# Adaptive Fixed-Time Terminal Sliding Mode Control on SE(3) for Coupled Spacecraft Tracking Maneuver

Authors: Kejie Gong; Ying Liao; Yong Wang
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

In this paper, a reaching law-based adaptive fixed-time terminal sliding mode control law, which is used for coupled spacecraft tracking maneuver in the presence of large inertia parametric uncertainties and external disturbances, is proposed. The coupled 6-DOF kinematics and dynamics for spacecraft motion are modeled on Lie group SE(3). The relative configuration is expressed by a local coordinate (exponential coordinate) of SE(3). In order to estimate the inertia parameters and external disturbances, we also propose a novel adaptive update law, which can make the control law be applied without the inertia parameters of the spacecraft a priori. Fixed-time convergence property of the closed-loop feedback system is proved in the framework of Lyapunov. Numerical simulations are performed to demonstrate the performances of the proposed control scheme for coupled spacecraft tracking maneuver.


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## Body

## 1. Introduction

Spacecraft proximity operation plays an important role in a variety of space missions such as space debris removal, space station maintenance and installation, on-orbit spacecraft service, large structure assembly, and spacecraft networking [1, 2], in which position and attitude tracking maneuvers are the key steps requiring to be performed with high control precision. So far, the rotational and translational maneuvers of spacecraft were easy to implement separately by ignoring their coupling, but difficult to provide fast and accurate control to meet the requirements of future space missions [3]. Different from the separated control strategy, spacecraft proximity operation needs a precise integral control strategy of coupled 6-DOF motion for the chaser spacecraft [4]. Modeling of coupled rotational and translational relative motions of rigid spacecraft has been receiving great interests in recent years based on different forms, such as dual quaternion [4–7], three-dimensional (3-D) Euclidean space (Lie group SE(3)) [8–11], 6-DOF Euler-Lagrange form [12] (translational motion described in LVLH (local-vertical-local-horizontal) frame and rotational motion described by MRPs (modified Rodrigues parameters)) or other forms [13–18] (translational motion described in body-fixed frame or LOS (line-of-sight) frame and rotational motion described by MRPs). However, dual quaternion inherits the ambiguities intrinsic of quaternion, which may cause unwinding problem. In addition, MRPs describing attitude in Ref. [12] are nonglobal and nonunique. In this study, the kinematics and dynamics of spacecraft relative motion are modeled in unified forms on Lie group SE(3). Lie group SE(3) is the set of positions and attitudes of a rigid body in a 3-D Euclidean space [19, 20] and can represent the spacecraft’s motion in a unique and nonsingularity way [21]. Control problem of the 6-DOF spacecraft maneuver modeled on SE(3) has been studied mainly by Lee et al. [8, 22], Sanyal et al. [23], Zhang et al. [11], Nazari et al. [24], Holguin et al. [25], and Jiang et al. [26]. However, they tackled the 6-DOF control problem with sliding mode or terminal sliding mode in the frame of asymptotic or finite-time convergence. The spacecraft control problems with fixed-time stabilization were studied in Refs. [16–18, 27–30]. The attitude control problem with fixed-time stabilization is studied in Ref. [27–29]. Hu et al. [30] considered the fixed-time convergence of position control for the spacecraft rendezvous mission. For spacecraft 6-DOF fixed-time control problem, the motion is modeled in a nonunified form (LOS frame to describe translational motion and MRPs to describe rotational motion) [16–18]. Different from the aforementioned researches, this study proposes a fixed-time control for spacecraft 6-DOF motion modeled on SE(3) using terminal sliding mode, which achieves a faster convergence rate than asymptotic or finite-time control. Lie group SE(3) is used for analyzing the relative motion. It provides a global analysis for the system, and thus, it is suitable for the relative motion of the spacecraft with any type of orbits, including parabolic or hyperbolic trajectories [31]. While Huang and Jia [16–18] treated the 6-DOF modeling based on a nonglobal analysis method where the model of relative position motion is approximate. In order to represent the translational and rotational motions in a unified form, exponential coordinates are used for representing the relative pose (including attitude and position) via logarithm from SE(3) to se3, where se3 is isomorphism with R6 [8]. Most researches apply the exponential coordinate of SE(3) at the kinematical level. Jiang et al. [26] first derived the second-order derivative of exponential coordinates at the dynamical level and designed an asymptotic control law for the spacecraft hovering over an asteroid.A spacecraft moving in the outer space is affected by a variety of disturbance forces and torques, including environmental and nonenvironmental influences. Moreover, the inertia parameters of spacecraft may not be known accurately but must be considered in the design of a control scheme. The existence of inertia parametric uncertainties and external disturbances makes the controller design more complicated [31]. Wu et al. [32] developed an adaptive sliding mode control scheme for 6-DOF synchronized control problem, which guarantees fast convergence rate, accurate control results, and no chattering, in the presence of control input constraints, unknown model system parameters, and external disturbances. Lv et al. [33] treated the 6-DOF synchronized control problem of space craft formation flying (SFF) task in the presence of uncertainties, external disturbances, and control saturation constraint. Hu et al. [34, 35] investigated the spacecraft attitude control problem with actuator saturation in the presence of no information about inertia and external disturbances. BoÏskovic et al. [36] proposed a continuous and globally stable tracking control algorithm for spacecraft attitude control problem in the presence of parametric uncertainties, external disturbances, and control input saturation. Filipe and Tsiotras [6] proposed a nonlinear adaptive controller for satellite position and attitude tracking in proximity operations, and the proposed controller requires no prior information about the inertia parameters of the chaser satellite. The gravity-gradient torque, the perturbing forces caused by Earth’s oblateness, and persistent external disturbances (but otherwise unknown) are all taken into account. Huang and Jia [16–18] investigated the fixed-time 6-DOF tracking control problem of a spacecraft fly-around task for a noncooperative target in the presence of parametric uncertainties and external disturbances.In this study, inspired by the adaptive asymptotic stabilization of position and attitude tracking problem that modeled by dual quaternion in Ref. [6], we extend the asymptotic stable control law with dual quaternion to the adaptive fixed-time terminal sliding mode control law on SE(3). In general, to develop the adaptive update law for inertia parameters, the products of the inertia matrix and other vectors are converted to the products of the regression matrices and inertia vector like in Ref. [12], while the update law for inertia parameters in this study requires no regression matrices, but just a simple operator for vectors related to inertia matrix. A fixed-time-type terminal sliding mode surface using the exponential coordinate of the pose is defined. And a fixed-time-type reaching law is applied to drive the tracking errors to converge to the equilibrium points in fixed time. The proposed adaptive control law and update law can be applied without inertia parameters and upper bounds of external disturbances, which guarantees the relative pose and velocities converging to the origins in fixed time from any given initial state, except that the initial relative error of principal rotation angle is exactly π radian. The proposed control scheme for coupled tracking maneuver guarantees the coupled rotational and translational tracking errors almost global fixed-time convergence over SE3⋉R6. Thus, the desired pose and velocities can be obtained in fixed time by performing the proposed maneuvering scheme in the presence of inertia parametric uncertainties and external disturbances.The contributions of this research are as follows: An adaptive modified fixed-time terminal sliding control law (AMFixed) is proposed for coupled spacecraft tracking maneuver that modeled on SE(3), so that the desired pose and velocities are obtained in fixed time in the presences of large inertia parametric uncertainties and external disturbances. The fixed-time control for spacecraft modeled on SE(3) in this study is firstly proposed, and the adaptive update law for inertial parameters is firstly extended from dual quaternion to SE(3). The modified fixed-time terminal sliding controller achieves a faster convergence rate than finite-time one (AFinite) for spacecraft relative motion in [9–12]. Moreover, the proposed modified controller (AMFixed) in this study can reach higher accuracy than the conventional one (AFixed) even the inertia parameters and disturbances are not accurately identified.The rest of this paper is as follows: The second section presents the rigid body dynamics of the spacecraft on SE(3). The third section explains the design and proofs of the adaptive fixed-time terminal sliding mode control on SE(3). The fourth section shows the simulation results of the proposed control laws. The last section concludes a summary of this research.Throughout the paper, some notations need to be predefined. SuperscriptT and -1 denote the transposition operation and inverse operation of a matrix, respectively. ⋅ means the absolute value. ⋅ denotes the 2-norm of a vector. Vector x=x1,x2,⋯,xnT and constant γ define the function sigxγ=x1γsgnx1,x1γsgnx2,⋯,x1γsgnxnT, where sgn⋅ is the signum function.


