Research ArticleExtended-State-Observer-Based Terminal Sliding Mode TrackingControl for Synchronous Fly-Aroundwith Space Tumbling Target

Zhijun Chen1 Yong Zhao 1 Yuzhu Bai1 Dechao Ran2 and Liang He1

1College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China2National Innovation Institute of Defense Technology Chinese Academy of Military Science Beijing 100091 China

Correspondence should be addressed to Yong Zhao zhaoyongnudteducn

Received 23 June 2019 Revised 4 August 2019 Accepted 11 August 2019 Published 5 November 2019

Academic Editor Alberto Cavallo

Copyright copy 2019 Zhijun Chen et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(is paper presents a robust controller with an extended state observer to solve the Synchronous Fly-Around problem of a chaserspacecraft approaching a tumbling target in the presence of unknown uncertainty and bounded external disturbance (erotational motion and time-varying docking trajectory of tumbling target are given in advance and referred as the desired trackingobjective Based on dual quaternion framework a six-degree-of-freedom coupled relative motion between two spacecrafts ismodeled in which the coupling effect model uncertainties and external disturbances are considered More specially a novelnonsingular terminal sliding mode is designed to ensure the convergence to the desired trajectory in finite time Based on thesecond-order sliding mode an extended state observer is employed to the controller to compensate the closed-loop system Bytheoretical analysis it is proved that the modified extended-state-observer-based controller guarantees the finite-time stabili-zation Numerical simulations are taken to show the effectiveness and superiority of the proposed control scheme FinallySynchronous Fly-Around maneuvers can be accomplished with fast response and high accuracy

1 Introduction

Synchronous Fly-Around (SFA) technique represents theconcept of proximity operation for driving a chaserspacecraft to fly around a space tumbling target with attitudesynchronized With the number of failed spacecraft andspace debris increasing on-orbit service operations areurgently required to extend life of the failed spacecrafts suchas assembly repairing module replacement detumblingrefueling and orbital debris removal [1 2] Under the in-fluence of terrestrial gravitational perturbation most of thefailed spacecrafts have evolved into space tumbling targetsdeveloping the noncooperative feature [3] (e SFA ap-proach (see Figure 1) can be identified as a significanttechnology for on-orbit service missions since it performshigh accuracy in close-range proximity to noncooperativetumbling target Some studies have been performed with theintention to solve the SFA problem [4ndash7]

As pointed out in [8] it also introduces a series ofproblems and challenges in SFA process Different fromtraditional cooperative space missions rendezvous and

docking (RVD) and spacecraft formation flying (SFF) thenoncooperative characteristics of SFA will result in de-ficiency of shape structure and quality information whichcauses low accuracy of navigation in close proximity An-other problem is complicated modeling It is obvious thatSFA is a six-degree-of-freedom (6-DOF) relative motion inwhich both the translational motion and rotational one areincluded Mutual coupling between the two motions willalso lead to complexity in modeling (erefore high precisemodeling is an essential step to overcome the highly coupleddrawback during SFA mission Ma et al [9] used C-Wdynamics equation to establish the relative translationalmotion and adopted modified Rodrigues parameters todescribe the relative rotational dynamics Similar modelingmethod has also been performed in [10] More recently Xuet al [11] considered the dynamical coupling inside the 6-DOF model and presented a nonlinear suboptimal controllaw for SFA problem However all the abovementionedstudies gave the two-part relative motion and ignored thenonlinear coupling effect [12] such that it would cause theproblem that rotational and translational control will not

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5791579 15 pageshttpsdoiorg10115520195791579

actuate simultaneously (erefore it is significant to es-tablish integrated 6-DOF relative motion considering cou-pled term Dual quaternion [13ndash15] gives us inspiration tosolve SFA problem since it has been an effective tool formotion description in physical problems and mathematicalcalculation It combines traditional Euler quaternions withdual numbers and inherits elegant properties of both Basedon dual quaternion the coupled 6-DOF dynamics motionbetween two spacecrafts can be derived successfully Wanget al [12] made attempt to use dual quaternion to describethe coupled dynamics of rigid spacecraft and investigated thecoordinated control problem for SFF problem Wu et al[16 17] proposed a nonlinear suboptimal control based ondual quaternion for synchronized attitude-position controlproblem Under dual-quaternion framework Filipe andTsiotras [18] proposed an adaptive tracking controller forspacecraft formation problem and ensured global asymp-totical stability in the presence of unknown disturbance Forthe safety problem of RVD Dong et al [19] proposed a dual-quaternion-based artificial potential function (APF) controlto guarantee the arrival at the docking port of the target withdesired attitude (e above research results illustrate theeffectiveness of 6-DOF modeling using dual-quaternionframework and also provide the motivation of this paper

(e guidance navigation and control (GNC) systemsfor space proximity operations should be taken into accountduring on-orbit SFAmissions [20] Due to the nonlinear andhighly coupled dynamics designing a controller for theaccurate SFA operation is still an open problem In additionto the preceding interests in 6-DOF modeling uncertaintiesand disturbances are another key issue that should beaddressed in the control system which will severely reducethe performance of the controller or lead to instability of theclosed-loop system(erefore it is urgent to realize accurateand fast SFA tracking control under strong coupling effectsand multisource interferences in practical process In view ofthis problem various nonlinear control methods have beencarried out to estimate the disturbance and compensate thesystem such as state-dependent Riccati equation (SDRE)control [21] adaptive control [22] and suboptimal control[11] Among these methods terminal sliding mode (TSM)control is an effective technique due to its robustness tosystem uncertainties and can provide finite-time conver-gence Wang and Sun [23] carried out an adaptive TSM

control method for spacecraft formation flying problemwithin dual-quaternion framework However the majordisadvantage of initial TSM is the singularity problem Inorder to eliminate this problem Zou et al [24] proposed anonsingular TSM control (NTSMC) to realize the finite-timestability of satellite attitude control Extended-state-observer(ESO) methods have recently been introduced to com-pensate for unmodeled dynamics and system with externaldisturbance and uncertainty Conventional ESO has shownfine performance in disturbance rejection Based on a newtechnique known as the second-order sliding mode (SOSM)[25] a novel ESO can guarantee high accuracy and ro-bustness of the estimation with finite-time convergenceSome studies have recently combined conventional TSMand ESO for spacecraft control problem [26 27] Moreoverthe combination of NTSMC and ESO is also widely usedespecially in aerospace engineering such as attitude trackingcontrol [28 29] underactuated spacecraft hovering [30] andreusable launch vehicle [31] In Reference [28] an ESO-based third-order NTSMC was proposed for the attitudetracking control problem Ran et al [32] proposed anadaptive SOSM-based ESO with fault-tolerant NTSMC forthe model uncertainty external disturbance and limitedactuator control during spacecraft attitude control Zhanget al [33] presented an adaptive fast NTSMC using esti-mated information by SOSM-based ESO to solve theproblem of spacecraft 6-DOF coupled tracking maneuver inthe presence of model uncertainties and actuator mis-alignment Moreno and Osori [25] gave a Lyapunov finite-time stability proof for a closed-loop system combiningSOSM controller and ESO To the best of the authorsrsquoknowledge ESO-based NTSMC is still an open problem forSFA mission within dual-quaternion framework

(is paper aims to investigate the Synchronous Fly-Around maneuver problem in the presence of model un-certainties and external disturbances Compared to existingresults the major contributions and differences of this studyare summarized as follows

(1) To date it is the first time to use dual quaternion todescribe 6-DOF relative dynamics for tumblingtarget-capturing problem in which the nonlinearcoupled effect model uncertainties and externaldisturbances are taken into account Moreover theerror kinematics and dynamics are given to improvemodeling accuracy significantly

(2) Motivated by the concept of SOSM a novel ESO isproposed to reconstruct the model uncertainties andexternal disturbances with finite-time convergenceBased on the estimated information of ESO a fastNTSMC is carried out to eliminate the total dis-turbances and ensure the finite-time stable of theclosed-loop system (e proposed method can gainfaster response and higher accuracy than existingmethods

(is paper is organized as follows Some necessarymathematical preliminaries of dual quaternion and usefullemmas are given and coupled kinematics and dynamics arederived for the relative motion problem in Section 2 In

ω1

ω2

Chaser satellite

Tumbling target

Tumbling motion

Figure 1 Synchronous fly-around for a chaser approaching atumbling target

2 Mathematical Problems in Engineering

Section 3 NTSMC and ESO strategies are elaborated andthe finite-time stability analysis of the system is performed inSection 4 Following that numerical simulations are illus-trated and discussed in Section 5 to show the effectiveness ofthe proposed method Finally some appropriate conclusionsare presented in Section 6

2 Mathematical Preliminaries

(is study considers the leader-follower spacecraft forma-tion including a chaser and a tumbling target (Figure 2) Inorder to facilitate the description of the model the followingcoordinate system is used OI minus XIYIZI is the Earth-cen-tered inertial coordinate system with the Cartesian right-hand reference frame Ol minus xlylzl is LVLH (local-verticallocal-horizontal) reference frame with the origin at thecenter of mass of target xl is pointing to the spacecraftradially outward zl is the axis normal to the target orbitalplane and yl is the axis established by right-hand rule Of minus

xfyfzf (respectively the tumbling target Ol minus xlylzl) is thebody-fixed coordinate system of the chaser with its origin inthe center of mass of the chaser and axes pointing to the axisof inertial Pl minus xlylzl is the desired tracking reference framewith the capturing point Pl of the tumbling target (econtrol objective is to control the chaser (Of minus xfyfzf )synchronize with desired state (Pl minus xlylzl) so that the on-orbit service can be guaranteed in the next step

21 Dual Quaternion Dual quaternion can be regarded as aquaternion whose element is dual numbers and can bedefined as [14 15 23]

1113954q [1113954η 1113954ξ] q + εqprime (1)

where 1113954η is the dual scalar part 1113954ξ [1113954ξ1 1113954ξ2 1113954ξ3] is the dualvector part q and qprime are the traditional Euler quaternionsrepresenting the real part and dual part of dual quaternionrespectively and ε is the called dual unit with ε2 0 andεne 0 In this paper N is employed to denote the set of dualquaternions and R denotes the set of dual numbers(erefore it is supposed that 1113954q isin N4 1113954ξ isin R3 and 1113954η isin R

Similar to the traditional quaternion and dual vector (seeAppendix) the basic operations for dual quaternions aregiven as follows

1113954q1 + 1113954q2 1113954η1 + 1113954η2 1113954ξ1 + 1113954ξ21113960 1113961

λ1113954q [λ1113954η λ1113954ξ]

1113954qlowast

[1113954η minus 1113954ξ]

1113954q1 otimes 1113954q2 1113954η11113954η2 minus 1113954ξ11113954ξ2 1113954η11113954ξ2 + 1113954η21113954ξ1 + 1113954ξ1 times 1113954ξ21113960 1113961

(2)

where 1113954q1 [1113954η1 1113954ξ1] and 1113954q2 [1113954η2 1113954ξ2] λ is a scalar and 1113954qlowast

represents the conjugate of 1113954qUsing dual quaternion the frame Of minus xfyfzf with re-

spect to frame Ol minus xlylzl can be defined as

1113954qfl 1113954qlowastl otimes 1113954qf qfl + ε

12plfl otimes qfl qfl + ε

12qfl otimesp

ffl (3)

where 1113954ql and 1113954qf represent the dual quaternion of frame of thetarget and the chaser respectively qfl is the relative qua-ternion plfl and pffl are the relative position vectors of thechaser relative to the target expressed in frames Ol minus xlylzland Of minus xfyfzf respectively

22 Relative Kinetics and Coupled Dynamics Referring to[12 18 23] the kinematics and coupled dynamics equationof Ol minus xlylzl with respect to Of minus xfyfzf using dual qua-ternion representation can be described as follows

2 _1113954qfl 1113954qfl otimes 1113954ωffl

_1113954ωffl minus 1113954Mminus 1

f 1113954ωffl + 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 1113873 times 1113954Mf 1113954ωffl + 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 11138731113960 1113961

+ 1113954Mminus 1f

1113954Fff minus 1113954qlowastfl otimes _1113954ω

ll otimes 1113954qfl + 1113954ωf

fl times 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 1113873

(4)

where 1113954ωffl ωf

fl + ε( _pffl + ωf

fl times pffl) isin N3 is the relative ve-

locity-expressed frame Of minus xfyfzf ωffl is the relative

angular velocity and _pffl is the relative linear velocity

between two frames As can be seen from the definition1113954ωffl can also be written as 1113954ωf

fl 1113954ωff minus 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl Besidesthe force 1113954F

ff 1113954F

fu + 1113954F

fg + 1113954F

fd is the total dual force 1113954F

fg

ffg + ετfg is the dual gravity 1113954F

fu ff

u + ετfu is the dualcontrol force 1113954F

fd ff

d + ετfd is the dual disturbance and ffg

is the dual gravity force and τfg is the dual gravity torquereferred to the chaser (1113954F

fu ff

u + ετfu and 1113954Ffd ff

d + ετfdrespectively) 1113954Mf mf(ddε)I + εJf is the dual inertiamatrix of chaser in which mf and Jf are the mass andmoment of inertia of chaser

23 Lemmas (e lemmas useful for this study are in-troduced as follows Firstly the following system isconsidered

_x f(x u t)

f(0 t) 0(5)

Chaser

ylxl

pfl

zlrl

XI

YI

OE

ZI

Target

Orbit of target

rf

Orbit of chaser

xfyf

zf

Ol

Of

Pl

Figure 2 Coordinate system definition

Mathematical Problems in Engineering 3

where x isin Rn is the system status and u isin Rm is the controlinput

Lemma 1 (see [34]) Suppose Φ is a positive definite functionand there exists scalar υgt 0 0lt αlt 1 and satisfies

