of chaser.


## 2.3. Lemmas

The lemmas useful for this study are introduced as follows. Firstly, the following system is considered:(5)x˙=fx,u,t,f0,t=0,where x∈Rn is the system status and u∈Rm is the control input.Lemma 1.
(see [34]). Suppose Φ is a positive definite function and there exists scalar υ>0, 0<α<1, and satisfies(6)Φ˙x+υΦαx≤0,x∈U0,then there is an field U0⊂Rn so that the Φx can reach equilibrium point in finite time. And the convergence time is satisfied as(7)T1≤Φ01−αυ−υα.Lemma 2.
(see [35]). Suppose Θ is a positive definite function and there are scalars ρ1>0 and ρ2>0, 0<β<1, and satisfies(8)Θ˙x≤−ρ1Θx−ρ2Θβx,x∈U1,then there is a field U1⊂Rn so that the Θx can reach equilibrium point in finite time. And the convergence time is satisfied as(9)T2≤1ρ11−βlnρ1Θ01−β+ρ2ρ2.


## 2.4. Problem Formulation

The control objective of this paper is to design a controller so that the relative motion stateq^flt,ω^flft of the chaser converges to the desired state q^dt,ω^dt in finite time. In order to describe the relative motion between the chaser and the capturing point of tumbling target, the relative motion between the reference frame Pl−xlylzl and frame Of−xfyfzf is established. Firstly, the description of target’s attitude is given which is considered as the general Euler attitude motion. The kinematics and dynamic are shown as follows, respectively:(10)2q˙l=ql∘ωll,Jlω˙ll=−ωll×Jlωll+τd,where ql is the quaternion of the tumbling target; ωll is the angular velocity expressed in Ol−xlylzl; Jl is the moment of inertia of tumbling target; and τd is the total disturbance torque acting on target. The operation ∘ represents the multiplication of traditional quaternion.The position vector of reference pointPl with respect to the centroid Ol on the tumbling target in the frame Of−xfyfzf is represented as follows:(11)pflf=qfl∗∘Pl∘qfl,where qfl is the attitude quaternion of the chaser relative to the target and Pl is the position of the noncentroid point Pl with respect to the centroid Ol at the initial moment.Remark 1.
Referring to [36], some information of relative motion can be measured. The model of 3D reconstruction which is obtained by the mathematical alignment of the image plane and binocular stereo matching is established to determine the relative position and pose of the target’s capturing point. In such a way, relative rotational and translational motion between two spacecrafts can be obtained.
Considering the external disturbances and model uncertainty of the spacecraft, the error kinematics and dynamics equations based onq^e and ω^e can be obtained as(12)2q^˙e=q^e⊗ω^e,M^0ω^˙e=−ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e−M^0q^e∗⊗ω^˙d⊗q^e+M^0ω^e×q^e∗⊗ω^d⊗q^e+F^u+F^g0+F^d+ΔF^g+ΔΣ^,where ω^e=ω^flf−q^e∗⊗ω^d⊗q^e=ωe+ενe and q^e=q^d−1⊗q^fl=qe+εpe are the dual tracking error; qe and pe are the rotation error and translation error, respectively; ωe and νe are the angular velocity error and linear velocity error, respectively; M^0 and ΔM^ are the nominal and uncertain parts of the dual matrix inertia matrix; F^d is the bounded disturbance; F^g0 and ΔF^g are the dual gravity associated with M^0 and ΔM^; and ΔΣ^ is the uncertainty of the dual inertia matrix and the value of which is(13)ΔΣ^=−M^0ω^˙e−ω^e+q^e∗⊗ω^d⊗q^e×ΔM^ω^e+q^e∗⊗ω^d⊗q^e−ΔM^q^e∗⊗ω^˙d⊗q^e+ΔM^ω^e×q^e∗⊗ω^d⊗q^e.


## 3. Controller Design

This section mainly studies the control problem, and modified controller of ESO-NTSMC is given. By defining the dual quaternion tracking error asΩ^e=Ωe+εpe with Ωe=2lnqe, the controller is designed to make the tracking error Ω^e,ω^e converge to 0^,0^ in finite time.


