# Adaptive Finite-Time Control on SE(3) for Spacecraft Final Proximity Maneuvers with Input Quantization

Authors: Sai Zhang; Zhen Yang
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

In this paper, a bounded finite-time control strategy is developed for the final proximity maneuver of spacecraft rendezvous and docking exposed to external disturbance and input quantization. To realize the integrated control for spacecraft final proximity operation, the coupling kinematics and dynamics of attitude and position are modeled by feat of Lie groupSE3. With a view to improving the convergence rate and reducing the chattering, an adaptive finite-time controller is proposed for the error tracking system with one-step theoretical proof of stability. Meanwhile, the hysteresis logarithmic quantizer is implemented to effectively reduce the frequency of data transmission and the quantization errors are reduced under the proposed controller. The algorithms outlined above are based on an integrated model expressed by SE3 and denoted by uniform motion states, which can simplify the design progress and improve control precision. Finally, simulations are provided to exhibit the effectiveness and advantages of the designed strategy.


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## Body

## 1. Introduction

Spacecraft rendezvous and docking (RVD) technology has been successfully applied in many space missions, including space station installation and operation and deep space exploration [1–3]. Up to now, most of the RVD missions are cooperative. However, with the rapid development of space technology, noncooperative RVD is expected to serve future space activities, such as space debris removals and on-orbit services. In noncooperative RVD missions, the final proximity maneuver is severely affected by a coupled model of rotational and translational relative motions. In addition, the complicated space environment renders the maneuver control of spacecraft RVD difficult work. To overcome these two obstacles, the robust control of the spacecraft is worth more attention from the space community in future engineering missions, and some of the studies [4–7] show positive results.In most of the existing papers, the modeling and control of relative translation and rotation in spacecraft proximity maneuvers are always supposed to be independent [5–7]. Nevertheless, the strong coupling of orbit and attitude motions makes the close-range control of noncooperative RVD extremely difficult to fulfill. Consequently, an integrated description of the relative translation channel and rotation channel is essential. Methods like dual quaternion [8–10], modified Rodrigues parameters (MRPs) [11–13], and Lie group SE3 [14–16] were researched to build a connection between these two channels. Lie group SE3 is the set of positions and orientations in the 3-D Euclidean space. Since Lie group SE3 can provide a unique and nonsingularity description of a rigid body’s motion [16], it has been applied in multibody systems [17, 18], heavy-duty industrial robots [19], and underwater vehicles [20]. In the field of orbit and attitude tracking control, methods using Lie group SE(3) have been proposed in [21–25].Conventional linear control methods, such as PD or PID, were widely used in practical applications. These methods are deemed inapplicable to missions with complex dynamics or uncertain inertias. To promote, a great number of modified control methods, including sliding mode control (SMC) [6, 26], feedback control [27, 28], and backstepping control [7, 29], have been developed. Among these methods, the SMC enjoys the superiority of high robustness against uncertainty and external disturbance in a complex nonlinear system. To enhance the finite-time stability of the system, the terminal SMC (TSMC) was first proposed by Venkataraman and Gulati [30]. It is worth noting that TSMC suffers from the singularity problem, which may result in instability. Additionally, the chattering problem inherent in SMC may cause an adverse effect on actuators. To overcome these two problems, two different fast terminal sliding mode surfaces (TSMS) were proposed in [31, 32] to tackle the attitude tracking control problem of the spacecraft. Inspired by [31, 32] and based on the integrated model parameters, a modified nonsingular finite-time TSMS is proposed in this paper.Besides the coupling of models and external disturbance mentioned above, the diminution of severe data transmission pressure requires further investigation from the perspective of the system’s high performance and low consumption. In general, the interspacecraft communication devices mounted on the spacecraft are quite power-limited as the load of the spacecraft is limited, which results in the limited communication bandwidth [33]. Additionally, as the control modules of the spacecraft comprise digital processors, the actual control signals from the controller to the actuators must be transformed from continuous ones to discrete ones [34]. Taking these two problems into consideration, the quantizers, such as uniform quantizer, logarithmic quantizer, and hysteretic quantizer, were studied and the superiorities had been validated in [34–36]. As an improvement of the uniform quantizer, a hysteretic logarithmic quantizer (HLQ) with the property of extra quantization levels was investigated in [37], which can handle the effect of chattering. Inspired by [33–36], HLQ is implemented in this paper to reduce the communication loads and improve the control accuracy. However, the methods of [34, 36, 37] can only ensure asymptotic convergence. As the system tends to be stable as time goes to infinity [38], asymptotic convergence cannot meet the needs of rapid, steady, and accurate control in missions with special requirements.To the best of the authors’ knowledge, few studies have been conducted from the perspective of constructing a quantized controller for spacecraft final proximity maneuver with finite-time convergence. Although Li et al. [39] proposed an adaptive controller based on sliding mode methodology and logarithmic quantizer for the RVD problem, the controller has a defect of asymptotic convergence. Accordingly, a quantized control method in conjunction with finite-time and nonsingular TSMS for spacecraft final proximity maneuver is needed.Motivated by the above observations, it is essential but challenging to tackle the final proximity maneuver control of spacecraft RVD subject to external disturbance and quantized input. Therefore, a finite-time control algorithm is proposed in this paper based on a modified TSMS and the HLQ. The main contributions of this study can be summarized as follows:(i)
A new finite-time TSMC strategy is proposed. The finite-time stability can be analyzed in two aspects. On the one hand, the modified TSMS with a switched function can improve the performance of system stability and avoid singularity. On the other hand, the specially designed control input in Equation (39) can provide fast finite-time stability and avoid the chattering phenomenon(ii)
By using the proposed TSMC strategy, the finite-time stability of the system can be proved directly without the upper bound value of the adaptive parameters, which is more convenient than the methods in [5–7, 32](iii)
The quantized control is first applied to the final proximity maneuver of spacecraft RVD modeled by Lie groupSE3. In comparison to the traditional time-consuming methods in [2–4], the method in this paper has sufficient merit of reducing the transmitting of tracking information and requires low communication resourcesThe remainder of this paper is presented as follows. The coupled model for spacecraft RVD maneuver on Lie groupSE3 is presented in Section 2. In Section 3, the control objective is given and a quantized TSMC algorithm is developed. Simulations are presented to validate the effectiveness of the proposed controller in Section 4, and the conclusions have been drawn in Section 5.


## 2. Spacecraft Dynamics Models on SE(3)

This section presents the kinematics and dynamics models of a spacecraft’s motion around the earth and obtains the relative motion model of attitude and position tracking systems denoted by Lie groupSE3. Additionally, notations used in this paper are presented in this section. Three orthogonal coordinate systems are used to describe the relative motion dynamics: the Earth-centered inertial (ECI) reference frame, denoted as E=Ex,Ey,Ez and the body-fixed frames of the chaser and the target, denoted as B=bx,by,bz and Bt=btx,bty,btz, respectively. As shown in Figure 1. By the way, only rigid spacecraft are considered in this paper.Figure 1
Illustration of coordinates for spacecraft proximity.In this paper,⋅T expresses the transpose of a matrix and ⋅ denotes the Euclidean norm of a vector. For a vector η=η1,η2,⋯,ηnT, it has sigηβ=sigη1β⋯sigηnβT with sigηiβ=signηiηiβ.The special group is denoted asSE3 and the special orthogonal group is denoted as SO3. so3 is the Lie algebra of SO3 and se3 denotes the Lie algebra of SE3.


### 2.1. Dynamics and Kinematics Models of a Rigid Spacecraft

In general, the kinematics model of a rigid spacecraft can be established as [4]
(1)ḃ=Rv,Ṙ=RΩ×,where b=bIbJbKT∈ℝ3 is the inertial position of the spacecraft in E. v∈ℝ3 and Ω∈ℝ3 are translational and angular velocities of the spacecraft, respectively. R∈SO3 denotes the transfer matrix between the body-fixed frame and ECI. The operator ⋅×:ℝ3⟶so3 is the cross-product, which can be expressed by a 3×3 skew-symmetric matrix:
(2)a×=0−a3a2a30−a1−a2a10,with a=a1,a2,a3T. Let Φc:SE3×se3⟶ℝ3 and τc:SE3×se3⟶ℝ3 denote the feedback control force and the feedback control torque, respectively. The dynamics model of a rigid spacecraft is then given as follows [4]:
(3)JΩ̇=JΩ×Ω+Mgb,R+τcb,R,v,Ω+τd,mv̇=mv×Ω+Fgb,R+mRTaJ2b+Φcb,R,v,Ω+Φd,where m and J denote the mass and moment of inertia matrix in its body frame. τd∈ℝ3 and Φd∈ℝ3 denote the unknown moment and force acting on the chaser spacecraft, respectively. Fg and Mg denote the gravity force and gravity gradient moment, respectively, which are as follows:
(4)Mg=3μb5p×Jp,Fg=−mμpb3−3μLpb5+15μpTJpp2b7,where p=RTb, L=1/2traceJI3+J. μ=398600.44km3s−2 is the gravitational constant of the Earth. The perturbation caused by Earth’s oblateness and J2 in E is
(5)aJ2=−3μJ2R⊕2bI2b51−5bK2b3−3μJ2R⊕2bJ2b51−5bK2b3−3μJ2R⊕2bK2b53−5bK2b3,where J2=1.08623×10−3 and R⊕=6378×103km.In order to tackle the problem of an integrated description of relative translation and rotation, the configuration space of rigid spacecraft motions can be denoted by the special Euclidean groupSE3. SE3 is the semiproduct of the special orthogonal group SO3 and three-dimensional position vector ℝ3 of the spacecraft, which is expressed as SE3=ℝ3×SO3.According to the expression in [2], the state space of the target is SE3×se3≃SE3×ℝ6 and b,R,v,Ω expresses its motion states. The configuration of the spacecraft on SE3 can be rewritten as
(6)g=Rb01×31∈SE3,where 01×3 means a 1×3 null matrix.Here, it defines the angular velocityΩ and translational velocity v in a unified form as ξ=ΩTvTT∈ℝ6. Thus, the kinematics model in Equation (1) is reconstructed as
(7)ġ=gξ∨where ξ∨=Ω×v01×30∈se3 is the velocity mapping. The denotation φg is introduced to express the vector of gravitational force and moment and Ξ denotes the mass and inertia of spacecraft, respectively. φg and Ξ are defined as
(8a)φg=MgFg+mRTaJ2∈ℝ6,Ξ=J03×303×3mI3∈ℝ6×6.Similar to the definition of adjoint action between a Lie group and its corresponding Lie algebra, the adjoint operator and coadjoint representation ofSE3 on se3 can be given by [21]:
(9)Adg=R03×3b×RR∈ℝ6×6s.t.AdgX∧=gXg−1∧,where·∧:se3⟶ℝ6 is the inverse of ⋅∨:ℝ6⟶se3. In matrix form, the adjoint representation of se3 is expressed in (10a) and the coadjoint representation is given in (10b).
(10a)adξ=Ω×03×3v×Ω×∈ℝ6×6,(10b)adξ∗=adξT=−Ω×−v×03×3−Ω×.In virtue of the coadjoint operator, the dynamics model in Equation (3) can be expressed in a compact form [21]:
(11)Ξξ̇=adξ∗Ξξ+φg+φc+φd,where φc=τcΦc and φd=τdΦd. Specifically, φc∈ℝ6 denotes the control input and φd∈ℝ6 denotes the external disturbance.Similarly, denoteRt∈SO3 to be the transfer matrix between Bt and E. The inertial position, translational velocity, and the angular velocity of the target can be denoted as bt=btIbtJbtKT∈ℝ3, vt∈ℝ3 and Ωt∈ℝ3, respectively. mt and Jt denote the target’s mass and moment of inertia matrix in its body frame. The configuration of the target on SE3 can be given as follows:
(12)gt=Rtbt01×31∈SE3.The dynamics model of the target on Lie GroupSE3 is
(13)Ξtξ̇t=adξt∗Ξtξt+φtg,with ξt=ΩtTvtTT∈ℝ6 and Ξt=Jt03×303×3mtI3. The inertial state is gtt0=ξtt0 at time t0. At the meantime, the state trajectory of the target for t≥t0 can be generated by using Equations (12) and (13).Remark 1.
The kinematics and dynamics for a normal rigid spacecraft are defined by the equations presented previously (see Equation (7) and (11)). It needs to emphasize that the target spacecraft is assumed to maneuver in the gravity field, and no external disturbance or control input is acting on the target. It is important not to confuse the chaser and the target.