## 2. Rigid Body Dynamics of Spacecraft on SE(3)

This section presents the 6-DOF kinematics and dynamics of a spacecraft moving on orbit around the earth and relative 6-DOF dynamics of the spacecraft tracking system. First of all, to describe the relative motion dynamics between two spacecrafts, three orthogonal coordinate systems are defined. The Earth-Centered Inertial (ECI) reference frame is denoted byE=Ex,Ey,Ez, which is usually used for describing absolute motion of a spacecraft around the earth. The leader and follower spacecrafts are both assumed to be rigid bodies. B0=Bx0,By0,Bz0 and B=Bx,By,Bz are used for denoting the body-fixed frames with respect to the two spacecrafts. Spacecraft coupled translational and rotational maneuver for the proximity operation is shown in Figure 1.Figure 1
Spacecraft coupled translational and rotational maneuvers.


### 2.1. Rigid Motion on Lie Group SE(3) and Kinematics of the s121 Spacecraft

The special Euclidean group (Lie group SE(3)) is the configuration space for a rigid body motion in a 3-D Euclidean space, including rotational and translational motions. Lie group SE(3) is the semiproduct ofR3 and SO(3), which is expressed by SE3=R3⋉SO3. R3 is 3-D real Euclidean space denoting the position of the spacecraft, and SO(3) is the special orthogonal group describing the rotation of the spacecraft in a 3-D space. Thus, the configuration of the spacecraft motion is expressed as
(1)g=Rp01×31,where R denotes the attitude matrix rotated from the spacecraft body-fixed frame B to the inertial frame E, p represents the position of the spacecraft expressed in the ECI frame, and 0m×n denotes the all-zero matrix with m rows and n columns.The angular velocityω and translational velocity v expressed in the spacecraft’s body-fixed frame are defined in a unified form as
(2)ξ=ωv∈R6.To describe the kinematics and dynamics of a spacecraft, the following mapping operations need to be defined. The adjoint operator ofg∈SE3 is defined as
(3)Adg=R03×3p×RR∈R6×6.Theoperator⋅× is the skew-symmetric transformation of a vector in R3 or cross-product. The velocity mapping from R6 to se3 is
(4)ξ∨=ω×v01×30∈se3,where se3 denotes the Lie algebra of Lie group SE(3) which is also called the tangent space at identity. The adjoint operation of se3 is presented in a block matrix form as
(5)adξ=ω×03×3v×ω×∈R6×6.Here follows the coadjoint operation ofse3 which can be expressed by a 6-by-6 matrix as
(6)adξ∗=adξT=−ω×v×03×3−ω×.In addition, the kinematics of the spacecraft can be expressed as follows:(7)Ṙ=Rω×,ṗ=Rv,where Ṙ and ṗ are the angular velocity and translational velocity expressed in the inertial frame, respectively. Then kinematics (7) can be rewritten in a compact form as
(8)ġ=gξ∨.


### 2.2. Dynamics of a Spacecraft

The dynamics equations of rotational and translational motions for a rigid spacecraft are presented as(9)JΩ̇=JΩ×Ω+Mgp,R+Tcp,R,v,Ω+Td,mv̇=mv×Ω+Fgp,R+mRTaJ2pFcpp,R,v,Ω+Fd,where Tc∈R3 and Fc∈R3 denote the control torque and control force performed by the spacecraft acting on itself, respectively; Mg∈R3 and Fg∈R3 denote the gravity gradient moment and gravity force, respectively; and Td∈R3 and Fd∈R3 denote the unknown torques and forces on the spacecraft, respectively, which are mainly caused by radiation pressure, atmosphere drag, and other bounded uncertain disturbances.
(10)Mg=3μp5b×Jb,Fg=−mμp3b,where b=RTp and µ=398,600.44km3s−2 is the gravitational constant of the Earth. The perturbation caused by the Earth’s oblateness J2 expressed in ECI frame is given as follows [37]:
(11)aJ2=−3μJ2Re22p5px1−5pz2p2py1−5pz2p2pz3−5pz2p2,where px,py,pz are the components of p, J2=0.00108263, and Re=6378km is the equatorial radius of the Earth.By employing the coadjoint operator of se(3), combining the gravity moment and force on the spacecraft given by (10), and combining the moment of inertia and mass, the rotational and translational dynamics equation (9) can be expressed in a compact form as [8]
(12)Iξ̇=adξ∗Iξ+τg+τc+τd.(13)τg=MgFg+mRTaJ2,τc=TcFc,τd=TdFd,I=J03×303×3mI3,where τg∈R6 is a unified vector which consists of gravity gradient moment and force (including perturbation cause by J2), τc∈R6 is a unified vector which consists of control torque and force performed by the spacecraft actuator, and τd∈R6 is a unified vector which consists of external disturbance torques and forces acting on the spacecraft.The variables presented at previous are used for describing the kinematics and dynamics of the follower spacecraft. By adding superscript 0, all the kinematics and dynamics variables switch to denote variables of the leader spacecraft. In this study, the leader spacecraft is supposed to move in the gravity field and subject to no control inputs and no external disturbances. Thus, kinematics and dynamics equations of the leader spacecraft in compact forms are(14)ġ0=gξ0∨,(15)I0ξ̇0=adξ0∗I0ξ0+τg0.These two equations can be used for generating trajectories when initial statesg0t0,ξ0t0 at time t0 are known.


### 2.3. Relative Coupled Rotational and Translational Dynamics of Spacecraft

This subsection presents the relative coupled rotational and translational dynamics on SE(3). Leth∈SE3 represent the actual relative pose between the two spacecrafts and hd∈SE3 represents the desired relative pose. The desired relative pose is assumed to be a fixed configuration, which means that the desired relative velocities are all zeros. Then the configuration tracking error can be described by using h and hd. The actual relative configuration is given by h=g0−1g, and then the pose tracking error is given by
(16)he=hd−1h=hd−1g0−1g.The exponential coordinate is used for expressing the configuration tracking error of the follower spacecraft, which can be expressed as(17)η~∨=logmhd−1h,where logm is the logarithm map from SE(3) to se3. The vectorized exponential coordinate of the pose tracking error is expressed as
(18)η~=Θ~β~∈R6,where Θ~∈R3 and β~∈R3 are attitude tracking error and position tracking error, respectively, which are expressed by the exponential coordinate vectors as
(19)Θ~×=0,θ=0,θ2sinθR~−R~T,θ∈−ππ,θ≠0.β~=S−1Θ~p~,where R~ is the attitude error matrix and p~ is the position error vector. SΘ~ is expressed in Ref. [38]. The logarithm map (logm) of matrix from SE(3) to se3 is bijective when Θ<π.By taking the time derivative of equation (16) and substituting (14) into (8), the velocity tracking error of the follower spacecraft with respect to the leader spacecraft, which is expressed in the follower’s body-fixed frame, is obtained
(20)ξ~=ξ−Adh−1ξ0,where Adh−1ξ0 denotes the leader’s unified velocity expressed in the follower spacecraft body-fixed frame.The kinematics equation expressed by local coordinates is as follows [38]:
(21)η~̇=Gη~ξ~.The computation method forGη~ is described as follows: From Ref. [38], Gη∈R6×6 is expanded as
(22)Gη=I6×6+12adη+aθadη2+bθadη2,where
(23)θ2aθ=2−34θcotθ2−θ28csc2θ2,θ2bθ=1−14θcotθ2−θ28csc2θ2.Sinceη=ΘT,βTT∈R6 is presented before, here θ=Θ. Note that η is not uniquely defined when Θ is exactly π rad.The relative accelerationξ~̇ can be written as
(24)ξ~̇=ξ̇+adξAdh−1ξ0−Adh−1ξ̇0.Substituting the follower’s dynamics equation (12) into (24) yields the relative acceleration as
(25)Iξ~̇=adξ∗Iξ+IadξAdh−1ξ0−Adh−1ξ̇0+τg+τc+τd.Therefore, the coupled translational and rotational error tracking system is composed of the relative kinematics expressed by exponential coordinate (21) and the relative acceleration equation (25).


