# Robust Adaptive Relative Position and Attitude Control for Noncooperative Spacecraft Hovering under Coupled Uncertain Dynamics

Authors: Jianghui Liu; Haiyang Li; YaKun Zhang; Jianyong Zhou; Lin Lu; Fuqi Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The control of body-fixed hovering over noncooperative target, as one of the key problems of relative motion control between spacecrafts, is studied in the paper. The position of the chaser in the noncooperative target’s body coordinate system is required to remain unchanged, and the attitude of the chaser and the target must be synchronized at the same time. Initially, a six-degrees-of-freedom-coupled dynamic model of a chaser relative to a target is established, and relative attitude dynamics is described through using modified Rodrigues parameters (MRP). Considering the model uncertainty and external disturbances of the noncooperative target system, an adaptive nonsingular terminal sliding mode (NTSM) controller is designed. Adaptive tuning method is used to overcome the effects of the model uncertainty and external disturbances. The upper bounds of the model uncertainty and external disturbances are not required to be known in advance. The actual control law is continuous and chatter-free, which is obtained by integrating the discontinuous derivative control signal. Finally, these theoretical results are verified by numerical simulation.


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## Body

## 1. Introduction

On-orbit spacecraft is of great value. When breaking down in space, on-orbit repair, component replacement, and refueling can significantly prevent further cost of replacing a new one [1–6]. Therefore, it has drawn much attention from researchers. Relative hovering usually indicates that a spacecraft stays fixed in its position and attitude in the body coordinate system of another spacecraft. The state of relative hovering greatly facilitates space surveillance and inspections.Historically, related research studies have mainly focused on the hovering over asteroids. Scheeres first proposed the concept of hovering orbit of the spacecraft relative to asteroids in 1999 [7]. In general, hovering over an asteroid mainly includes inertial hovering and body-fixed hovering [8, 9]. Broschart simulated the hovering control of an asteroid under slight gravity and determined the stable region of inertial hovering [8]. In the control of hovering over asteroids, the spacecraft needs to continuously apply the control thrust to counteract the gravity and rotational acceleration to maintain the desired position [9, 10]. This method is feasible on asteroids due to low nominal acceleration of the spacecraft [7, 8, 11]. Zeng proposed the solar sail spacecraft’s hover over an asteroid, which greatly extended the hover time and hover range without fuel consumption [12]. It should be pointed out that the research on spacecraft hovering over an asteroid in the early stage mainly concentrated on the relative position control of the spacecraft and asteroid irrespective of the relative attitude control [7–9, 11, 13–15]. Lee et al. put forward passive tracking control of the relative position and relative attitude of hovering over an asteroid in the framework of geometric mechanics [16]. Aiming at the orbit and attitude control of hovering over a rotating asteroid at a low speed, Lee et al. designed a continuous finite-time convergence control scheme [17, 18].Strictly speaking, the situation of a spacecraft hovering over an asteroid is different from that of hovering over the target spacecraft. This is because the former is influenced by the gravity from both the asteroid and the sun, while the latter is only influenced by Earth’s gravity. Therefore, the dynamic equations in the two cases are also different.In this paper, we mainly study the hovering control of one spacecraft relative to another spacecraft. Recently, some researchers have also published many findings concerning hovering control between two spacecrafts. Tan analyzed and solved the relative motions of the two spacecrafts in the near circle and elliptical orbits, respectively. The controller was designed using generalized inverse matrix transformation, and the guidance deviation was obtained by model prediction [19, 20]. Xue has established a hybrid system model for relative hovering between the two spacecrafts [21]. Based on the description of the state transfer matrix between the two spacecrafts, Cheng used a multipulse control method to study the relative hovering motion [22]. To solve the problem of hovering between the two spacecrafts at superclose distance, Xu proposed the control method of line of sight pointing tracking on the basis of relative orbit control and realized the joint control of the relative orbit and attitude [23]. Song studied the hovering closed-loop control method based on Hill equation [24]. Dang established a precise analytical solution for hovering between the two spacecrafts and deduced the minimum force and minimum fuel positions during the orbital period [25]. Huang studied the problem of finite-time hovering control in the absence of the radial or in-track thrust [26].However, previous research studies on hovering control mainly focused on the control of the relative position between the two spacecrafts. There are few research studies on relative attitude control, and hovering control is mostly in the form of open-loop control. Even if the relative attitude control problem is considered, it is restricted to the condition of the stable attitude or slow attitude change of the target, and the relative position and relative attitude are also controlled separately, ignoring their coupling. Further complicating things, the model parameters of the spacecraft are not likely to be precisely known during hovering operations, and the spacecraft is always subject to external environment disturbances [27–29]. Sun investigates relative position and attitude control for spacecraft rendezvous and proximity operations subject to input saturation, kinematic couplings, parametric uncertainties, and unknown external disturbances. State feedback control method [30], disturbance-observer-based robust nonlinear control scheme [31], a six-degrees-of-freedom integrated adaptive fuzzy nonlinear control method [32], and robust adaptive control approach [33] are proposed. The terminal sliding mode control (TSMC) method based on the conventional sliding mode control (SMC) method has the advantage of finite-time convergence [34]. Its sliding mode is independent of the model parameters and external disturbances, and thus has been widely used in nonlinear control [35, 36]. However, the disadvantage is that the nonlinear term used in the TSMC may cause a singularity problem leading to a control magnitude to become unbounded. In order to solve this problem, Feng proposed the nonsingular terminal sliding mode control (NTSMC) method [37, 38]. Nonsingular terminal sliding mode controllers are now widely used in a variety of application areas like robotics [39–41], aerospace, and process control [42, 43].Considering the limitations of previous studies, it is necessary to further study hovering control between the two spacecrafts. In this paper, control problem of the chaser hovering over the target with relatively rapid attitude change in space is studied. A nonlinear six-degrees-of-freedom-coupled dynamic model is established in the chaser’s body coordinate system. A chattering-free adaptive nonsingular terminal sliding mode (NTSM) controller is designed. Firstly, the conventional sliding surface is established, and then the nonsingular terminal sliding surface is constructed on this basis. The adaptive tuning method is used to deal with the model uncertainty and external disturbances. The upper bounds of the model uncertainty and external disturbances are not required to be known in advance [44–46]. The actual control law is continuous and chatter-free, which is obtained by integrating the discontinuous derivative control signal.The rest of the sections are as follows. Section2 introduces dynamics for the chaser and the target. Section 3 presents the designs of the adaptive nonsingular terminal sliding mode (NTSM) controller and demonstrates the stability of the closed-loop system. Simulation studies performed on the relative position and attitude control of the hovering example are presented in Section 4. Section 5 draws conclusions.