### 2.2. Relative Dynamical Model for Spacecraft Proximity Operation

It defines the relative configuration and desired relative configuration between the chaser and the target to beh∈SE3 and hf∈SE3. According to the tracking trajectory created by Equations (6) and (12), the desired states of the chaser spacecraft are as follows:
(14)gd=gthf,ξd=Adhf−1ξtfort≥0.And the configuration error between the chaser and target is(15)h=gt−1gfort≥0,whereAdhf−1 denotes the adjoint action of hf. In view of the logarithm map, the configuration error can be expressed in exponential coordinate, that is
(16)η~∨=logSE3hf−1h=logSE3gd−1gwhere logSE3:SE3⟶se3 is the logarithm map (logm). Thus, the relative configuration between the actual and desired configuration can be defined by the exponential coordinate vector η~ of the chaser, which is
(17)η~=Θ~β~T∈ℝ6,where Θ~∈ℝ3 and β~∈ℝ3 represent the attitude tracking error and position tracking error, respectively.Remark 2.
The exponential mapexpSE3 from se3 to SE3is a surjective map and a local diffeomorphism. expSE3 is bijective when Θ≤π. The inverse expression of expSE3 is a logarithm map denoted as logSE3. The standard computations of expSE3 and logSE3 are presented in [40]. According to Equations (14)–(17), one can conclude that η~∨=Θ~β~∨=logSE3hf−1h=logSE3he and expSE3Θ~β~∨=he=Rebe01×31. The solution process can be found in [40]. Apart from that, for any initial relative attitude and any initial relative principal rotation axis, the norm of Re is 1, which has been proven in Appendix A. As a result, the exponential coordinate vector Θ~,β~ has a strict relationship with Re,be and can represent the attitude tracking error and position tracking error.With the aid of Equations (7) and (12), the relative velocity is expressed as
(18)ξ~=ξ−Adh−1ξt=ξ−Adhf−1ξd,where ξ~ represents the relative velocity of the chaser with respect to the target. The first-order derivative of ξ~ is
(19)ξ~̇=ξ̇+adξAdh−1ξt−Adh−1ξ̇t.It defines thath0=ht0, η~0=ηt0, ξ0=ξt0, and ξ~0=ξ~t0 to be the initial states at time t0, which are assumed to be known as long as the initial states of the chaser spacecraft are known. According to [21], the kinematics model of exponential coordinates is
(20)η~̇=Gη~ξ~where Gη~ is a block-triangular matrix. Given that η=ΘTβTT∈ℝ6 denotes the actual configuration, the function Gη can be defined as
(21)Gη=AΘ03×3TΘ,βAΘ,where AΘ=I3+1/2Θ×+1/θ2−1+cosθ/2θsinθΘ×2, SΘ=I3+1−cosθ/θ2Θ×+θ−sinθ/θ3Θ×2, and TΘ,β=1/θ2−1+cosθ/2θsinθΘβT+ΘTβAΘ+1/2SΘβ×AΘ+1+cosθθ+sinθ/2θ3sin2θ−2/θ4ΘTβΘΘT−1+cosθθ−sinθ/2θsin2θSΘβΘT with θ=Θ and RΘ=expSO3Θ=I3+sinθ/θΘ×+1−cosθ/θ2Θ×2.According to [21], Gη satisfies this relation:
(22)Gηη=η.Equation (22) has been proved in [21]. Thus, Gη is a positive definite matrix when η≠0 and Gη=I when η=0.Substituting Equations (11) and (13) into Equation (19) leads to the first-order derivative of relative acceleration:
(23)Ξξ~̇=adξ∗Ξξ+φg+φc+φd+ΞadξAdh−1ξt−Adh−1ξ̇t.By combining the relative kinematics expressed by exponential coordinate in (20) and the relative acceleration equation in Equation (23), the coupled error tracking system of spacecraft final proximity maneuvers is established.


## 2.1. Dynamics and Kinematics Models of a Rigid Spacecraft

In general, the kinematics model of a rigid spacecraft can be established as [4]
(1)ḃ=Rv,Ṙ=RΩ×,where b=bIbJbKT∈ℝ3 is the inertial position of the spacecraft in E. v∈ℝ3 and Ω∈ℝ3 are translational and angular velocities of the spacecraft, respectively. R∈SO3 denotes the transfer matrix between the body-fixed frame and ECI. The operator ⋅×:ℝ3⟶so3 is the cross-product, which can be expressed by a 3×3 skew-symmetric matrix:
(2)a×=0−a3a2a30−a1−a2a10,with a=a1,a2,a3T. Let Φc:SE3×se3⟶ℝ3 and τc:SE3×se3⟶ℝ3 denote the feedback control force and the feedback control torque, respectively. The dynamics model of a rigid spacecraft is then given as follows [4]:
(3)JΩ̇=JΩ×Ω+Mgb,R+τcb,R,v,Ω+τd,mv̇=mv×Ω+Fgb,R+mRTaJ2b+Φcb,R,v,Ω+Φd,where m and J denote the mass and moment of inertia matrix in its body frame. τd∈ℝ3 and Φd∈ℝ3 denote the unknown moment and force acting on the chaser spacecraft, respectively. Fg and Mg denote the gravity force and gravity gradient moment, respectively, which are as follows:
(4)Mg=3μb5p×Jp,Fg=−mμpb3−3μLpb5+15μpTJpp2b7,where p=RTb, L=1/2traceJI3+J. μ=398600.44km3s−2 is the gravitational constant of the Earth. The perturbation caused by Earth’s oblateness and J2 in E is
(5)aJ2=−3μJ2R⊕2bI2b51−5bK2b3−3μJ2R⊕2bJ2b51−5bK2b3−3μJ2R⊕2bK2b53−5bK2b3,where J2=1.08623×10−3 and R⊕=6378×103km.In order to tackle the problem of an integrated description of relative translation and rotation, the configuration space of rigid spacecraft motions can be denoted by the special Euclidean groupSE3. SE3 is the semiproduct of the special orthogonal group SO3 and three-dimensional position vector ℝ3 of the spacecraft, which is expressed as SE3=ℝ3×SO3.According to the expression in [2], the state space of the target is SE3×se3≃SE3×ℝ6 and b,R,v,Ω expresses its motion states. The configuration of the spacecraft on SE3 can be rewritten as
(6)g=Rb01×31∈SE3,where 01×3 means a 1×3 null matrix.Here, it defines the angular velocityΩ and translational velocity v in a unified form as ξ=ΩTvTT∈ℝ6. Thus, the kinematics model in Equation (1) is reconstructed as
(7)ġ=gξ∨where ξ∨=Ω×v01×30∈se3 is the velocity mapping. The denotation φg is introduced to express the vector of gravitational force and moment and Ξ denotes the mass and inertia of spacecraft, respectively. φg and Ξ are defined as
(8a)φg=MgFg+mRTaJ2∈ℝ6,Ξ=J03×303×3mI3∈ℝ6×6.Similar to the definition of adjoint action between a Lie group and its corresponding Lie algebra, the adjoint operator and coadjoint representation ofSE3 on se3 can be given by [21]:
(9)Adg=R03×3b×RR∈ℝ6×6s.t.AdgX∧=gXg−1∧,where·∧:se3⟶ℝ6 is the inverse of ⋅∨:ℝ6⟶se3. In matrix form, the adjoint representation of se3 is expressed in (10a) and the coadjoint representation is given in (10b).
(10a)adξ=Ω×03×3v×Ω×∈ℝ6×6,(10b)adξ∗=adξT=−Ω×−v×03×3−Ω×.In virtue of the coadjoint operator, the dynamics model in Equation (3) can be expressed in a compact form [21]:
(11)Ξξ̇=adξ∗Ξξ+φg+φc+φd,where φc=τcΦc and φd=τdΦd. Specifically, φc∈ℝ6 denotes the control input and φd∈ℝ6 denotes the external disturbance.Similarly, denoteRt∈SO3 to be the transfer matrix between Bt and E. The inertial position, translational velocity, and the angular velocity of the target can be denoted as bt=btIbtJbtKT∈ℝ3, vt∈ℝ3 and Ωt∈ℝ3, respectively. mt and Jt denote the target’s mass and moment of inertia matrix in its body frame. The configuration of the target on SE3 can be given as follows:
(12)gt=Rtbt01×31∈SE3.The dynamics model of the target on Lie GroupSE3 is
(13)Ξtξ̇t=adξt∗Ξtξt+φtg,with ξt=ΩtTvtTT∈ℝ6 and Ξt=Jt03×303×3mtI3. The inertial state is gtt0=ξtt0 at time t0. At the meantime, the state trajectory of the target for t≥t0 can be generated by using Equations (12) and (13).Remark 1.
The kinematics and dynamics for a normal rigid spacecraft are defined by the equations presented previously (see Equation (7) and (11)). It needs to emphasize that the target spacecraft is assumed to maneuver in the gravity field, and no external disturbance or control input is acting on the target. It is important not to confuse the chaser and the target.