_Φ(x) + υΦα(x)le 0 x isin U0 (6)

then there is an field U0 sub Rn so that the Φ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T1 leΦ1minus α

0υ minus υα

(7)

Lemma 2 (see [35]) Suppose Θ is a positive definite functionand there are scalars ρ1 gt 0 and ρ2 gt 0 0lt βlt 1 and satisfies

_Θ(x)le minus ρ1Θ(x) minus ρ2Θβ(x) x isin U1 (8)

then there is a field U1 sub Rn so that the Θ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T2 le1

ρ1(1 minus β)lnρ1Θ

1minus β0 + ρ2ρ2

(9)

24 Problem Formulation (e control objective of thispaper is to design a controller so that the relative motionstate [1113954qfl(t) 1113954ωf

fl(t)] of the chaser converges to the desiredstate [1113954qd(t) 1113954ωd(t)] in finite time In order to describe therelative motion between the chaser and the capturing pointof tumbling target the relative motion between the referenceframe Pl minus xlylzl and frame Of minus xfyfzf is establishedFirstly the description of targetrsquos attitude is given which isconsidered as the general Euler attitude motion (e kine-matics and dynamic are shown as follows respectively

2 _ql ql ∘ωll

Jl _ωll minus ωl

l times Jlωll + τd

(10)

where ql is the quaternion of the tumbling target ωll is the

angular velocity expressed in Ol minus xlylzl Jl is the moment ofinertia of tumbling target and τd is the total disturbancetorque acting on target (e operation ∘ represents themultiplication of traditional quaternion

(e position vector of reference point Pl with respect tothe centroid Ol on the tumbling target in the frame Of minus

xfyfzf is represented as follows

pffl qlowastfl ∘Pl ∘ qfl (11)

where qfl is the attitude quaternion of the chaser relative tothe target and Pl is the position of the noncentroid point Plwith respect to the centroid Ol at the initial moment

Remark 1 Referring to [36] some information of relativemotion can be measured (e model of 3D reconstruction

which is obtained by the mathematical alignment of theimage plane and binocular stereo matching is established todetermine the relative position and pose of the targetrsquoscapturing point In such a way relative rotational andtranslational motion between two spacecrafts can beobtained

Considering the external disturbances and model un-certainty of the spacecraft the error kinematics and dy-namics equations based on 1113954qe and 1113954ωe can be obtained as

2 _1113954qe 1113954qe otimes 1113954ωe

1113954M0_1113954ωe minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fu + 1113954Fg0 + 1113954Fd + Δ1113954Fg + Δ1113954Σ(12)

where 1113954ωe 1113954ωffl minus 1113954qlowaste otimes 1113954ωd otimes 1113954qe ωe + ε]e and

1113954qe 1113954qminus 1d otimes 1113954qfl qe + εpe are the dual tracking error qe and

pe are the rotation error and translation error respectivelyωe and ]e are the angular velocity error and linear velocityerror respectively 1113954M0 and Δ 1113954M are the nominal and un-certain parts of the dual matrix inertia matrix 1113954Fd is thebounded disturbance 1113954Fg0 and Δ1113954Fg are the dual gravityassociated with 1113954M0 and Δ 1113954M and Δ1113954Σ is the uncertainty of thedual inertia matrix and the value of which isΔ1113954Σ minus 1113954M0

_1113954ωe minus 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times Δ 1113954M 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus Δ 1113954M 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + Δ 1113954M1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

(13)

3 Controller Design

(is section mainly studies the control problem andmodified controller of ESO-NTSMC is given By defining thedual quaternion tracking error as 1113954Ωe Ωe + εpe withΩe 2 ln(qe) the controller is designed tomake the trackingerror ( 1113954Ωe 1113954ωe) converge to (11139540 11139540) in finite time

31 Nonsingular Terminal Sliding Mode Controller DesignIn order to achieve the control objective a fast terminalsliding surface is defined as

s _x + α1x + α2xp (14)

where x isin R and p is a constant which satisfies 05ltplt 1and α1 and α2 are both positive normal numbers (e timederivative of the sliding surface s is

_s eurox + α1 _x + pα2xpminus 1

_x (15)

As can be seen when x 0 _xne 0 and p minus 1lt 0 thesingularity may occur in equation (15) Motivated by[24 37] an alternative form of sliding surface is proposed asfollows to avoid singularity

s _x + α1x + α2β(x) (16)

4 Mathematical Problems in Engineering

where β(x) is

β(x) sgn(x)|x|p if s 0 or sne 0 |x|ge μ

c1x + c2sgn(x)x2 if sne 0 |x|le μ

⎧⎨

⎩ (17)

In the above formula c1 (2 minus p) μpminus 1 and c2 (p minus

1) μpminus 2 are the parameters s _x + α1x + α2sgn(x)|x|p is thenominal sliding surface and μ is a positive small constant(erefore the time derivative of sliding surface (16) is

_s eurox + α1 _x + pα2sgn(x)|x|pminus 1 _x if s 0 or sne 0 |x|ge μ

eurox + α1 _x + α2 c1 _x + 2c2sgn(x)x _x( 1113857 if sne 0 |x|le μ

⎧⎪⎨

⎪⎩

(18)

Remark 2 (e conventional nonsingular sliding mode(NTSM) in Reference [38]was proposed as

s x + ξ _xmn

(19)

where x isin R ξ gt 0 and q and p are the positive odd integerssatisfying 1lt (mn)lt 2 In Reference [39] it has beenproved that the NTSM (19) is still singular when a dualsliding mode surface control method is designed Further-more as explained in Reference [24] the NTSM (19) hasslower convergence rate than the linear sliding mode whenthe sliding mode s 0 is reached (erefore in order toovercome this disadvantage the fast NTSM is proposed inthis paper (e singularity of TSM is avoided by switchingfrom terminal to general sliding manifold which is functionc1x + c2sgn(x)x2 in equation (17)

Equation (18) is continuous and singularity is avoidedAccording to above derivation the paper propose a fastNTSM surface as

1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ χ 1113954Ωe1113872 1113873 (20)

where 1113954β β + εβprime isin R3 is the positive dual constant withβgt 0 and βprime gt 0 so does 1113954c c + εcprimeisin R3 the operation of ⊙is defined in Appendix χ( 1113954Ωe) [χ( 1113954Ωe1) χ( 1113954Ωe2) χ( 1113954Ωe3)]

T

is designed as follows

χ 1113954Ωei1113872 1113873

sigα 1113954Ωei1113872 1113873 if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c11113954Ωei + c2sgn 1113954Ωei1113872 1113873 1113954Ωei

111386811138681113868111386811138681113868111386811138682 if 1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(21)

where sigα( 1113954Ωei) sgn(Ωei)|Ωei|α + εsgn(pei)|pei|

α (12)ltαlt 1 ε0 is a positive value c1 (2 minus α)μαminus 1 and c2 (α minus 1)

μαminus 2 and 1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ sigα( 1113954Ωe)Taking derivation of the sliding surface 1113954s with respect to

time yields

_1113954s _1113954ωe + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873 (22)

where _χ( 1113954Ωei) is

_χ 1113954Ωei1113872 1113873

α 1113954Ωe1113868111386811138681113868

1113868111386811138681113868αminus 1 _1113954Ωe if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c1_1113954Ωei + 2c2sgn 1113954Ωei1113872 1113873 1113954Ωei

11138681113868111386811138681113868111386811138681113868

_1113954Ωei if 1113954sne 0 1113954Ωei1113868111386811138681113868

1113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(23)

(e NTSMC method is

1113954Ffu 1113954Fe + 1113954Fs (24)

where 1113954Fs minus 1113954k⊙ sigp1(1113954s) is the switching term 1113954k isin R3 with1113954ki ki + εki

prime i 1 2 3 satisfying ki gt 0 and kiprime gt 0 p1 gt 0

and 1113954Fe is the equivalent term and the value of which is

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ 1113954M0_1113954Ωe minus 1113954c⊙ 1113954M0 _χ 1113954Ωe1113872 1113873

(25)

32 Extended-State-Observer Design (e ESO methodshows very fine performance in disturbance rejection andcompensation of uncertainties that the accuracy of thecontroller can be improved Based on the SOSM methodmentioned in References [25 32] and by combining thelinear and nonlinear correction terms a novel extended stateobserver is proposed as follows

1113954e1 1113954z1 minus 1113954s

1113954e2 1113954z2 minus 1113954D

_1113954z1 1113954Mminus 10

1113954Fu + 1113954Mminus 10 S 1113954ωe 1113954qe( 1113857 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857 + 1113954z2

_1113954z2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

where 1113954z1 isin N3 and 1113954z2 isin N3 are the outputs of the observer1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total disturbance and also theextended state variable 1113954e1 and 1113954e2 represent the observermeasurement p2 isin [05 1) 1113954κ 1113954η 1113954ρ 1113954λ isin R3 are the observerdesigned parameters respectively and S(1113954ωe 1113954qe) is

S 1113954ωe 1113954qe( 1113857 1113954Mminus 111138741113954Fd + 1113954Fg minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

times 1113954M 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 minus 1113954M 1113954q

lowaste otimes _1113954ωd otimes 1113954qe1113872 1113873

+ 1113954M1113954ωe times 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 11138571113875

(27)

(e proposed ESO (26) can not only obtain the feature ofSOSM such as finite-time convergence but also weaken thechattering of conventional first-order sliding mode also Inaddition it inherits the best properties of linear and non-linear correction terms (en the error dynamics can beestablished from equation (26) in the scalar form

Mathematical Problems in Engineering 5

_1113954e1 1113954e2 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857

_1113954e2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857 minus 1113954gi(t)

⎧⎨

⎩ (28)

where 1113954gi(t) is the derivative of 1113954D and the amplitude of 1113954gi(t)

is assumed to be bounded by a positive value g that is|1113954gi(t)|leg

Based on the NTSMC and ESO designed above theintegral controller is designed as

1113954Ffu 1113954Fe + 1113954Fs + 1113954Fτ (29)

where 1113954Fe 1113954Fs and 1113954Fτ are given by

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M0 1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ _1113954Ωe minus 1113954c⊙ _χ 1113954Ωe1113872 1113873

1113954Fs minus 1113954k⊙ sigp1(1113954s)

1113954Fτ minus 1113954z2

(30)

Under the proposed controller in equations (29) and(30) the control objective can be achieved for trackingtumbling target (e closed-loop system for coupled rota-tional and translational control is shown in Figure 3

4 Stability Analysis

In this section the stability analysis is divided into two partsIn the first part ((eorem 1) the finite-time convergence ofESO is proved which implies that the proposed ESO canestimate the exact state and total disturbances within a fixedtime In the second part ((eorem 2) the finite-time stabilityof the closed-loop system is proved which indicates that thefinite-time convergence of the tracking error ( 1113954Ωe 1113954ωe) can beobtained (e proofs of finite-time convergence of ESO andtotal closed-loop system will be given using Lyapunovstability theory

Theorem 1 Suppose the parameters of ESO in equation (26)satisfy the following

p11113954λ1113954ρgt 1113954κ21113954λ + p2 p2 + 1( 1113857

21113954η21113954κ2 (31)

(en the observer errors will converge to the followingregion in finite-time

Ω1 ϑ | ϑle δ gR

λmin Q1( 11138571113888 1113889

p22p2minus 1⎧⎨

⎩

⎫⎬

⎭ (32)

where ϑ [1113954e1ip2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T R 1113954η 1113954κ minus 21113858 1113859 and thematrix Q1 satisfies the following expression

Q1 η1113954λ + p11113954η2 0 minus p11113954η

0 1113954ρ + 2 + p1( 11138571113954κ2 minus p1 + 1( 11138571113954κminus p11113954η minus p1 + 1( 11138571113954κ p1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (33)

(e convergence time is upperbounded by tm le 2θ1ln((θ1V

(12)0 + θ2)θ2) where V0 is the initial Lyapunov state

and θ1 and θ2 are the constants depending on the gains

Proof To verify the finite-convergence performance inaccordance with the idea of Moreno and Osorio [25] aLyapunov candidate is proposed as

V1 12

1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙1113954e1i minus 1113954e2i1113872 1113873

T

middot 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙ 1113954e1i minus 1113954e2i1113872 1113873 +

1113954λp1⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2

+ 1113954ρ⊙ 1113954eT1i1113954e1i +

121113954e

T2i1113954e2i

ϑTΓϑ

(34)

where ϑ [|1113954e1i|p2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T and the matrix Γ is de-fined as

Γ 12

21113954λp1 + 1113954η2 1113954η1113954κ minus 1113954η

1113954η1113954κ 21113954ρ + 1113954κ2 minus 1113954κ

minus 1113954η minus 1113954κ 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

It is obvious that V1 is positive definite and radiallyunbounded if 1113954λ and 1113954ρ are positive (e range of V1 is ob-tained as follows

λmin(Γ)ϑ2 leV1 le λmax(Γ)ϑ

2 (36)

where ϑ2 |1113954e1i|2p2 + 1113954eT1i1113954e1i + 1113954eT2i1113954e2i is the Euclidean norm of

ϑ and λmin(middot) and λmax(middot) represent the minimum andmaximum eigenvalues of the matrix respectively

(e time derivative of V1 can be obtained as_V1 21113954λ + p11113954η21113872 1113873⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2minus 1 _1113954e1isgn 1113954e1i( 1113857 + 21113954ρ + 1113954κ21113872 1113873⊙ 1113954e

T1i

_1113954e1i

+ 21113954eT2i

_1113954e2i + p1 + 1( 11138571113954η1113954κ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 _1113954e1isgn 1113954e1i( 1113857

minus p11113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 _1113954e

T1i1113954e2i minus 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857_1113954e2i

minus 1113954κ⊙ _1113954eT1i1113954e2i minus 1113954κ⊙1113954e

T1i

_1113954e2i

(37)

qd

qe

q

FuNTSMCequation (30)