## 2. Dynamics for the Chaser and the Target

In this paper, the skew-symmetric matrixSζ∈ℝ3×3 derived from a vector ζ=ζ1ζ2ζ3T∈ℝ3 is defined as(1)Sζ=0−ζ3ζ2ζ30−ζ1−ζ2ζ10,and it satisfies ζTSζ=0, Sζξ=−Sξζ, ξTSζξ=0 for any ξ=ξ1ξ2ξ3T∈ℝ3.sgn
s
=
sgn
s
1
,
⋯
,
sgn
s
n
T for any s=s1,⋯,snT∈ℝn, where(2)sgnsi=−1,si<0,0,si=0,1,si>0.


### 2.1. Dynamics for Relative Orbit

The relative motion scenario between the chaser and the target is shown in Figure1. FE=OEXEYEZE is the Earth-centered inertial frame, with its coordinate origin at the Earth center and nonrotating with respect to the stars. The OEXE axis points in the direction of the Earth's vernal equinox. The OEZE axis points to the direction of North Pole. The direction of OEYE axis is determined by the right-hand rule. FT=OTXTBYTBZTB and FC=OCXCBYCBZCB are the body frames of the target and the chaser, respectively. The body frame is fixed onto the spacecraft body and rotates with it. rc,re is the position vectors in the frame FC, and rt,rd,hf is the position vectors in the frame FT.Figure 1
Relative position and attitude motion coordinate system.The position and attitude dynamic model of the chaser relative to the frameFE is [47](3)r˙c=υc−Sωcrc,σ˙c=141−σcTσcI3+2σcσcT+2Sσcωc,mcυ˙c+mcSωcυc=Fc+Fd,Jcω˙c+SωcJcωc=τc+τd,where υc∈ℝ3 and ωc∈ℝ3 are the chaser’s velocity and angular velocity, respectively. rc∈ℝ3 is the chaser’s position, and σc∈ℝ3 is the modified Rodrigues parameter (MRP) vector to describe the attitude of frame FC with respect to frame FE. Jc∈ℝ3×3 and mc∈ℝ are the chaser’s inertial matrix and mass, respectively. Fc∈ℝ3 and τc∈ℝ3 are the chaser’s control force and control torque, respectively. Fd∈ℝ3 and τd∈ℝ3 are unknown bounded disturbance force and unknown bounded disturbance torque of the chaser, respectively. They are all expressed in the frame FC.Ignoring the external forces and torques, the position and attitude dynamic model of the noncooperative target relative to the frameFE is [31](4)r˙t=υt−Sωtrt,σ˙t=141−σtTσtI3+2σtσtT+2Sσtωt,υ˙t+Sωtυt=0,Jtω˙t+SωtJtωt=0,where υt∈ℝ3 and ωt∈ℝ3 are the target’s velocity and angular velocity, respectively. rt∈ℝ3 is the target’s position, and σt∈ℝ3 is the attitude of frame FT with respect to frame FE. Jt∈ℝ3×3 and mt∈ℝ are the target’s inertial matrix and mass, respectively. They are all expressed in the frame FT.