## 2.2. Relative Dynamical Model for Spacecraft Proximity Operation

It defines the relative configuration and desired relative configuration between the chaser and the target to beh∈SE3 and hf∈SE3. According to the tracking trajectory created by Equations (6) and (12), the desired states of the chaser spacecraft are as follows:
(14)gd=gthf,ξd=Adhf−1ξtfort≥0.And the configuration error between the chaser and target is(15)h=gt−1gfort≥0,whereAdhf−1 denotes the adjoint action of hf. In view of the logarithm map, the configuration error can be expressed in exponential coordinate, that is
(16)η~∨=logSE3hf−1h=logSE3gd−1gwhere logSE3:SE3⟶se3 is the logarithm map (logm). Thus, the relative configuration between the actual and desired configuration can be defined by the exponential coordinate vector η~ of the chaser, which is
(17)η~=Θ~β~T∈ℝ6,where Θ~∈ℝ3 and β~∈ℝ3 represent the attitude tracking error and position tracking error, respectively.Remark 2.
The exponential mapexpSE3 from se3 to SE3is a surjective map and a local diffeomorphism. expSE3 is bijective when Θ≤π. The inverse expression of expSE3 is a logarithm map denoted as logSE3. The standard computations of expSE3 and logSE3 are presented in [40]. According to Equations (14)–(17), one can conclude that η~∨=Θ~β~∨=logSE3hf−1h=logSE3he and expSE3Θ~β~∨=he=Rebe01×31. The solution process can be found in [40]. Apart from that, for any initial relative attitude and any initial relative principal rotation axis, the norm of Re is 1, which has been proven in Appendix A. As a result, the exponential coordinate vector Θ~,β~ has a strict relationship with Re,be and can represent the attitude tracking error and position tracking error.With the aid of Equations (7) and (12), the relative velocity is expressed as
(18)ξ~=ξ−Adh−1ξt=ξ−Adhf−1ξd,where ξ~ represents the relative velocity of the chaser with respect to the target. The first-order derivative of ξ~ is
(19)ξ~̇=ξ̇+adξAdh−1ξt−Adh−1ξ̇t.It defines thath0=ht0, η~0=ηt0, ξ0=ξt0, and ξ~0=ξ~t0 to be the initial states at time t0, which are assumed to be known as long as the initial states of the chaser spacecraft are known. According to [21], the kinematics model of exponential coordinates is
(20)η~̇=Gη~ξ~where Gη~ is a block-triangular matrix. Given that η=ΘTβTT∈ℝ6 denotes the actual configuration, the function Gη can be defined as
(21)Gη=AΘ03×3TΘ,βAΘ,where AΘ=I3+1/2Θ×+1/θ2−1+cosθ/2θsinθΘ×2, SΘ=I3+1−cosθ/θ2Θ×+θ−sinθ/θ3Θ×2, and TΘ,β=1/θ2−1+cosθ/2θsinθΘβT+ΘTβAΘ+1/2SΘβ×AΘ+1+cosθθ+sinθ/2θ3sin2θ−2/θ4ΘTβΘΘT−1+cosθθ−sinθ/2θsin2θSΘβΘT with θ=Θ and RΘ=expSO3Θ=I3+sinθ/θΘ×+1−cosθ/θ2Θ×2.According to [21], Gη satisfies this relation:
(22)Gηη=η.Equation (22) has been proved in [21]. Thus, Gη is a positive definite matrix when η≠0 and Gη=I when η=0.Substituting Equations (11) and (13) into Equation (19) leads to the first-order derivative of relative acceleration:
(23)Ξξ~̇=adξ∗Ξξ+φg+φc+φd+ΞadξAdh−1ξt−Adh−1ξ̇t.By combining the relative kinematics expressed by exponential coordinate in (20) and the relative acceleration equation in Equation (23), the coupled error tracking system of spacecraft final proximity maneuvers is established.


## 3. Quantized Finite-Time TSMC Scheme Design and Stability Analysis

In this section, a finite-time controller is proposed for spacecraft RVD considering external disturbance and input quantization. Firstly, it gives the preliminaries. To better show the aim and design process of this paper, the schematic diagram and the control objective are given. Finally, in conjunction with adaptive algorithms and a switched TSMS, a quantized finite-time TSMC algorithm is proposed. Via the proposed control scheme, the finite-time stability of the system can be proved directly.


### 3.1. Preliminaries

A quantizer refers to the existing results in [37]; the HLQ is introduced here to replace the traditional time-consuming methods:
(24)Qu=uisgnu,ui1+δ<u≤ui,u̇<0orui<u≤ui1−δ,u̇>0,ui1+δsgnu,ui<u≤ui1−δ,u̇<0orui1−δ<u≤ui1+δ1−δ,u̇>0,0,0≤u<umin1+δ,u̇<0orumin1+δ<u≤umin,u̇>0,Qut−,u̇=0,where ui=ρ1−iumini=1,2,⋯, umin>0 denotes the range of the hysteresis zone for Qu, and δ determines the transmitting rate of the communication channel and satisfies δ=1−ρ/1+ρ with 0<ρ<1. Obviously, Qu is in the set of U=0,±ui,±ui1+δ.The HLQ can be decomposed into a continuous part and a discontinuous part [41], which is
(25)Qu=κuu+Eu,where κu=Qu/u,Qu≠01,Qu=0 and Eu=0,Qu≠0−u,Qu=0.Remark 3.
The control signals transmitted to the actuators are quantized in the system with an HLQ. The quantizer here can convert continuous signals into discrete signals numerically, which can reduce the burden of information transmission. In this study,umin and δ of each dimension of control input are supposed to be the same.To promote the design, the following lemmas and assumptions are essential for the establishment of the control algorithm.Lemma 4 (see [41]).
It is easy to find thatκu is continuous and Eu is discontinuous; thus, the following inequality holds:
(26)1−δ≤κu≤1+δ,Eu≤umin,where δ and umin are design parameters of HLQ.Lemma 5.
For anyε>0 and x∈ℝ, the following inequality holds:
(27)0≤x−x2x2+ε2≤x−x2x+ε<ε.Lemma 6.
For any1>b>0 and xi∈ℝi=1,⋯,n, the following inequality holds:
(28)x1+⋯+xnb≤x1b+⋯+xnb.Lemma 7.
Based on Equations (27) and (28), the following inequation can be obtained for a vector x=x1x2⋯xnT:
(29)−xTx2α+1x2α+1+ε2≤−xα+1+ε,where x2α+1=x12α+1x22α+1⋯xn2α+1T, and α is defined as
(30)α=−m2n+1,where 0<2m<2n+1 with m,n∈ℕ+. It is obvious that −1/2<α<0.Lemma 8.
According to Young’s inequality, for a constantμ, if μ^+μ~=μ holds, one has
(31)μ^μ~≤−12μ~2+12μ2.Lemma 9 (see [42]).
For the system in Equation (23), if a Lyapunov function V exists and satisfies
(32)V̇≤−αVγ+σwith 1>γ>0, α>0, and 0≤σ<∞, then the trajectory of Equation (23) is practical finite-time stable.Lemmas5 and 6 are easy to be proved by virtue of mathematical expansion. The proof of Lemma 7 is given in Appendix B.Remark 10.
There is a condition:α=−m/2n+1 with 0<2m<2n+1. Hence, for any constant x∈ℝ, xα, and x1+2α will make sense. By the way, the value of α can ensure finite-time convergence for system states, which will be illustrated in the following text.Assumption 11.
The inertial matrixJ and mass mp of the chaser spacecraft are known. The inertial matrix Jt and mass mt of the target are unknown. Both J and Jt are constant positive-definite symmetric matrices. mp and mt are bounded constants.Assumption 12.
All the disturbances are unknown but bounded with an upper bound. That is,φd≤D with D being an uncertain positive constant.Remark 13.
With the help of onboard devices equipped on spacecraft, the information of the chaser can be interactive. Thus, Assumption11 is reasonable. As for Assumption 12, it is believed that environmental interference is limited and the unknown disturbance can be strictly dominated by control inputs.For the control objectives, in this paper, we denote to designing a finite-time TSMC scheme for spacecraft RVD subject to external disturbance and input quantization. The objective is to design the control input commandφc for the error system in Equation (23) under Assumptions 11 and 12, such that all closed-loop signals are guaranteed to be bounded and the global system is finite-time stable. For a better illustration, the architecture of the controller synthesis is shown in Figure 2.Figure 2
Block diagram of the control architecture for spacecraft rendezvous motion.