ω

[q ω]

Modelequation (12)

[z1 z2] Disturbancesobserver

equation (26)

ndash

Figure 3 Closed-loop system of NTSMC-ESO for spacecraftcoupled control

6 Mathematical Problems in Engineering

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Section 3 NTSMC and ESO strategies are elaborated andthe finite-time stability analysis of the system is performed inSection 4 Following that numerical simulations are illus-trated and discussed in Section 5 to show the effectiveness ofthe proposed method Finally some appropriate conclusionsare presented in Section 6

2 Mathematical Preliminaries

(is study considers the leader-follower spacecraft forma-tion including a chaser and a tumbling target (Figure 2) Inorder to facilitate the description of the model the followingcoordinate system is used OI minus XIYIZI is the Earth-cen-tered inertial coordinate system with the Cartesian right-hand reference frame Ol minus xlylzl is LVLH (local-verticallocal-horizontal) reference frame with the origin at thecenter of mass of target xl is pointing to the spacecraftradially outward zl is the axis normal to the target orbitalplane and yl is the axis established by right-hand rule Of minus

xfyfzf (respectively the tumbling target Ol minus xlylzl) is thebody-fixed coordinate system of the chaser with its origin inthe center of mass of the chaser and axes pointing to the axisof inertial Pl minus xlylzl is the desired tracking reference framewith the capturing point Pl of the tumbling target (econtrol objective is to control the chaser (Of minus xfyfzf )synchronize with desired state (Pl minus xlylzl) so that the on-orbit service can be guaranteed in the next step

21 Dual Quaternion Dual quaternion can be regarded as aquaternion whose element is dual numbers and can bedefined as [14 15 23]

1113954q [1113954η 1113954ξ] q + εqprime (1)

where 1113954η is the dual scalar part 1113954ξ [1113954ξ1 1113954ξ2 1113954ξ3] is the dualvector part q and qprime are the traditional Euler quaternionsrepresenting the real part and dual part of dual quaternionrespectively and ε is the called dual unit with ε2 0 andεne 0 In this paper N is employed to denote the set of dualquaternions and R denotes the set of dual numbers(erefore it is supposed that 1113954q isin N4 1113954ξ isin R3 and 1113954η isin R

Similar to the traditional quaternion and dual vector (seeAppendix) the basic operations for dual quaternions aregiven as follows

1113954q1 + 1113954q2 1113954η1 + 1113954η2 1113954ξ1 + 1113954ξ21113960 1113961

λ1113954q [λ1113954η λ1113954ξ]

1113954qlowast

[1113954η minus 1113954ξ]

1113954q1 otimes 1113954q2 1113954η11113954η2 minus 1113954ξ11113954ξ2 1113954η11113954ξ2 + 1113954η21113954ξ1 + 1113954ξ1 times 1113954ξ21113960 1113961

(2)

where 1113954q1 [1113954η1 1113954ξ1] and 1113954q2 [1113954η2 1113954ξ2] λ is a scalar and 1113954qlowast

represents the conjugate of 1113954qUsing dual quaternion the frame Of minus xfyfzf with re-

spect to frame Ol minus xlylzl can be defined as

1113954qfl 1113954qlowastl otimes 1113954qf qfl + ε

12plfl otimes qfl qfl + ε

12qfl otimesp

ffl (3)

where 1113954ql and 1113954qf represent the dual quaternion of frame of thetarget and the chaser respectively qfl is the relative qua-ternion plfl and pffl are the relative position vectors of thechaser relative to the target expressed in frames Ol minus xlylzland Of minus xfyfzf respectively

22 Relative Kinetics and Coupled Dynamics Referring to[12 18 23] the kinematics and coupled dynamics equationof Ol minus xlylzl with respect to Of minus xfyfzf using dual qua-ternion representation can be described as follows

2 _1113954qfl 1113954qfl otimes 1113954ωffl

_1113954ωffl minus 1113954Mminus 1

f 1113954ωffl + 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 1113873 times 1113954Mf 1113954ωffl + 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 11138731113960 1113961

+ 1113954Mminus 1f

1113954Fff minus 1113954qlowastfl otimes _1113954ω

ll otimes 1113954qfl + 1113954ωf

fl times 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl1113872 1113873

(4)

where 1113954ωffl ωf

fl + ε( _pffl + ωf

fl times pffl) isin N3 is the relative ve-

locity-expressed frame Of minus xfyfzf ωffl is the relative

angular velocity and _pffl is the relative linear velocity

between two frames As can be seen from the definition1113954ωffl can also be written as 1113954ωf

fl 1113954ωff minus 1113954qlowastfl otimes 1113954ωl

l otimes 1113954qfl Besidesthe force 1113954F

ff 1113954F

fu + 1113954F

fg + 1113954F

fd is the total dual force 1113954F

fg

ffg + ετfg is the dual gravity 1113954F

fu ff

u + ετfu is the dualcontrol force 1113954F

fd ff

d + ετfd is the dual disturbance and ffg

is the dual gravity force and τfg is the dual gravity torquereferred to the chaser (1113954F

fu ff

u + ετfu and 1113954Ffd ff

d + ετfdrespectively) 1113954Mf mf(ddε)I + εJf is the dual inertiamatrix of chaser in which mf and Jf are the mass andmoment of inertia of chaser

23 Lemmas (e lemmas useful for this study are in-troduced as follows Firstly the following system isconsidered

_x f(x u t)

f(0 t) 0(5)

Chaser

ylxl

pfl

zlrl

XI

YI

OE

ZI

Target

Orbit of target

rf

Orbit of chaser

xfyf

zf

Ol

Of

Pl

Figure 2 Coordinate system definition

Mathematical Problems in Engineering 3

where x isin Rn is the system status and u isin Rm is the controlinput

Lemma 1 (see [34]) Suppose Φ is a positive definite functionand there exists scalar υgt 0 0lt αlt 1 and satisfies

_Φ(x) + υΦα(x)le 0 x isin U0 (6)

then there is an field U0 sub Rn so that the Φ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T1 leΦ1minus α

0υ minus υα

(7)

Lemma 2 (see [35]) Suppose Θ is a positive definite functionand there are scalars ρ1 gt 0 and ρ2 gt 0 0lt βlt 1 and satisfies

_Θ(x)le minus ρ1Θ(x) minus ρ2Θβ(x) x isin U1 (8)

then there is a field U1 sub Rn so that the Θ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T2 le1

ρ1(1 minus β)lnρ1Θ

1minus β0 + ρ2ρ2

(9)

24 Problem Formulation (e control objective of thispaper is to design a controller so that the relative motionstate [1113954qfl(t) 1113954ωf

fl(t)] of the chaser converges to the desiredstate [1113954qd(t) 1113954ωd(t)] in finite time In order to describe therelative motion between the chaser and the capturing pointof tumbling target the relative motion between the referenceframe Pl minus xlylzl and frame Of minus xfyfzf is establishedFirstly the description of targetrsquos attitude is given which isconsidered as the general Euler attitude motion (e kine-matics and dynamic are shown as follows respectively

2 _ql ql ∘ωll

Jl _ωll minus ωl

l times Jlωll + τd

(10)

where ql is the quaternion of the tumbling target ωll is the

angular velocity expressed in Ol minus xlylzl Jl is the moment ofinertia of tumbling target and τd is the total disturbancetorque acting on target (e operation ∘ represents themultiplication of traditional quaternion

(e position vector of reference point Pl with respect tothe centroid Ol on the tumbling target in the frame Of minus

xfyfzf is represented as follows

pffl qlowastfl ∘Pl ∘ qfl (11)

where qfl is the attitude quaternion of the chaser relative tothe target and Pl is the position of the noncentroid point Plwith respect to the centroid Ol at the initial moment

Remark 1 Referring to [36] some information of relativemotion can be measured (e model of 3D reconstruction

which is obtained by the mathematical alignment of theimage plane and binocular stereo matching is established todetermine the relative position and pose of the targetrsquoscapturing point In such a way relative rotational andtranslational motion between two spacecrafts can beobtained

Considering the external disturbances and model un-certainty of the spacecraft the error kinematics and dy-namics equations based on 1113954qe and 1113954ωe can be obtained as

2 _1113954qe 1113954qe otimes 1113954ωe

1113954M0_1113954ωe minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fu + 1113954Fg0 + 1113954Fd + Δ1113954Fg + Δ1113954Σ(12)

where 1113954ωe 1113954ωffl minus 1113954qlowaste otimes 1113954ωd otimes 1113954qe ωe + ε]e and

1113954qe 1113954qminus 1d otimes 1113954qfl qe + εpe are the dual tracking error qe and

pe are the rotation error and translation error respectivelyωe and ]e are the angular velocity error and linear velocityerror respectively 1113954M0 and Δ 1113954M are the nominal and un-certain parts of the dual matrix inertia matrix 1113954Fd is thebounded disturbance 1113954Fg0 and Δ1113954Fg are the dual gravityassociated with 1113954M0 and Δ 1113954M and Δ1113954Σ is the uncertainty of thedual inertia matrix and the value of which isΔ1113954Σ minus 1113954M0

_1113954ωe minus 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times Δ 1113954M 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus Δ 1113954M 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + Δ 1113954M1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

(13)

3 Controller Design

(is section mainly studies the control problem andmodified controller of ESO-NTSMC is given By defining thedual quaternion tracking error as 1113954Ωe Ωe + εpe withΩe 2 ln(qe) the controller is designed tomake the trackingerror ( 1113954Ωe 1113954ωe) converge to (11139540 11139540) in finite time

31 Nonsingular Terminal Sliding Mode Controller DesignIn order to achieve the control objective a fast terminalsliding surface is defined as

s _x + α1x + α2xp (14)

where x isin R and p is a constant which satisfies 05ltplt 1and α1 and α2 are both positive normal numbers (e timederivative of the sliding surface s is

_s eurox + α1 _x + pα2xpminus 1

_x (15)

As can be seen when x 0 _xne 0 and p minus 1lt 0 thesingularity may occur in equation (15) Motivated by[24 37] an alternative form of sliding surface is proposed asfollows to avoid singularity

s _x + α1x + α2β(x) (16)

4 Mathematical Problems in Engineering

where β(x) is

β(x) sgn(x)|x|p if s 0 or sne 0 |x|ge μ

c1x + c2sgn(x)x2 if sne 0 |x|le μ

⎧⎨

⎩ (17)

In the above formula c1 (2 minus p) μpminus 1 and c2 (p minus

1) μpminus 2 are the parameters s _x + α1x + α2sgn(x)|x|p is thenominal sliding surface and μ is a positive small constant(erefore the time derivative of sliding surface (16) is

_s eurox + α1 _x + pα2sgn(x)|x|pminus 1 _x if s 0 or sne 0 |x|ge μ

eurox + α1 _x + α2 c1 _x + 2c2sgn(x)x _x( 1113857 if sne 0 |x|le μ

⎧⎪⎨

⎪⎩

(18)

Remark 2 (e conventional nonsingular sliding mode(NTSM) in Reference [38]was proposed as

s x + ξ _xmn

(19)

where x isin R ξ gt 0 and q and p are the positive odd integerssatisfying 1lt (mn)lt 2 In Reference [39] it has beenproved that the NTSM (19) is still singular when a dualsliding mode surface control method is designed Further-more as explained in Reference [24] the NTSM (19) hasslower convergence rate than the linear sliding mode whenthe sliding mode s 0 is reached (erefore in order toovercome this disadvantage the fast NTSM is proposed inthis paper (e singularity of TSM is avoided by switchingfrom terminal to general sliding manifold which is functionc1x + c2sgn(x)x2 in equation (17)

Equation (18) is continuous and singularity is avoidedAccording to above derivation the paper propose a fastNTSM surface as

1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ χ 1113954Ωe1113872 1113873 (20)

where 1113954β β + εβprime isin R3 is the positive dual constant withβgt 0 and βprime gt 0 so does 1113954c c + εcprimeisin R3 the operation of ⊙is defined in Appendix χ( 1113954Ωe) [χ( 1113954Ωe1) χ( 1113954Ωe2) χ( 1113954Ωe3)]

T

is designed as follows

χ 1113954Ωei1113872 1113873

sigα 1113954Ωei1113872 1113873 if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c11113954Ωei + c2sgn 1113954Ωei1113872 1113873 1113954Ωei

111386811138681113868111386811138681113868111386811138682 if 1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(21)

where sigα( 1113954Ωei) sgn(Ωei)|Ωei|α + εsgn(pei)|pei|

α (12)ltαlt 1 ε0 is a positive value c1 (2 minus α)μαminus 1 and c2 (α minus 1)

μαminus 2 and 1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ sigα( 1113954Ωe)Taking derivation of the sliding surface 1113954s with respect to

time yields

_1113954s _1113954ωe + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873 (22)

where _χ( 1113954Ωei) is

_χ 1113954Ωei1113872 1113873

α 1113954Ωe1113868111386811138681113868

1113868111386811138681113868αminus 1 _1113954Ωe if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c1_1113954Ωei + 2c2sgn 1113954Ωei1113872 1113873 1113954Ωei

11138681113868111386811138681113868111386811138681113868

_1113954Ωei if 1113954sne 0 1113954Ωei1113868111386811138681113868

1113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(23)

(e NTSMC method is

1113954Ffu 1113954Fe + 1113954Fs (24)

where 1113954Fs minus 1113954k⊙ sigp1(1113954s) is the switching term 1113954k isin R3 with1113954ki ki + εki

prime i 1 2 3 satisfying ki gt 0 and kiprime gt 0 p1 gt 0

and 1113954Fe is the equivalent term and the value of which is

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ 1113954M0_1113954Ωe minus 1113954c⊙ 1113954M0 _χ 1113954Ωe1113872 1113873

(25)