### 2.2. Relative Dynamic Model

The relative attitudeσe∈ℝ3 of frame FC with respect to frame FT described by MRP is [48](5)σe=σc1−σtTσt+σtσcTσc−1−2Sσtσc1+2σtTσc+σtTσtσcTσc.The corresponding attitude transfer matrixMe∈ℝ3×3 is(6)Me=I3+8SσeSσe−41−σeTσeSσe1+σeTσe2,where I3∈ℝ3×3 is a unit matrix. It can be seen from Figure 1 that the position and velocity of the hovering point H described in the frame FT are(7)rd=rt+hf,(8)υd=υt+Sωthf.The relative angular velocity, relative position, and relative velocity of the two spacecrafts described in the frameFC are(9)ωe=ωc−Meωt,(10)re=rc−Mert,(11)υe=υc−Meυt.Substituting equations (9)∼(11) into equation (3) and using the equations M˙e=−SωeMe, r˙d=υd−Sωtrd and Me−1=MeT, the relative motion equations described in the frame FC can be derived:(12)r˙e=υe−Sωcre,σ˙e=Gσeωe,mcυ˙e=−mcSωcυc+Meυ˙d−Sωeυc−υe+Fc+Fd,Jcω˙e=−SωcJcωc−JcMeω˙t+Sωcωe+τc+τd,where Gσe=1−σeTσeI3+2σeσeT+2Sσe/4 is a nonsingular matrix. From equations (4) and (8)–(11) and MeSζ=−SMeζMe, Meυ˙d can be derived as(13)Meυ˙d=Meυ˙t+Sω˙thf=−MeSωtυt−MeShfω˙t=−SMeωtMeυd−MeSωthf−MeShfω˙t=−Sωc−ωeυc−υe−Sωc−ωeMehf−MeShfω˙t.Also,ω˙t in equation (13) can be calculated by equations (4) and (9):(14)ω˙t=−Jt−1SωtJtωt=−Jt−1SMeTωc−ωeJtMeTωc−ωe.Therefore, equation (12) can be rewritten as(15)r˙e=υe−Sωcr,σ˙e=Gσeωe,mcυ˙e=−mcP1−P2+Fc+Fd,Jcω˙e=−SωcJcωc−JcSωcωe+P3+τc+τd,where(16)P1=Sωcυe+S2ωc−ωeMehf,P2=mcMeShfJt−1SMeTωc−ωeJtMeTωc−ωe,P3=JcMeJt−1SMeTωc−ωeJtMeTωc−ωe.


### 2.3. Integrated Dynamic Model

Defining state variablese1=reT,σeTT and e2=υeT,ωeTT from equation (15) yields(17)e˙1=A1e1+A2e2,(18)Me˙2=B1+B2+u+d,(19)A1=−Sωc030303,A2=I30303Gσe,M=mcI30303Jc,B1=−mcP1−SωcJcωc−JcSωcωe,B2=P2P3,u=Fcτc,d=Fdτd.Remark 1.
A
1 in model (17) and B1 in model (18) reflect that the relative position motion between the two spacecrafts is affected by the relative attitude motion, which indicates that there is a strong coupling effect between relative attitude and relative position.
This paper aims to design a controller so that the chaser reaches a predetermined hovering pointH, and the relative attitude between the two spacecrafts is σe=σe0, σe0 is a constant. In order to facilitate the analysis, the relative attitude is σe0=03×1, that is, the attitude between the two spacecrafts is synchronized. From equations (5), (9)–(11), (17), and (18), we can see that the control goal is equivalent to designing the control input u to satisfy limt⟶∞e1t≤ϕ1 and limt⟶∞e2t≤ϕ2, where ϕ1 and ϕ2 are any small positive numbers.