### 3.2. Quantized Finite-Time TSMC Design

Inspired by [37], the actual control input is the quantized control torque, and as a result, Equation (23) is rewritten as
(33)Ξξ~̇=adξ∗Ξξ+φg+Qφ+φd+ΞadξAdh−1ξt−Adh−1ξ̇t,where Qφ=Qϕ1Qϕ2Qϕ3Qϕ4Qϕ5Qϕ6∈ℝ6 is the quantized value of the designed control input εu. The quantized input Qφ can be written as
(34)Qφ=κuεu+EuI1,with I1=111111T. κu and Eu satisfy the inequality in Equation (26) with δ and umin being design parameters.By using the Sigmoid function, a modified switched TSMSS=S1S2S3S4S5S6T∈ℝ6 is designed as follows:
(35)S=ξ~+k1η~+k2φη~,where k1=diagk11,k12,k13,k14,k15,k16 and k2=diagk21,k22,k23,k24,k25,k26 with k1ii=1,2,3,4,5,6 and k2ii=1,2,3,4,5,6 being positive constants. In particular, φη~=φη~1φη~2φη~3φη~4φη~5φη~6T is a piecewise smooth function and φη~i is defined as
(36)φη~i=sigη~iβ,η~i>ϑ,2ϑπsinπ2ϑη~i+βϑβ−1η~i,otherwise,where 0<β<1, and ϑ is a small positive constant, which is defined as
(37)ϑ=π21−κ1/1−κ,with 0<κ<1. The derivative of φη~ is φ̇η~=φ̇η~1φ̇η~2φ̇η~3φ̇η~4φ̇η~5φ̇η~6T with φ̇η~i being
(38)φ̇η~i=βη~iβ−1η~̇i,η~i>ϑ,cosπ2ϑη~iη~̇i+βϑβ−1η~̇i,otherwise.The actual control inputQφ is determined by εu, which is designed as
(39a)εu=11−δu,(39b)u=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4SαS2α+1+ε12−ΓS,with Sα=S12α+1S22α+1S32α+1S42α+1S52α+1S62α+1T. k3 and k4 are diagonal matrixes denoted as k3=diagk31,k32,k33,k34,k35,k36 and k4=diagk41,k42,k43,k44,k45,k46 with k3ii=1,2,3,4,5,6 and k4ii=1,2,3,4,5,6 being positive constants, which are humanly designed and will not change. Γ=Λφ2/Λφ2S2+ε12+D∧2/D∧2S2+ε12+u2/D∧2S2+ε12 is a time-varying state variable with Λφ=adξ∗Ξξ+φg+ΞadξAdh−1ξt−Adh−1ξ̇t. u is a constant satisfying umin≤u. D^ is the estimation of D.The adaptive law ofD is designed as follows:
(40)D^̇=c1S−c2D^,where c1 and c2 are positive constants. The estimated error of D is defined as
(41)D~=D−D^.Remark 14.
Three components comprise the quantized finite-time controller: the adaptive law, the finite-time input, and the quantizer. The finite-time input is made up of a switched TSMS (Equations (35)– (37)) and a modified finite-time convergent term (Equation (39b)). Combined with the adaptive law, the basic controller is obtained. Additionally, an HLQ is implemented to transfer the control signal, and, meanwhile, the basic controller is then altered by the quantizer parameter (Equation (39a)). By applying these steps, the quantized finite-time controller is fulfilled.The design process of the quantized finite-time TSMC is shown in Figure3. More detailed analysis about the proposed method can be seen in Remarks 15–17.Remark 15.
Inspired by [32], a modified switched TSMS is designed in Equation (35) with the aid of a nonlinear piecewise smooth function. As has been illustrated in [32], ϑ=π/21−κ1/1−κ is essential to ensure φη~i to be continuous and differentiable at point η~i=ϑ. The sliding mode variable S implemented here has two merits. Firstly, the terminal sliding mode variable can improve the convergence rate and guarantee the finite-time stability of the global system with improved robustness. In addition, the term φη~i defined by Equation (36) can avoid the singularity problem. Compared with the existing SMC methods in [6, 27, 28], the proposed S possesses the advantage of finite-time convergence with nonsingularity.Remark 16.
The control input in Equation (39) is designed to guarantee the system with finite-time convergence and convincing robustness. The designed control inputεu is derived by u and δ. More specifically, k3S is designed to guarantee the global stability of the system and Sα in Equation (39) is a finite-time convergent term which is different from the hyperbolic tangent function-based control input in [5, 6] and the Sigmoid function-based control input in [31, 32]. Γ is composed of state parameters and adaptive parameters, which enjoys the property of improving system robustness and ensuring the convergence of adaptive parameters. It is worth noting that u2/D∧2S2+ε12 in Γ can reduce the quantization errors. The introduction of Sα and Γ can improve the performance of system stability and avoid inherent chattering phenomenon.Remark 17.
There are seven control parameters in the finite-time controller, e.g.,kii=1,2,3,4, cii=1,2, and ε1, which have an effect on the control performance of the final proximity maneuver system. Although the criterion to select appropriate parameters mainly consists in the rule of thumb, there are rules to follow. More specifically, a bigger value of kii=1,2,3,4 means higher control accuracy and bigger overshoot of control force. The convergence rate of ω~,Θ~ will increase as k1 and k2 increase while the settling time of r~,v~ mainly depends on the value of k3 and k4. However, increasing k2 will improve the settling time of r~,v~ and increasing k4 will aggravate the chattering phenomenon. Additionally, estimation errors mainly depend on cii=1,2 and a smaller ε1 means higher accuracy. Given the above empirical rules, the approximate optimal solution can be found by trying various combination. In brief, the performance of the control system is determined by the controller, but also is largely affected by the design parameters.Figure 3
Block diagram of the controller design process.Letkimin be the minimum of each dimension of ki, that is, kjmin=minkji with i=1,2,3,4,5,6 and j=1,2,3,4.


### 3.3. Stability Analysis

Theorem 18.
Consider the relative accelerations of two spacecraft in RVD under Assumptions11 and 12. The error dynamics model is modeled by Equation (23). The control schemes are designed as Equations (35)–(41) under kij≥0i=1,⋯,4,j=1,⋯,6 and c1>0,c2>0. Under the proposed controller, the following conclusions can be obtained:
(i)
The global system is finite-time stable, and all signals in the closed-loop system are guaranteed to be bounded. In particular, the TSMSS will convergence to a small region ΔS=SS<2/λΞΔ1/ρ1/1+α in finite time(ii)
The lumped errorsη~i and ξ~i are convergent and will converge to the small region Δηi and Δξi in finite time, where Δη=max2Δ2/ρ21/α+1,ϑ and Δξi=maxϑ1i,ϑ2i with ϑ1i=ΔS+k1i+2k2i/πϑ+k2iβϑβ and ϑ2i=ΔS+k1iϑ0+k2iϑ0β+1Proof.
Define the Lyapunov function (LF) candidate:(42)V1=12STΞS.
Taking the time derivative of Equation (42) yields
(43)V̇1=STΞṠ=STΞξ~̇+k1η~̇+k2φ̇η~=STadξ∗Ξξ+φg+Qφ+φd+ΞadξAdh−1ξt−Adh−1ξ̇t+STk1ΞGη~ξ~+k2φ̇η~=STQφ+φd+Λφ+k1ΞGη~ξ~+k2φ̇η~.
Based on Equation (34), STQφ can be extended to be
(44)STQφ=STκuεu+EuI1=ST−κu1−δu+EuI1≤−STu+uminSTI1.
Substituting Equation (44), the derivative of V1 becomes
(45)V̇1≤−STu+uminSTI1+STφd+Λφ+k1ΞGη~ξ~+k2φ̇η~≤−STk1ΞGη~ξ~+k2φ̇η~−STk3S−STk4SαS2α+1+ε12+uminSTI1+STφd+Λφ+k1ΞGη~ξ~+k2φ̇η~−Λφ2STSΛφ2S2+ε12−D∧2STSD∧2S2+ε12−u2STSD∧2S2+ε12≤−k3minS2−k4minSα+1−SΛφ−D^S−uS+μminS+DS+STΛφ+k4minε1+ε1+ε1+ε1≤−k3minS2−k4minSα+1+D~S+3+k4minε1.
Define the overall LF candidate:(46)V2=V1+12c1D~2.
Taking the time derivative of Equation (46) yields
(47)V̇2≤−k3minS2−k4minSα+1+D~S+1c1D~D~̇+3+k4minε1=−k3minS2−k4minSα+1+D~S−1c1D~D^̇+3+k4minε1=−k3minS2−k4minSα+1+D~S−D~S−c2D^+3+k4minε1≤−k3minS2−k4minSα+1+c2D~D^+3+k4minε1.
In consideration of Lemma8 and α defined in Equation (30), the following equation holds:
(48)D~D^=D~D−D~≤−12D~2+12D2=−12D~2+12D2=−14D~2−14D~−D~α2+14D~2α−12D~1+α+12D2≤−12D~1+α+12D2−14D~2+14D~2α≤−12D~1+α+12D2+1−α4αα/1−α.
Substituting Equation (48) into Equation (47) yields
(49)V̇2≤−k3minS2−k4minSα+1+3+k4minε1+c2−12D~1+α+12D2+1−α4αα/1−α=−k3minS2−k4minSα+1−c22D~1+α+Δ1,where Δ1=c2/22D2+1−α/αα/1−α+3+k4minε1.Through mathematical transformation, Equation (49) can be rewritten as
(50)V̇2≤−k4minSTS1+α/2−c22D~21+α/2+Δ1=−k4minSTS1+α/2−c22D~21+α/2+Δ1=−k4min2λmaxΞ1+α/212STΞS1+α/2−c222c11+α/212c1D~21+α/2+Δ1≤−ρV21+α/2+Δ1,where ρ=mink4min2/λΞ1+α/2,c2/22c11+α/2 with λΞ being the minimum of each dimension of Ξ. It is obvious that ρ>0,Δ1≥0.Remark 19.
As mentioned in Remark10, it is known that α=−m/2n+1 with 0<2m<2n+1. As a result, D~α and D~1+2α will make sense. It concludes that D~1+α=D~21+α/2=D~21+α/2. Furthermore, 1/4<1+α/2<1/2 is tenable such that V̇2≤−ρV21+α/2+Δ1 in Equation (50) satisfies the requirement in Lemma 9. As a consequence, the system is finite-time stable.Equation (50) can be reconstructed to be
(51)V̇2≤−ρV21+α/2+Δ1=−ρ−Δ1V21+α/2V21+α/2.As the result in Equation (50) has proved that the global system is stable, thus, V21+α/2≥Δ1/ρ holds until the system turning to be stable. Consequently, it yields the convergence region of S: ΔS=SS<2/λΞΔ1/ρ1/1+α..Thus, point (i) has been proven.Remark 20.
Through the design of the controller and the proof of the effectiveness, the result in Equation (50) indicates that the system is globally stable with finite-time convergence. Significantly, V2 is only the LF candidate formed by S and D~. The stability of system states (ξ~ and η~) is not clear. Thus, a further developed analysis and proof are needed.SinceS=S1S2S3S4S5S6T satisfies S≤ΔS, one has Si=ξ~i+k1iη~i+k2iφη~i=ϑS with ϑS∈−ΔS,ΔS.Whenη~i≤ϑ, ξ~i+k1iη~i+k2i2ϑ/πsinπ/2ϑη~i+βϑβ−1η~i≤ΔS holds all time, which implies
(52)ξ~i≤ΔS+k1iη~i+k2i2ϑπsinπ2ϑη~i+βϑβ−1η~i≤ΔS+k1iϑ+k2i2ϑπsinπ2ϑϑ+βϑβ−1ϑ=ΔS+k1i+2k2iπϑ+k2iβϑβ=ϑ1i.Whenη~i>ϑ, the LF is chosen as
(53)V0=∑i=1612η~i2.Taking the time derivate ofV0 yields
(54)V̇0=∑i=16η~iGη~iξ~i=∑i=16η~iϑS−k1iη~i−k2iφη~i=∑i=16ϑSη~i−k1iη~i2−k2iη~iβ+1≤∑i=16−k1iη~i2−k2iη~iβ+1+k1iη~i2+14k1iϑS2=∑i=16−k2iη~iβ+1+14k1iϑS2≤−ρ1V0β+1/2+Δ2,with Δ2=∑i=16ϑS2/4k1i and ρ1=k2min2β+1/2, which implies that the LF V0 is stable when η~i>ϑ. Through the result in Equation (54), one has
(55)V̇0≤−ρ1−Δ2V0β+1/2V0β+1/2.Based on Equation (55), if ρ1−Δ2/V0β+1/2≥0 holds, η~i will converge to the region η~i<2Δ2/ρ11/β+1 in finite time. Consequently, η~i will converge to the region Δη.Letϑ0=2Δ2/ρ21/β+1. According to Equation (51), ξ~i+k1iη~i+k2iη~iβ+1≤ΔS holds, which implies
(56)ξ~i≤ΔS+k1iη~i+k2iη~iβ+1≤ΔS+k1iϑ0+k2iϑ0β+1=ϑ2i.Consequently,ξ~i will converge to the region Δξi=maxϑ1i,ϑ2i with ϑ1i=ΔS+k1i+2k2i/πϑ+k2iβϑβ and ϑ2i=ΔS+k1iϑ0+k2iϑ0β+1.Thus, point (ii) has been proven.Finally, Theorem18 has been proven.