32 Extended-State-Observer Design (e ESO methodshows very fine performance in disturbance rejection andcompensation of uncertainties that the accuracy of thecontroller can be improved Based on the SOSM methodmentioned in References [25 32] and by combining thelinear and nonlinear correction terms a novel extended stateobserver is proposed as follows

1113954e1 1113954z1 minus 1113954s

1113954e2 1113954z2 minus 1113954D

_1113954z1 1113954Mminus 10

1113954Fu + 1113954Mminus 10 S 1113954ωe 1113954qe( 1113857 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857 + 1113954z2

_1113954z2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

where 1113954z1 isin N3 and 1113954z2 isin N3 are the outputs of the observer1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total disturbance and also theextended state variable 1113954e1 and 1113954e2 represent the observermeasurement p2 isin [05 1) 1113954κ 1113954η 1113954ρ 1113954λ isin R3 are the observerdesigned parameters respectively and S(1113954ωe 1113954qe) is

S 1113954ωe 1113954qe( 1113857 1113954Mminus 111138741113954Fd + 1113954Fg minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

times 1113954M 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 minus 1113954M 1113954q

lowaste otimes _1113954ωd otimes 1113954qe1113872 1113873

+ 1113954M1113954ωe times 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 11138571113875

(27)

(e proposed ESO (26) can not only obtain the feature ofSOSM such as finite-time convergence but also weaken thechattering of conventional first-order sliding mode also Inaddition it inherits the best properties of linear and non-linear correction terms (en the error dynamics can beestablished from equation (26) in the scalar form

Mathematical Problems in Engineering 5

_1113954e1 1113954e2 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857

_1113954e2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857 minus 1113954gi(t)

⎧⎨

⎩ (28)

where 1113954gi(t) is the derivative of 1113954D and the amplitude of 1113954gi(t)

is assumed to be bounded by a positive value g that is|1113954gi(t)|leg

Based on the NTSMC and ESO designed above theintegral controller is designed as

1113954Ffu 1113954Fe + 1113954Fs + 1113954Fτ (29)

where 1113954Fe 1113954Fs and 1113954Fτ are given by

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M0 1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ _1113954Ωe minus 1113954c⊙ _χ 1113954Ωe1113872 1113873

1113954Fs minus 1113954k⊙ sigp1(1113954s)

1113954Fτ minus 1113954z2

(30)

Under the proposed controller in equations (29) and(30) the control objective can be achieved for trackingtumbling target (e closed-loop system for coupled rota-tional and translational control is shown in Figure 3

4 Stability Analysis

In this section the stability analysis is divided into two partsIn the first part ((eorem 1) the finite-time convergence ofESO is proved which implies that the proposed ESO canestimate the exact state and total disturbances within a fixedtime In the second part ((eorem 2) the finite-time stabilityof the closed-loop system is proved which indicates that thefinite-time convergence of the tracking error ( 1113954Ωe 1113954ωe) can beobtained (e proofs of finite-time convergence of ESO andtotal closed-loop system will be given using Lyapunovstability theory

Theorem 1 Suppose the parameters of ESO in equation (26)satisfy the following

p11113954λ1113954ρgt 1113954κ21113954λ + p2 p2 + 1( 1113857

21113954η21113954κ2 (31)

(en the observer errors will converge to the followingregion in finite-time

Ω1 ϑ | ϑle δ gR

λmin Q1( 11138571113888 1113889

p22p2minus 1⎧⎨

⎩

⎫⎬

⎭ (32)

where ϑ [1113954e1ip2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T R 1113954η 1113954κ minus 21113858 1113859 and thematrix Q1 satisfies the following expression

Q1 η1113954λ + p11113954η2 0 minus p11113954η

0 1113954ρ + 2 + p1( 11138571113954κ2 minus p1 + 1( 11138571113954κminus p11113954η minus p1 + 1( 11138571113954κ p1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (33)

(e convergence time is upperbounded by tm le 2θ1ln((θ1V

(12)0 + θ2)θ2) where V0 is the initial Lyapunov state

and θ1 and θ2 are the constants depending on the gains

Proof To verify the finite-convergence performance inaccordance with the idea of Moreno and Osorio [25] aLyapunov candidate is proposed as

V1 12

1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙1113954e1i minus 1113954e2i1113872 1113873

T

middot 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙ 1113954e1i minus 1113954e2i1113872 1113873 +

1113954λp1⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2

+ 1113954ρ⊙ 1113954eT1i1113954e1i +

121113954e

T2i1113954e2i

ϑTΓϑ

(34)

where ϑ [|1113954e1i|p2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T and the matrix Γ is de-fined as

Γ 12

21113954λp1 + 1113954η2 1113954η1113954κ minus 1113954η

1113954η1113954κ 21113954ρ + 1113954κ2 minus 1113954κ

minus 1113954η minus 1113954κ 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

It is obvious that V1 is positive definite and radiallyunbounded if 1113954λ and 1113954ρ are positive (e range of V1 is ob-tained as follows

λmin(Γ)ϑ2 leV1 le λmax(Γ)ϑ

2 (36)

where ϑ2 |1113954e1i|2p2 + 1113954eT1i1113954e1i + 1113954eT2i1113954e2i is the Euclidean norm of

ϑ and λmin(middot) and λmax(middot) represent the minimum andmaximum eigenvalues of the matrix respectively

(e time derivative of V1 can be obtained as_V1 21113954λ + p11113954η21113872 1113873⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2minus 1 _1113954e1isgn 1113954e1i( 1113857 + 21113954ρ + 1113954κ21113872 1113873⊙ 1113954e

T1i

_1113954e1i

+ 21113954eT2i

_1113954e2i + p1 + 1( 11138571113954η1113954κ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 _1113954e1isgn 1113954e1i( 1113857

minus p11113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 _1113954e

T1i1113954e2i minus 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857_1113954e2i

minus 1113954κ⊙ _1113954eT1i1113954e2i minus 1113954κ⊙1113954e

T1i

_1113954e2i

(37)

qd

qe

q

FuNTSMCequation (30)

ω

[q ω]

Modelequation (12)

[z1 z2] Disturbancesobserver

equation (26)

ndash

Figure 3 Closed-loop system of NTSMC-ESO for spacecraftcoupled control

6 Mathematical Problems in Engineering

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

where x isin Rn is the system status and u isin Rm is the controlinput

Lemma 1 (see [34]) Suppose Φ is a positive definite functionand there exists scalar υgt 0 0lt αlt 1 and satisfies

_Φ(x) + υΦα(x)le 0 x isin U0 (6)

then there is an field U0 sub Rn so that the Φ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T1 leΦ1minus α

0υ minus υα

(7)

Lemma 2 (see [35]) Suppose Θ is a positive definite functionand there are scalars ρ1 gt 0 and ρ2 gt 0 0lt βlt 1 and satisfies

_Θ(x)le minus ρ1Θ(x) minus ρ2Θβ(x) x isin U1 (8)

then there is a field U1 sub Rn so that the Θ(x) can reachequilibrium point in finite time And the convergence time issatisfied as

T2 le1

ρ1(1 minus β)lnρ1Θ

1minus β0 + ρ2ρ2

(9)

24 Problem Formulation (e control objective of thispaper is to design a controller so that the relative motionstate [1113954qfl(t) 1113954ωf

fl(t)] of the chaser converges to the desiredstate [1113954qd(t) 1113954ωd(t)] in finite time In order to describe therelative motion between the chaser and the capturing pointof tumbling target the relative motion between the referenceframe Pl minus xlylzl and frame Of minus xfyfzf is establishedFirstly the description of targetrsquos attitude is given which isconsidered as the general Euler attitude motion (e kine-matics and dynamic are shown as follows respectively

2 _ql ql ∘ωll

Jl _ωll minus ωl

l times Jlωll + τd

(10)

where ql is the quaternion of the tumbling target ωll is the

angular velocity expressed in Ol minus xlylzl Jl is the moment ofinertia of tumbling target and τd is the total disturbancetorque acting on target (e operation ∘ represents themultiplication of traditional quaternion

(e position vector of reference point Pl with respect tothe centroid Ol on the tumbling target in the frame Of minus

xfyfzf is represented as follows

pffl qlowastfl ∘Pl ∘ qfl (11)

where qfl is the attitude quaternion of the chaser relative tothe target and Pl is the position of the noncentroid point Plwith respect to the centroid Ol at the initial moment

Remark 1 Referring to [36] some information of relativemotion can be measured (e model of 3D reconstruction

which is obtained by the mathematical alignment of theimage plane and binocular stereo matching is established todetermine the relative position and pose of the targetrsquoscapturing point In such a way relative rotational andtranslational motion between two spacecrafts can beobtained

Considering the external disturbances and model un-certainty of the spacecraft the error kinematics and dy-namics equations based on 1113954qe and 1113954ωe can be obtained as

2 _1113954qe 1113954qe otimes 1113954ωe

1113954M0_1113954ωe minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fu + 1113954Fg0 + 1113954Fd + Δ1113954Fg + Δ1113954Σ(12)

where 1113954ωe 1113954ωffl minus 1113954qlowaste otimes 1113954ωd otimes 1113954qe ωe + ε]e and

1113954qe 1113954qminus 1d otimes 1113954qfl qe + εpe are the dual tracking error qe and

pe are the rotation error and translation error respectivelyωe and ]e are the angular velocity error and linear velocityerror respectively 1113954M0 and Δ 1113954M are the nominal and un-certain parts of the dual matrix inertia matrix 1113954Fd is thebounded disturbance 1113954Fg0 and Δ1113954Fg are the dual gravityassociated with 1113954M0 and Δ 1113954M and Δ1113954Σ is the uncertainty of thedual inertia matrix and the value of which isΔ1113954Σ minus 1113954M0

_1113954ωe minus 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times Δ 1113954M 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus Δ 1113954M 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 + Δ 1113954M1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

(13)

3 Controller Design

(is section mainly studies the control problem andmodified controller of ESO-NTSMC is given By defining thedual quaternion tracking error as 1113954Ωe Ωe + εpe withΩe 2 ln(qe) the controller is designed tomake the trackingerror ( 1113954Ωe 1113954ωe) converge to (11139540 11139540) in finite time

31 Nonsingular Terminal Sliding Mode Controller DesignIn order to achieve the control objective a fast terminalsliding surface is defined as

s _x + α1x + α2xp (14)

where x isin R and p is a constant which satisfies 05ltplt 1and α1 and α2 are both positive normal numbers (e timederivative of the sliding surface s is

_s eurox + α1 _x + pα2xpminus 1

_x (15)

As can be seen when x 0 _xne 0 and p minus 1lt 0 thesingularity may occur in equation (15) Motivated by[24 37] an alternative form of sliding surface is proposed asfollows to avoid singularity

s _x + α1x + α2β(x) (16)

4 Mathematical Problems in Engineering

where β(x) is

β(x) sgn(x)|x|p if s 0 or sne 0 |x|ge μ

c1x + c2sgn(x)x2 if sne 0 |x|le μ

⎧⎨

⎩ (17)

In the above formula c1 (2 minus p) μpminus 1 and c2 (p minus

1) μpminus 2 are the parameters s _x + α1x + α2sgn(x)|x|p is thenominal sliding surface and μ is a positive small constant(erefore the time derivative of sliding surface (16) is

_s eurox + α1 _x + pα2sgn(x)|x|pminus 1 _x if s 0 or sne 0 |x|ge μ

eurox + α1 _x + α2 c1 _x + 2c2sgn(x)x _x( 1113857 if sne 0 |x|le μ

⎧⎪⎨

⎪⎩

(18)

Remark 2 (e conventional nonsingular sliding mode(NTSM) in Reference [38]was proposed as

s x + ξ _xmn

(19)

where x isin R ξ gt 0 and q and p are the positive odd integerssatisfying 1lt (mn)lt 2 In Reference [39] it has beenproved that the NTSM (19) is still singular when a dualsliding mode surface control method is designed Further-more as explained in Reference [24] the NTSM (19) hasslower convergence rate than the linear sliding mode whenthe sliding mode s 0 is reached (erefore in order toovercome this disadvantage the fast NTSM is proposed inthis paper (e singularity of TSM is avoided by switchingfrom terminal to general sliding manifold which is functionc1x + c2sgn(x)x2 in equation (17)

Equation (18) is continuous and singularity is avoidedAccording to above derivation the paper propose a fastNTSM surface as

1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ χ 1113954Ωe1113872 1113873 (20)

where 1113954β β + εβprime isin R3 is the positive dual constant withβgt 0 and βprime gt 0 so does 1113954c c + εcprimeisin R3 the operation of ⊙is defined in Appendix χ( 1113954Ωe) [χ( 1113954Ωe1) χ( 1113954Ωe2) χ( 1113954Ωe3)]

T

is designed as follows

χ 1113954Ωei1113872 1113873

sigα 1113954Ωei1113872 1113873 if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c11113954Ωei + c2sgn 1113954Ωei1113872 1113873 1113954Ωei

111386811138681113868111386811138681113868111386811138682 if 1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(21)

where sigα( 1113954Ωei) sgn(Ωei)|Ωei|α + εsgn(pei)|pei|

α (12)ltαlt 1 ε0 is a positive value c1 (2 minus α)μαminus 1 and c2 (α minus 1)

μαminus 2 and 1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ sigα( 1113954Ωe)Taking derivation of the sliding surface 1113954s with respect to

time yields

_1113954s _1113954ωe + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873 (22)

where _χ( 1113954Ωei) is

_χ 1113954Ωei1113872 1113873

α 1113954Ωe1113868111386811138681113868

1113868111386811138681113868αminus 1 _1113954Ωe if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c1_1113954Ωei + 2c2sgn 1113954Ωei1113872 1113873 1113954Ωei

11138681113868111386811138681113868111386811138681113868

_1113954Ωei if 1113954sne 0 1113954Ωei1113868111386811138681113868

1113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(23)