## 2.1. Dynamics for Relative Orbit

The relative motion scenario between the chaser and the target is shown in Figure1. FE=OEXEYEZE is the Earth-centered inertial frame, with its coordinate origin at the Earth center and nonrotating with respect to the stars. The OEXE axis points in the direction of the Earth's vernal equinox. The OEZE axis points to the direction of North Pole. The direction of OEYE axis is determined by the right-hand rule. FT=OTXTBYTBZTB and FC=OCXCBYCBZCB are the body frames of the target and the chaser, respectively. The body frame is fixed onto the spacecraft body and rotates with it. rc,re is the position vectors in the frame FC, and rt,rd,hf is the position vectors in the frame FT.Figure 1
Relative position and attitude motion coordinate system.The position and attitude dynamic model of the chaser relative to the frameFE is [47](3)r˙c=υc−Sωcrc,σ˙c=141−σcTσcI3+2σcσcT+2Sσcωc,mcυ˙c+mcSωcυc=Fc+Fd,Jcω˙c+SωcJcωc=τc+τd,where υc∈ℝ3 and ωc∈ℝ3 are the chaser’s velocity and angular velocity, respectively. rc∈ℝ3 is the chaser’s position, and σc∈ℝ3 is the modified Rodrigues parameter (MRP) vector to describe the attitude of frame FC with respect to frame FE. Jc∈ℝ3×3 and mc∈ℝ are the chaser’s inertial matrix and mass, respectively. Fc∈ℝ3 and τc∈ℝ3 are the chaser’s control force and control torque, respectively. Fd∈ℝ3 and τd∈ℝ3 are unknown bounded disturbance force and unknown bounded disturbance torque of the chaser, respectively. They are all expressed in the frame FC.Ignoring the external forces and torques, the position and attitude dynamic model of the noncooperative target relative to the frameFE is [31](4)r˙t=υt−Sωtrt,σ˙t=141−σtTσtI3+2σtσtT+2Sσtωt,υ˙t+Sωtυt=0,Jtω˙t+SωtJtωt=0,where υt∈ℝ3 and ωt∈ℝ3 are the target’s velocity and angular velocity, respectively. rt∈ℝ3 is the target’s position, and σt∈ℝ3 is the attitude of frame FT with respect to frame FE. Jt∈ℝ3×3 and mt∈ℝ are the target’s inertial matrix and mass, respectively. They are all expressed in the frame FT.


## 2.2. Relative Dynamic Model

The relative attitudeσe∈ℝ3 of frame FC with respect to frame FT described by MRP is [48](5)σe=σc1−σtTσt+σtσcTσc−1−2Sσtσc1+2σtTσc+σtTσtσcTσc.The corresponding attitude transfer matrixMe∈ℝ3×3 is(6)Me=I3+8SσeSσe−41−σeTσeSσe1+σeTσe2,where I3∈ℝ3×3 is a unit matrix. It can be seen from Figure 1 that the position and velocity of the hovering point H described in the frame FT are(7)rd=rt+hf,(8)υd=υt+Sωthf.The relative angular velocity, relative position, and relative velocity of the two spacecrafts described in the frameFC are(9)ωe=ωc−Meωt,(10)re=rc−Mert,(11)υe=υc−Meυt.Substituting equations (9)∼(11) into equation (3) and using the equations M˙e=−SωeMe, r˙d=υd−Sωtrd and Me−1=MeT, the relative motion equations described in the frame FC can be derived:(12)r˙e=υe−Sωcre,σ˙e=Gσeωe,mcυ˙e=−mcSωcυc+Meυ˙d−Sωeυc−υe+Fc+Fd,Jcω˙e=−SωcJcωc−JcMeω˙t+Sωcωe+τc+τd,where Gσe=1−σeTσeI3+2σeσeT+2Sσe/4 is a nonsingular matrix. From equations (4) and (8)–(11) and MeSζ=−SMeζMe, Meυ˙d can be derived as(13)Meυ˙d=Meυ˙t+Sω˙thf=−MeSωtυt−MeShfω˙t=−SMeωtMeυd−MeSωthf−MeShfω˙t=−Sωc−ωeυc−υe−Sωc−ωeMehf−MeShfω˙t.Also,ω˙t in equation (13) can be calculated by equations (4) and (9):(14)ω˙t=−Jt−1SωtJtωt=−Jt−1SMeTωc−ωeJtMeTωc−ωe.Therefore, equation (12) can be rewritten as(15)r˙e=υe−Sωcr,σ˙e=Gσeωe,mcυ˙e=−mcP1−P2+Fc+Fd,Jcω˙e=−SωcJcωc−JcSωcωe+P3+τc+τd,where(16)P1=Sωcυe+S2ωc−ωeMehf,P2=mcMeShfJt−1SMeTωc−ωeJtMeTωc−ωe,P3=JcMeJt−1SMeTωc−ωeJtMeTωc−ωe.


## 2.3. Integrated Dynamic Model

Defining state variablese1=reT,σeTT and e2=υeT,ωeTT from equation (15) yields(17)e˙1=A1e1+A2e2,(18)Me˙2=B1+B2+u+d,(19)A1=−Sωc030303,A2=I30303Gσe,M=mcI30303Jc,B1=−mcP1−SωcJcωc−JcSωcωe,B2=P2P3,u=Fcτc,d=Fdτd.Remark 1.
A
1 in model (17) and B1 in model (18) reflect that the relative position motion between the two spacecrafts is affected by the relative attitude motion, which indicates that there is a strong coupling effect between relative attitude and relative position.
This paper aims to design a controller so that the chaser reaches a predetermined hovering pointH, and the relative attitude between the two spacecrafts is σe=σe0, σe0 is a constant. In order to facilitate the analysis, the relative attitude is σe0=03×1, that is, the attitude between the two spacecrafts is synchronized. From equations (5), (9)–(11), (17), and (18), we can see that the control goal is equivalent to designing the control input u to satisfy limt⟶∞e1t≤ϕ1 and limt⟶∞e2t≤ϕ2, where ϕ1 and ϕ2 are any small positive numbers.