## 2.1. Rigid Motion on Lie Group SE(3) and Kinematics of the s121 Spacecraft

The special Euclidean group (Lie group SE(3)) is the configuration space for a rigid body motion in a 3-D Euclidean space, including rotational and translational motions. Lie group SE(3) is the semiproduct ofR3 and SO(3), which is expressed by SE3=R3⋉SO3. R3 is 3-D real Euclidean space denoting the position of the spacecraft, and SO(3) is the special orthogonal group describing the rotation of the spacecraft in a 3-D space. Thus, the configuration of the spacecraft motion is expressed as
(1)g=Rp01×31,where R denotes the attitude matrix rotated from the spacecraft body-fixed frame B to the inertial frame E, p represents the position of the spacecraft expressed in the ECI frame, and 0m×n denotes the all-zero matrix with m rows and n columns.The angular velocityω and translational velocity v expressed in the spacecraft’s body-fixed frame are defined in a unified form as
(2)ξ=ωv∈R6.To describe the kinematics and dynamics of a spacecraft, the following mapping operations need to be defined. The adjoint operator ofg∈SE3 is defined as
(3)Adg=R03×3p×RR∈R6×6.Theoperator⋅× is the skew-symmetric transformation of a vector in R3 or cross-product. The velocity mapping from R6 to se3 is
(4)ξ∨=ω×v01×30∈se3,where se3 denotes the Lie algebra of Lie group SE(3) which is also called the tangent space at identity. The adjoint operation of se3 is presented in a block matrix form as
(5)adξ=ω×03×3v×ω×∈R6×6.Here follows the coadjoint operation ofse3 which can be expressed by a 6-by-6 matrix as
(6)adξ∗=adξT=−ω×v×03×3−ω×.In addition, the kinematics of the spacecraft can be expressed as follows:(7)Ṙ=Rω×,ṗ=Rv,where Ṙ and ṗ are the angular velocity and translational velocity expressed in the inertial frame, respectively. Then kinematics (7) can be rewritten in a compact form as
(8)ġ=gξ∨.


## 2.2. Dynamics of a Spacecraft

The dynamics equations of rotational and translational motions for a rigid spacecraft are presented as(9)JΩ̇=JΩ×Ω+Mgp,R+Tcp,R,v,Ω+Td,mv̇=mv×Ω+Fgp,R+mRTaJ2pFcpp,R,v,Ω+Fd,where Tc∈R3 and Fc∈R3 denote the control torque and control force performed by the spacecraft acting on itself, respectively; Mg∈R3 and Fg∈R3 denote the gravity gradient moment and gravity force, respectively; and Td∈R3 and Fd∈R3 denote the unknown torques and forces on the spacecraft, respectively, which are mainly caused by radiation pressure, atmosphere drag, and other bounded uncertain disturbances.
(10)Mg=3μp5b×Jb,Fg=−mμp3b,where b=RTp and µ=398,600.44km3s−2 is the gravitational constant of the Earth. The perturbation caused by the Earth’s oblateness J2 expressed in ECI frame is given as follows [37]:
(11)aJ2=−3μJ2Re22p5px1−5pz2p2py1−5pz2p2pz3−5pz2p2,where px,py,pz are the components of p, J2=0.00108263, and Re=6378km is the equatorial radius of the Earth.By employing the coadjoint operator of se(3), combining the gravity moment and force on the spacecraft given by (10), and combining the moment of inertia and mass, the rotational and translational dynamics equation (9) can be expressed in a compact form as [8]
(12)Iξ̇=adξ∗Iξ+τg+τc+τd.(13)τg=MgFg+mRTaJ2,τc=TcFc,τd=TdFd,I=J03×303×3mI3,where τg∈R6 is a unified vector which consists of gravity gradient moment and force (including perturbation cause by J2), τc∈R6 is a unified vector which consists of control torque and force performed by the spacecraft actuator, and τd∈R6 is a unified vector which consists of external disturbance torques and forces acting on the spacecraft.The variables presented at previous are used for describing the kinematics and dynamics of the follower spacecraft. By adding superscript 0, all the kinematics and dynamics variables switch to denote variables of the leader spacecraft. In this study, the leader spacecraft is supposed to move in the gravity field and subject to no control inputs and no external disturbances. Thus, kinematics and dynamics equations of the leader spacecraft in compact forms are(14)ġ0=gξ0∨,(15)I0ξ̇0=adξ0∗I0ξ0+τg0.These two equations can be used for generating trajectories when initial statesg0t0,ξ0t0 at time t0 are known.


## 2.3. Relative Coupled Rotational and Translational Dynamics of Spacecraft

This subsection presents the relative coupled rotational and translational dynamics on SE(3). Leth∈SE3 represent the actual relative pose between the two spacecrafts and hd∈SE3 represents the desired relative pose. The desired relative pose is assumed to be a fixed configuration, which means that the desired relative velocities are all zeros. Then the configuration tracking error can be described by using h and hd. The actual relative configuration is given by h=g0−1g, and then the pose tracking error is given by
(16)he=hd−1h=hd−1g0−1g.The exponential coordinate is used for expressing the configuration tracking error of the follower spacecraft, which can be expressed as(17)η~∨=logmhd−1h,where logm is the logarithm map from SE(3) to se3. The vectorized exponential coordinate of the pose tracking error is expressed as
(18)η~=Θ~β~∈R6,where Θ~∈R3 and β~∈R3 are attitude tracking error and position tracking error, respectively, which are expressed by the exponential coordinate vectors as
(19)Θ~×=0,θ=0,θ2sinθR~−R~T,θ∈−ππ,θ≠0.β~=S−1Θ~p~,where R~ is the attitude error matrix and p~ is the position error vector. SΘ~ is expressed in Ref. [38]. The logarithm map (logm) of matrix from SE(3) to se3 is bijective when Θ<π.By taking the time derivative of equation (16) and substituting (14) into (8), the velocity tracking error of the follower spacecraft with respect to the leader spacecraft, which is expressed in the follower’s body-fixed frame, is obtained
(20)ξ~=ξ−Adh−1ξ0,where Adh−1ξ0 denotes the leader’s unified velocity expressed in the follower spacecraft body-fixed frame.The kinematics equation expressed by local coordinates is as follows [38]:
(21)η~̇=Gη~ξ~.The computation method forGη~ is described as follows: From Ref. [38], Gη∈R6×6 is expanded as
(22)Gη=I6×6+12adη+aθadη2+bθadη2,where
(23)θ2aθ=2−34θcotθ2−θ28csc2θ2,θ2bθ=1−14θcotθ2−θ28csc2θ2.Sinceη=ΘT,βTT∈R6 is presented before, here θ=Θ. Note that η is not uniquely defined when Θ is exactly π rad.The relative accelerationξ~̇ can be written as
(24)ξ~̇=ξ̇+adξAdh−1ξ0−Adh−1ξ̇0.Substituting the follower’s dynamics equation (12) into (24) yields the relative acceleration as
(25)Iξ~̇=adξ∗Iξ+IadξAdh−1ξ0−Adh−1ξ̇0+τg+τc+τd.Therefore, the coupled translational and rotational error tracking system is composed of the relative kinematics expressed by exponential coordinate (21) and the relative acceleration equation (25).


## 3. Adaptive Fixed-Time Terminal Sliding Mode Controller Design

This section proposes a novel fixed-time convergence adaptive control scheme inspired by the asymptotic convergence one. Some assumptions, definitions, and lemmas are given before proposing the controller.