(e NTSMC method is

1113954Ffu 1113954Fe + 1113954Fs (24)

where 1113954Fs minus 1113954k⊙ sigp1(1113954s) is the switching term 1113954k isin R3 with1113954ki ki + εki

prime i 1 2 3 satisfying ki gt 0 and kiprime gt 0 p1 gt 0

and 1113954Fe is the equivalent term and the value of which is

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ 1113954M0_1113954Ωe minus 1113954c⊙ 1113954M0 _χ 1113954Ωe1113872 1113873

(25)

32 Extended-State-Observer Design (e ESO methodshows very fine performance in disturbance rejection andcompensation of uncertainties that the accuracy of thecontroller can be improved Based on the SOSM methodmentioned in References [25 32] and by combining thelinear and nonlinear correction terms a novel extended stateobserver is proposed as follows

1113954e1 1113954z1 minus 1113954s

1113954e2 1113954z2 minus 1113954D

_1113954z1 1113954Mminus 10

1113954Fu + 1113954Mminus 10 S 1113954ωe 1113954qe( 1113857 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857 + 1113954z2

_1113954z2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

where 1113954z1 isin N3 and 1113954z2 isin N3 are the outputs of the observer1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total disturbance and also theextended state variable 1113954e1 and 1113954e2 represent the observermeasurement p2 isin [05 1) 1113954κ 1113954η 1113954ρ 1113954λ isin R3 are the observerdesigned parameters respectively and S(1113954ωe 1113954qe) is

S 1113954ωe 1113954qe( 1113857 1113954Mminus 111138741113954Fd + 1113954Fg minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

times 1113954M 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 minus 1113954M 1113954q

lowaste otimes _1113954ωd otimes 1113954qe1113872 1113873

+ 1113954M1113954ωe times 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 11138571113875

(27)

(e proposed ESO (26) can not only obtain the feature ofSOSM such as finite-time convergence but also weaken thechattering of conventional first-order sliding mode also Inaddition it inherits the best properties of linear and non-linear correction terms (en the error dynamics can beestablished from equation (26) in the scalar form

Mathematical Problems in Engineering 5

_1113954e1 1113954e2 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857

_1113954e2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857 minus 1113954gi(t)

⎧⎨

⎩ (28)

where 1113954gi(t) is the derivative of 1113954D and the amplitude of 1113954gi(t)

is assumed to be bounded by a positive value g that is|1113954gi(t)|leg

Based on the NTSMC and ESO designed above theintegral controller is designed as

1113954Ffu 1113954Fe + 1113954Fs + 1113954Fτ (29)

where 1113954Fe 1113954Fs and 1113954Fτ are given by

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M0 1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ _1113954Ωe minus 1113954c⊙ _χ 1113954Ωe1113872 1113873

1113954Fs minus 1113954k⊙ sigp1(1113954s)

1113954Fτ minus 1113954z2

(30)

Under the proposed controller in equations (29) and(30) the control objective can be achieved for trackingtumbling target (e closed-loop system for coupled rota-tional and translational control is shown in Figure 3

4 Stability Analysis

In this section the stability analysis is divided into two partsIn the first part ((eorem 1) the finite-time convergence ofESO is proved which implies that the proposed ESO canestimate the exact state and total disturbances within a fixedtime In the second part ((eorem 2) the finite-time stabilityof the closed-loop system is proved which indicates that thefinite-time convergence of the tracking error ( 1113954Ωe 1113954ωe) can beobtained (e proofs of finite-time convergence of ESO andtotal closed-loop system will be given using Lyapunovstability theory

Theorem 1 Suppose the parameters of ESO in equation (26)satisfy the following

p11113954λ1113954ρgt 1113954κ21113954λ + p2 p2 + 1( 1113857

21113954η21113954κ2 (31)

(en the observer errors will converge to the followingregion in finite-time

Ω1 ϑ | ϑle δ gR

λmin Q1( 11138571113888 1113889

p22p2minus 1⎧⎨

⎩

⎫⎬

⎭ (32)

where ϑ [1113954e1ip2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T R 1113954η 1113954κ minus 21113858 1113859 and thematrix Q1 satisfies the following expression

Q1 η1113954λ + p11113954η2 0 minus p11113954η

0 1113954ρ + 2 + p1( 11138571113954κ2 minus p1 + 1( 11138571113954κminus p11113954η minus p1 + 1( 11138571113954κ p1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (33)

(e convergence time is upperbounded by tm le 2θ1ln((θ1V

(12)0 + θ2)θ2) where V0 is the initial Lyapunov state

and θ1 and θ2 are the constants depending on the gains

Proof To verify the finite-convergence performance inaccordance with the idea of Moreno and Osorio [25] aLyapunov candidate is proposed as

V1 12

1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙1113954e1i minus 1113954e2i1113872 1113873

T

middot 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙ 1113954e1i minus 1113954e2i1113872 1113873 +

1113954λp1⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2

+ 1113954ρ⊙ 1113954eT1i1113954e1i +

121113954e

T2i1113954e2i

ϑTΓϑ

(34)

where ϑ [|1113954e1i|p2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T and the matrix Γ is de-fined as

Γ 12

21113954λp1 + 1113954η2 1113954η1113954κ minus 1113954η

1113954η1113954κ 21113954ρ + 1113954κ2 minus 1113954κ

minus 1113954η minus 1113954κ 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

It is obvious that V1 is positive definite and radiallyunbounded if 1113954λ and 1113954ρ are positive (e range of V1 is ob-tained as follows

λmin(Γ)ϑ2 leV1 le λmax(Γ)ϑ

2 (36)

where ϑ2 |1113954e1i|2p2 + 1113954eT1i1113954e1i + 1113954eT2i1113954e2i is the Euclidean norm of

ϑ and λmin(middot) and λmax(middot) represent the minimum andmaximum eigenvalues of the matrix respectively

(e time derivative of V1 can be obtained as_V1 21113954λ + p11113954η21113872 1113873⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2minus 1 _1113954e1isgn 1113954e1i( 1113857 + 21113954ρ + 1113954κ21113872 1113873⊙ 1113954e

T1i

_1113954e1i

+ 21113954eT2i

_1113954e2i + p1 + 1( 11138571113954η1113954κ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 _1113954e1isgn 1113954e1i( 1113857

minus p11113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 _1113954e

T1i1113954e2i minus 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857_1113954e2i

minus 1113954κ⊙ _1113954eT1i1113954e2i minus 1113954κ⊙1113954e

T1i

_1113954e2i

(37)

qd

qe

q

FuNTSMCequation (30)

ω

[q ω]

Modelequation (12)

[z1 z2] Disturbancesobserver

equation (26)

ndash

Figure 3 Closed-loop system of NTSMC-ESO for spacecraftcoupled control

6 Mathematical Problems in Engineering

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

where β(x) is

β(x) sgn(x)|x|p if s 0 or sne 0 |x|ge μ

c1x + c2sgn(x)x2 if sne 0 |x|le μ

⎧⎨

⎩ (17)

In the above formula c1 (2 minus p) μpminus 1 and c2 (p minus

1) μpminus 2 are the parameters s _x + α1x + α2sgn(x)|x|p is thenominal sliding surface and μ is a positive small constant(erefore the time derivative of sliding surface (16) is

_s eurox + α1 _x + pα2sgn(x)|x|pminus 1 _x if s 0 or sne 0 |x|ge μ

eurox + α1 _x + α2 c1 _x + 2c2sgn(x)x _x( 1113857 if sne 0 |x|le μ

⎧⎪⎨

⎪⎩

(18)

Remark 2 (e conventional nonsingular sliding mode(NTSM) in Reference [38]was proposed as

s x + ξ _xmn

(19)

where x isin R ξ gt 0 and q and p are the positive odd integerssatisfying 1lt (mn)lt 2 In Reference [39] it has beenproved that the NTSM (19) is still singular when a dualsliding mode surface control method is designed Further-more as explained in Reference [24] the NTSM (19) hasslower convergence rate than the linear sliding mode whenthe sliding mode s 0 is reached (erefore in order toovercome this disadvantage the fast NTSM is proposed inthis paper (e singularity of TSM is avoided by switchingfrom terminal to general sliding manifold which is functionc1x + c2sgn(x)x2 in equation (17)

Equation (18) is continuous and singularity is avoidedAccording to above derivation the paper propose a fastNTSM surface as

1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ χ 1113954Ωe1113872 1113873 (20)

where 1113954β β + εβprime isin R3 is the positive dual constant withβgt 0 and βprime gt 0 so does 1113954c c + εcprimeisin R3 the operation of ⊙is defined in Appendix χ( 1113954Ωe) [χ( 1113954Ωe1) χ( 1113954Ωe2) χ( 1113954Ωe3)]

T

is designed as follows

χ 1113954Ωei1113872 1113873

sigα 1113954Ωei1113872 1113873 if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c11113954Ωei + c2sgn 1113954Ωei1113872 1113873 1113954Ωei

111386811138681113868111386811138681113868111386811138682 if 1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(21)

where sigα( 1113954Ωei) sgn(Ωei)|Ωei|α + εsgn(pei)|pei|

α (12)ltαlt 1 ε0 is a positive value c1 (2 minus α)μαminus 1 and c2 (α minus 1)

μαminus 2 and 1113954s 1113954ωe + 1113954β⊙ 1113954Ωe + 1113954c⊙ sigα( 1113954Ωe)Taking derivation of the sliding surface 1113954s with respect to

time yields

_1113954s _1113954ωe + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873 (22)

where _χ( 1113954Ωei) is

_χ 1113954Ωei1113872 1113873

α 1113954Ωe1113868111386811138681113868

1113868111386811138681113868αminus 1 _1113954Ωe if 1113954s 0 or1113954sne 0 1113954Ωei

11138681113868111386811138681113868111386811138681113868ge ε0

c1_1113954Ωei + 2c2sgn 1113954Ωei1113872 1113873 1113954Ωei

11138681113868111386811138681113868111386811138681113868

_1113954Ωei if 1113954sne 0 1113954Ωei1113868111386811138681113868

1113868111386811138681113868le ε0

i 1 2 3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(23)

(e NTSMC method is

1113954Ffu 1113954Fe + 1113954Fs (24)

where 1113954Fs minus 1113954k⊙ sigp1(1113954s) is the switching term 1113954k isin R3 with1113954ki ki + εki

prime i 1 2 3 satisfying ki gt 0 and kiprime gt 0 p1 gt 0

and 1113954Fe is the equivalent term and the value of which is

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

minus 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M01113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ 1113954M0_1113954Ωe minus 1113954c⊙ 1113954M0 _χ 1113954Ωe1113872 1113873

(25)

32 Extended-State-Observer Design (e ESO methodshows very fine performance in disturbance rejection andcompensation of uncertainties that the accuracy of thecontroller can be improved Based on the SOSM methodmentioned in References [25 32] and by combining thelinear and nonlinear correction terms a novel extended stateobserver is proposed as follows

1113954e1 1113954z1 minus 1113954s

1113954e2 1113954z2 minus 1113954D

_1113954z1 1113954Mminus 10

1113954Fu + 1113954Mminus 10 S 1113954ωe 1113954qe( 1113857 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857 + 1113954z2

_1113954z2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

where 1113954z1 isin N3 and 1113954z2 isin N3 are the outputs of the observer1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total disturbance and also theextended state variable 1113954e1 and 1113954e2 represent the observermeasurement p2 isin [05 1) 1113954κ 1113954η 1113954ρ 1113954λ isin R3 are the observerdesigned parameters respectively and S(1113954ωe 1113954qe) is

S 1113954ωe 1113954qe( 1113857 1113954Mminus 111138741113954Fd + 1113954Fg minus 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

times 1113954M 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 minus 1113954M 1113954q

lowaste otimes _1113954ωd otimes 1113954qe1113872 1113873

+ 1113954M1113954ωe times 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 11138571113875

(27)

(e proposed ESO (26) can not only obtain the feature ofSOSM such as finite-time convergence but also weaken thechattering of conventional first-order sliding mode also Inaddition it inherits the best properties of linear and non-linear correction terms (en the error dynamics can beestablished from equation (26) in the scalar form

Mathematical Problems in Engineering 5

_1113954e1 1113954e2 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857

_1113954e2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857 minus 1113954gi(t)

⎧⎨

⎩ (28)

where 1113954gi(t) is the derivative of 1113954D and the amplitude of 1113954gi(t)

is assumed to be bounded by a positive value g that is|1113954gi(t)|leg

Based on the NTSMC and ESO designed above theintegral controller is designed as

1113954Ffu 1113954Fe + 1113954Fs + 1113954Fτ (29)

where 1113954Fe 1113954Fs and 1113954Fτ are given by

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M0 1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ _1113954Ωe minus 1113954c⊙ _χ 1113954Ωe1113872 1113873

1113954Fs minus 1113954k⊙ sigp1(1113954s)

1113954Fτ minus 1113954z2

(30)

Under the proposed controller in equations (29) and(30) the control objective can be achieved for trackingtumbling target (e closed-loop system for coupled rota-tional and translational control is shown in Figure 3

4 Stability Analysis

In this section the stability analysis is divided into two partsIn the first part ((eorem 1) the finite-time convergence ofESO is proved which implies that the proposed ESO canestimate the exact state and total disturbances within a fixedtime In the second part ((eorem 2) the finite-time stabilityof the closed-loop system is proved which indicates that thefinite-time convergence of the tracking error ( 1113954Ωe 1113954ωe) can beobtained (e proofs of finite-time convergence of ESO andtotal closed-loop system will be given using Lyapunovstability theory

Theorem 1 Suppose the parameters of ESO in equation (26)satisfy the following

p11113954λ1113954ρgt 1113954κ21113954λ + p2 p2 + 1( 1113857

21113954η21113954κ2 (31)