## 3. Controller Design and Stability Analysis

Considering the model uncertainty of the system, from equation (18) yields(20)Md+ΔMe˙2=Bd1+ΔB1+Bd2+ΔB2+u+d,where Md∈ℝ6×6, Bd1∈ℝ6, and Bd2∈ℝ6 are the nominal model of the system. ΔM∈ℝ6×6, ΔB1∈ℝ6, and ΔB2∈ℝ6 are model uncertainty terms of the system. Set d¯∈ℝ6 as system compound disturbances including model uncertainty and external disturbances, and its expression is(21)d¯=−ΔMe˙2+ΔB1+ΔB2+d.Assumption 1.
The chaser can obtain its own motion informationrc,υc,σc,ωc and relative motion information re,υe,σe,ωe through its own measurement device. The measured relative motion and relative attitude information are smooth and bounded.Assumption 2.
System compound disturbancesd¯∈ℝ6 and its time derivative d¯˙∈ℝ6 are unknown but bounded.
The system compound disturbancesd¯∈ℝ6 satisfies the following equation [33]:(22)d¯<k0+k1e1+k2e22,where k0>0, k1>0, and k2>0 are constants, and ⋅ represents the 2 norm of the vector.
Substituting equation (21) into (20) yields(23)Mde˙2=Bd1+Bd2+u+d¯.
From equation (23) yields(24)e˙2=MpBd1+Bd2+Mpu+Mpd¯,where Mp∈ℝ6×6 is the generalized inverse matrix of Md∈ℝ6×6. The expression of Mp is(25)Mp=MdTMd−1MdT.
The time derivative of equation (17) gives(26)e¨1=A˙1e1+A1e˙1+A˙2e2+A2e˙2.
The time derivative of equation (24) gives(27)e¨2=dMpBd1+MpBd2dt+M˙pu+Mpu˙+D1,where D1=M˙pd¯+Mpd¯˙.
The sliding mode function is designed as(28)s=e2+ce1,where c∈ℝ6×6 is a sixth-order positive-definite diagonal matrix, c=diagc1,⋯,c6, ci>0,i=1,2,⋯,6. From equation (28), it can be obtained that when s⟶0, then e1⟶0 and e2⟶0 as well.
The first derivative of the sliding mode function is(29)s˙=e˙2+ce˙1.
Substituting equations (17) and (24) into equation (29) yields(30)s˙=MpBd1+Bd2+u+d¯+cA1e1+A2e2.
In this paper, the second-order nonsingular terminal sliding mode (NTSM) control theory is used to design the control law. In order to design a nonsingular terminal sliding mode (NTSM) controller, the nonsingular terminal sliding mode (NTSM) surfaceσ is defined as(31)σ=s+β−1s˙p/q,where β=diagβ1,⋯,β6, βi>0,i=1,2,⋯,6. p>0 and q>0 are odd integers and satisfy the condition 1<p/q<2. In order to facilitate subsequent derivation, the following definition is made.Definition 1.
G
j
H
=
G
1
j
H
1
,
G
2
j
H
2
,
…
,
G
m
j
H
m
T for any two vectors G=G1,G2,…,GmT and H=H1,H2,…,HmT.
The time derivative of equation (31) gives(32)σ˙=s˙+p/qβ−1s˙p/q−1s¨=p/qβ−1s˙p/q−1s¨+q/pβs˙2−p/q.
The time derivative of equation (29) gives(33)s¨=e¨2+ce¨1=dMpBd1+MpBd2dt+M˙pu+Mpu˙+D1+cA˙1e1+A1e˙1+A˙2e2+A2e˙2=dMpBd1+MpBd2dt+M˙pu+Mpu˙+cA1A1e1+A2e2+A2MpBd1+MpBd2+Mpu+Mpd¯+cA˙1e1+A˙2e2+D1=dMpBd1+MpBd2dt+M˙p+cA2Mpu+Mpu˙+cA1A1e1+A2e2+cA˙1e1+A˙2e2+cA2MpBd1+MpBd2+D,where D=cA2Mpd¯+D1. According to equation (22) and Assumption 2 yield(34)D=cA2Mpd¯+D1<D0+D1e1+D2e22,where D0>0, D1>0, and D2>0 are constants.Proof.
Lyapunov functionV1 is expressed as(35)V1=12σTσ+λ0D˜02+λ1D˜12+λ2D˜22,where λ0>0, λ1>0, and λ2>0 are constant parameters, D˜0=D^0−D0, D˜1=D^1−D1, and D˜2=D^2−D2 are adaptation errors, and D^0, D^1, and D^2 are the estimations of D0, D1, and D2, respectively.
The time derivative of equation (35) gives(36)V˙1=σTσ˙+λ0D˜0D˜˙0+λ1D˜1D˜˙1+λ2D˜2D˜˙2=σTp/qβ−1s˙p/q−1s¨+q/pβs˙2−p/q+λ0D^0−D0D˜˙0+λ1D^1−D1D˜˙1+λ2D^2−D2D˜˙2=σTp/qβ−1s˙p/q−1dMpBd1+MpBd2dt+M˙p+cA2Mpu+Mpu˙+cA2MpBd1+MpBd2+D+cA1A1e1+A2e2+cA˙1e1+A˙2e2+q/pβs˙2−p/q+λ0D^0−D0D^˙0+λ1D^1−D1D^˙1+λ2D^2−D2D^˙2.
The control lawu is designed as(37)u=∫0tu˙dt,(38)u˙=−Mp−1dMpBd1+MpBd2dt+M˙p+cA2Mpu+cA1A1e1+A2e2+cA˙1e1+A˙2e2+cA2MpBd1+MpBd2+q/pβs˙2−p/q+D^0+D^1e1+D^2e22sgnσ+γσ,where γ=diagγ1,⋯,γ6, γi>0,i=1,2,⋯,6.
The adaptation lawD^˙0, D^˙1, and D^˙2 is designed as(39)D^˙0=ε0p/qβ−1s˙p/q−1σT,(40)D^˙1=ε1p/qβ−1s˙p/q−1σTe1,(41)D^˙2=ε2p/qβ−1s˙p/q−1σTe22,where ε0>0, ε1>0, and ε2>0 are the tuning parameters.
Substituting equations (38)∼(41) into equation (36) yields(42)V˙1=σTp/qβ−1s˙p/q−1D−D^0+D^1e1+D^2e22sgnσ−γσ+λ0D^0−D0ε0p/qβ−1s˙p/q−1σT+λ1D^1−D1ε1p/qβ−1s˙p/q−1σTe1+λ2D^2−D2ε2p/qβ−1s˙p/q−1σTe22,then(43)V˙1≤p/qβ−1s˙p/q−1Dσ−γσ2−D^0+D^1e1+D^2e22σ+λ0ε0D^0−D0σ+λ1ε1D^1−D1σe1+λ2ε2D^2−D2σe22≤p/qβ−1s˙p/q−1Dσ−γσ2−D^0+D^1e1+D^2e22σ+D0+D1e1+D2e22σ−D0+D1e1+D2e22σ+λ0ε0D^0−D0σ+λ1ε1D^1−D1σe1+λ2ε2D^2−D2σe22≤−p/qβ−1s˙p/q−1D0+D1e1+D2e22−Dσ−1−λ0ε0D^0−D0σ−1−λ1ε1D^1−D1σe1−1−λ2ε2D^2−D2σe22≤−h1σ−h2D^0−D0−h3D^1−D1−h4D^2−D2≤−h12σ2−h22λ0D˜02λ0−h32λ1D˜12λ1−h42λ2D˜22λ2,where(44)h1=p/qβ−1s˙p/q−1D0+D1e1+D2e22−D,h2=p/qβ−1s˙p/q−11−λ0ε0σ,h3=p/qβ−1s˙p/q−11−λ1ε1σe1,h4=p/qβ−1s˙p/q−11−λ2ε2σe22.
So,(45)V˙1≤−min2h1,2/λ0h2,2/λ1h3,2/λ2h4σ2+λ02D˜0+λ12D˜1+λ22D˜2≤−hV1,where h=min2h1,2/λ0h2,2/λ1h3,2/λ2h4. According to equation (34), it can be known that h1>0. In order to ensure h2>0, h3>0, and h4>0, the condition 1−λ0ε0>0, 1−λ1ε1>0, and 1−λ2ε2>0 must be satisfied when designing parameters.
s
˙
p
/
q
−
1
>
0 for any s˙≠0 and s˙p/q−1=0 only when s˙=0. According to the accessibility condition of the sliding mode, the system can reach σ=0 from any initial state σ0≠0 within a finite time tr. Then, it will be proven that the sliding mode function s will also reach the s=0 plane in the finite time ts on the nonsingular terminal sliding mode surface σ=0. First, the lemma for the stability of a nonlinear system is given [49].Lemma 1.
If there is a continuous functionVt, the following conditions are met:(1)
Vt is positive(2)