### 3.1. Nonsingular Terminal Sliding Mode Controller Design

In order to achieve the control objective, a fast terminal sliding surface is defined as(14)s=x˙+α1x+α2xp,where x∈R and p is a constant which satisfies 0.5<p<1 and α1 and α2 are both positive normal numbers. The time derivative of the sliding surface s is(15)s˙=x¨+α1x˙+pα2xp−1x˙.As can be seen, whenx=0, x˙≠0, and p−1<0, the singularity may occur in equation (15). Motivated by [24, 37], an alternative form of sliding surface is proposed as follows to avoid singularity:(16)s=x˙+α1x+α2βx,where βx is(17)βx=sgnxxp,ifs¯=0ors¯≠0,x≥μ,γ1x+γ2sgnxx2,ifs¯≠0,x≤μ.In the above formula,γ1=2−pμp−1 and γ2=p−1μp−2 are the parameters; s¯=x˙+α1x+α2sgnxxp is the nominal sliding surface; and μ is a positive small constant. Therefore, the time derivative of sliding surface (16) is(18)s˙=x¨+α1x˙+pα2sgnxxp−1x˙,ifs¯=0ors¯≠0,x≥μ,x¨+α1x˙+α2γ1x˙+2γ2sgnxxx˙,ifs¯≠0,x≤μ.Remark 2.
The conventional nonsingular sliding mode (NTSM) in Reference [38]was proposed as(19)s=x+ξx˙m/n,where x∈R; ξ>0; and q and p are the positive odd integers satisfying 1<m/n<2. In Reference [39], it has been proved that the NTSM (19) is still singular when a dual sliding mode surface control method is designed. Furthermore, as explained in Reference [24], the NTSM (19) has slower convergence rate than the linear sliding mode when the sliding mode s=0 is reached. Therefore, in order to overcome this disadvantage, the fast NTSM is proposed in this paper. The singularity of TSM is avoided by switching from terminal to general sliding manifold, which is function γ1x+γ2sgnxx2 in equation (17).
Equation (18) is continuous, and singularity is avoided. According to above derivation, the paper propose a fast NTSM surface as(20)s^=ω^e+β^⊙Ω^e+c^⊙χΩ^e,where β^=β+εβ′∈ℝ3 is the positive dual constant, with β>0 and β′>0, so does c^=c+εc′∈ℝ3; the operation of ⊙ is defined in Appendix; χΩ^e=χΩ^e1,χΩ^e2,χΩ^e3T is designed as follows:(21)χΩ^ei=sigαΩ^ei,ifs^¯=0ors^¯≠0,Ω^ei≥ε0,γ1Ω^ei+γ2sgnΩ^eiΩ^ei2,ifs^¯≠0,Ω^ei≤ε0,i=1,2,3,where sigαΩ^ei=sgnΩeiΩeiα+εsgnpeipeiα; 1/2<α<1; ε0 is a positive value; γ1=2−αμα−1 and γ2=α−1μα−2; and s^¯=ω^e+β^⊙Ω^e+c^⊙sigαΩ^e.
Taking derivation of the sliding surfaces^ with respect to time yields(22)s^˙=ω^˙e+β^⊙Ω^˙e+c^⊙χ˙Ω^e,where χ˙Ω^ei is(23)χ˙Ω^ei=αΩ^eα−1Ω^˙e,ifs^¯=0ors^¯≠0,Ω^ei≥ε0,γ1Ω^˙ei+2γ2sgnΩ^eiΩ^eiΩ^˙ei,ifs^¯≠0,Ω^ei≤ε0,i=1,2,3.
The NTSMC method is(24)F^uf=F^e+F^s,where F^s=−k^⊙sigp1s^ is the switching term, k^∈ℝ3 with k^i=ki+εki′,i=1,2,3, satisfying ki>0andki′>0; p1>0; and F^e is the equivalent term and the value of which is(25)F^e=ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e−M^0q^e∗⊗ω^˙d⊗q^e−M^0ω^e×q^e∗⊗ω^d⊗q^e+F^g0−β^⊙M^0Ω^˙e−c^⊙M^0χ˙Ω^e.