(en the observer errors will converge to the followingregion in finite-time

Ω1 ϑ | ϑle δ gR

λmin Q1( 11138571113888 1113889

p22p2minus 1⎧⎨

⎩

⎫⎬

⎭ (32)

where ϑ [1113954e1ip2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T R 1113954η 1113954κ minus 21113858 1113859 and thematrix Q1 satisfies the following expression

Q1 η1113954λ + p11113954η2 0 minus p11113954η

0 1113954ρ + 2 + p1( 11138571113954κ2 minus p1 + 1( 11138571113954κminus p11113954η minus p1 + 1( 11138571113954κ p1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (33)

(e convergence time is upperbounded by tm le 2θ1ln((θ1V

(12)0 + θ2)θ2) where V0 is the initial Lyapunov state

and θ1 and θ2 are the constants depending on the gains

Proof To verify the finite-convergence performance inaccordance with the idea of Moreno and Osorio [25] aLyapunov candidate is proposed as

V1 12

1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙1113954e1i minus 1113954e2i1113872 1113873

T

middot 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙ 1113954e1i minus 1113954e2i1113872 1113873 +

1113954λp1⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2

+ 1113954ρ⊙ 1113954eT1i1113954e1i +

121113954e

T2i1113954e2i

ϑTΓϑ

(34)

where ϑ [|1113954e1i|p2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T and the matrix Γ is de-fined as

Γ 12

21113954λp1 + 1113954η2 1113954η1113954κ minus 1113954η

1113954η1113954κ 21113954ρ + 1113954κ2 minus 1113954κ

minus 1113954η minus 1113954κ 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

It is obvious that V1 is positive definite and radiallyunbounded if 1113954λ and 1113954ρ are positive (e range of V1 is ob-tained as follows

λmin(Γ)ϑ2 leV1 le λmax(Γ)ϑ

2 (36)

where ϑ2 |1113954e1i|2p2 + 1113954eT1i1113954e1i + 1113954eT2i1113954e2i is the Euclidean norm of

ϑ and λmin(middot) and λmax(middot) represent the minimum andmaximum eigenvalues of the matrix respectively

(e time derivative of V1 can be obtained as_V1 21113954λ + p11113954η21113872 1113873⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2minus 1 _1113954e1isgn 1113954e1i( 1113857 + 21113954ρ + 1113954κ21113872 1113873⊙ 1113954e

T1i

_1113954e1i

+ 21113954eT2i

_1113954e2i + p1 + 1( 11138571113954η1113954κ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 _1113954e1isgn 1113954e1i( 1113857

minus p11113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 _1113954e

T1i1113954e2i minus 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857_1113954e2i

minus 1113954κ⊙ _1113954eT1i1113954e2i minus 1113954κ⊙1113954e

T1i

_1113954e2i

(37)

qd

qe

q

FuNTSMCequation (30)

ω

[q ω]

Modelequation (12)

[z1 z2] Disturbancesobserver

equation (26)

ndash

Figure 3 Closed-loop system of NTSMC-ESO for spacecraftcoupled control

6 Mathematical Problems in Engineering

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

_1113954e1 1113954e2 minus 1113954κ⊙ 1113954e1 minus 1113954η⊙ sgnp2 1113954e1( 1113857

_1113954e2 minus 1113954ρ⊙ 1113954e1 minus 1113954λ⊙ sgn2p2minus 1 1113954e1( 1113857 minus 1113954gi(t)

⎧⎨

⎩ (28)

where 1113954gi(t) is the derivative of 1113954D and the amplitude of 1113954gi(t)

is assumed to be bounded by a positive value g that is|1113954gi(t)|leg

Based on the NTSMC and ESO designed above theintegral controller is designed as

1113954Ffu 1113954Fe + 1113954Fs + 1113954Fτ (29)

where 1113954Fe 1113954Fs and 1113954Fτ are given by

1113954Fe 1113954ωe + 1113954qlowaste otimes 1113954ωd otimes 1113954qe( 1113857 times 1113954M0 1113954ωe + 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954M0 1113954qlowaste otimes _1113954ωd otimes 1113954qe1113872 1113873 minus 1113954M0 1113954ωe times 1113954q

lowaste otimes 1113954ωd otimes 1113954qe( 1113857

+ 1113954Fg0 minus 1113954β⊙ _1113954Ωe minus 1113954c⊙ _χ 1113954Ωe1113872 1113873

1113954Fs minus 1113954k⊙ sigp1(1113954s)

1113954Fτ minus 1113954z2

(30)

Under the proposed controller in equations (29) and(30) the control objective can be achieved for trackingtumbling target (e closed-loop system for coupled rota-tional and translational control is shown in Figure 3

4 Stability Analysis

In this section the stability analysis is divided into two partsIn the first part ((eorem 1) the finite-time convergence ofESO is proved which implies that the proposed ESO canestimate the exact state and total disturbances within a fixedtime In the second part ((eorem 2) the finite-time stabilityof the closed-loop system is proved which indicates that thefinite-time convergence of the tracking error ( 1113954Ωe 1113954ωe) can beobtained (e proofs of finite-time convergence of ESO andtotal closed-loop system will be given using Lyapunovstability theory

Theorem 1 Suppose the parameters of ESO in equation (26)satisfy the following

p11113954λ1113954ρgt 1113954κ21113954λ + p2 p2 + 1( 1113857

21113954η21113954κ2 (31)

(en the observer errors will converge to the followingregion in finite-time

Ω1 ϑ | ϑle δ gR

λmin Q1( 11138571113888 1113889

p22p2minus 1⎧⎨

⎩

⎫⎬

⎭ (32)

where ϑ [1113954e1ip2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T R 1113954η 1113954κ minus 21113858 1113859 and thematrix Q1 satisfies the following expression

Q1 η1113954λ + p11113954η2 0 minus p11113954η

0 1113954ρ + 2 + p1( 11138571113954κ2 minus p1 + 1( 11138571113954κminus p11113954η minus p1 + 1( 11138571113954κ p1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (33)

(e convergence time is upperbounded by tm le 2θ1ln((θ1V

(12)0 + θ2)θ2) where V0 is the initial Lyapunov state

and θ1 and θ2 are the constants depending on the gains

Proof To verify the finite-convergence performance inaccordance with the idea of Moreno and Osorio [25] aLyapunov candidate is proposed as

V1 12

1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙1113954e1i minus 1113954e2i1113872 1113873

T

middot 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 + 1113954κ⊙ 1113954e1i minus 1113954e2i1113872 1113873 +

1113954λp1⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2

+ 1113954ρ⊙ 1113954eT1i1113954e1i +

121113954e

T2i1113954e2i

ϑTΓϑ

(34)

where ϑ [|1113954e1i|p2 sgn(1113954e1i) 1113954e1i 1113954e2i]

T and the matrix Γ is de-fined as

Γ 12

21113954λp1 + 1113954η2 1113954η1113954κ minus 1113954η

1113954η1113954κ 21113954ρ + 1113954κ2 minus 1113954κ

minus 1113954η minus 1113954κ 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

It is obvious that V1 is positive definite and radiallyunbounded if 1113954λ and 1113954ρ are positive (e range of V1 is ob-tained as follows

λmin(Γ)ϑ2 leV1 le λmax(Γ)ϑ

2 (36)

where ϑ2 |1113954e1i|2p2 + 1113954eT1i1113954e1i + 1113954eT2i1113954e2i is the Euclidean norm of

ϑ and λmin(middot) and λmax(middot) represent the minimum andmaximum eigenvalues of the matrix respectively

(e time derivative of V1 can be obtained as_V1 21113954λ + p11113954η21113872 1113873⊙ 1113954e1i

111386811138681113868111386811138681113868111386811138682p2minus 1 _1113954e1isgn 1113954e1i( 1113857 + 21113954ρ + 1113954κ21113872 1113873⊙ 1113954e

T1i

_1113954e1i

+ 21113954eT2i

_1113954e2i + p1 + 1( 11138571113954η1113954κ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 _1113954e1isgn 1113954e1i( 1113857

minus p11113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 _1113954e

T1i1113954e2i minus 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857_1113954e2i

minus 1113954κ⊙ _1113954eT1i1113954e2i minus 1113954κ⊙1113954e

T1i

_1113954e2i

(37)

qd

qe

q

FuNTSMCequation (30)

ω

[q ω]

Modelequation (12)

[z1 z2] Disturbancesobserver

equation (26)

ndash

Figure 3 Closed-loop system of NTSMC-ESO for spacecraftcoupled control

6 Mathematical Problems in Engineering

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Submit your manuscripts atwwwhindawicom

Substituting equation (28) into equation (37) yields

_V1 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1

1113876 minus 1113954ηλ + 1113954ηp21113954η21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 minus 1113954η1113954ρ + 2 + p2( 11138571113954η1113954κ21113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682

minus p11113954η⊙1113954eT2i1113954e2i + 2 p2 + 1( 11138571113954η1113954κ⊙ 1113954e

T1i1113954e2i

+ 2p21113954η2 ⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 11138571113954e2i1113877 + 1113876 minus 1113954κ⊙ 1113954e

T2i1113954e2i minus 1113954κ1113954ρ + 1113954κ31113872 1113873

⊙1113954eT2i1113954e2i minus 1113954κ1113954λ + 21113954κp21113954η2 + 1113954η21113954κ1113872 1113873 1113954e1i

111386811138681113868111386811138681113868111386811138682p2 + 21113954κ2 ⊙1113954e

T1i1113954e2i

+ 1113954η⊙ 1113954e1i

11138681113868111386811138681113868111386811138681113868p2 sgn 1113954e1i( 1113857 minus 21113954e2i + 1113954κ1113954e1i1113960 11139611113954gi(t)

minus 1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1ϑTQ1ϑ minus ϑTQ2ϑ + 1113954gi(t)Rϑ

(38)

where the matrices Q1 Q2 and R are given by

Q1 1113954η

1113954λ + p21113954η2 0 minus p21113954η

0 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 11138571113954κ

minus p21113954η minus p2 + 1( 11138571113954κ p2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2 1113954κ

1113954λ + 2p2 + 1( 11138571113954η2 0 0

0 1113954ρ + 1113954κ2 minus 1113954κ

0 minus 1113954κ 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R 1113954η 1113954κ minus 21113858 1113859 (39)

It is obvious thatQ2 is a symmetric matrix and if λ and κare positive it is positive definite In addition calculate thedeterminant of Q1 as

Q1

p21113954λ + p21113954η21113872 1113873 1113954ρ + 2 + p2( 11138571113954κ2 minus p2 + 1( 1113857

21113954κ21113960 11139611113966 1113967

minus p221113954η2 1113954ρ + 2 + p2( 11138571113954κ21113960 1113961

1113954λ p21113954ρ + p2 2 + p2( 11138571113954κ2 minus p2 + 1( 111385721113954κ21113960 1113961

minus p21113954η2 p2 + 1( 111385721113954κ2

1113954λ p21113954ρ minus 1113954κ21113872 1113873 minus p21113954η2 p2 + 1( 111385721113954κ2

(40)

Substituting equation (31) into equation (40) it yieldsQ1gt 0 (erefore Q1 is the positive definite matrixAccording to the definition of ϑ it can be obtained that

1113954e1i

11138681113868111386811138681113868111386811138681113868p2minus 1 ge ϑ

p2minus 1p2( ) (41)

According to equations (34)ndash(41) one obtains_V1 minus e1i

11138681113868111386811138681113868111386811138681113868p2minus 1λmin Q1( 1113857ϑ

2minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus λmin Q1( 1113857ϑp2minus 1p2( )ϑ minus λmin Q2( 1113857ϑ

2+ gRϑ

le minus c1ϑ2p2minus 1p2( ) minus c31113874 1113875V

(12)1 minus c2V1

(42)

where c1 λmin(Q1)λmax(Γ)

1113968 c2 λmin(Q2)λmax(Γ) and

c3 gRλmin(Γ)

1113968 Based on equation (36) we assume

that V1 ge (c3c1)(2p22p2minus 1)λmax(Γ) which implies that

c1ϑ(2p2minus 1p2) minus c3 ge 0 And it is obvious that c2 gt 0 Bysubstituting θ1 c1ϑ(2p2minus 1p2) minus c3 and θ2 c2 intoequation (42) one obtains

_V1 le minus θ1V(12)1 minus θ2V1 (43)

(erefore according to Lemma 2 it is obvious that _V1 isdefinitely negative and V1 is positive (erefore observererror will converge to the small set Ω1 in finite time and theconvergence time satisfies tm le 2θ1 ln(θ1V

(12)0 + θ2)(θ2)

(e proof of (eorem 1 is completed

Remark 3 As seen in (eorem 1 and its proof the re-construction error of the model uncertainties and externaldisturbances can be driven to the neighborhoods of theorigin in finite time (e parameter (p22p2 minus 1) gets largerand the region Ω1 ϑ | ϑleδ (gRλmin(Q1))

(p22p2minus 1)1113966 1113967

will be smaller by choosing adequate ESO parameters p2 1113954κ1113954η 1113954ρ and 1113954λ which implies that the proposed ESO has highaccuracy In this way the output of observer will not in-fluence the controller (erefore the finite-time conver-gence of the closed-loop system mentioned in(eorem 2 isproved on the basis of (eorem 1 Some studies containingESO-NTSMC analyzed the total finite-time convergence ofthe closed-loop system in the same way [32 33]

Theorem 2 Based on the estimated state in observer (26)and with the application of the controller in equations (29)and (30) the close-loop system states will be driven to aneighborhood of the sliding surface 1113954s 11139540 in finite timeMoreover the tracking errors 1113954Ωe and 1113954ωe will converge to asmall region in finite time

Proof (e proof includes two consecutive stepsFirstly it is to be proved that the sliding surface will

converge to 1113954s 11139540 in finite time Define the form of theLyapunov function as follows

V2 121113954sT 1113954M1113954s (44)