There is a real numberχ>0 and an open neighborhood n∈0,1 so that(46)V˙t+χVnt≤0.Then, the system is stable in finite time, and the convergence timetf satisfies(47)tf≤1χ1−nV1−n0.Lyapunov functionV2 is defined as(48)V2=12sTs.On the nonsingular terminal sliding surfaceσ=0, the following equation can be obtained:(49)σ=s+β−1s˙p/q=0,i.e.,(50)s˙=−βsq/p.The time derivative of equation (48) gives(51)V˙2=sTs˙.Substituting equation (50) in (51) yields(52)V˙2=−sTβq/psp/q≤−βminq/p2s2p+q/p=−βminq/p2p+q/2pV2p+q/2p=−χsV2nst,i.e.,(53)V˙2+χsV2nst≤0,where βmin is the smallest absolute component of β, χs=βminq/p2p+q/2p>0, ns=p+q/2p∈0,1.Let thes converge to 0 within finite time ts. It can be deduced from Lemma 1 that when t≥ts, st=0 and satisfies(54)ts=tr+βmin−q/ppp−qstrp−q/p.Since the sliding mode functions converges to 0 within finite time ts, according to equation (49), σ will also converge to 0 within finite time ts. It can be seen from equation (28) that if s⟶0, then e1⟶0 and e2⟶0. This case completes the proof.The control system diagram with the adaptive nonsingular terminal sliding mode controller is shown in Figure2.From equation (38) yields(55)u˙+Mp−1M˙p+cA2Mpu=−Mp−1dMpBd1+MpBd2dt+cA1A1e1+A2e2+cA˙1e1+A˙2e2+cA2MpBd1+MpBd2+q/pβs˙2−p/q+D^0+D^1e1+D^2e22sgnσ+γσ.Figure 2
Control system diagram.Remark 2.
Equation (55) is essentially a low-pass filter with the right side of the equation as the input and u as the output. Therefore, the adaptive nonsingular terminal sliding mode (ANTSM) controller designed in this paper can eliminate the chattering phenomenon of the conventional sliding mode.Remark 3.
The parametersε0, ε1, and ε2 determine the rate at which the estimated values D^0, D^1, and D^2 converge to their respective boundaries. Larger ε0, ε1, and ε2 ensure that the estimated values D^0, D^1, and D^2 quickly converge to the actual boundary D0, D1, and D2. Considering λ0ε0<1, λ1ε1<1, and λ2ε2<1, the actual value is to weigh the convergence rate and constraints.Remark 4.
The parameterγ determines the convergence rate of the sliding surface. Larger γ means faster convergence rate, which requires larger input. In reality, the thrust of the spacecraft is limited. Therefore, the value of γ needs to be weighed between thrust condition and convergence rate.Remark 5.
The proposed methodology is applicable whenσ is reachable. However, σ cannot become exactly 0 in a finite time due to nonlinear characteristics of the system and the switching delays. Therefore, the adaptive parameters D^0, D^1, and D^2 may become boundless. A simple method is to use the dead zone to overcome the above difficulties, and the adaptive law D^˙0, D^˙1, and D^˙2 of equations (39)∼(41) are modified as(56)D^˙0=ε0p/qβ−1s˙p/q−1σT,σ≥μ,0,σ<μ,D^˙1=ε1p/qβ−1s˙p/q−1σTe1,σ≥μ,0,σ<μ,D^˙2=ε2p/qβ−1s˙p/q−1σTe22,σ≥μ,0,σ<μ.


## 4. Simulation Example

This section verifies the effectiveness of the designed controller through simulation of a chaser’s superclose distance hovering over a noncooperative target. The mass of the chaser ismc=58.2kg. The moments of inertia of the target and chaser, Jt and Jc, are [30](57)Jt=3336.3−135.4−154.2−135.43184.5−148.5−154.2−148.52423.7kg⋅m2,Jc=598.3−22.5−51.5−22.5424.4−27−51.5−27263.6kg⋅m2.The model uncertainties areΔmc=0.5kg and ΔJc=1.721.52.54.431.5−273.6kg⋅m2.The hovering position in the frameFT is hf. The initial position and velocity are rct0 and υct0, respectively. The initial attitude and angular velocity are σct0 and ωct0, respectively. The initial relative position and relative velocity are ret0 and υet0, respectively. The initial relative attitude and relative angular velocity are σet0 and ωet0, respectively. The values of the above parameters are listed in Table 1.Table 1
Simulation parameters.
Variable
Value
h
f
0,5,0
T
r
c
t
0
/
m
−
2
.
0449,4
.
6646,4
.
4429
T
×
10
6
υ
c
t
0
/
m
/
s
−
4.6536
,
−
5.1599
,
3.2906
T
×
10
3
σ
c
t
0
0,0,0
T
ω
c
t
0
/
rad
/
s
0,0,0
T
r
e
t
0
/
m
30,0,25
T
υ
e
t
0
/
m
/
s
0.8
,
−
0.6
,
0.3
T
σ
e
t
0
0.4
,
−
0.5
,
0.6
T
ω
e
t
0
/
rad
/
s
−
0
.
0207
,
0
.
0125
,
0
.
0258
TDisturbance torqueτd and disturbance force Fd are(58)τd=1−3sinω0t−2cosω0t3+2sinω0t+cosω0t2−3sinω0t+3cosω0t×10−5Nm,Fd=2−3sinω0t+cosω0t1+3sinω0t+2cosω0t3−2sinω0t−2cosω0t×10−4N,where ωo=μ/rc3 is the chaser’s average orbital angular velocity, and μ=3.986×1014m3/s2 is the gravitational constant of the Earth.The controller parameters are selected asc=diag0.1,0.1,0.1,0.1,0.1,0.1, p=5, q=3, β=diag0.2,0.2,0.2,0.1,0.1,0.1, γ=diag90,90,90,3,3,3, ε0=1×10−5, ε1=3×10−5, ε2=3×10−5, D^00=0, D^10=0, D^20=0, and μ=0.001. The simulation results are presented in Figures 3 and 4.Figure 3
Relative motion, control torque and, control force.Figure 4
Estimated parametersD^0, D^1, and D^2 using the adaptive tuning method.Figure3 shows relative motion, control torque, and control force. The relative attitude drops from 0.4,−0.5,0.6T to 03×1 after 58 s; the relative angular velocity increased rapidly from −0.0207,0.0125,0.0258Trad/s and then decreased to 03×1rad/s after reaching the peak value, which indicates that the chaser completes attitude capture for the noncooperative target and then maintains the attitude synchronization with the target. After 71 s, the relative position drops from 30,−7,18Tm to 03×1m and the relative velocity drops from 0.776,−0.485,0.186Tm/s to 03×1m/s. It is indicated that the chaser’s center of mass has reached the desired hovering position in the frame FT and has remained unchanged thereafter. The control torque required by the chaser is minimal after 73 s, and the maximum control torque in each direction does not exceed 0.05Nm. The control force required by the chaser is minimal after 48 s, and the maximum control force in each direction does not exceed 0.2N. The chaser's control torque and control force curve are smooth throughout the entire process, indicating no chattering occurs. Figure 4 shows estimated parameters D^0, D^1, and D^2 using the adaptive tuning method. D^0, D^1, and D^2 finally converge to 0.0085, 0.3925, and 0.0419.In order to verify the effectiveness of the proposed controller, it is necessary to compare it with the conventional sliding mode controller. The conventional sliding mode controller is chosen as [50](59)u=−MdcA1e1+A2e2−Bd1−Bd2−Ksgns.The parameter of the conventional sliding mode controller isK=diag0.2,0.2,0.2,0.2,0.2,0.2, and all other simulation parameters are the same as those of the proposed sliding mode controller. The simulation results are presented in Figure 5. By further comparing Figure 5 with Figure 3, it can be concluded that under the effect of the conventional sliding mode controller, the chaser finishes the attitude synchronization with the target and reaches the hovering point need 59 s and 74 s , respectively, which is 1 s and 3 s longer than the proposed sliding mode controller. After reaching the hovering position and completing the attitude synchronization, the maximum control torque and maximum control force required by the chaser in each direction are still as high as 0.5Nm and 0.8 N, respectively. Chattering phenomenon occurs in both control torque and control force of the conventional sliding mode controller.Figure 5
Relative motion, control torque, and control force under conventional sliding mode control.In addition, for the mean performance error indexMPE=1/T∫0TXe2dt, the tracking error is Xe=reT,veT,σeT,ωeTT and the simulation time is T=100s. It can be calculated that the proposed controller is MPE1=1.2550, and the conventional sliding mode controller is MPE2=1.4340. Obviously, the proposed controller effectively improves the closed-loop system dynamic response performance.In order to investigate the robustness of the proposed controller, the external bounded disturbance forceFd and torque τd are increased to 100Fd and 100τd, respectively. Other simulation parameters are the same. The simulation results are presented in Figures 6 and 7. By comparing Figures 6 and 7 with Figures 3 and 4, it can be found that the chaser finishes the position synchronization at the hovering point after 71 s and finishes the attitude synchronization with the target after 58 s. The estimated parameters D^0, D^1, and D^2 finally converge to 0.0085, 0.3926, and 0.0419. The control torque and control force curve are smooth without chattering, and the mean performance error index is MPE3=1.2551. Compared with the small disturbance, the difference in control performance is very small, thus proving the strong robustness of the proposed sliding mode controller.Figure 6
Relative motion, control torque, and control force under large disturbance.Figure 7
Estimated parametersD^0, D^1, and D^2 using the adaptive tuning method under large disturbance.


## 5. Conclusions

In this paper, an adaptive nonsingular terminal sliding mode controller is designed based on the measurable information when the chaser hovers over a noncooperative target, and the six-degrees-of-freedom-coupled relative position and relative attitude control are realized. Instead of the conventional control input, its time derivative is used in the controller. The derivative control contains the discontinuous sign function, and the actual control input is obtained by integration, so it is continuous and chatter-free. The adaptive tuning method is used to deal with the model uncertainty and external disturbances. Simulation results show that the controller can effectively overcome the model uncertainty and the influence of external disturbance factors. The chaser quickly reaches the hovering position of the noncooperative target, and both the hovering position error and the hovering velocity error converge to a smaller range. The required control torque and control force are minimally, continuously, and smoothly produced with no chattering occurring.



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*Source: 8678473-2019-12-28.xml*