### 3.1. Assumptions, Definitions, and Lemmas

Assumption 1.
Assuming that the external disturbanceTd and Fd are constrained by some known constant τcs∈R6, i.e., each component of τd must satisfy the following inequality:
(26)τdi≤τcsi.Assumption 2.
Assume that the configuration (g0 and g) and velocities (ξ0 and ξ) of the two spacecrafts can be obtained for designing the controller for the follower spacecraft through onboard navigation devices.Assumption 3.
Each component of the control inputs is bounded by specified saturation values, expressed as follows:(27)τci≤Tcmax,i=1,2,3,τci≤Fcmax,i=4,5,6.Assumption 4.
The control inputs strictly dominate the unknown disturbance, given by(28)τdit≤Tdmax<Tcmax,i=1,2,3,τdit≤Fdmax<Fcmax,i=4,5,6.Lemma 5.
For anyx∈Rn,a∈R, the following equations hold:
(29)dxa+1dt=a+1diagsigxaẋ,dsigxa+1dt=a+1diagxaẋ.Definition 6.
Considering a nonlinear system represented by the following differential equation:(30)ẋt=fxt,x0=0,f0=0,where x∈Rn. The origin of (30) is said to be a finite-time stable equilibrium point if there exists Tx that satisfies xt≡0,∀t>Tx.Definition 7.
Considering the system of differential equation (30): if the origin of (30) is globally finite-time stable with bounded settling time function Tx0, that is, ∃Tmax>0 such that Tx0<Tmax. We conclude that it is a fixed-time stable equilibrium point. In other words, the convergence time of a fixed-time stable system is independent of the initial states of the system.Lemma 8 [16].
Forxi∈R,i=1,2,⋯,n,0<r1≤1,r2>1, the following inequalities hold:
(31)∑i=1nxir1≤∑i=1nxir1,∑i=1nxir2≤nr2−1∑i=1nxir2.Lemma 9 [39].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(32)V̇t+λ1Vt+λ2Vσt≤0,∀t>0,we conclude that Vt will converge to the equilibrium point in finite time bounded by
(33)T≤1λ11−σlnλ1V1−σt0+λ2λ2,where λ1>0,λ2>0 and 0<σ<1.Lemma 10 [40].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(34)V̇t+λ1Vtσ2+λ2Vtσ2≤0,∀t>0,where λ1>0,λ2>0 and 1<σ1<∞,0<σ2<1, we conclude that Vt will converge to the equilibrium point in fixed time bounded by
(35)T≤1λ1σ1−1+1λ21−σ2.Lemma 11 [41].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(36)V̇t+λ1Vt1/2+1/2σ1+1/2σ1−1/2sgnVt−1+λ2Vtσ2≤0,∀t>0,where λ1>0,λ2>0 and 1<σ1<∞,0<σ2<1, we conclude that Vt will converge to the equilibrium point in fixed time bounded by
(37)T≤1λ1σ1−1+1λ11−σ2ln1+λ1λ2.Since the inequalityln1+λ1/λ2≤λ1/λ2 holds, the convergence time proposed in Lemma 11 is shorter than that in Lemma 10.


### 3.2. Fixed-Time Terminal Sliding Mode Surface

The fixed-time type sliding surface consists of exponential coordinate of configuration tracking error and relative velocity tracking error, and is defined as:(38)s=ξ~+C1sigη~σ1+C2sigη~σ2,where s=s1,⋯,s6T∈R6 is the sliding surface, ξ~=ξ1,⋯ξ6,T, sigη~σ=sigη1σ,⋯,sigη6σT, C1=diagc11,⋯,c16 and C2=diagc21,⋯,c26 are both positive definite matrices, and 1<σ1<∞,0<σ2<1.


### 3.3. Adaptive Controller

Inspired by Ref. [6], the main results of this research are the adaptive finite-time and fixed-time control laws for spacecraft 6-DOF tracking maneuver. The adaptive control laws need no information about the inertia parameters and disturbances. In addition, it does not need to know the bounds of the eigenvalues of the inertia matrix and mass. The following theorem will propose an adaptive fixed-time type controller. Moreover, it will be proved that the controller guarantees the rotational and translational tracking errors to converge to the equilibrium points in fixed time.To design a fixed-time terminal sliding mode control law, an adaptive fixed-time TSM-type reaching law is defined as [41]
(39)ṡt=−Ks1sigsσ1−Ks2sigsσ1,where Ks1 and Ks1 are both positive definite matrices. The reaching law-based control scheme is able to ensure stabilization of a closed-loop system. By tuning the coefficient matrices Ks1 and Ks1, chattering will be attenuated.Considering the error tracking kinematics and dynamics shown in (21) and (25), respectively, the unified feedback control law is given as follows (named as AFixed in abbreviation):
(40)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~−Ks1sigsσ1−Ks2sigsσ2−KgGTη~η~,where
(41)s=ξ~+C1sigη~σ1+C2sigη~σ2,(42)Λ1=diagσ1C1η~σ1−1,(43)Λ2=diagσ2C2η~σ2−1.τ^g is the estimated unified gravity force vector denoted as follows which takes a same form as previous:
(44)τ^g=M^gF^g+m^RTaJ2,M^g=3μp5b×J^b,F^g=−m^μp3b.I^ is an estimated vector of the inertia parameters and updated by the following equation:
(45)ddtvI^=Kihs,adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~+agE+aJ2E−Kihadsξ,ξ+hadsbμE,bE,where Ki∈R6×6 is a positive definite matrix, agE=01×3,−μbT/p3T, aJ2E=01×3,aJ2TRT, bμE=3μbT/p5,01×3T, bE=bT,01×3T, and vI=J11,J12,J13,J22,J23,J33,mT is a vectorized version of the inertia matrix I; the function h⋅,⋅ is defined as aTIb=ha,bTvI=vITha,b,
(46)ha,b=a1b1,a2b1+a1b2,a3b1+a1b3,a2b2,a3b2+a2b3,a4b4+a5b5+a6b6T,where τ^d is the disturbance estimations updated by
(47)ddtτ^d=Kjs,where Kj∈R6×6 is a positive definite matrix.Theorem 12.
Under Assumption1 and Assumption 3, we apply the proposed control law (40) and the adaptive update laws (45) and (47) to the 6-DOF-coupled relative dynamics model. Then the Si will converge to region Si≤Os in fixed time. Also the tracking error ξ~ and η~ will converge into the regions ξ~i≤Oξ~ and η~i≤Oη~ in fixed time, respectively.
(48)Os=min2λminIΔ′κ11/1+σ1,2λminIΔ′κ21/1+σ2,Oη~=minOsλminC1σ1,OsλminC2σ2,Oξ~=Os+λmaxC1Oη~σ1+λmaxC2Oη~σ2,where λmin⋅ and λmax⋅ denote the minimum and maximum eigenvalues of a matrix, respectively. Si denotes the ith component of s.Proof.
Step 1. This step proves that a certain component of sliding surfaceSi will converge to the region Os in fixed time that satisfying Si≤Os.
First, the inertia matrix (consists of the moment of inertia matrix and mass) and external disturbance estimated errors are defined as(49)I~=I^−I,τ~d=τ^d−τd.
Note thatη~=06×1, s=06×1, vI~=07×1, and τ~d=06×1 are the equilibrium points of the closed-loop system represented by equation (9). A candidate Lyapunov function is designed as follows:
(50)Vη~,s,vI~,τ~d=12η~TKgη~+12sTIs+12vI~TKi−1vI~+12τ~dTKj−1τ~d.
V is said to be a valid Lyapunov function since Vη~=0,s=0,vI~=0,τ~d=0=0 and Vη~,s,vI~,τ~d>0, for all η~=0,s=0,vI~=0,τ~d=0∈R6×R6×R7×R606×1,06×1,07×1,06×1. Taking the derivative of V yields
(51)V̇=η~TKgη~̇+sTIṡ+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
According to equations (21) and (41), we have Gη~s=Gη~ξ~+Gη~C1sigη~σ1+Gη~C2sigη~σ2, then
(52)η~̇=Gη~ξ~=Gη~s−Gη~C1sigη~σ1−Gη~C2sigη~σ2.
Taking the time derivative of the sliding mode surface (41), together with equation (52), then plug them into V̇ yields
(53)V̇=η~TKgGη~s−Gη~C1sigη~σ1−Gη~C2sigη~σ2+sTIξ~̇+sTIΛ1η~̇+Λ2η~̇+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Applying equations (21) and (25) to the above equation,
(54)V̇=η~TKgGη~s−η~TKgGη~C1η~σ1−η~TKgGη~C2η~σ2+sTadξ∗Iξ+τg+τc+τd+Iadξadh−1ξ0−adh−1ξ̇0+Λ1η~̇+Λ2η~̇+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Plugging the feedback control law (40) yields
(55)V̇=η~TKgGη~s−η~TKgGη~C1η~σ1−η~TKgGη~C2η~σ2−sTadξ∗I~ξ+τ~g+τ~d+I~adξadh−1ξ0−adh−1ξ̇0+Λ1η~̇+Λ2η~̇−sTKs1sigsσ1+Ks2sigsσ2+KgGTη~η~+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Sincev̇I~ and τ~̇ are defined by equations (45) and (47), respectively, it shows that
(56)V̇=−η~TKgGη~C1sigη~σ1−η~TKgGη~C2sigη~σ2−sTKs1sigsσ1−sTKs2sigsσ2.
According to Lemma8, the following inequality holds:
(57)V̇≤−61−σ1/2λmin2λmaxI1−σ1/212sTIs1−σ1/2−λminKs22λmaxI1−σ2/212sTIs1−σ2/2−61−σ1/2λminGC121−σ1/212η~TKgη~1−σ1/2−λminGC221−σ2/212η~TKgη~1−σ2/2.
Considering the last two terms ofV̇
(58)V̇≤−61−σ1/2λminKs12λmaxI1+σ1/212sTIs1+σ1/2−λminKs22λmaxI1+σ2/212sTIs1+σ2/2−61−σ1/2λminGC121+σ1/212η~TKgη~1+σ1/2−λminGC221+σ2/212η~TKgη~1+σ2/2−12vI~TKi−1vI~1+σ1/2−12vI~TKi−1vI~1+σ2/2−12τ~dTKj−1τ~d1+σ1/2−12τ~dTKj−1τ~d1+σ2/2+Δ,where Δ is defined as
(59)Δ=12vI~TKi−1vI~1+σ1/2+12vI~TKi−1vI~1+σ2/2+12τ~dTKj−1τ~d1+σ1/2+12τ~dTKj−1τ~d1+σ2/2.
Assuming that there exist unknown constantsχ1,χ2 satisfying vI~≤χ1 and τ~d≤χ2, respectively. Then Δ satisfies the following inequality
(60)Δ≤12λmaxKi−1χ121+σ1/2+12λmaxKi−1χ121+σ2/2+12λmaxKj−1χ221+σ1/2+12λmaxKjχ121+σ2/2=Δ′.
It is obvious thatΔ′ is bounded. We define κ1,κ1, as follows:
(61)κ1=41−σ1/2min−61−σ1/2λminKs12λmaxI1+σ1/2,−61−σ1/2λminGC121+σ1/2,1,κ2=minλminKs22λmaxI1+σ2/2,λminGC221+σ2/2,1.
Then the inequality (58) can be simplified as
(62)V̇≤−κ1V1+σ1/2−κ2V1+σ2/2+Δ′.
HenceV will converge into the region OV=minΔ′/κ12/1+σ1,Δ′/κ22/1+σ2 in fixed time.
The following inequalities hold:(63)12sTIs≤V,12η~TKgη~≤V.
Then we can conclude thatSi will converge to the region Os=2/λminIOV in fixed time and ηi will converge to Oη~=2/λminKgOV simultaneously.
Step 2. In this step, we prove that the tracking errorsξ~i and η~i will converge into the small regions Oξ~ and Oη~ in fixed time, respectively. Then, for Si≤Os in Step 1, it follows the two cases given as follows:
Case 1. IfSi=0 is reached, ξ~i=0∈Oξ~ and η~i=0∈Oη~ will be reached in fixed time according to Lemma 10.
Case 2. IfSi≠0, according to the sliding surface, it means that the tracking error1 η~i has converged into the region η~i≤Oη~. Based on equation (38), we have ξ~i+C1isigη~iσ1+C2isigη~iσ2=ψi,ψi≤Os. It follows that ξ~i≤ψi+C1isigη~iσ1+C2isigη~iσ2≤Oξ~. This inequality shows that η~i has converged into the region ξ~i≤Oξ~. According to Lemma 10, we can conclude that the tracking errors η~i and ξ~i will converge to the regions η~i≤Oη~ and ξ~i≤Oξ~ in fixed time, respectively.


### 3.4. Other Sliding Surfaces and Reaching Laws

If the powerσ1 in sliding surface (38) and reaching law (39) is replaced by a power that takes the form as
(64)12+12σ112σ1−12sgnx−1,i.e., when si<1 in the reaching phase and η~i<1 in the sliding phase are satisfied, the power σ1 switches to 1. According to Lemma 11, the modified power will lead to a faster convergence rate. To compare different controllers’ performances, we present the following sliding surfaces and reaching laws, as well as the corresponding control laws.
(i)
Adaptive fixed-time terminal sliding mode control law (AFixed): the corresponding sliding surface (38), reaching law (39), and control law (40) are presented previously
(ii)
Adaptive modified fixed-time terminal sliding mode control law (AMFixed):(65)s=ξ~+C1sigη~1/2+1/2σ1+1/2σ1−1/2signη~−1+C2sigη~σ2,(66)ṡt=−Ks1sigs1/2+1/2σ1+1/2σ1−1/2signs−1−Ks2sigs~σ2,(67)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ1Gη~ξ~−Ks1sigs1/2+1/2σ1+1/2σ1−1/2signs−1+Ks2sigsσ2+KgGTη~η~
(iii)
Adaptive finite-time terminal sliding mode control law (AFinite):(68)s=ξ~+C1η~+C2sigη~σ2,(69)ṡt=−Ks1s−Ks2sigs~σ2,(70)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~−Ks1s+Ks2sigsσ2+GTη~η~Let abbreviations AFixed, AMFixed, and AFinite represent the three controllers, respectively. Since the sliding surfaces and reaching laws have similar forms, the feedback control laws take similar forms as equation (40). The convergence properties of the feedback control laws are easy to prove and the proofs will not go into details here. In the simulation section, different feedback control laws will be applied to the error tracking control system.The proposed control laws and adaptive update laws take similar forms as ones in Ref. [6]. As extensions of adaptive control law with asymptotic convergence property proposed in Ref. [6], this study proposes fixed-time and finite-time convergent adaptive control laws for 6-DOF tracking maneuver of a rigid body spacecraft. The control laws (40), (67), and (70) can be performed without accurate inertia parameters and information about external disturbances. Thus, the proposed control laws and update laws can be applied to the coupled spacecraft 6-DOF tracking system, although there exist large inertia parametric uncertainties and external disturbances.Remark 13.
In this study, the follower’s states and the leader’s states are simulated simultaneously. Kinematics equations of the two take the same forms as that of relative motion, which are(71)η̇0=Gη0ξ0,(72)η̇=Gηξ.The leader’s differential equations (71) and (15) and the follower’s differential equations (12) and (72) can be integrated by the Runge-Kutta method which is suitable for the Euclidean space, since η0t,ξ0t and ηt,ξt are vectors in R6. Then the poses of the two spacecrafts are obtained through the exponential map from se(3) to SE(3). Thereafter, the relative states can be computed by using equations in the second section, which will be applied to the control laws.


## 3.1. Assumptions, Definitions, and Lemmas

Assumption 1.
Assuming that the external disturbanceTd and Fd are constrained by some known constant τcs∈R6, i.e., each component of τd must satisfy the following inequality:
(26)τdi≤τcsi.Assumption 2.
Assume that the configuration (g0 and g) and velocities (ξ0 and ξ) of the two spacecrafts can be obtained for designing the controller for the follower spacecraft through onboard navigation devices.Assumption 3.
Each component of the control inputs is bounded by specified saturation values, expressed as follows:(27)τci≤Tcmax,i=1,2,3,τci≤Fcmax,i=4,5,6.Assumption 4.
The control inputs strictly dominate the unknown disturbance, given by(28)τdit≤Tdmax<Tcmax,i=1,2,3,τdit≤Fdmax<Fcmax,i=4,5,6.Lemma 5.
For anyx∈Rn,a∈R, the following equations hold:
(29)dxa+1dt=a+1diagsigxaẋ,dsigxa+1dt=a+1diagxaẋ.Definition 6.
Considering a nonlinear system represented by the following differential equation:(30)ẋt=fxt,x0=0,f0=0,where x∈Rn. The origin of (30) is said to be a finite-time stable equilibrium point if there exists Tx that satisfies xt≡0,∀t>Tx.Definition 7.
Considering the system of differential equation (30): if the origin of (30) is globally finite-time stable with bounded settling time function Tx0, that is, ∃Tmax>0 such that Tx0<Tmax. We conclude that it is a fixed-time stable equilibrium point. In other words, the convergence time of a fixed-time stable system is independent of the initial states of the system.Lemma 8 [16].
Forxi∈R,i=1,2,⋯,n,0<r1≤1,r2>1, the following inequalities hold:
(31)∑i=1nxir1≤∑i=1nxir1,∑i=1nxir2≤nr2−1∑i=1nxir2.Lemma 9 [39].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(32)V̇t+λ1Vt+λ2Vσt≤0,∀t>0,we conclude that Vt will converge to the equilibrium point in finite time bounded by
(33)T≤1λ11−σlnλ1V1−σt0+λ2λ2,where λ1>0,λ2>0 and 0<σ<1.Lemma 10 [40].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(34)V̇t+λ1Vtσ2+λ2Vtσ2≤0,∀t>0,where λ1>0,λ2>0 and 1<σ1<∞,0<σ2<1, we conclude that Vt will converge to the equilibrium point in fixed time bounded by
(35)T≤1λ1σ1−1+1λ21−σ2.Lemma 11 [41].
Assuming thatVt is a continuous positive definite function and satisfies the inequality as follows:
(36)V̇t+λ1Vt1/2+1/2σ1+1/2σ1−1/2sgnVt−1+λ2Vtσ2≤0,∀t>0,where λ1>0,λ2>0 and 1<σ1<∞,0<σ2<1, we conclude that Vt will converge to the equilibrium point in fixed time bounded by
(37)T≤1λ1σ1−1+1λ11−σ2ln1+λ1λ2.Since the inequalityln1+λ1/λ2≤λ1/λ2 holds, the convergence time proposed in Lemma 11 is shorter than that in Lemma 10.


## 3.2. Fixed-Time Terminal Sliding Mode Surface

The fixed-time type sliding surface consists of exponential coordinate of configuration tracking error and relative velocity tracking error, and is defined as:(38)s=ξ~+C1sigη~σ1+C2sigη~σ2,where s=s1,⋯,s6T∈R6 is the sliding surface, ξ~=ξ1,⋯ξ6,T, sigη~σ=sigη1σ,⋯,sigη6σT, C1=diagc11,⋯,c16 and C2=diagc21,⋯,c26 are both positive definite matrices, and 1<σ1<∞,0<σ2<1.


## 3.3. Adaptive Controller

Inspired by Ref. [6], the main results of this research are the adaptive finite-time and fixed-time control laws for spacecraft 6-DOF tracking maneuver. The adaptive control laws need no information about the inertia parameters and disturbances. In addition, it does not need to know the bounds of the eigenvalues of the inertia matrix and mass. The following theorem will propose an adaptive fixed-time type controller. Moreover, it will be proved that the controller guarantees the rotational and translational tracking errors to converge to the equilibrium points in fixed time.To design a fixed-time terminal sliding mode control law, an adaptive fixed-time TSM-type reaching law is defined as [41]
(39)ṡt=−Ks1sigsσ1−Ks2sigsσ1,where Ks1 and Ks1 are both positive definite matrices. The reaching law-based control scheme is able to ensure stabilization of a closed-loop system. By tuning the coefficient matrices Ks1 and Ks1, chattering will be attenuated.Considering the error tracking kinematics and dynamics shown in (21) and (25), respectively, the unified feedback control law is given as follows (named as AFixed in abbreviation):
(40)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~−Ks1sigsσ1−Ks2sigsσ2−KgGTη~η~,where
(41)s=ξ~+C1sigη~σ1+C2sigη~σ2,(42)Λ1=diagσ1C1η~σ1−1,(43)Λ2=diagσ2C2η~σ2−1.τ^g is the estimated unified gravity force vector denoted as follows which takes a same form as previous:
(44)τ^g=M^gF^g+m^RTaJ2,M^g=3μp5b×J^b,F^g=−m^μp3b.I^ is an estimated vector of the inertia parameters and updated by the following equation:
(45)ddtvI^=Kihs,adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~+agE+aJ2E−Kihadsξ,ξ+hadsbμE,bE,where Ki∈R6×6 is a positive definite matrix, agE=01×3,−μbT/p3T, aJ2E=01×3,aJ2TRT, bμE=3μbT/p5,01×3T, bE=bT,01×3T, and vI=J11,J12,J13,J22,J23,J33,mT is a vectorized version of the inertia matrix I; the function h⋅,⋅ is defined as aTIb=ha,bTvI=vITha,b,
(46)ha,b=a1b1,a2b1+a1b2,a3b1+a1b3,a2b2,a3b2+a2b3,a4b4+a5b5+a6b6T,where τ^d is the disturbance estimations updated by
(47)ddtτ^d=Kjs,where Kj∈R6×6 is a positive definite matrix.Theorem 12.
Under Assumption1 and Assumption 3, we apply the proposed control law (40) and the adaptive update laws (45) and (47) to the 6-DOF-coupled relative dynamics model. Then the Si will converge to region Si≤Os in fixed time. Also the tracking error ξ~ and η~ will converge into the regions ξ~i≤Oξ~ and η~i≤Oη~ in fixed time, respectively.
(48)Os=min2λminIΔ′κ11/1+σ1,2λminIΔ′κ21/1+σ2,Oη~=minOsλminC1σ1,OsλminC2σ2,Oξ~=Os+λmaxC1Oη~σ1+λmaxC2Oη~σ2,where λmin⋅ and λmax⋅ denote the minimum and maximum eigenvalues of a matrix, respectively. Si denotes the ith component of s.Proof.
Step 1. This step proves that a certain component of sliding surfaceSi will converge to the region Os in fixed time that satisfying Si≤Os.
First, the inertia matrix (consists of the moment of inertia matrix and mass) and external disturbance estimated errors are defined as(49)I~=I^−I,τ~d=τ^d−τd.
Note thatη~=06×1, s=06×1, vI~=07×1, and τ~d=06×1 are the equilibrium points of the closed-loop system represented by equation (9). A candidate Lyapunov function is designed as follows:
(50)Vη~,s,vI~,τ~d=12η~TKgη~+12sTIs+12vI~TKi−1vI~+12τ~dTKj−1τ~d.
V is said to be a valid Lyapunov function since Vη~=0,s=0,vI~=0,τ~d=0=0 and Vη~,s,vI~,τ~d>0, for all η~=0,s=0,vI~=0,τ~d=0∈R6×R6×R7×R606×1,06×1,07×1,06×1. Taking the derivative of V yields
(51)V̇=η~TKgη~̇+sTIṡ+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
According to equations (21) and (41), we have Gη~s=Gη~ξ~+Gη~C1sigη~σ1+Gη~C2sigη~σ2, then
(52)η~̇=Gη~ξ~=Gη~s−Gη~C1sigη~σ1−Gη~C2sigη~σ2.
Taking the time derivative of the sliding mode surface (41), together with equation (52), then plug them into V̇ yields
(53)V̇=η~TKgGη~s−Gη~C1sigη~σ1−Gη~C2sigη~σ2+sTIξ~̇+sTIΛ1η~̇+Λ2η~̇+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Applying equations (21) and (25) to the above equation,
(54)V̇=η~TKgGη~s−η~TKgGη~C1η~σ1−η~TKgGη~C2η~σ2+sTadξ∗Iξ+τg+τc+τd+Iadξadh−1ξ0−adh−1ξ̇0+Λ1η~̇+Λ2η~̇+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Plugging the feedback control law (40) yields
(55)V̇=η~TKgGη~s−η~TKgGη~C1η~σ1−η~TKgGη~C2η~σ2−sTadξ∗I~ξ+τ~g+τ~d+I~adξadh−1ξ0−adh−1ξ̇0+Λ1η~̇+Λ2η~̇−sTKs1sigsσ1+Ks2sigsσ2+KgGTη~η~+vI~TKi−1v̇I~+τ~dTKj−1τ~̇d.
Sincev̇I~ and τ~̇ are defined by equations (45) and (47), respectively, it shows that
(56)V̇=−η~TKgGη~C1sigη~σ1−η~TKgGη~C2sigη~σ2−sTKs1sigsσ1−sTKs2sigsσ2.
According to Lemma8, the following inequality holds:
(57)V̇≤−61−σ1/2λmin2λmaxI1−σ1/212sTIs1−σ1/2−λminKs22λmaxI1−σ2/212sTIs1−σ2/2−61−σ1/2λminGC121−σ1/212η~TKgη~1−σ1/2−λminGC221−σ2/212η~TKgη~1−σ2/2.
Considering the last two terms ofV̇
(58)V̇≤−61−σ1/2λminKs12λmaxI1+σ1/212sTIs1+σ1/2−λminKs22λmaxI1+σ2/212sTIs1+σ2/2−61−σ1/2λminGC121+σ1/212η~TKgη~1+σ1/2−λminGC221+σ2/212η~TKgη~1+σ2/2−12vI~TKi−1vI~1+σ1/2−12vI~TKi−1vI~1+σ2/2−12τ~dTKj−1τ~d1+σ1/2−12τ~dTKj−1τ~d1+σ2/2+Δ,where Δ is defined as
(59)Δ=12vI~TKi−1vI~1+σ1/2+12vI~TKi−1vI~1+σ2/2+12τ~dTKj−1τ~d1+σ1/2+12τ~dTKj−1τ~d1+σ2/2.
Assuming that there exist unknown constantsχ1,χ2 satisfying vI~≤χ1 and τ~d≤χ2, respectively. Then Δ satisfies the following inequality
(60)Δ≤12λmaxKi−1χ121+σ1/2+12λmaxKi−1χ121+σ2/2+12λmaxKj−1χ221+σ1/2+12λmaxKjχ121+σ2/2=Δ′.
It is obvious thatΔ′ is bounded. We define κ1,κ1, as follows:
(61)κ1=41−σ1/2min−61−σ1/2λminKs12λmaxI1+σ1/2,−61−σ1/2λminGC121+σ1/2,1,κ2=minλminKs22λmaxI1+σ2/2,λminGC221+σ2/2,1.
Then the inequality (58) can be simplified as
(62)V̇≤−κ1V1+σ1/2−κ2V1+σ2/2+Δ′.
HenceV will converge into the region OV=minΔ′/κ12/1+σ1,Δ′/κ22/1+σ2 in fixed time.
The following inequalities hold:(63)12sTIs≤V,12η~TKgη~≤V.
Then we can conclude thatSi will converge to the region Os=2/λminIOV in fixed time and ηi will converge to Oη~=2/λminKgOV simultaneously.
Step 2. In this step, we prove that the tracking errorsξ~i and η~i will converge into the small regions Oξ~ and Oη~ in fixed time, respectively. Then, for Si≤Os in Step 1, it follows the two cases given as follows:
Case 1. IfSi=0 is reached, ξ~i=0∈Oξ~ and η~i=0∈Oη~ will be reached in fixed time according to Lemma 10.
Case 2. IfSi≠0, according to the sliding surface, it means that the tracking error1 η~i has converged into the region η~i≤Oη~. Based on equation (38), we have ξ~i+C1isigη~iσ1+C2isigη~iσ2=ψi,ψi≤Os. It follows that ξ~i≤ψi+C1isigη~iσ1+C2isigη~iσ2≤Oξ~. This inequality shows that η~i has converged into the region ξ~i≤Oξ~. According to Lemma 10, we can conclude that the tracking errors η~i and ξ~i will converge to the regions η~i≤Oη~ and ξ~i≤Oξ~ in fixed time, respectively.


## 3.4. Other Sliding Surfaces and Reaching Laws

If the powerσ1 in sliding surface (38) and reaching law (39) is replaced by a power that takes the form as
(64)12+12σ112σ1−12sgnx−1,i.e., when si<1 in the reaching phase and η~i<1 in the sliding phase are satisfied, the power σ1 switches to 1. According to Lemma 11, the modified power will lead to a faster convergence rate. To compare different controllers’ performances, we present the following sliding surfaces and reaching laws, as well as the corresponding control laws.
(i)
Adaptive fixed-time terminal sliding mode control law (AFixed): the corresponding sliding surface (38), reaching law (39), and control law (40) are presented previously
(ii)
Adaptive modified fixed-time terminal sliding mode control law (AMFixed):(65)s=ξ~+C1sigη~1/2+1/2σ1+1/2σ1−1/2signη~−1+C2sigη~σ2,(66)ṡt=−Ks1sigs1/2+1/2σ1+1/2σ1−1/2signs−1−Ks2sigs~σ2,(67)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ1Gη~ξ~−Ks1sigs1/2+1/2σ1+1/2σ1−1/2signs−1+Ks2sigsσ2+KgGTη~η~
(iii)
Adaptive finite-time terminal sliding mode control law (AFinite):(68)s=ξ~+C1η~+C2sigη~σ2,(69)ṡt=−Ks1s−Ks2sigs~σ2,(70)τc=−τ^g+τ^d+adξ∗I^ξ+I^adξadh−1ξ0−adh−1ξ̇0+Λ1Gη~ξ~+Λ2Gη~ξ~−Ks1s+Ks2sigsσ2+GTη~η~Let abbreviations AFixed, AMFixed, and AFinite represent the three controllers, respectively. Since the sliding surfaces and reaching laws have similar forms, the feedback control laws take similar forms as equation (40). The convergence properties of the feedback control laws are easy to prove and the proofs will not go into details here. In the simulation section, different feedback control laws will be applied to the error tracking control system.The proposed control laws and adaptive update laws take similar forms as ones in Ref. [6]. As extensions of adaptive control law with asymptotic convergence property proposed in Ref. [6], this study proposes fixed-time and finite-time convergent adaptive control laws for 6-DOF tracking maneuver of a rigid body spacecraft. The control laws (40), (67), and (70) can be performed without accurate inertia parameters and information about external disturbances. Thus, the proposed control laws and update laws can be applied to the coupled spacecraft 6-DOF tracking system, although there exist large inertia parametric uncertainties and external disturbances.Remark 13.
In this study, the follower’s states and the leader’s states are simulated simultaneously. Kinematics equations of the two take the same forms as that of relative motion, which are(71)η̇0=Gη0ξ0,(72)η̇=Gηξ.The leader’s differential equations (71) and (15) and the follower’s differential equations (12) and (72) can be integrated by the Runge-Kutta method which is suitable for the Euclidean space, since η0t,ξ0t and ηt,ξt are vectors in R6. Then the poses of the two spacecrafts are obtained through the exponential map from se(3) to SE(3). Thereafter, the relative states can be computed by using equations in the second section, which will be applied to the control laws.


## 4. Numerical Simulation Results

The initial states are given, and the proposed controllers are simulated in this section. The functionblkdiagM1,M2,⋯,Mn in Table 1 denotes the block diagonal matrix composed of submatrices M1,M2,⋯,Mn. The parameters in Table 1 satisfy the equalities: C1=C2,Ks1=Ks2. The three controllers for coupled spacecraft tracking maneuver are simulated, mainly to demonstrate the effectiveness of AMFixed (67). AMFixed will achieve better control performances theoretically according to Lemma 11. The other control laws, AFixed (40) and AFinite (70), are also applied to the closed-loop feedback system to compare their performances with AMFixed (67). The leader spacecraft moves on a Molniya orbit which would increase complexity to the maneuver task, and the follower moves on the neighboring orbit. Table 2 presents the leader’s initial orbital elements, supposing that the leader’s initial attitude is to align its body-fixed frame with its LVLH frame. The leader is assumed to move freely in the gravity field, and not be affected by any external disturbance or maneuver. The purpose of the control is to drive the actual relative states to the desired relative states by performing maneuvers. In other words, the follower spacecraft should achieve the desired relative pose hd and keep the attitude in synchronizing with the leader. Here the desired position and velocity expressed in the leader’s body-fixed frame are 500Tm and 000Tm/s, respectively. The initial relative position, attitude, and velocities are given in Table 3. The inertia parameters of the leader are defined as follows:
(73)m0=110kg,J0=21.7−0.2−0.5−0.222.3−0.3−0.5−0.325.5kg⋅m2.Table 1
Design parameters.
Design parameters
Values
σ1,σ2
5/4, 4/5
C1,C2
diag0.12,0.12,0.12,0.084,0.066,0.063
Ks1,Ks2
blkdiag30I3,100I3
Ki
blkdiag103I6,2
Kj
0.005I6
Kg
0.1I6Table 2
Initial orbital elements of the leader.
Orbital element
Value
Semimajor axis (km)
26628
Eccentricity
0.7417
Inclination (°)
63.4
Argument of perigee (°)
270
RAAN (°)
0
True anomaly (°)
120Table 3
Initial states of the follower spacecraft relative to the leader.
Initial states
Values
Leader’s angular velocity (rad/s)
0,0,0.0011T
Relative position (m)
15,15,15T
Relative attitude (°)
120
Relative principal rotation axis
−2,−2,−3T
Relative translational velocity (m/s)
−0.051,−0.247,−0.075T
Relative angular velocity (rad/s)
0.009,5.98,−9.31T×10−4The nominal inertia properties of the follower spacecraft are defined as follows:(74)m=100kg,J=23.7−0.2−0.5−0.225.3−0.3−0.5−0.322.5kg⋅m2.The follower spacecraft is assumed to be able to perform continuous control maneuvers. Constraints of control torque and force are denoted byTci≤Tcmax=1N⋅m and Fci≤Fcmax=10N, respectively. The control parameters are presented in Table 1. The initial values of parameter estimations are chosen as τ^d0=0,0,0,0,0,0T, vI0=0,0,0,0,0,0,0T. The fourth-order Runge-Kutta method with MATLAB/SIMULINK is used for the numerical simulations. The step size is 0.01 s, and the simulation time is tf=100s. Meanwhile, the external disturbances on the follower spacecraft are modeled as follows:Td=0.005sin0.1t,cos0.1t,−sin0.1tT, Fd=0.05sin0.1t,cos0.1t,−sin0.1tT. Figures 2 and 3 show rotational and translational tracking errors, respectively. Here we describe the control performances of AMFixed in details. The magnitude of attitude tracking error θ~=Θ~ in Figure 2(a) converges to less than 0.0017° within 25 s, while the norm of the position tracking error β~=β~t in Figure 3(a) corresponding to AMFixed converges to less than 0.028 m within 37 s. Figure 2(b) shows that the magnitude of angular velocity tracking error ω~ corresponding to AMFixed converges to less than 0.00061°/s within 25 s, while Figure 3(b) shows that the magnitude of translational velocity tracking error v~ corresponding to AMFixed converge to less than 0.0056 m/s within 37 s.Figure 2
Rotational tracking errors.
(a)
Attitude
(b)
Angular velocityFigure 3
Translational tracking errors.
(a)
Position
(b)
VelocityFigure4(a) shows estimations of inertia parameters. It is observed that the inertia estimations are not exactly identified as their actual values. Because the control inputs in this case do not satisfy the persistent excitation condition. However, the adaptive control law does not need accurate values [42]. The disturbance estimations are shown in Figure 4(b), which also converge to constant values.Figure 4
Estimated parameters.
(a)
Estimated inertia parameters
(b)
Estimated disturbancesFrom the simulation results in Figures2 and 3, we can know that attitude and position tracking errors of the three controllers are different. AMFixed and AFinite hold almost the same steady-state errors, and AFixed keeps the maximum steady-state errors in the three controllers. The convergence rates of the three controllers in attitude control are almost the same. When zooming in Figure 2(a), it can be found that the convergence rates of AMFixed and AFixed are slightly faster than that of AFinite in the midstage of attitude control. In Figure 3, it shows that position and velocity tracking errors of AMFixed and AFixed converge significantly faster than that of AFinite. The velocity tracking steady-state errors of the three controllers are almost the same.Figure5 shows control inputs (torques and forces) varying with time. Control torques and forces of the three control laws are shown in Figures 5(a) and 5(b), respectively. And each component of the control inputs is bounded by ±1 N·m (control torque) or ±10 N (control force), respectively. Control inputs on the follower spacecraft are saturate or relatively large in the early stage and then decrease quickly. The reaching law-based terminal sliding mode control laws in this study largely attenuate the chattering caused by the signum function-based sliding mode control. The performances of the three controllers are shown in Table 4. Let tst in Table 4 denote the time to reach the steady state.Figure 5
Control torque and force.
(a)
Torque
(b)
ForceTable 4
Control performance comparisons.
Performance indices
AMFixed
AFixed
AFinite
θ~tf∘
0.0017
0.002
0.0017
β~tfm
0.028
0.036
0.028
ω~tf∘/s
0.00061
0.00077
0.0061
v~tfm/s
0.0056
0.0056
0.0056
∫0tfTcdtN⋅s
16.95
16.95
16.66
∫0tfFcdtN⋅m⋅s
382.5
381.5
332
tstsattitude
24.5
31
24.5
tstsposition
37
38
51From Figures5 and 6, some evident features can be found. In attitude control processes of the three controllers, the control torque inputs varying with time are slightly different, while in position control processes of the three controllers, the control force inputs are quite different. In both attitude and position control processes, control inputs of AMFixed and AFixed decrease later than that of AFinite. The control force inputs of the two fixed-time control cases begin to decrease from about 20 s, and the control force inputs of the finite-time control case decrease from about 10 s. This is due to the power greater than 1 of the fixed-time control. Fixed-time control generates larger control inputs than the finite time control in the early stage. It means that the fixed-time control accelerates and brakes faster in the phase with relatively large error. The control torque effort (Integration of control torque) of AFinite is less than those AMFixed and AFixed, although the differences between them are small, while control force efforts (Integration of control force) of AFinite is significantly less than those of the other two. The faster accelerating and braking processes lead to more considerable control effort naturally. The difference between the three controllers is just the switching threshold with the magnitude of 1. The initial attitude error is relatively small compared to 1. This is the reason that the convergence rates and control efforts of the three control laws hold just slight differences between each other in their attitude control processes.Figure 6
Integration of control torque and force.
(a)
Integration of control torque
(b)
Integration of control forceIn summary, the adaptive modified fixed-time control law (AMFixed) performs with higher control accuracy comparing to the conventional adaptive fixed-time control law (AFixed) and with faster convergence rate comparing to the adaptive finite-time control law (AFinite). These can be demonstrated clearly again in Table4.


## 5. Conclusion

The control problem of spacecraft coupled rotational and translational maneuver tasks in the presence of large parametric uncertainties and external disturbances is studied. Based on the spacecraft relative coupled kinematics and dynamics equations on SE(3), a reaching law-based adaptive fixed-time terminal sliding mode control law and the modified one are designed. The two control laws are able to drive the states of the closed-loop feedback tacking system to converge to the equilibrium points in fixed time. The adaptive finite-time control law is proposed simultaneously to compare the control performances with each other. The states of the closed-loop system are almost globally convergent over the spaceSE3×R6. Numerical simulation results demonstrate that the follower spacecraft can reach the desired pose and velocities in fixed time by applying proposed fixed-time control laws. Furthermore, it shows that the proposed adaptive modified fixed-time terminal sliding mode control law is superior to the conventional one with almost the same control effort. The two fixed-time type control laws drive the states with faster convergence rates than the finite-time one but cost more energy. There are some drawbacks related to the proposed inertia parameter update law and disturbance update law. The former cannot identify the actual inertia parameters when the persistent excitation condition is not satisfied. The latter cannot exactly identify the true external disturbances, and the estimations always converge to constant values.



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*Source: 3085495-2020-02-04.xml*