## 3.1. Preliminaries

A quantizer refers to the existing results in [37]; the HLQ is introduced here to replace the traditional time-consuming methods:
(24)Qu=uisgnu,ui1+δ<u≤ui,u̇<0orui<u≤ui1−δ,u̇>0,ui1+δsgnu,ui<u≤ui1−δ,u̇<0orui1−δ<u≤ui1+δ1−δ,u̇>0,0,0≤u<umin1+δ,u̇<0orumin1+δ<u≤umin,u̇>0,Qut−,u̇=0,where ui=ρ1−iumini=1,2,⋯, umin>0 denotes the range of the hysteresis zone for Qu, and δ determines the transmitting rate of the communication channel and satisfies δ=1−ρ/1+ρ with 0<ρ<1. Obviously, Qu is in the set of U=0,±ui,±ui1+δ.The HLQ can be decomposed into a continuous part and a discontinuous part [41], which is
(25)Qu=κuu+Eu,where κu=Qu/u,Qu≠01,Qu=0 and Eu=0,Qu≠0−u,Qu=0.Remark 3.
The control signals transmitted to the actuators are quantized in the system with an HLQ. The quantizer here can convert continuous signals into discrete signals numerically, which can reduce the burden of information transmission. In this study,umin and δ of each dimension of control input are supposed to be the same.To promote the design, the following lemmas and assumptions are essential for the establishment of the control algorithm.Lemma 4 (see [41]).
It is easy to find thatκu is continuous and Eu is discontinuous; thus, the following inequality holds:
(26)1−δ≤κu≤1+δ,Eu≤umin,where δ and umin are design parameters of HLQ.Lemma 5.
For anyε>0 and x∈ℝ, the following inequality holds:
(27)0≤x−x2x2+ε2≤x−x2x+ε<ε.Lemma 6.
For any1>b>0 and xi∈ℝi=1,⋯,n, the following inequality holds:
(28)x1+⋯+xnb≤x1b+⋯+xnb.Lemma 7.
Based on Equations (27) and (28), the following inequation can be obtained for a vector x=x1x2⋯xnT:
(29)−xTx2α+1x2α+1+ε2≤−xα+1+ε,where x2α+1=x12α+1x22α+1⋯xn2α+1T, and α is defined as
(30)α=−m2n+1,where 0<2m<2n+1 with m,n∈ℕ+. It is obvious that −1/2<α<0.Lemma 8.
According to Young’s inequality, for a constantμ, if μ^+μ~=μ holds, one has
(31)μ^μ~≤−12μ~2+12μ2.Lemma 9 (see [42]).
For the system in Equation (23), if a Lyapunov function V exists and satisfies
(32)V̇≤−αVγ+σwith 1>γ>0, α>0, and 0≤σ<∞, then the trajectory of Equation (23) is practical finite-time stable.Lemmas5 and 6 are easy to be proved by virtue of mathematical expansion. The proof of Lemma 7 is given in Appendix B.Remark 10.
There is a condition:α=−m/2n+1 with 0<2m<2n+1. Hence, for any constant x∈ℝ, xα, and x1+2α will make sense. By the way, the value of α can ensure finite-time convergence for system states, which will be illustrated in the following text.Assumption 11.
The inertial matrixJ and mass mp of the chaser spacecraft are known. The inertial matrix Jt and mass mt of the target are unknown. Both J and Jt are constant positive-definite symmetric matrices. mp and mt are bounded constants.Assumption 12.
All the disturbances are unknown but bounded with an upper bound. That is,φd≤D with D being an uncertain positive constant.Remark 13.
With the help of onboard devices equipped on spacecraft, the information of the chaser can be interactive. Thus, Assumption11 is reasonable. As for Assumption 12, it is believed that environmental interference is limited and the unknown disturbance can be strictly dominated by control inputs.For the control objectives, in this paper, we denote to designing a finite-time TSMC scheme for spacecraft RVD subject to external disturbance and input quantization. The objective is to design the control input commandφc for the error system in Equation (23) under Assumptions 11 and 12, such that all closed-loop signals are guaranteed to be bounded and the global system is finite-time stable. For a better illustration, the architecture of the controller synthesis is shown in Figure 2.Figure 2
Block diagram of the control architecture for spacecraft rendezvous motion.


## 3.2. Quantized Finite-Time TSMC Design

Inspired by [37], the actual control input is the quantized control torque, and as a result, Equation (23) is rewritten as
(33)Ξξ~̇=adξ∗Ξξ+φg+Qφ+φd+ΞadξAdh−1ξt−Adh−1ξ̇t,where Qφ=Qϕ1Qϕ2Qϕ3Qϕ4Qϕ5Qϕ6∈ℝ6 is the quantized value of the designed control input εu. The quantized input Qφ can be written as
(34)Qφ=κuεu+EuI1,with I1=111111T. κu and Eu satisfy the inequality in Equation (26) with δ and umin being design parameters.By using the Sigmoid function, a modified switched TSMSS=S1S2S3S4S5S6T∈ℝ6 is designed as follows:
(35)S=ξ~+k1η~+k2φη~,where k1=diagk11,k12,k13,k14,k15,k16 and k2=diagk21,k22,k23,k24,k25,k26 with k1ii=1,2,3,4,5,6 and k2ii=1,2,3,4,5,6 being positive constants. In particular, φη~=φη~1φη~2φη~3φη~4φη~5φη~6T is a piecewise smooth function and φη~i is defined as
(36)φη~i=sigη~iβ,η~i>ϑ,2ϑπsinπ2ϑη~i+βϑβ−1η~i,otherwise,where 0<β<1, and ϑ is a small positive constant, which is defined as
(37)ϑ=π21−κ1/1−κ,with 0<κ<1. The derivative of φη~ is φ̇η~=φ̇η~1φ̇η~2φ̇η~3φ̇η~4φ̇η~5φ̇η~6T with φ̇η~i being
(38)φ̇η~i=βη~iβ−1η~̇i,η~i>ϑ,cosπ2ϑη~iη~̇i+βϑβ−1η~̇i,otherwise.The actual control inputQφ is determined by εu, which is designed as
(39a)εu=11−δu,(39b)u=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4SαS2α+1+ε12−ΓS,with Sα=S12α+1S22α+1S32α+1S42α+1S52α+1S62α+1T. k3 and k4 are diagonal matrixes denoted as k3=diagk31,k32,k33,k34,k35,k36 and k4=diagk41,k42,k43,k44,k45,k46 with k3ii=1,2,3,4,5,6 and k4ii=1,2,3,4,5,6 being positive constants, which are humanly designed and will not change. Γ=Λφ2/Λφ2S2+ε12+D∧2/D∧2S2+ε12+u2/D∧2S2+ε12 is a time-varying state variable with Λφ=adξ∗Ξξ+φg+ΞadξAdh−1ξt−Adh−1ξ̇t. u is a constant satisfying umin≤u. D^ is the estimation of D.The adaptive law ofD is designed as follows:
(40)D^̇=c1S−c2D^,where c1 and c2 are positive constants. The estimated error of D is defined as
(41)D~=D−D^.Remark 14.
Three components comprise the quantized finite-time controller: the adaptive law, the finite-time input, and the quantizer. The finite-time input is made up of a switched TSMS (Equations (35)– (37)) and a modified finite-time convergent term (Equation (39b)). Combined with the adaptive law, the basic controller is obtained. Additionally, an HLQ is implemented to transfer the control signal, and, meanwhile, the basic controller is then altered by the quantizer parameter (Equation (39a)). By applying these steps, the quantized finite-time controller is fulfilled.The design process of the quantized finite-time TSMC is shown in Figure3. More detailed analysis about the proposed method can be seen in Remarks 15–17.Remark 15.
Inspired by [32], a modified switched TSMS is designed in Equation (35) with the aid of a nonlinear piecewise smooth function. As has been illustrated in [32], ϑ=π/21−κ1/1−κ is essential to ensure φη~i to be continuous and differentiable at point η~i=ϑ. The sliding mode variable S implemented here has two merits. Firstly, the terminal sliding mode variable can improve the convergence rate and guarantee the finite-time stability of the global system with improved robustness. In addition, the term φη~i defined by Equation (36) can avoid the singularity problem. Compared with the existing SMC methods in [6, 27, 28], the proposed S possesses the advantage of finite-time convergence with nonsingularity.Remark 16.
The control input in Equation (39) is designed to guarantee the system with finite-time convergence and convincing robustness. The designed control inputεu is derived by u and δ. More specifically, k3S is designed to guarantee the global stability of the system and Sα in Equation (39) is a finite-time convergent term which is different from the hyperbolic tangent function-based control input in [5, 6] and the Sigmoid function-based control input in [31, 32]. Γ is composed of state parameters and adaptive parameters, which enjoys the property of improving system robustness and ensuring the convergence of adaptive parameters. It is worth noting that u2/D∧2S2+ε12 in Γ can reduce the quantization errors. The introduction of Sα and Γ can improve the performance of system stability and avoid inherent chattering phenomenon.Remark 17.
There are seven control parameters in the finite-time controller, e.g.,kii=1,2,3,4, cii=1,2, and ε1, which have an effect on the control performance of the final proximity maneuver system. Although the criterion to select appropriate parameters mainly consists in the rule of thumb, there are rules to follow. More specifically, a bigger value of kii=1,2,3,4 means higher control accuracy and bigger overshoot of control force. The convergence rate of ω~,Θ~ will increase as k1 and k2 increase while the settling time of r~,v~ mainly depends on the value of k3 and k4. However, increasing k2 will improve the settling time of r~,v~ and increasing k4 will aggravate the chattering phenomenon. Additionally, estimation errors mainly depend on cii=1,2 and a smaller ε1 means higher accuracy. Given the above empirical rules, the approximate optimal solution can be found by trying various combination. In brief, the performance of the control system is determined by the controller, but also is largely affected by the design parameters.Figure 3
Block diagram of the controller design process.Letkimin be the minimum of each dimension of ki, that is, kjmin=minkji with i=1,2,3,4,5,6 and j=1,2,3,4.