Taking the time derivative of 1113954s in equation (20) it yields

_1113954s minus 1113954Mminus 1 1113954Fu + 1113954D1113872 1113873 minus 1113954Mminus 1S 1113954ωe 1113954qe( 1113857 + 1113954β⊙ _1113954Ωe + 1113954c⊙ _χ 1113954Ωe1113872 1113873

(45)

With the application of equations (29) and (30) _1113954s can beobtained as

_1113954s 1113954Mminus 1minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113872 1113873 (46)

Take the time derivative of V2 yields_V2 1113954s

T 1113954M_1113954s

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954z2 + 1113954D1113960 1113961

1113954sT

minus 1113954k⊙ sigp1(1113954s) minus 1113954e21113960 1113961

(47)

where 1113954D Δ1113954Fg + 1113954Fd + Δ1113954Σ is the total external disturbanceand model uncertainty and 1113954e2 1113954z2 minus 1113954D is the error state ofESO Recalling from the stability in (eorem 1 the ESOerrors can enter the small setΩ in fixed time tm After finite-time tm the ESO errors satisfy 1113954e2lt δ In consequence itcan be obtained as

Mathematical Problems in Engineering 7

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Submit your manuscripts atwwwhindawicom

_V2 minus 1113954sT 1113954k⊙ sigp1(1113954s) + 1113954e21113960 1113961le minus 1113954s

T 1113954k⊙ sigp1(1113954s) minus δ1113960 1113961

le minus 1113944

3

i1ki minus δ si

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 si

11138681113868111386811138681113868111386811138681113868p1+1

+ kiprime minus δ siprime

11138681113868111386811138681113868111386811138681113868minus p11113872 1113873 siprime

11138681113868111386811138681113868111386811138681113868p1+1

1113960 1113961

le minus θ3 sp1+1

+ sprime

p1+1

1113874 1113875

(48)

where θ3 min[(ki minus δ|si|minus p1) (kiprime minus δ|siprime|minus p1)] i 1 2 3 is

the positive scalar For equation (48) if θ3 gt 0 that is ki minus

δ|si|minus p1 gt 0 or ki

prime minus δ|siprime|minus p1 gt 0 then one can obtain

_V2 le minus θ4Vp1+(12)( )

2 (49)

where θ4 θ32max σmax(Jf ) mf1113864 1113865

1113969is a positive number

(erefore it can be proved by Lemma 1 that the slidingsurface 1113954s will converge to the set Ω2 1113954s|1113954sleΔs 1113864

(1113954kδ)p1 in finite timeSecondly when the close-loop system state reaches the

neighborhood of 1113954s 11139540 it should be proved that the trackingerror 1113954Ωe and 1113954ωe will converge into small regions containingthe origin (11139540 11139540) in finite time Due to different conditions of1113954s and 1113954Ωe in (23) the following cases should be taken intoconsideration

Case 1 If 1113954s 0 equation (20) becomes

1113954ωei minus 1113954βi ⊙ 1113954Ωei minus 1113954ci ⊙ χ 1113954Ωei1113872 1113873 i 1 2 3 (50)

Consider the following Lyapunov function

V3 lang 1113954Ωe1113954Ωerang (51)

Taking the time derivative of V3 yields

_V3 lang 1113954Ωe_1113954Ωerang lang 1113954Ωe 1113954ωerang

1113884 1113954Ωe minus 1113954βi ⊙ 1113954Ωe minus 1113954c⊙ sgn Ωe( 1113857 Ωe1113868111386811138681113868

1113868111386811138681113868α

+ εsgn pe( 1113857 pe1113868111386811138681113868

1113868111386811138681113868α

1113960 11139611113885

minus 1113954βlang 1113954Ωe1113954Ωerang minus 1113944

3

i1ci Ωei

11138681113868111386811138681113868111386811138681113868α+1

+ ciprime pei1113868111386811138681113868

1113868111386811138681113868α+1

1113872 1113873

le minus θ5V3 minus θ6V(1+α2)3

(52)

where θ5 1113954β and θ6 min(ci ciprime) i 1 2 3 are the

parameters(erefore according to Lemma 2 the error states 1113954Ωe and

1113954ωe will converge to (11139540 11139540) in finite time

Case 2 If 1113954sne 0 | 1113954Ωei|le ε0 which indicates that 1113954Ωei con-verged into the set | 1113954Ωei|le ε0 One can obtain

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ 1113954c1i ⊙ 1113954Ωei + 1113954c2i ⊙ sgn 1113954Ωei1113872 1113873 1113954Ωei1113868111386811138681113868

11138681113868111386811138682

1113874 1113875 δs

δs

11138681113868111386811138681113868111386811138681113868ltΔs

(53)

(erefore 1113954ωei will eventually converge to the region asfollows

1113954ωei le δs

11138681113868111386811138681113868111386811138681113868 + 1113954βiε0 + 1113954ci ⊙ 1113954c1iε0 + 1113954c2iε

201113872 1113873 (54)

Case 3 If 1113954sne 0 | 1113954Ωei|ge ε0 it implies that

1113954ωei + 1113954βi ⊙ 1113954Ωei + 1113954ci ⊙ sigα 1113954Ωei1113872 1113873 δs δs

11138681113868111386811138681113868111386811138681113868ltΔs

1113954ωei + 1113954βi minusδs

2 1113954Ωei1113888 1113889⊙ 1113954Ωei + 1113954ci minus

δs

2 sigα 1113954Ωei1113872 1113873⎛⎝ ⎞⎠⊙ sigα 1113954Ωei1113872 1113873 0 δs

11138681113868111386811138681113868111386811138681113868ltΔs

(55)

Once 1113954βi gt δs2 1113954Ωei and 1113954ci gt δs2 sigα( 1113954Ωei) then 1113954s is stillkept in the form of terminal sliding mode which impliesthat | 1113954Ωei| will be bounded as | 1113954Ωei|le ε1 min (Δs2βi)1113864

(Δs2ci)(1α)

In conclusion |1113954ωei| and | 1113954Ωei| will be bounded as |1113954ωei|le ς1and | 1113954Ωei|le ς2 in finite time where ς1 |δs| + 1113954βiε0 + 1113954ci ⊙

(1113954c1iε0 + 1113954c2iε20) and ς2 max ε0 ε11113864 1113865 Based on the analysisabove it can be concluded that the control objective can beaccomplished in finite time (e proof of (eorem 2 iscompleted

Remark 4 (e total finite-time stability of closed-loop isachieved on the basis of the above proof With the

Table 1 Simulation experiment parameters

Orbital elements ValuesSemimajor axis (km) 6778Eccentricity 002Inclination (deg) 45Right ascension of the ascending node (deg) 30Argument of perigee (deg) 25True anomaly (deg) 30

Table 2 Control gains for simulations

Controllers Controller gains

ESO-NTSMCα 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε05

1113954k 20 + ε10 1113954η 3 + ε1 1113954κ 10 + ε5 1113954λ 3 + ε091113954ρ 001 + ε001 p1 06 p2 06

LESO-NTSMC

α 067 μ 10minus 3 1113954β 015 + ε2 1113954c 02 + ε051113954k 20 + ε101113954κ 10 + ε5 1113954ρ 001 + ε001

p1 06 p2 06

ESO-TSMα 53 1113954β 05 + ε08 1113954k 20 + ε10 1113954η 3 + ε1

1113954κ 10 + ε5 1113954λ 3 + ε09 1113954ρ 001 + ε001p1 06 p2 06

8 Mathematical Problems in Engineering

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

application of ESO and its estimation of total disturbancesthe controller is reconstructed by adding the disturbancerejection term 1113954Fτ in equation (30) (erefore the robustcontroller can guarantee the finite-time stability of theoverall closed-loop system with high accuracy By choosingsuitable controller parameters 1113954k and p1 such thatΔs (1113954kδ)p1 is smaller enough the convergence setΩ2 canbe a sufficiently small set (erefore it can be seen thatunder the proposed controller method the closed-loop

system can achieve finite-time convergence with satisfactoryperformance

Remark 5 Compared with the existing NTSMC methodsthe structure of proposed controller scheme in this researchcombines ESO and NTSMC techniques which can obtainhigher tracking accuracy and achieve finite-time conver-gence (e overall controller is developed based on theestimated information of external disturbances and model

e21e22e23

times10ndash3

10 20 30 40 500Time (s)

ndash04

ndash02

0

02

04Re

cons

truc

tion

erro

r (N

)

ndash5

0

5

45 5040

(a)Re

cons

truc

tion

erro

r (N

middotm)

e24e25e26

times10ndash3

10 20 30 40 500Time (s)

ndash02

ndash01

0

01

02

ndash1

0

1

45 5040

(b)

Figure 4 Time responses of reconstruction error (a) Translational term (b) Rotational term

eϕeθeψ

50

times10ndash5

ndash5

0

5

40 45

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

(a)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash5

0

5

(b)

eϕeθeψ

times10ndash4

40 45 50

ndash002

0

002

004

006

008

01

Atti

tude

(rad

)

10 20 30 40 500Time (s)

ndash2

0

2

(c)

Figure 5 Time responses of relative rotation (a) ESO NTSMC (b) LESO NTSMC (c) ESO TSM

Mathematical Problems in Engineering 9

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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International Journal of

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

Volume 2018

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Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

uncertainties by ESO (erefore the controller does notneed the priori information of perturbations Moreoverbased on the proof in equation (48) the ESO-based con-troller can achieve smaller steady-state error when the es-timation 1113954z2 of ESO is applied to develop the robustcontroller which indicates that the system has better per-formance and higher accuracy

5 Simulation Results

In this section numerical simulations are given to illustratethe effectiveness of the proposed ESO-NTSMC for the chasertracking the tumbling target It is assumed that the orbitalelements of tumbling target are presented in Table 1 (etarget body-fixed frame completely coincides with the

exeyez

40 45 50ndash002

0

002

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6Po

sitio

n (m

)

(a)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(b)

exeyez

40 45 50

10 20 30 40 500Time (s)

ndash1

0

1

2

3

4

5

6

Posit

ion

(m)

ndash002

0

002

(c)

Figure 6 Time responses of relative translation (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

ewxewyewz

40 45 50

times10ndash5

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(a)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash1

0

1

(b)

ewxewyewz

40 45 50

times10ndash3

10 20 30 40 500Time (s)

ndash006

ndash004

ndash002

0

002

004

Ang

ular

vel

ocity

(rad

s)

ndash2

0

2

(c)

Figure 7 Time responses of relative angular velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10 Mathematical Problems in Engineering

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

orbital coordinate system (e control objective is to keepattitude of chaser consistent with the target and the relativeposition of two can be kept at a certain distance

(e rotational angular velocity of the tumbling targetis considered to be slow speed as ωt(0) [0 0 001]T

(rads) and the capturing point in Ol minus xlylzl is set aspfld [10 0 0]T m (e inertia matrix is supposed to be

measured as Jt [20 0 0 0 23 0 0 0 25] (kg middot m2) (einitial conditions of the relation motion between twospacecraft are given as pffl(0) [15 0 0]T m qfl(0)

[09991 0 0 00431]T and 1113954ωffl(0) 11139540

In addition proposed NTSMC with linear-ESO LESO-NTSMC designed in Reference [25] and proposed ESO withconventional NTSMC ESO-TSM designed in Reference [38]

Vel

ocity

(ms

)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

ndash5

0

5

(a)

evxevyevz

times10ndash4

40 45 50

10 20 30 40 500Time (s)

ndash06

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

Vel

ocity

(ms

)ndash5

0

5

(b)

evxevyevz

40 45 50

times10ndash4

10 20 30 40 500Time (s)

ndash08

ndash06

ndash04

ndash02

0

02

Vel

ocity

(ms

)

ndash5

0

5

(c)

Figure 8 Time responses of relative linear velocity (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(a)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Con

trol f

orce

(N)

(b)

fuxfuyfuz

10 20 30 40 500Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10C

ontro

l for

ce (N

)

(c)

Figure 9 Time responses of control torque (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

Mathematical Problems in Engineering 11

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

are also implemented for purpose of comparison(e gains ofproposed and above controllers are given in Table 2

(e preset uncertainties and disturbances are set inaccordance with Reference [12] (e real mass and inertiamatrix of the chaser are assumed to be mf 97 kg andJf [18 0 0 0 17 0 0 0 20] (kg middot m2) And the nominal

mass and inertia matrix of the chaser is assumed to bemf0 100 kg and Jf0 [22 0 0 0 20 0 0 0 23] (kg middot m2)with the model uncertainties bounded by |Δmf |le 3 kg and|Jf ii|le 3 (kg middot m2) i 1 2 3 respectively Moreover it isassumed that the disturbance forces and torques are setas

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(a)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(b)

Con

trol t

orqu

e (N

middotm)

τuxτuyτuz

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

(c)

Figure 10 Time responses of control force (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash1

0

1

45 5040

(a)

s1s2s3

times10ndash3

10 20 30 40 500Time (s)

ndash02

0

02

04

06

08

1

12

14

16

Slid

ing

surfa

ce (m

s)

ndash2

0

2

45 5040

(b)

s1s2s3

ndash1

0

1times10ndash3

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

1

15

2

25Sl

idin

g su

rface

(ms

)

45 5040

(c)

Figure 11 Time responses of sliding surface of translational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

12 Mathematical Problems in Engineering

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

ffd [005 cos(nt)007 cos(nt) minus 001 cos(nt)]T

τfd [minus 004 cos(nt)003 cos(nt) minus 005 cos(nt)]T(56)

where n πT0 is a target orbit period constant withT0 asymp 5711 s (e control force is limited to the range of|fu|le 10N and the control torque is limited to the range of|τu|le 1Nmiddotm