### 3.2. Extended-State-Observer Design

The ESO method shows very fine performance in disturbance rejection and compensation of uncertainties that the accuracy of the controller can be improved. Based on the SOSM method mentioned in References [25, 32], and by combining the linear and nonlinear correction terms, a novel extended state observer is proposed as follows:(26)e^1=z^1−s^,e^2=z^2−D^,z^˙1=M^0−1F^u+M^0−1Sω^e,q^e−κ^⊙e^1−η^⊙sgnp2e^1+z^2,z^˙2=−ρ^⊙e^1−λ^⊙sgn2p2−1e^1,where z^1∈ℕ3 and z^2∈ℕ3 are the outputs of the observer; D^=ΔF^g+F^d+ΔΣ^ is the total disturbance and also the extended state variable; e^1 and e^2 represent the observer measurement; p2∈0.5,1; κ^,η^,ρ^,λ^∈ℝ3 are the observer designed parameters, respectively; and Sω^e,q^e is(27)Sω^e,q^e=M^−1F^d+F^g−ω^e+q^e∗⊗ω^d⊗q^e×M^ω^e+q^e∗⊗ω^d⊗q^e−M^q^e∗⊗ω^˙d⊗q^e+M^ω^e×q^e∗⊗ω^d⊗q^e.The proposed ESO (26) can not only obtain the feature of SOSM, such as finite-time convergence but also weaken the chattering of conventional first-order sliding mode also. In addition, it inherits the best properties of linear and nonlinear correction terms. Then, the error dynamics can be established from equation (26) in the scalar form:(28)e^˙1=e^2−κ^⊙e^1−η^⊙sgnp2e^1,e^˙2=−ρ^⊙e^1−λ^⊙sgn2p2−1e^1−g^it,where g^it is the derivative of D^ and the amplitude of g^it is assumed to be bounded by a positive value g¯, that is g^it≤g¯.Based on the NTSMC and ESO designed above, the integral controller is designed as(29)F^uf=F^e+F^s+F^τ,where F^e, F^s, and F^τ are given by(30)F^e=ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e+M^0q^e∗⊗ω^˙d⊗q^e−M^0ω^e×q^e∗⊗ω^d⊗q^e+F^g0−β^⊙Ω^˙e−c^⊙χ˙Ω^e,F^s=−k^⊙sigp1s^,F^τ=−z^2.Under the proposed controller in equations (29) and (30), the control objective can be achieved for tracking tumbling target. The closed-loop system for coupled rotational and translational control is shown in Figure 3.Figure 3
Closed-loop system of NTSMC-ESO for spacecraft coupled control.


## 3.1. Nonsingular Terminal Sliding Mode Controller Design

In order to achieve the control objective, a fast terminal sliding surface is defined as(14)s=x˙+α1x+α2xp,where x∈R and p is a constant which satisfies 0.5<p<1 and α1 and α2 are both positive normal numbers. The time derivative of the sliding surface s is(15)s˙=x¨+α1x˙+pα2xp−1x˙.As can be seen, whenx=0, x˙≠0, and p−1<0, the singularity may occur in equation (15). Motivated by [24, 37], an alternative form of sliding surface is proposed as follows to avoid singularity:(16)s=x˙+α1x+α2βx,where βx is(17)βx=sgnxxp,ifs¯=0ors¯≠0,x≥μ,γ1x+γ2sgnxx2,ifs¯≠0,x≤μ.In the above formula,γ1=2−pμp−1 and γ2=p−1μp−2 are the parameters; s¯=x˙+α1x+α2sgnxxp is the nominal sliding surface; and μ is a positive small constant. Therefore, the time derivative of sliding surface (16) is(18)s˙=x¨+α1x˙+pα2sgnxxp−1x˙,ifs¯=0ors¯≠0,x≥μ,x¨+α1x˙+α2γ1x˙+2γ2sgnxxx˙,ifs¯≠0,x≤μ.Remark 2.
The conventional nonsingular sliding mode (NTSM) in Reference [38]was proposed as(19)s=x+ξx˙m/n,where x∈R; ξ>0; and q and p are the positive odd integers satisfying 1<m/n<2. In Reference [39], it has been proved that the NTSM (19) is still singular when a dual sliding mode surface control method is designed. Furthermore, as explained in Reference [24], the NTSM (19) has slower convergence rate than the linear sliding mode when the sliding mode s=0 is reached. Therefore, in order to overcome this disadvantage, the fast NTSM is proposed in this paper. The singularity of TSM is avoided by switching from terminal to general sliding manifold, which is function γ1x+γ2sgnxx2 in equation (17).
Equation (18) is continuous, and singularity is avoided. According to above derivation, the paper propose a fast NTSM surface as(20)s^=ω^e+β^⊙Ω^e+c^⊙χΩ^e,where β^=β+εβ′∈ℝ3 is the positive dual constant, with β>0 and β′>0, so does c^=c+εc′∈ℝ3; the operation of ⊙ is defined in Appendix; χΩ^e=χΩ^e1,χΩ^e2,χΩ^e3T is designed as follows:(21)χΩ^ei=sigαΩ^ei,ifs^¯=0ors^¯≠0,Ω^ei≥ε0,γ1Ω^ei+γ2sgnΩ^eiΩ^ei2,ifs^¯≠0,Ω^ei≤ε0,i=1,2,3,where sigαΩ^ei=sgnΩeiΩeiα+εsgnpeipeiα; 1/2<α<1; ε0 is a positive value; γ1=2−αμα−1 and γ2=α−1μα−2; and s^¯=ω^e+β^⊙Ω^e+c^⊙sigαΩ^e.
Taking derivation of the sliding surfaces^ with respect to time yields(22)s^˙=ω^˙e+β^⊙Ω^˙e+c^⊙χ˙Ω^e,where χ˙Ω^ei is(23)χ˙Ω^ei=αΩ^eα−1Ω^˙e,ifs^¯=0ors^¯≠0,Ω^ei≥ε0,γ1Ω^˙ei+2γ2sgnΩ^eiΩ^eiΩ^˙ei,ifs^¯≠0,Ω^ei≤ε0,i=1,2,3.
The NTSMC method is(24)F^uf=F^e+F^s,where F^s=−k^⊙sigp1s^ is the switching term, k^∈ℝ3 with k^i=ki+εki′,i=1,2,3, satisfying ki>0andki′>0; p1>0; and F^e is the equivalent term and the value of which is(25)F^e=ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e−M^0q^e∗⊗ω^˙d⊗q^e−M^0ω^e×q^e∗⊗ω^d⊗q^e+F^g0−β^⊙M^0Ω^˙e−c^⊙M^0χ˙Ω^e.


## 3.2. Extended-State-Observer Design

The ESO method shows very fine performance in disturbance rejection and compensation of uncertainties that the accuracy of the controller can be improved. Based on the SOSM method mentioned in References [25, 32], and by combining the linear and nonlinear correction terms, a novel extended state observer is proposed as follows:(26)e^1=z^1−s^,e^2=z^2−D^,z^˙1=M^0−1F^u+M^0−1Sω^e,q^e−κ^⊙e^1−η^⊙sgnp2e^1+z^2,z^˙2=−ρ^⊙e^1−λ^⊙sgn2p2−1e^1,where z^1∈ℕ3 and z^2∈ℕ3 are the outputs of the observer; D^=ΔF^g+F^d+ΔΣ^ is the total disturbance and also the extended state variable; e^1 and e^2 represent the observer measurement; p2∈0.5,1; κ^,η^,ρ^,λ^∈ℝ3 are the observer designed parameters, respectively; and Sω^e,q^e is(27)Sω^e,q^e=M^−1F^d+F^g−ω^e+q^e∗⊗ω^d⊗q^e×M^ω^e+q^e∗⊗ω^d⊗q^e−M^q^e∗⊗ω^˙d⊗q^e+M^ω^e×q^e∗⊗ω^d⊗q^e.The proposed ESO (26) can not only obtain the feature of SOSM, such as finite-time convergence but also weaken the chattering of conventional first-order sliding mode also. In addition, it inherits the best properties of linear and nonlinear correction terms. Then, the error dynamics can be established from equation (26) in the scalar form:(28)e^˙1=e^2−κ^⊙e^1−η^⊙sgnp2e^1,e^˙2=−ρ^⊙e^1−λ^⊙sgn2p2−1e^1−g^it,where g^it is the derivative of D^ and the amplitude of g^it is assumed to be bounded by a positive value g¯, that is g^it≤g¯.Based on the NTSMC and ESO designed above, the integral controller is designed as(29)F^uf=F^e+F^s+F^τ,where F^e, F^s, and F^τ are given by(30)F^e=ω^e+q^e∗⊗ω^d⊗q^e×M^0ω^e+q^e∗⊗ω^d⊗q^e+M^0q^e∗⊗ω^˙d⊗q^e−M^0ω^e×q^e∗⊗ω^d⊗q^e+F^g0−β^⊙Ω^˙e−c^⊙χ˙Ω^e,F^s=−k^⊙sigp1s^,F^τ=−z^2.Under the proposed controller in equations (29) and (30), the control objective can be achieved for tracking tumbling target. The closed-loop system for coupled rotational and translational control is shown in Figure 3.Figure 3
Closed-loop system of NTSMC-ESO for spacecraft coupled control.


## 4. Stability Analysis

In this section, the stability analysis is divided into two parts. In the first part (Theorem 1), the finite-time convergence of ESO is proved, which implies that the proposed ESO can estimate the exact state and total disturbances within a fixed time. In the second part (Theorem 2), the finite-time stability of the closed-loop system is proved, which indicates that the finite-time convergence of the tracking errorΩ^e,ω^e can be obtained. The proofs of finite-time convergence of ESO and total closed-loop system will be given using Lyapunov stability theory.Theorem 1.
Suppose the parameters of ESO in equation (26) satisfy the following:(31)p1λ^ρ^>κ^2λ^+p2p2+12η^2κ^2.Then, the observer errors will converge to the following region in finite-time:(32)Ω1=ϑϑ≤δ=g¯RλminQ1p2/2p2−1,where ϑ=e^1ip2sgne^1i,e^1i,e^2iT, R=η^κ^−2, and the matrix Q1 satisfies the following expression:(33)Q1=ηλ^+p1η^20−p1η^0ρ^+2+p1κ^2−p1+1κ^−p1η^−p1+1κ^p1.The convergence time is upperbounded bytm≤2/θ1lnθ1V01/2+θ2/θ2, where V0 is the initial Lyapunov state and θ1 and θ2 are the constants depending on the gains.Proof.
To verify the finite-convergence performance, in accordance with the idea of Moreno and Osorio [25], a Lyapunov candidate is proposed as(34)V1=12η^⊙e^1ip2sgne^1i+κ^⊙e^1i−e^2iTη^⊙e^1ip2sgne^1i+κ^⊙e^1i−e^2i+λ^p1⊙e^1i2p2+ρ^⊙e^1iTe^1i+12e^2iTe^2i,=ϑTΓϑ,where ϑ=e^1ip2sgne^1i,e^1i,e^2iT and the matrix Γ is defined as(35)Γ=122λ^/p1+η^2η^κ^−η^η^κ^2ρ^+κ^2−κ^−η^−κ^2.
It is obvious thatV1 is positive definite and radially unbounded if λ^ and ρ^ are positive. The range of V1 is obtained as follows:(36)λminΓϑ2≤V1≤λmaxΓϑ2,where ϑ2=e^1i2p2+e^1iTe^1i+e^2iTe^2i is the Euclidean norm of ϑ and λmin⋅ and λmax⋅ represent the minimum and maximum eigenvalues of the matrix, respectively.
The time derivative ofV1 can be obtained as(37)V˙1=2λ^+p1η^2⊙e^1i2p2−1e^˙1isgne^1i+2ρ^+κ^2⊙e^1iTe^˙1i+2e^2iTe^˙2i+p1+1η^κ^⊙e^1ip2e^˙1isgne^1i−p1η^⊙e^1ip2−1e^˙1iTe^2i−η^⊙e^1ip2sgne^1ie^˙2i−κ^⊙e^˙1iTe^2i−κ^⊙e^1iTe^˙2i.
Substituting equation (28) into equation (37) yields(38)V˙1=e^1ip2−1−η^λ+η^p2η^2e^1i2p2−η^ρ^+2+p2η^κ^2e^1i2−p1η^⊙e^2iTe^2i+2p2+1η^κ^⊙e^1iTe^2i+2p2η^2⊙e^1ip2sgne^1ie^2i+−κ^⊙e^2iTe^2i−κ^ρ^+κ^3⊙e^2iTe^2i−κ^λ^+2κ^p2η^2+η^2κ^e^1i2p2+2κ^2⊙e^1iTe^2i+η^⊙e^1ip2sgne^1i−2e^2i+κ^e^1ig^it,=−e^1ip2−1ϑTQ1ϑ−ϑTQ2ϑ+g^itRϑ,where the matrices Q1, Q2, and R are given by(39)Q1=η^λ^+p2η^20−p2η^0ρ^+2+p2κ^2−p2+1κ^−p2η^−p2+1κ^p2,Q2=κ^λ^+2p2+1η^2000ρ^+κ^2−κ^0−κ^1,R=η^κ^−2.
It is obvious thatQ2 is a symmetric matrix, and if λ and κ are positive, it is positive definite. In addition, calculate the determinant of Q1 as(40)Q1=p2λ^+p2η^2ρ^+2+p2κ^2−p2+12κ^2−p22η^2ρ^+2+p2κ^2,=λ^p2ρ^+p22+p2κ^2−p2+12κ^2−p2η^2p2+12κ^2,=λ^p2ρ^−κ^2−p2η^2p2+12κ^2.
Substituting equation (31) into equation (40), it yields Q1>0. Therefore, Q1 is the positive definite matrix. According to the definition of ϑ, it can be obtained that(41)e^1ip2−1≥ϑp2−1/p2.
According to equations (34)–(41), one obtains(42)V˙1=−e1ip2−1λminQ1ϑ2−λminQ2ϑ2+g¯Rϑ≤−λminQ1ϑp2−1/p2ϑ−λminQ2ϑ2+g¯Rϑ≤−c1ϑ2p2−1/p2−c3V11/2−c2V1,where c1=λminQ1/λmaxΓ, c2=λminQ2/λmaxΓ, and c3=g¯R/λminΓ. Based on equation (36), we assume that V1≥c3/c12p2/2p2−1λmaxΓ, which implies that c1ϑ2p2−1/p2−c3≥0. And it is obvious that c2>0. By substituting θ1=c1ϑ2p2−1/p2−c3 and θ2=c2 into equation (42), one obtains(43)V˙1≤−θ1V11/2−θ2V1.
Therefore, according to Lemma 2, it is obvious thatV˙1 is definitely negative and V1 is positive. Therefore, observer error will converge to the small set Ω1 in finite time and the convergence time satisfies tm≤2/θ1lnθ1V01/2+θ2/θ2. The proof of Theorem 1 is completed.Remark 3.
As seen in Theorem 1 and its proof, the reconstruction error of the model uncertainties and external disturbances can be driven to the neighborhoods of the origin in finite time. The parameterp2/2p2−1 gets larger, and the region Ω1=ϑϑ≤δ=g¯R/λminQ1p2/2p2−1 will be smaller by choosing adequate ESO parameters, p2, κ^, η^, ρ^, and λ^, which implies that the proposed ESO has high accuracy. In this way, the output of observer will not influence the controller. Therefore, the finite-time convergence of the closed-loop system, mentioned in Theorem 2, is proved on the basis of Theorem 1. Some studies containing ESO-NTSMC analyzed the total finite-time convergence of the closed-loop system in the same way [32, 33].Theorem 2.
Based on the estimated state in observer (26), and with the application of the controller in equations (29) and (30), the close-loop system states will be driven to a neighborhood of the sliding surface s^=0^ in finite time. Moreover, the tracking errors Ω^e and ω^e will converge to a small region in finite time.Proof.
The proof includes two consecutive steps:
Firstly, it is to be proved that the sliding surface will converge tos^=0^ in finite time. Define the form of the Lyapunov function as follows:(44)V2=12s^TM^s^.
Taking the time derivative ofs^ in equation (20), it yields(45)s^˙=−M^−1F^u+D^−M^−1Sω^e,q^e+β^⊙Ω^˙e+c^⊙χ˙Ω^e.
With the application of equations (29) and (30), s^˙ can be obtained as(46)s^˙=M^−1−k^⊙sigp1s^−z^2+D^.
Take the time derivative ofV2 yields(47)V˙2=s^TM^s^˙,=s^T−k^⊙sigp1s^−z^2+D^,=s^T−k^⊙sigp1s^−e^2,where D^=ΔF^g+F^d+ΔΣ^ is the total external disturbance and model uncertainty and e^2=z^2−D^ is the error state of ESO. Recalling from the stability in Theorem 1, the ESO errors can enter the small set Ω in fixed time tm. After finite-time tm, the ESO errors satisfy e^2<δ. In consequence, it can be obtained as(48)V˙2=−s^Tk^⊙sigp1s^+e^2≤−s^Tk^⊙sigp1s^−δ≤−∑i=13ki−δsi−p1sip1+1+ki′−δsi′−p1si′p1+1≤−θ3sp1+1+s′p1+1,where θ3=minki−δsi−p1,ki′−δsi′−p1,i=1,2,3, is the positive scalar. For equation (48), if θ3>0, that is, ki−δsi−p1>0 or ki′−δsi′−p1>0, then one can obtain(49)V˙2≤−θ4V2p1+1/2,where θ4=θ32/maxσmaxJf,mf is a positive number. Therefore, it can be proved by Lemma 1 that the sliding surface s^ will converge to the set Ω2=s^s^≤Δs=k^/δp1 in finite time.
Secondly, when the close-loop system state reaches the neighborhood ofs^=0^, it should be proved that the tracking error Ω^e and ω^e will converge into small regions containing the origin 0^,0^ in finite time. Due to different conditions of s^ and Ω^e in (23), the following cases should be taken into consideration.Case 1.
Ifs^¯=0, equation (20) becomes(50)ω^ei=−β^i⊙Ω^ei−c^i⊙χΩ^ei,i=1,2,3.
Consider the following Lyapunov function(51)V3=Ω^e,Ω^e.
Taking the time derivative ofV3 yields(52)V˙3=Ω^e,Ω^˙e=Ω^e,ω^e,=Ω^e,−β^i⊙Ω^e−c^⊙sgnΩeΩeα+εsgnpepeα,=−β^Ω^e,Ω^e−∑i=13ciΩeiα+1+ci′peiα+1≤−θ5V3−θ6V31+α/2,where θ5=β^ and θ6=minci,ci′,i=1,2,3 are the parameters.
Therefore, according to Lemma 2, the error statesΩ^e and ω^e will converge to 0^,0^ in finite time.Case 2.
Ifs^¯≠0,Ω^ei≤ε0, which indicates that Ω^ei converged into the set Ω^ei≤ε0. One can obtain(53)ω^ei+β^i⊙Ω^ei+c^i⊙γ^1i⊙Ω^ei+γ^2i⊙sgnΩ^eiΩ^ei2=δs,δs<Δs.
Therefore,ω^ei will eventually converge to the region as follows:(54)ω^ei≤δs+β^iε0+c^i⊙γ^1iε0+γ^2iε02.Case 3.
Ifs^¯≠0,Ω^ei≥ε0, it implies that(55)ω^ei+β^i⊙Ω^ei+c^i⊙sigαΩ^ei=δs,δs<Δs,ω^ei+β^i−δs2Ω^ei⊙Ω^ei+c^i−δs2sigαΩ^ei⊙sigαΩ^ei=0,δs<Δs.
Onceβ^i>δs/2Ω^ei and c^i>δs/2sigαΩ^ei, then s^ is still kept in the form of terminal sliding mode, which implies that Ω^ei will be bounded as Ω^ei≤ε1=minΔs/2βi,Δs/2ci1/α.
In conclusion,ω^ei and Ω^ei will be bounded as ω^ei≤ς1 and Ω^ei≤ς2 in finite time, where ς1=δs+β^iε0+c^i⊙γ^1iε0+γ^2iε02 and ς2=maxε0,ε1. Based on the analysis above, it can be concluded that the control objective can be accomplished in finite time. The proof of Theorem 2 is completed.Remark 4.
The total finite-time stability of closed-loop is achieved on the basis of the above proof. With the application of ESO and its estimation of total disturbances, the controller is reconstructed by adding the disturbance rejection termF^τ in equation (30). Therefore, the robust controller can guarantee the finite-time stability of the overall closed-loop system with high accuracy. By choosing suitable controller parameters k^ and p1 such that Δs=k^/δp1 is smaller enough, the convergence set Ω2 can be a sufficiently small set. Therefore, it can be seen that, under the proposed controller method, the closed-loop system can achieve finite-time convergence with satisfactory performance.Remark 5.
Compared with the existing NTSMC methods, the structure of proposed controller scheme in this research combines ESO and NTSMC techniques, which can obtain higher tracking accuracy and achieve finite-time convergence. The overall controller is developed based on the estimated information of external disturbances and model uncertainties by ESO. Therefore, the controller does not need the priori information of perturbations. Moreover, based on the proof in equation (48), the ESO-based controller can achieve smaller steady-state error when the estimation z^2 of ESO is applied to develop the robust controller, which indicates that the system has better performance and higher accuracy.


## 5. Simulation Results

In this section, numerical simulations are given to illustrate the effectiveness of the proposed ESO-NTSMC for the chaser tracking the tumbling target. It is assumed that the orbital elements of tumbling target are presented in Table1. The target body-fixed frame completely coincides with the orbital coordinate system. The control objective is to keep attitude of chaser consistent with the target, and the relative position of two can be kept at a certain distance.Table 1
Simulation experiment parameters.
Orbital elements
Values
Semimajor axis (km)
6778
Eccentricity
0.02
Inclination (deg)
45
Right ascension of the ascending node (deg)
30
Argument of perigee (deg)
25
True anomaly (deg)
30The rotational angular velocity of the tumbling target is considered to be slow speed asωt0=0,0,0.01Trad/s, and the capturing point in Ol−xlylzl is set as pfl,d=10,0,0Tm. The inertia matrix is supposed to be measured as Jt=20,0,0;0,23,0;0,0,25kg⋅m2. The initial conditions of the relation motion between two spacecraft are given as pflf0=15,0,0Tm; qfl0=0.9991,0,0,0.0431T; and ω^flf0=0^.In addition, proposed NTSMC with linear-ESO, LESO-NTSMC, designed in Reference [25], and proposed ESO with conventional NTSMC, ESO-TSM, designed in Reference [38] are also implemented for purpose of comparison. The gains of proposed and above controllers are given in Table 2.Table 2
Control gains for simulations.
Controllers
Controller gains
ESO-NTSMC
α
=
0.67; μ=10−3; β^=0.15+ε2; c^=0.2+ε0.5; k^=20+ε10; η^=3+ε1; κ^=10+ε5; λ^=3+ε0.9; ρ^=0.01+ε0.01; p1=0.6; p2=0.6
LESO-NTSMC
α
=
0.67; μ=10−3; β^=0.15+ε2; c^=0.2+ε0.5; k^=20+ε10;κ^=10+ε5; ρ^=0.01+ε0.01; p1=0.6; p2=0.6
ESO-TSM
α
=
5
/
3; β^=0.5+ε0.8; k^=20+ε10; η^=3+ε1; κ^=10+ε5; λ^=3+ε0.9; ρ^=0.01+ε0.01; p1=0.6; p2=0.6The preset uncertainties and disturbances are set in accordance with Reference [12]. The real mass and inertia matrix of the chaser are assumed to be mf=97kg and Jf=18,0,0;0,17,0;0,0,20kg⋅m2. And the nominal mass and inertia matrix of the chaser is assumed to be mf0=100kg and Jf0=22,0,0;0,20,0;0,0,23kg⋅m2, with the model uncertainties bounded by Δmf≤3kg and Jf,ii≤3kg⋅m2,i=1,2,3, respectively. Moreover, it is assumed that the disturbance forces and torques are set as(56)fdf=0.05cosnt0.07cosnt−0.01cosntT,τdf=−0.04cosnt0.03cosnt−0.05cosntT,where n=π/T0 is a target orbit period constant with T0≈5711s. The control force is limited to the range of fu≤10N, and the control torque is limited to the range of τu≤1N⋅m.Figures4–13 show the time responses of the simulation results using the proposed ESO-NTSMC and compared control schemes, LESO-NTSMC and ESO-TSM, respectively. The results shown in Figure 4 illustrate the effectiveness of the ESO where the observer reconstruction errors of translational term and rotational term converge to a small range in finite time. In such a situation, the accuracy of the system can be enhanced when the model uncertainties and bounded external disturbances are estimated by ESO and the system can be compensated. Figures 5(a) and 6(a) shows the time responses of the tracking error of relative translation and rotation, respectively. It can be seen that the rotation error falls to tolerance within 10 s and converges to the range of 5×10−5rad while the translation tracking error converges to the range of 0.02 m in around 20 s, which illustrates the performance of accuracy and rapidity of ESO-NTSMC. The time histories of relative velocity and angular velocity tracking errors are given in Figures 7(a) and 8(a), respectively. The steady-state behavior shows that the angular velocity error converges to 1×10−5rad/s in 10 s, and the velocity error converges to 5×10−4m/s at 20 s, which confirms the control objective of tracking the velocity of target accurately. It can be seen that, under the bounded sliding mode control force and torque shown in Figures 9(a) and 10(a), the sliding surfaces converge to the range of si≤1×10−3m/s and si′≤1×10−5rad/s in finite time successfully, which are shown in Figures 11(a) and 12(a).Figure 4
Time responses of reconstruction error. (a) Translational term. (b) Rotational term.
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(b)Figure 5
Time responses of relative rotation. (a) ESO NTSMC. (b) LESO NTSMC. (c) ESO TSM.
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(c)Figure 6
Time responses of relative translation. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 7
Time responses of relative angular velocity. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 8
Time responses of relative linear velocity. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 9
Time responses of control torque. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 10
Time responses of control force. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 11
Time responses of sliding surface of translational term. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 12
Time responses of sliding surface of rotational term. (a) ESO-NTSMC. (b) LESO-NTSMC. (c) ESO-TSM.
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(c)Figure 13
Trajectory of chaser tracking capturing point of tumbling target.The simulations of LESO-NTSMC are also carried out, which are shown in Figures5(b)–12(b). From these figures, it can be seen that LESO-NTSMC is also tolerant to model uncertainties and external disturbances when the system states can be driven to a neighborhood of origin. As shown in Figures 11(b) and 12(b), the sliding surface accuracy of LESO-NTSMC is much lower than of ESO-NTSMC, which can illustrate that the proposed method obtains better control ability and high control accuracy in comparison with LESO-NTSMC scheme. Figures 5(c)–12(c) show the simulation results using the ESO-based conventional NTSMC method. Severe chattering and large overshoot exist in ESO-TSM during the control process, and the convergence time of ESO-TSM is larger than the proposed one. As seen, the proposed ESO-NTSMC has faster convergence rate than ESO-TSM and can also avoid singularity. From the tracking trajectory shown in Figure 13, the chaser can successfully track the desired trajectory of docking port of tumbling target. In conclusion, it is obvious that the control object is achieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can be guaranteed.


## 6. Conclusion

In this study, a finite-time stabilization problem of a chaser spacecraft approaching a tumbling target in presence of unknown uncertainty and bounded external disturbance has been handled. Based on the 6-DOF coupled relative motion described by dual quaternion, a combination of NTSMC and ESO is proposed to drive the translational and rotational motion to the desired state. Meanwhile, the finite-time convergence and stability property of the proposed method is rigorously proven. It was shown that the tracking error can be driven to the equilibrium point with fast response and high accuracy. Numerical simulation results show the effectiveness and performance of the proposed controller and its robustness to the model uncertainty and external disturbance.



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*Source: 5791579-2019-11-05.xml*