## 3.3. Stability Analysis

Theorem 18.
Consider the relative accelerations of two spacecraft in RVD under Assumptions11 and 12. The error dynamics model is modeled by Equation (23). The control schemes are designed as Equations (35)–(41) under kij≥0i=1,⋯,4,j=1,⋯,6 and c1>0,c2>0. Under the proposed controller, the following conclusions can be obtained:
(i)
The global system is finite-time stable, and all signals in the closed-loop system are guaranteed to be bounded. In particular, the TSMSS will convergence to a small region ΔS=SS<2/λΞΔ1/ρ1/1+α in finite time(ii)
The lumped errorsη~i and ξ~i are convergent and will converge to the small region Δηi and Δξi in finite time, where Δη=max2Δ2/ρ21/α+1,ϑ and Δξi=maxϑ1i,ϑ2i with ϑ1i=ΔS+k1i+2k2i/πϑ+k2iβϑβ and ϑ2i=ΔS+k1iϑ0+k2iϑ0β+1Proof.
Define the Lyapunov function (LF) candidate:(42)V1=12STΞS.
Taking the time derivative of Equation (42) yields
(43)V̇1=STΞṠ=STΞξ~̇+k1η~̇+k2φ̇η~=STadξ∗Ξξ+φg+Qφ+φd+ΞadξAdh−1ξt−Adh−1ξ̇t+STk1ΞGη~ξ~+k2φ̇η~=STQφ+φd+Λφ+k1ΞGη~ξ~+k2φ̇η~.
Based on Equation (34), STQφ can be extended to be
(44)STQφ=STκuεu+EuI1=ST−κu1−δu+EuI1≤−STu+uminSTI1.
Substituting Equation (44), the derivative of V1 becomes
(45)V̇1≤−STu+uminSTI1+STφd+Λφ+k1ΞGη~ξ~+k2φ̇η~≤−STk1ΞGη~ξ~+k2φ̇η~−STk3S−STk4SαS2α+1+ε12+uminSTI1+STφd+Λφ+k1ΞGη~ξ~+k2φ̇η~−Λφ2STSΛφ2S2+ε12−D∧2STSD∧2S2+ε12−u2STSD∧2S2+ε12≤−k3minS2−k4minSα+1−SΛφ−D^S−uS+μminS+DS+STΛφ+k4minε1+ε1+ε1+ε1≤−k3minS2−k4minSα+1+D~S+3+k4minε1.
Define the overall LF candidate:(46)V2=V1+12c1D~2.
Taking the time derivative of Equation (46) yields
(47)V̇2≤−k3minS2−k4minSα+1+D~S+1c1D~D~̇+3+k4minε1=−k3minS2−k4minSα+1+D~S−1c1D~D^̇+3+k4minε1=−k3minS2−k4minSα+1+D~S−D~S−c2D^+3+k4minε1≤−k3minS2−k4minSα+1+c2D~D^+3+k4minε1.
In consideration of Lemma8 and α defined in Equation (30), the following equation holds:
(48)D~D^=D~D−D~≤−12D~2+12D2=−12D~2+12D2=−14D~2−14D~−D~α2+14D~2α−12D~1+α+12D2≤−12D~1+α+12D2−14D~2+14D~2α≤−12D~1+α+12D2+1−α4αα/1−α.
Substituting Equation (48) into Equation (47) yields
(49)V̇2≤−k3minS2−k4minSα+1+3+k4minε1+c2−12D~1+α+12D2+1−α4αα/1−α=−k3minS2−k4minSα+1−c22D~1+α+Δ1,where Δ1=c2/22D2+1−α/αα/1−α+3+k4minε1.Through mathematical transformation, Equation (49) can be rewritten as
(50)V̇2≤−k4minSTS1+α/2−c22D~21+α/2+Δ1=−k4minSTS1+α/2−c22D~21+α/2+Δ1=−k4min2λmaxΞ1+α/212STΞS1+α/2−c222c11+α/212c1D~21+α/2+Δ1≤−ρV21+α/2+Δ1,where ρ=mink4min2/λΞ1+α/2,c2/22c11+α/2 with λΞ being the minimum of each dimension of Ξ. It is obvious that ρ>0,Δ1≥0.Remark 19.
As mentioned in Remark10, it is known that α=−m/2n+1 with 0<2m<2n+1. As a result, D~α and D~1+2α will make sense. It concludes that D~1+α=D~21+α/2=D~21+α/2. Furthermore, 1/4<1+α/2<1/2 is tenable such that V̇2≤−ρV21+α/2+Δ1 in Equation (50) satisfies the requirement in Lemma 9. As a consequence, the system is finite-time stable.Equation (50) can be reconstructed to be
(51)V̇2≤−ρV21+α/2+Δ1=−ρ−Δ1V21+α/2V21+α/2.As the result in Equation (50) has proved that the global system is stable, thus, V21+α/2≥Δ1/ρ holds until the system turning to be stable. Consequently, it yields the convergence region of S: ΔS=SS<2/λΞΔ1/ρ1/1+α..Thus, point (i) has been proven.Remark 20.
Through the design of the controller and the proof of the effectiveness, the result in Equation (50) indicates that the system is globally stable with finite-time convergence. Significantly, V2 is only the LF candidate formed by S and D~. The stability of system states (ξ~ and η~) is not clear. Thus, a further developed analysis and proof are needed.SinceS=S1S2S3S4S5S6T satisfies S≤ΔS, one has Si=ξ~i+k1iη~i+k2iφη~i=ϑS with ϑS∈−ΔS,ΔS.Whenη~i≤ϑ, ξ~i+k1iη~i+k2i2ϑ/πsinπ/2ϑη~i+βϑβ−1η~i≤ΔS holds all time, which implies
(52)ξ~i≤ΔS+k1iη~i+k2i2ϑπsinπ2ϑη~i+βϑβ−1η~i≤ΔS+k1iϑ+k2i2ϑπsinπ2ϑϑ+βϑβ−1ϑ=ΔS+k1i+2k2iπϑ+k2iβϑβ=ϑ1i.Whenη~i>ϑ, the LF is chosen as
(53)V0=∑i=1612η~i2.Taking the time derivate ofV0 yields
(54)V̇0=∑i=16η~iGη~iξ~i=∑i=16η~iϑS−k1iη~i−k2iφη~i=∑i=16ϑSη~i−k1iη~i2−k2iη~iβ+1≤∑i=16−k1iη~i2−k2iη~iβ+1+k1iη~i2+14k1iϑS2=∑i=16−k2iη~iβ+1+14k1iϑS2≤−ρ1V0β+1/2+Δ2,with Δ2=∑i=16ϑS2/4k1i and ρ1=k2min2β+1/2, which implies that the LF V0 is stable when η~i>ϑ. Through the result in Equation (54), one has
(55)V̇0≤−ρ1−Δ2V0β+1/2V0β+1/2.Based on Equation (55), if ρ1−Δ2/V0β+1/2≥0 holds, η~i will converge to the region η~i<2Δ2/ρ11/β+1 in finite time. Consequently, η~i will converge to the region Δη.Letϑ0=2Δ2/ρ21/β+1. According to Equation (51), ξ~i+k1iη~i+k2iη~iβ+1≤ΔS holds, which implies
(56)ξ~i≤ΔS+k1iη~i+k2iη~iβ+1≤ΔS+k1iϑ0+k2iϑ0β+1=ϑ2i.Consequently,ξ~i will converge to the region Δξi=maxϑ1i,ϑ2i with ϑ1i=ΔS+k1i+2k2i/πϑ+k2iβϑβ and ϑ2i=ΔS+k1iϑ0+k2iϑ0β+1.Thus, point (ii) has been proven.Finally, Theorem18 has been proven.


## 4. Simulation Results

In this section, numerical simulations are developed to validate the effectiveness of the proposed scheme for spacecraft RVD maneuver control. A RVD scenario with a chaser and a target is adopted in this simulation. The target is assumed to maneuver freely in an elliptical orbit (the target will not maneuver or be affected by space interference). The chaser is forced to track the target to achieve the objective. According to [1], the inertia parameters of the target and the chaser are given as
(57)Chaser:m=100kg,J=23.7−0.2−0.5−0.225.3−0.3−0.5−0.322.5kg⋅m2,Target:mt=110kg,Jt=21.7−0.2−0.5−0.222.3−0.3−0.5−0.325.3kg⋅m2.The initial orbital elements of the target are given in Table1 [1, 43]. As the chaser is forced to rendezvous with the target, whose initial states relative to the target are essential and given in Table 2 [1, 43]. It assumes that Bt is in the same direction as LVLH of the target at the beginning of the mission. Consequently, the initial transfer matrix Rt0 is
(58)Rt0=cosΩcosω−sinΩcosisinω−cosΩsinω−sinΩcosicosωsinΩsinisinΩcosω+cosΩcosisinω−sinΩsinω+cosΩcosicosω−cosΩsinisinisinωsinicosωcosi.Table 1
Initial orbital elements of the target [1, 43].
Orbital elementValueSemimajor axisa (km)26628Eccentricitye0.7417Inclinationi (deg)63.4RAANΩ (deg)0Argument of perigeeω (deg)270The anomalyf (deg)120Table 2
Initial states of the chaser relative to the target [1, 43].
Initial relative statesValueTarget’s angular velocity (rad/s)000.0011TInitial relative position (m)151515TInitial relative velocity (m/s)−0.051−0.247−0.075TInitial relative attitude (rad)2π/3Initial relative principal rotation axis−2−2−3TInitial relative angular velocity (rad/s)0.0095.98−9.31T×10−4The time varying disturbanceφd=τdΦdT is defined as [1]:
(59)τd=0.005×sin0.1tcos0.1t−sin0.1tT,Φd=0.005×sin0.1tcos0.1t−sin0.1tT.The purpose of error tracking control is to drive the relative states to track the desired one. The desired states are given in Table3 [1, 43].Table 3
Desired states of the chaser relative to the target [1, 43].
Desired relative statesValueDesired relative position (m)500TDesired relative velocity (m/s)000TDesired relative attitude (rad)0Initial relative angular velocity (rad/s)000TThe simulation result includes two parts: (1) the comparison of three different kinds of SMC algorithms and (2) the verification of the proposed quantized TSMC scheme.


### 4.1. Simulation Results of Three Finite-Time Control Schemes without Input Quantization

To better show the superiority of the proposed control algorithm, three different controllers are simulated. Input quantization is not considered in this section.Inspired by the proposed controller in Equation (39), the newly designed control input isu=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4Sα/S2α+1+ε12−ΓS; more details can be seen in Table 4. Denote the proposed controller to be Sproposed. A Sigmoid function-based scheme (denote as Ssig) in [31, 32] and a hyperbolic tangent function-based scheme (denote as Stanh) in [5, 6] are adopted to form contrastive simulations. Three different SMSs and corresponding control inputs are given in Table 4. Table 5 presents the SMC gains, control gains, and adaptive gains. These three methods are all valid for error tracking of the spacecraft final proximity maneuver.Table 4
Three different kinds of SMC schemes.
ComponentSMSControl inputsSproposedS=ξ~+k1η~+k2φη~u=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4Sα/S2α+1+ε12−ΓS with Γ=Λφ2/Λφ2S2+ε12+D∧2/D∧2S2+ε12SsigSs=ξ~+b1η~+b2sigη~ζus=−b1ΞGη~ξ~−b2ζdsigη~−b3S−Λφ+DsigSαζ−b4sigSαζ with dsigη~=η~1ζ−1η~̇1⋯η~6ζ−1η~̇6TStanhSt=ξ~+l1η~+l2tanhη~ut=−l1ΞGη~ξ~−l2η~̇−l3St−l4tanhSt−Λφ+DtanhStTable 5
Gains of the three control schemes.
ComponentValueSMS gainsk1=0.1×diag0.52,0.52,0.52,3.6,3.6,3.6,k2=10−3×diag15,15,15,15,15,15.b1=10−2×diag3.6,3.6,3.6,12.5,12.5,12.5,b2=10−2×diag1.6,1.6,1.6,15,15,15.l1=10−2×diag2.2,2.2,2.2,27,27,27, l2=10−2×diag4,4,4,2,2,2.Control gainsk3=diag6.6,6.6,6.6,6,6,6, k4=diag10,10,10,6,6,6.b3=diag11.5,11.5,11.5,5.8,5.8,5.8, b4=diag11,11,11,10,10,10.l1=diag13.5,13.5,13.5,7,7,7, l2=diag12,12,12,15,15,15.Adaptive gainsc1=0.12,c2=4.Other input parametersα=−1/9, β=κ=0.6, ε1=0.1, ζ=0.6.As input quantization is not considered, the control inputs in Table4 act on the error system in Equation (23) directly. Besides, Γ in Table 4 is different from that in Equation (39b), where u2/D∧2S2+ε12 is ignored.To improve the credibility of simulations, the power consumptions (the integration of control force) of three control schemes are designed to be approximately equal. For each control scheme, the SMS gains and control gains in Table5 are approximately optimal schemes under the constraint of power consumption. And simulation results represent the best control performance of three control schemes.The simulation results ofSproposed are presented in Figures 4–7. And the contrastive results between Sproposed, Ssig, and Stanh are shown in Figures 8–13.Figure 4
Relative configuration error curves.Figure 5
Relative velocity error curves.Figure 6
Control torque and force curves.Figure 7
Estimated parameter curve.Figure 8
Control force curves.Figure 9
Curves of the integration of control force.Figure 10
Relative distance error curves.Figure 11
Relative attitude error curves.Figure 12
Relative translational velocity error curves.Figure 13
Relative angular velocity error curves.Figures4 and 5 indicate the relative configuration error and relative velocity error under the proposed control scheme, respectively. In Figure 4, the curves of i=1,2,3 denote the attitude tracking error and curves of i=4,5,6 denote the position tracking error. According to Figure 4, the attitude tracking error converges to less than 10−3×567Trad in 70 s and the position tracking error converges to less than 10−3×334Tm in 60 s. In Figure 5, the curves of i=1,2,3 denote the angular velocity tracking error and curves of i=4,5,6 denote the translational velocity tracking error. According to Figure 5, the angular velocity tracking error converges to less than 10−3×558Trad/s in 40 s and the translational velocity tracking error converges to less than 10−3×558Tm/s in 53 s. In Figure 6, the curves of i=1,2,3 denote the feedback control torque and curves of i=4,5,6 denote the feedback control force. After 70 s, the change of control force in a small scale exists because of the existence of a disturbance, especially the time-varying disturbance φd. Figure 7 implies that the estimated value of D is stable.Figures4–7 indicate that the proposed control algorithm is effective, which can provide the system with high performance and stability. The contrastive results are shown in Figures 8–13.To better show the contrastive results, several notations are defined.F=normu represents the control force, and W=∑Fdt is the integration of F. eii=1,2,3,4 is defined to express the tracking error of system states. In detail, e1=normb~ with b~=SΘβ~, e2=θ~/π×180 with θ~=normΘ~, e3=normv~ with v~=η~4η~5η~6T, and e4=normΩ~ with Ω~=η~1η~2η~3T.According to Figure9, the power consumptions of three control schemes are approximately equal after 60 s. Consequently, the comparative simulations satisfy the premise of equal power consumption. Figures 10–13 indicates that the convergence performances are roughly the same on the whole. More details about the stability can be found in Table 6.Table 6
Control precision comparisons.
ComponentSproposedStanhSsigRelative distance errorAccuracy (m)5×10−39×10−39×10−3Time (s)605550Relative attitude errorAccuracy (deg)0.60.90.6Time (s)708070Relative translational velocity errorAccuracy (m/s)6×10−38×10−37×10−3Time (s)555050Relative angular velocity errorAccuracy (deg/s)7×10−37×10−38×10−3Time (s)455050More information can be found in Figures10–13: (1) the chattering phenomenon is obvious in Ssig. (2) The final convergence region of Stanh is bigger than that of Ssig and Sproposed, which means that the control accuracy of Stanh is lower. Take Figure 10 for example, the position tracking error under Ssig converges to less than 1×10−7m in 90 s, but with a severely chattering, which can be seen in Figure 10(c). The position tracking error under Stanh converges to less than 1.6×10−3m in 90 s while that of Sproposed converges to less than 1.6×10−4m in 90 s (see Figures 10(a) and 10(b)). As a result, two prominent advantages of Sproposed can be found by means of the steady-state analysis. Firstly, in comparison to Ssig, the chattering problem is reduced by virtue of the chattering-free control input in Sproposed. Another merit is that Sproposed has a better error tracking accuracy than Stanh.Two conclusions can be drawn through the above results: (1) the proposed method can guarantee the system with finite-time stability and has a better performance of system robustness and control accuracy. (2) The chattering phenomenon in SMC can be effectively reduced by the proposed method.


### 4.2. Simulation Results of Adaptive Quantized TSMC Scheme

An HLQ is adopted in this paper to validate the feasibility of the proposed quantized TSMC scheme. The quantizer parameters are chosen to beu=0.0008 and δ=0.33. The SMS and control inputs of Sproposed is implemented here but with no change in SMS gains and control gains in Table 5. Simulation results of the quantized control are presented in Figures 14–17. Figures 18–23 show the comparisons between normal control and quantized control.Figure 14
Relative configuration error curves.Compared with Figures3 and 4, similar simulation results are obtained in Figures 14 and 15, which implies that quantized control is effective for error tracking.Figure 15
Relative velocity error curves.As the designed continuous control inputs in Equation (39) are quantized, the change of control forces is discontinuous in Figure16. Figures 24 and 25 show the dwell times of quantized control torque and force, from which it concludes that data transmission frequency is considerably reduced.Figure 16
Control torque and force curves.Figure 17
Estimated parameter.Figure 18
Relative distance error curves.Figure 19
Relative attitude error curves.Figure 20
Relative translational velocity error curves.Figure 21
Relative angular velocity error curves.Figure 22
Control force curves.Figure 23
Curves of integration of control force.Figure 24
The dwell time of control torque.Figure 25
The dwell time of control force.Thus, the HLQ implemented here is feasible and effective for the coupled relative motions control of spacecraft RVD. More contrastive results are shown in Figures18–23.Figures22 and 23 indicate that a small increment comes to the control force of the quantized control scheme. Though the convergence rate is faster than normal control without quantization seemingly, there is a small delay (about 2-5 s) in convergence time of quantized control, which can be seen in Figures 18–21. Apart from the improvement of convergence rate, the tiny increase of power consumption leads to better control accuracy. The changes in control accuracy are small, and a distinct result can be seen in Figure 18.In general, the quantized TSMC scheme can guarantee the error tracking system with high stability and control accuracy, which is similar to that of normal control. Meanwhile, the information transmission is reduced by means of an HLQ.


## 4.1. Simulation Results of Three Finite-Time Control Schemes without Input Quantization

To better show the superiority of the proposed control algorithm, three different controllers are simulated. Input quantization is not considered in this section.Inspired by the proposed controller in Equation (39), the newly designed control input isu=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4Sα/S2α+1+ε12−ΓS; more details can be seen in Table 4. Denote the proposed controller to be Sproposed. A Sigmoid function-based scheme (denote as Ssig) in [31, 32] and a hyperbolic tangent function-based scheme (denote as Stanh) in [5, 6] are adopted to form contrastive simulations. Three different SMSs and corresponding control inputs are given in Table 4. Table 5 presents the SMC gains, control gains, and adaptive gains. These three methods are all valid for error tracking of the spacecraft final proximity maneuver.Table 4
Three different kinds of SMC schemes.
ComponentSMSControl inputsSproposedS=ξ~+k1η~+k2φη~u=−k1ΞGη~ξ~−k2φ̇η~−k3S−k4Sα/S2α+1+ε12−ΓS with Γ=Λφ2/Λφ2S2+ε12+D∧2/D∧2S2+ε12SsigSs=ξ~+b1η~+b2sigη~ζus=−b1ΞGη~ξ~−b2ζdsigη~−b3S−Λφ+DsigSαζ−b4sigSαζ with dsigη~=η~1ζ−1η~̇1⋯η~6ζ−1η~̇6TStanhSt=ξ~+l1η~+l2tanhη~ut=−l1ΞGη~ξ~−l2η~̇−l3St−l4tanhSt−Λφ+DtanhStTable 5
Gains of the three control schemes.
ComponentValueSMS gainsk1=0.1×diag0.52,0.52,0.52,3.6,3.6,3.6,k2=10−3×diag15,15,15,15,15,15.b1=10−2×diag3.6,3.6,3.6,12.5,12.5,12.5,b2=10−2×diag1.6,1.6,1.6,15,15,15.l1=10−2×diag2.2,2.2,2.2,27,27,27, l2=10−2×diag4,4,4,2,2,2.Control gainsk3=diag6.6,6.6,6.6,6,6,6, k4=diag10,10,10,6,6,6.b3=diag11.5,11.5,11.5,5.8,5.8,5.8, b4=diag11,11,11,10,10,10.l1=diag13.5,13.5,13.5,7,7,7, l2=diag12,12,12,15,15,15.Adaptive gainsc1=0.12,c2=4.Other input parametersα=−1/9, β=κ=0.6, ε1=0.1, ζ=0.6.As input quantization is not considered, the control inputs in Table4 act on the error system in Equation (23) directly. Besides, Γ in Table 4 is different from that in Equation (39b), where u2/D∧2S2+ε12 is ignored.To improve the credibility of simulations, the power consumptions (the integration of control force) of three control schemes are designed to be approximately equal. For each control scheme, the SMS gains and control gains in Table5 are approximately optimal schemes under the constraint of power consumption. And simulation results represent the best control performance of three control schemes.The simulation results ofSproposed are presented in Figures 4–7. And the contrastive results between Sproposed, Ssig, and Stanh are shown in Figures 8–13.Figure 4
Relative configuration error curves.Figure 5
Relative velocity error curves.Figure 6
Control torque and force curves.Figure 7
Estimated parameter curve.Figure 8
Control force curves.Figure 9
Curves of the integration of control force.Figure 10
Relative distance error curves.Figure 11
Relative attitude error curves.Figure 12
Relative translational velocity error curves.Figure 13
Relative angular velocity error curves.Figures4 and 5 indicate the relative configuration error and relative velocity error under the proposed control scheme, respectively. In Figure 4, the curves of i=1,2,3 denote the attitude tracking error and curves of i=4,5,6 denote the position tracking error. According to Figure 4, the attitude tracking error converges to less than 10−3×567Trad in 70 s and the position tracking error converges to less than 10−3×334Tm in 60 s. In Figure 5, the curves of i=1,2,3 denote the angular velocity tracking error and curves of i=4,5,6 denote the translational velocity tracking error. According to Figure 5, the angular velocity tracking error converges to less than 10−3×558Trad/s in 40 s and the translational velocity tracking error converges to less than 10−3×558Tm/s in 53 s. In Figure 6, the curves of i=1,2,3 denote the feedback control torque and curves of i=4,5,6 denote the feedback control force. After 70 s, the change of control force in a small scale exists because of the existence of a disturbance, especially the time-varying disturbance φd. Figure 7 implies that the estimated value of D is stable.Figures4–7 indicate that the proposed control algorithm is effective, which can provide the system with high performance and stability. The contrastive results are shown in Figures 8–13.To better show the contrastive results, several notations are defined.F=normu represents the control force, and W=∑Fdt is the integration of F. eii=1,2,3,4 is defined to express the tracking error of system states. In detail, e1=normb~ with b~=SΘβ~, e2=θ~/π×180 with θ~=normΘ~, e3=normv~ with v~=η~4η~5η~6T, and e4=normΩ~ with Ω~=η~1η~2η~3T.According to Figure9, the power consumptions of three control schemes are approximately equal after 60 s. Consequently, the comparative simulations satisfy the premise of equal power consumption. Figures 10–13 indicates that the convergence performances are roughly the same on the whole. More details about the stability can be found in Table 6.Table 6
Control precision comparisons.
ComponentSproposedStanhSsigRelative distance errorAccuracy (m)5×10−39×10−39×10−3Time (s)605550Relative attitude errorAccuracy (deg)0.60.90.6Time (s)708070Relative translational velocity errorAccuracy (m/s)6×10−38×10−37×10−3Time (s)555050Relative angular velocity errorAccuracy (deg/s)7×10−37×10−38×10−3Time (s)455050More information can be found in Figures10–13: (1) the chattering phenomenon is obvious in Ssig. (2) The final convergence region of Stanh is bigger than that of Ssig and Sproposed, which means that the control accuracy of Stanh is lower. Take Figure 10 for example, the position tracking error under Ssig converges to less than 1×10−7m in 90 s, but with a severely chattering, which can be seen in Figure 10(c). The position tracking error under Stanh converges to less than 1.6×10−3m in 90 s while that of Sproposed converges to less than 1.6×10−4m in 90 s (see Figures 10(a) and 10(b)). As a result, two prominent advantages of Sproposed can be found by means of the steady-state analysis. Firstly, in comparison to Ssig, the chattering problem is reduced by virtue of the chattering-free control input in Sproposed. Another merit is that Sproposed has a better error tracking accuracy than Stanh.Two conclusions can be drawn through the above results: (1) the proposed method can guarantee the system with finite-time stability and has a better performance of system robustness and control accuracy. (2) The chattering phenomenon in SMC can be effectively reduced by the proposed method.


## 4.2. Simulation Results of Adaptive Quantized TSMC Scheme

An HLQ is adopted in this paper to validate the feasibility of the proposed quantized TSMC scheme. The quantizer parameters are chosen to beu=0.0008 and δ=0.33. The SMS and control inputs of Sproposed is implemented here but with no change in SMS gains and control gains in Table 5. Simulation results of the quantized control are presented in Figures 14–17. Figures 18–23 show the comparisons between normal control and quantized control.Figure 14
Relative configuration error curves.Compared with Figures3 and 4, similar simulation results are obtained in Figures 14 and 15, which implies that quantized control is effective for error tracking.Figure 15
Relative velocity error curves.As the designed continuous control inputs in Equation (39) are quantized, the change of control forces is discontinuous in Figure16. Figures 24 and 25 show the dwell times of quantized control torque and force, from which it concludes that data transmission frequency is considerably reduced.Figure 16
Control torque and force curves.Figure 17
Estimated parameter.Figure 18
Relative distance error curves.Figure 19
Relative attitude error curves.Figure 20
Relative translational velocity error curves.Figure 21
Relative angular velocity error curves.Figure 22
Control force curves.Figure 23
Curves of integration of control force.Figure 24
The dwell time of control torque.Figure 25
The dwell time of control force.Thus, the HLQ implemented here is feasible and effective for the coupled relative motions control of spacecraft RVD. More contrastive results are shown in Figures18–23.Figures22 and 23 indicate that a small increment comes to the control force of the quantized control scheme. Though the convergence rate is faster than normal control without quantization seemingly, there is a small delay (about 2-5 s) in convergence time of quantized control, which can be seen in Figures 18–21. Apart from the improvement of convergence rate, the tiny increase of power consumption leads to better control accuracy. The changes in control accuracy are small, and a distinct result can be seen in Figure 18.In general, the quantized TSMC scheme can guarantee the error tracking system with high stability and control accuracy, which is similar to that of normal control. Meanwhile, the information transmission is reduced by means of an HLQ.


## 5. Conclusion

This paper presents a finite-time control scheme on SE(3) for the final proximity maneuver of spacecraft RVD subject to external disturbance and input quantization. By feat of Lie group SE(3), the integrated model of translational and rotational motion can simplify the controller design and avoid coupling. The proposed TSMC algorithm can drive the states of the error tracking system to the equilibrium points in finite-time, which is proved rigorously with no knowledge of the upper bound values of adaptive parameters. The frequency of information transmission is effectively reduced by the quantized controller. Meanwhile, the proposed control algorithm possesses a superiority of better control precision, reduced chattering, and nonsingularity. Finally, simulation results demonstrate the effectiveness of the proposed scheme.


### 5.1. Future Work

Besides these advantages, there are two drawbacks in this paper. Firstly, the presented control scheme requires a precise model of the chaser, which may be unfeasible in missions with uncertain inertia. Secondly, the quantizer parameters for different dimensions of control torque and force are assumed to be the same. Adaptive methods or neural networks can be referenced to tackle these two limitations in future work.


## 5.1. Future Work

Besides these advantages, there are two drawbacks in this paper. Firstly, the presented control scheme requires a precise model of the chaser, which may be unfeasible in missions with uncertain inertia. Secondly, the quantizer parameters for different dimensions of control torque and force are assumed to be the same. Adaptive methods or neural networks can be referenced to tackle these two limitations in future work.



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*Source: 7423919-2021-09-15.xml*