Figures 4ndash13 show the time responses of the simulationresults using the proposed ESO-NTSMC and comparedcontrol schemes LESO-NTSMC and ESO-TSM re-spectively (e results shown in Figure 4 illustrate the ef-fectiveness of the ESO where the observer reconstructionerrors of translational term and rotational term converge toa small range in finite time In such a situation the accuracyof the system can be enhanced when the model uncertaintiesand bounded external disturbances are estimated by ESOand the system can be compensated Figures 5(a) and 6(a)shows the time responses of the tracking error of relativetranslation and rotation respectively It can be seen that therotation error falls to tolerance within 10 s and converges tothe range of 5 times 10minus 5 rad while the translation tracking errorconverges to the range of 002m in around 20 s which il-lustrates the performance of accuracy and rapidity of ESO-NTSMC (e time histories of relative velocity and angularvelocity tracking errors are given in Figures 7(a) and 8(a)respectively (e steady-state behavior shows that the an-gular velocity error converges to 1 times 10minus 5 (rads) in 10 s andthe velocity error converges to 5 times 10minus 4 (ms) at 20 s whichconfirms the control objective of tracking the velocity oftarget accurately It can be seen that under the boundedsliding mode control force and torque shown in Figures 9(a)and 10(a) the sliding surfaces converge to the range of

|si|le 1 times 10minus 3 (ms) and |siprime|le 1 times 10minus 5 (rads) in finite time

successfully which are shown in Figures 11(a) and 12(a)(e simulations of LESO-NTSMC are also carried out

which are shown in Figures 5(b)ndash12(b) From these figuresit can be seen that LESO-NTSMC is also tolerant to modeluncertainties and external disturbances when the systemstates can be driven to a neighborhood of origin As shownin Figures 11(b) and 12(b) the sliding surface accuracy ofLESO-NTSMC is much lower than of ESO-NTSMC whichcan illustrate that the proposed method obtains better

s4s5s6

times10ndash5

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash1

0

1

45 5040

(a)

s4s5s6

times10ndash4

10 20 30 40 500Time (s)

ndash005

0

005

01

015

02

Slid

ing

surfa

ce (r

ads

)

ndash5

0

5

45 5040

(b)

s4s5s6

times10ndash3

10 20 30 40 500Time (s)

ndash001

0

001

002

003

004

005

006

Slid

ing

surfa

ce (r

ads

)

45 5040ndash2

0

2

(c)

Figure 12 Time responses of sliding surface of rotational term (a) ESO-NTSMC (b) LESO-NTSMC (c) ESO-TSM

10

15Y (m)

X (m)

0 100ndash10

ndash15ndash10

TargetInitial posTrajectory

ndash10

0

10

Z (m

)

Figure 13 Trajectory of chaser tracking capturing point of tum-bling target

Mathematical Problems in Engineering 13

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

control ability and high control accuracy in comparison withLESO-NTSMC scheme Figures 5(c)ndash12(c) show the sim-ulation results using the ESO-based conventional NTSMCmethod Severe chattering and large overshoot exist in ESO-TSM during the control process and the convergence timeof ESO-TSM is larger than the proposed one As seen theproposed ESO-NTSMC has faster convergence rate thanESO-TSM and can also avoid singularity From the trackingtrajectory shown in Figure 13 the chaser can successfullytrack the desired trajectory of docking port of tumblingtarget In conclusion it is obvious that the control object isachieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can beguaranteed

6 Conclusion

In this study a finite-time stabilization problem of a chaserspacecraft approaching a tumbling target in presence ofunknown uncertainty and bounded external disturbance hasbeen handled Based on the 6-DOF coupled relative motiondescribed by dual quaternion a combination of NTSMC andESO is proposed to drive the translational and rotationalmotion to the desired state Meanwhile the finite-timeconvergence and stability property of the proposed methodis rigorously proven It was shown that the tracking error canbe driven to the equilibrium point with fast response andhigh accuracy Numerical simulation results show the ef-fectiveness and performance of the proposed controller andits robustness to the model uncertainty and externaldisturbance

Appendix

For a given quaternion q [η ξ] η and ξ are the scalar partand the vector part respectively (e conjugate of q is de-fined as qlowast [η minus ξ] and its norm is calculated byq

q ∘ qlowast

1113968 It is called unit quaternion if q 1 Its

logarithm is defined as

ln q 0arc cos η2

1 minus η2

1113968 ξ1113890 1113891 (A1)

(e equivalent of the unit quaternion referred to Eulerangle ϕ and the unit vector n is defined as

ln q 0ϕ2

n1113890 1113891 0leϕle 2π (A2)

As seen when the angle ϕ 0 there is q+ [1 0 0 0]and when ϕ 2π correspondingly qminus [minus 1 0 0 0] Be-cause of ln q+ ln qminus (0 0 0) it is obvious that a unitquaternion can be transformed to a three-dimensionalvector by the logarithm operation

(e definition of dual numbers is1113954a a + εaprime (A3)

where a and aprime are respectively called real part and dualpart ε is called the dual unit satisfying ε2 0 and εne 0 Andthe operation of the dual numbers is defined as

1113954a1 + 1113954a2 a1 + a2 + ε a1prime + a2prime( 1113857

λ1113954a λa + ελaprime

1113954a11113954a2 a1a2 + ε a2a1prime + a1a2prime( 1113857

(A4)

where λ is a scalar(e dual vector is that both the dual part and real part are

vectors For two dual vectors 1113954v1 v1 + εv1prime and 1113954v2 v2 + εv2primetheir cross multiplication is

1113954v1 times 1113954v2 v1 times v2 + ε v1 times v2prime + v1prime times v2( 1113857

lang1113954v1 1113954v2rang vT1 v1prime + v

T2 v2prime

(A5)

In addition to this dual number 1113954a and dual vector 1113954v havethe following operation

1113954a⊙ 1113954v av + εaprimevprime (A6)

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the Major Program of NationalNatural Science Foundation of China under Grant nos61690210 and 61690213

References

[1] A E White and H G Lewis ldquo(e many futures of activedebris removalrdquoActa Astronautica vol 95 pp 189ndash197 2014

[2] R A Gangapersaud G Liu and A H J De RuiterldquoDetumbling of a non-cooperative target with unknown in-ertial parameters using a space robotrdquo Advances in SpaceResearch vol 63 no 12 pp 3900ndash3915 2019

[3] B W Barbee J R Carpenter S Heatwole et al ldquoA guidanceand navigation strategy for rendezvous and proximity op-erations with a noncooperative spacecraft in geosynchronousorbitrdquoRe Journal of the Astronautical Sciences vol 58 no 3pp 389ndash408 2011

[4] N FitzCoy and M Liu ldquoModified proportional navigationscheme for rendezvous and docking with tumbling targets theplanar caserdquo in Proceedings of the Symposium on FlightMechanicsEstimation Reory pp 243ndash252 Greenbelt MDUSA May 1995

[5] S Nakasuka and T Fujiwara ldquoNew method of capturingtumbling object in space and its control aspectsrdquo in Pro-ceedings of the presented at the IEEE International Conferenceon Control Applications Kohala Coast HI USA August 1999

[6] Y Tsuda and S Nakasuka ldquoNew attitude motion followingcontrol algorithm for capturing tumbling object in spacerdquoActa Astronautica vol 53 no 11 pp 847ndash861 2003

[7] G Boyarko O Yakimenko and M Romano ldquoOptimalrendezvous trajectories of a controlled spacecraft and atumbling objectrdquo Journal of Guidance Control and Dy-namics vol 34 no 4 pp 1239ndash1252 2011

14 Mathematical Problems in Engineering

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

[8] W Fehse Automated Rendezvous and Docking of SpacecraftRe Drivers for the Approach Strategy pp 8ndash74 CambridgeUniversity Press London UK 2003

[9] Z Ma O Ma and B Shashikanth ldquoOptimal control forspacecraft to rendezvous with a tumbling satellite in a closerangerdquo in Proceedings of the presented at the 2006 IEEERSJInternational Conference on Intelligent Robots and SystemsBeijing China October 2006

[10] M Wilde M Ciarcia A Grompone and M RomanoldquoExperimental characterization of inverse dynamics guidancein docking with a rotating targetrdquo Journal of GuidanceControl and Dynamics vol 39 no 6 pp 1173ndash1187 2016

[11] Z Xu X Chen Y Huang Y Bai and W Yao ldquoNonlinearsuboptimal tracking control of spacecraft approaching atumbling targetrdquo Chinese Physics B vol 27 no 9 Article ID090501 2018

[12] J Wang H Liang Z Sun S Zhang and M Liu ldquoFinite-timecontrol for spacecraft formation with dual-number-baseddescriptionrdquo Journal of Guidance Control and Dynamicsvol 35 no 3 pp 950ndash962 2012

[13] I S Fisher Dual-Number Methods in Kinetics Statics andDynamics p 66 CRC Press Boca Raton FL USA 1999

[14] A T Yang Application of Quaternion Algebra and DualNumbers to the Analysis of Spital Mechanisms ColumbiaUniversity New York NY USA 1964

[15] V Brodsky and M Shoham ldquoDual numbers representation ofrigid body dynamicsrdquo Mechanism and Machine Reoryvol 34 no 5 pp 693ndash718 1999

[16] J Wu D Han K Liu and J Xiang ldquoNonlinear suboptimalsynchronized control for relative position and relative attitudetracking of spacecraft formation flyingrdquo Journal of ReFranklin Institute vol 352 no 4 pp 1495ndash1520 2015

[17] J Wu K Liu and D Han ldquoAdaptive sliding mode control forsix-DOF relative motion of spacecraft with input constraintrdquoActa Astronautica vol 87 pp 64ndash76 2013

[18] N Filipe and P Tsiotras ldquoAdaptive position and attitude-tracking controller for satellite proximity operations usingdual quaternionsrdquo Journal of Guidance Control and Dy-namics vol 38 no 4 pp 566ndash577 2015

[19] H Dong Q Hu and M R Akella ldquoDual-quaternion-basedspacecraft autonomous rendezvous and docking under six-degree-of-freedom motion constraintsrdquo Journal of Guid-ance Control and Dynamics vol 41 no 5 pp 1150ndash11622018

[20] G A Jr and L S Martins-Filho ldquoGuidance and control ofposition and attitude for rendezvous and dockberthing with anoncooperativetarget spacecraftrdquo Mathematical Problems inEngineering vol 2014 Article ID 508516 8 pages 2014

[21] Q Ni Y Huang and X Chen ldquoNonlinear control ofspacecraft formation flying with disturbance rejection andcollision avoidancerdquo Chinese Physics B vol 26 no 1 ArticleID 014502 2017

[22] P Razzaghi E Al Khatib and S Bakhtiari ldquoSliding mode andSDRE control laws on a tethered satellite system to de-orbitspace debrisrdquo Advances in Space Research vol 64 no 1pp 18ndash27 2019

[23] J Wang and Z Sun ldquo6-DOF robust adaptive terminal slidingmode control for spacecraft formation flyingrdquo Acta Astro-nautica vol 73 pp 76ndash87 2012

[24] A-M Zou K D Kumar Z-G Hou and L Xi ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and Chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Part B-(Cybernetics) vol 41no 4 pp 950ndash963 2011

[25] J A Moreno and M Osorio ldquoA Lyapunov approach tosecond-order sliding mode controllers and observersrdquo inProceedings of the 47th IEEE Conference on Decision andControl vol 16 no 5 pp 2856ndash2861 Cancun MexicoDecember 2008

[26] S Wu R Wang G Radice and Z Wu ldquoRobust attitudemaneuver control of spacecraft with reaction wheel low-speedfriction compensationrdquo Aerospace Science and Technologyvol 43 pp 213ndash218 2015

[27] R Yan and Z Wu ldquoNonlinear disturbance observer basedspacecraft attitude control subject to disturbances and actu-ator faultsrdquo in Proceedings of the American Institute of PhysicsConference Series vol 1834 no 1 Athens Greece April 2017

[28] C Pukdeboon ldquoExtended state observer-based third-ordersliding mode finite-time attitude tracking controller for rigidspacecraftrdquo Science China Information Sciences vol 62 no 1pp 1ndash16 2019

[29] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[30] X Huang Y Yan and Y Zhou ldquoNonlinear control ofunderactuated spacecraft hoveringrdquo Journal of GuidanceControl and Dynamics vol 39 no 3 pp 685ndash694 2015

[31] L Zhang C Wei R Wu and N Cui ldquoFixed-time extendedstate observer based non-singular fast terminal sliding modecontrol for a VTVL reusable launch vehiclerdquo AerospaceScience and Technology vol 82-83 pp 70ndash79 2018

[32] D Ran X Chen A de Ruiter and B Xiao ldquoAdaptive ex-tended-state observer-based fault tolerant attitude control forspacecraft with reaction wheelsrdquo Acta Astronautica vol 145pp 501ndash514 2018

[33] J Zhang D Ye Z Sun and C Liu ldquoExtended state observerbased robust adaptive control on SE(3) for coupled spacecrafttracking maneuver with actuator saturation and mis-alignmentrdquo Acta Astronautica vol 143 pp 221ndash233 2018

[34] E Jin and Z Sun ldquoRobust controllers design with finite timeconvergence for rigid spacecraft attitude tracking controlrdquoAerospace Science andTechnology vol12 no 4 pp 324ndash330 2008

[35] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[36] H D He D Y Wang and H Gao ldquoVision-based position andpose determination of non-cooperative target for on-orbitservicingrdquo Multimedia Tools and Applications pp 1ndash14 2018

[37] L Wang T Chai and L Zhai ldquoNeural-network-based ter-minal sliding mode control of robotic manipulators includingactuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[38] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38no 12 pp 2159ndash2167 2002

[39] X Huang Y Yan and Z Huang ldquoFinite-time control ofunderactuated spacecraft hoveringrdquo Control EngineeringPractice vol 68 pp 46ndash62 2017

Mathematical Problems in Engineering 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom