research article finite-time reentry attitude...

Research ArticleFinite-Time Reentry Attitude Control Using Time-VaryingSliding Mode and Disturbance Observer

Xuzhong Wu Shengjing Tang Jie Guo and Yao Zhang

Key Laboratory of Dynamics and Control of Flight Vehicle Ministry of Education School of Aerospace EngineeringBeijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Shengjing Tang tsj8678126com

Received 5 June 2014 Revised 16 September 2014 Accepted 11 October 2014

Academic Editor Rongni Yang

Copyright copy 2015 Xuzhong Wu et al This 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

This paper presents the finite-time attitude control problem for reentry vehicle with redundant actuators in consideration of planetuncertainties and external disturbances Firstly feedback linearization technique is used to cancel the nonlinearities of equationsof motion to construct a basic mode for attitude controller Secondly two kinds of time-varying sliding mode control methodswith disturbance observer are integrated with the basic mode in order to enhance the control performance and system robustnessOne method is designed based on boundary layer technique and the other is a novel second-order sliding model control methodThe finite-time stability analyses of both resultant closed-loop systems are carried out Furthermore after attitude controllerproduces the torque commands an optimization control allocation approach is introduced to allocate them into aerodynamicsurface deflections and on-off reaction control system thrusts Finally the numerical simulation results demonstrate that bothof the time-varying sliding mode control methods are robust to uncertainties and disturbances without chattering phenomenonMoreover the proposed second-order sliding mode control method possesses better control accuracy

1 Introduction

Covering from outer space into earthrsquos atmosphere reentryflight is a critical phase of the operation for reentry vehicles(RVs) [1] Since the flight conditions change rapidly in thereentry phase reentry attitude control is always in face ofwide range of planet uncertainties and external disturbancesOn the other hand aerodynamic surfaces come into the eyesof engineers firstly with their advantage of saving energyHowever the density of atmosphere can be so low in begin-ning of reentry flight that the desired control torque may beunachievable with the employment of aerodynamic surfacesalone because of poor aerodynamic maneuverability As aresult RVs have to rely on reaction control system (RCS) jetsin addition to aerodynamic surfaces In this case controlallocation among redundant actuators becomes necessarywhich further raises the difficulties in attitude control designMeanwhile a robust attitude control system for RVs withredundant actuators is desirable

The conventional attitude control method for RVs is gainscheduling (GS) [2 3] This method linearizes the systemwith a set of trimmed points designs individual gains at eachpoint and then interpolates those gains online with respectto system parameters such as dynamic pressure or Machnumber Nevertheless the conventional GS involves the lackof guaranteed global robustness and stability [4] The reentryflight conditions change rapidly which makes this methodimpractical [5] Moreover the point designs of gain schedul-ing are manpower intensive and highly time consuming [6]To conclude GS is weak at performing linearity

Compared to GS feedback linearization (FBL) [7ndash10]can exactly cancel the model nonlinearities and replaceundesirable dynamics with desirable dynamic using non-linear coordinate transformation However FBL relies onthe knowledge of the exact model dynamics which severelyinfluences FBLrsquos practicality because uncertainties and distur-bances exist inevitably To improve the flight control perfor-mance systematically on the basis of FBL Rahideh et al [7]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 262753 19 pageshttpdxdoiorg1011552015262753

2 Mathematical Problems in Engineering

incorporated neural network (NN) based compensation inthe FBL design Van Soest et al [8] combined FBL with con-strained linear model predictive control (MPC) method Xuet al [9] utilized the combination of FBL and adaptive slidingmode control (SMC) method

Among the various upgraded nonlinear control methodsSMC outstands with many advantages such as simplicityof implementation fast dynamic response good transientbehavior exponential stability insensitivity to parameter var-iations and robustness to plant uncertainties and externaldisturbances [11ndash14] Therefore SMC has been successfullyapplied to a variety of complex engineering systems [15]Barambones Caramazana et al [14] develop a sliding modeposition control incorporating a flux estimator for high-performance real-time applications of induction motors Wuet al [15] investigate the key problems of SMC of Markovianjump singular time-delay systems Shtessel et al [16ndash19] stud-ied the application of SMCmethod to reusable launch vehicle(RLV) in launch and reentrymode and amultiple-time-scaleSMC strategy is proposed in [18] for RLV in ascent phase

Generally SMC design consists of two steps [20 21] (1)select a sliding surface as a function of the system statesso that the system trajectories along the surface meet thedesired performance such as stability and tracking capability(2) design a suitable control law to drive the states onto thepredefined sliding surface in finite timeWhen it comes to thedesign of conventional SMC there are two major problemsconcerned One is its unguaranteed global robustness andthe other is chattering phenomenonThe conventional slidingsurfaces [16ndash18] employ linear function of tracking errorswhich results in the fact that the transient dynamics of SMCconsists of reaching phase and sliding phase However theSMC method can only ensure the robustness against planetuncertainties and external disturbances in sliding phaseTherefore the conventional sliding surfaces do not possessthe property of global robustness Several studies are dedi-cated to global robustness of SMC Sun et al [22] introducedan integral sliding mode control (ISMC) method to solvethe longitudinal control problem of air-breathing hypersonicvehicle (AHV) Shtessel et al [19] proposed a two-loop con-troller that utilized a time-varying sliding mode control(TVSMC) method to achieve fault tolerance for RLV attitudecontrol With the elimination of reaching phase both ISMCand TVSMC can keep the system states on the sliding surfacefrom the initial time so that global robustness against planetuncertainties and external disturbances is guaranteed

As to the chattering phenomenon it is assumed that thecontrol can be switched from one structure to anotherinfinitely fast in the design of SMC [23] However it is impos-sible to achieve high-speed switching control because of theinevitable switching delay computation and the limitation ofthe physical actuatorsThe existence of time delay introducesinstability oscillation and poor performance [24] High con-trol gains of SMC lead to high frequency oscillations knownas chattering phenomenon This harmful phenomenon mayerode the performance to gain robustness decrease the con-trol accuracy and damage the actuators [25]There are essen-tially two ways to alleviate the chattering phenomenon [23]one way is boundary layermethod [17ndash19 26] and the other is

higher order sliding mode control (HOSMC) The boundarylayer method replaces the sign function (discontinuous con-trol) with smooth approximations such as high-gain satura-tion function or sigmoid function Nevertheless this methodno longer drives the system state to the origin and cannotguarantee the robustness and accuracy within the boundarylayer [23] HOSMC was proposed by Levant [27] Instead ofinfluencing the first order time derivative the discontinuouscontrol acts on sliding variablersquos higher order derivative Asa special case of HOSMC second-order sliding mode control(SOSMC) is themost popular approach in engineeringThereare many kinds of SOSMC such as twisting algorithm [28]super-twisting algorithm [29] suboptimal algorithm [30 31]and prescribed convergence law algorithm [28]

Disturbance observer (DO) is an effective way to enhancesystem robustness The disturbance estimation is used forcompensation DO was first proposed by Ohishi et al [32]Hall and Shtessel [33] combined SMC and sliding modedisturbance observer (SMDO) which estimates the boundeduncertainties and disturbances effectively to improve RLVattitude control Shtessel et al [34] proposed a homogeneousDO based on the standard robust exact differentiator to solvethe missile guidance problem

Inspired by previous work this paper proposes twoTVSMC methods to solve the finite-time attitude controlproblem by incorporating the disturbance observer One isBTVSMCDO which is the abbreviation for boundary layermethod based time-varying sliding mode controller withdisturbance observer and the other is SOTVSMCDO whichmeans the second-order time-varying sliding mode con-troller with disturbance observer With the same dedicationto systematically enhance robustness and suppress controlchattering the two methods adopt different ways to alleviatechattering The former is designed based on boundary layertechnique and the latter utilizes a novel SOSMC The maincontributions of this paper are summarized as follows

(1) This paper incorporates a novel reaching law basedon SOSMCwith the time-varying sliding function Inorder to enhance the robustness of the method a DObased on the standard robust exact differentiator isemployed to estimate the systemrsquos uncertainties anddisturbances in finite time In addition the finite-time convergence of time-varying sliding function forresulted method is proved via Lyapunov theory andconsequently the asymptotical stability of the closed-loop nonlinear system is proved according to thedefinition of the time-varying sliding function

(2) Since RVs deploy both aerodynamic surfaces andRCS jets this paper introduces a control allocationapproach to assign control responsibility amongstredundant actuators The nonlinear programmingproblem is established and solved by optimiza-tion method and the pulse-width-pulse-frequency(PWPF) is employed tomodulate the on-off thrusters

(3) The proposed control methods are applied to finite-time attitude control problem for RVs Numericalsimulation results confirm the validity and superiorperformance of the proposed control methods by

Mathematical Problems in Engineering 3

comparing them with other conventional controlmethods The comparison between boundary layermethod and SOSMC is also presented

The major contents of the following part in this paperare as follows Section 2 describes the rotational equationsof motion and formulates problems of attitude controllerand control allocation In Section 3 feedback linearizationtechnique is employed to the equations of motion Section 4presents two TVSMC methods as well as the correspondingstability analysis A control allocation method is introducedin Section 5 In Section 6 the performances of proposedcontrol methods are assessed by numerical tests FinallySection 7 summarizes and lists the conclusions

2 Preliminary

21 The Rotational Equations of Motion Reentry guidance isconcerned with steering the vehicle from entry interface (EI)to the designated target point in prescribed condition whilesatisfying necessary path constraints such as heating rate con-straint aerodynamic load constraint and dynamic pressureconstraint [35 36] The steering commands are defined interms of angle of attack (AOA) 120572

119888 sideslip angle 120573

119888 and bank

angle 120590119888 Furthermore to prevent excessive heat buildup 120573

119888

is kept around zero under the application of back-to-turn(BTT) control policy [37] The subsequent reentry controlsystem tracks these three attitude commands And the objec-tive of the reentry control system is to determine the actuatorcommand vector 120575 so that the reentry vehicle can follow theattitude commands that are specified by guidance system

The motion of reentry vehicle can be divided into trans-lational motion and rotational motion Since the focus of thispaper is about control system the translational equations ofmotion utilized in guidance system are not presented Thereentry dynamics are governed by a group of nonlinear dif-ferential equations [38] The kinematic equations of reentryvehicle are defined as [37]

119889120572

119889119905= minus 119901 cos120572 tan120573 + 119902 minus 119903 sin120572 tan120573

+sin120590

cos120573( cos 120574 minus 120601 sin120595 sin 120574

+ ( 120579 + 120596119890) (cos120601 cos120595 sin 120574 minus sin120601 cos 120574))

minuscos120590cos120573

( 120574 minus 120601 cos120595 minus ( 120579 + 120596119890) cos120601 sin120595)

119889120573

119889119905= 119901 sin120572 minus 119903 cos120572

+ sin120590 ( 120574 minus 120601 cos120595 minus ( 120579 + 120596119890) cos120601 sin120595)

+ cos120590 ( cos 120574 minus 120601 sin120595 sin 120574


119889120590

119889119905= minus 119901 cos120572 cos120573 minus 119902 sin120573 minus 119903 sin120572 cos120573

+ sin120573 minus sin 120574 minus 120601 sin120595 cos 120574

+ ( 120579 + 120596119890) (cos120601 cos120595 cos 120574 minus sin120601 sin 120574)

(1)

where 120572 120573 and 120590 are AOA sideslip angle and bank anglerespectively 119901 119902 and 119903 are the rates of roll pitch and yawrespectively 120574 denotes flight path angle and 120595 denotes head-ing angle 120579 and120601 are longitude and latitude of reentry vehicle120596119890is the angular rate of Earth rotationIn order to simplify the online calculation this paper

obtains the kinetics of reentry vehicle under the followingassumption

Assumption 1 (1) The reentry vehicle is a rigid body theterms impacted by elastic effects are not considered (2) Thereentry vehicle has a longitudinal symmetry plane whichmeans the products of inertia 119868

119909119910= 119868119910119911

= 0 (3)The vehicle isunpowered during reentry

Hence the kinetics of reentry vehicle can be expressed as[37]

d119901d119905

=119868119911119872119897+ 119868119911119909

119872119899+ 119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682

119911minus 1198682

119911119909) 119902119903

119868119909119868119911minus 1198682119911119909

d119902d119905

=119872119898

+ (119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

d119903d119905

=119868119911119909

119872119897+ 119868119909119872119899+ (minus119868

119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

(2)

where 119872119897 119872119898 and 119872

119899are three control torques defined in

the body frame roll pitch and yaw respectively 119868119909 119868119910 and 119868

119911

denote the moments of inertia and 119868119911119909

denotes the productof inertia

The control-oriented model can be developed for controldesign based on (1) and (2) Since the rotational motions aremuch faster than translational motions and the motion ofEarth the translational terms and angular velocity of earthcan be neglected that is 120574 = = 120579 = 120601 = 0 120596

119890= 0

Therefore the rotational equations of motions (1) and (2) canbe further simplified as follows

Ω = R120596 + Δf

= minus Iminus1120596timesI120596 + Iminus1M119888+ Δd

(3)


whereΩ = [120572 120573 120590]119879 is the attitude angle vector120596 = [119901 119902 119903]

119879

is the attitude angular rate vector and M119888= [119872119897119872119898119872119899]119879

is the command control torque vector R isin R3times3 is thecoordinate-transformation matrix Δf = [Δ119891

1 Δ1198912 Δ1198913 ]119879

denotes the unknown bounded uncertainties caused by themodel reduction 120596times isin R3times3 stands for the skew-symmetricmatrix operator on vector120596 I isin R3times3 denotes the symmetricpositive definite inertia matrix of reentry vehicle and Δd isin

R3 denotes the bounded uncertain termR120596times I and Δd aregiven by

R =[[

[

minus cos120572 tan120573 1 minus sin120572 tan120573

sin120572 0 minus cos120572minus cos120572 cos120573 minus sin120573 minus sin120572 cos120573

]]

]

120596times

=[[

[

0 minus119903 119902

119903 0 minus119901

119902 119901 0

]]

]

I =[[

[

119868119909

0 minus119868119911119909

0 119868119910

0

minus119868119911119909

0 119868119911

]]

]

(4)

Δd = Iminus1 [minusΔI minus 120596times

ΔI120596 + ΔM] (5)

where ΔI isin R3times3 denotes unknown bounded inertiavariations and ΔM isin R3 stands for the bounded externaldisturbance moment

22 Problem Formulation As shown in Figure 1 the controlproblem for reentry vehicle with redundant actuators can besolved in two steps They are

(1) specifying the control torque vector M119888

= [119872119897119872119898

119872119899]119879 in equation set (3) which leads the output vector Ω to

track the attitude commandΩ119888= [120572119888 120573119888 120590119888]119879 in a finite time

lim119905gt119905119865

e = lim119905gt119905119865

1003817100381710038171003817Ω minusΩ119888

1003817100381710038171003817 = 0 (6)

where e = Ω minusΩ119888is the tracking error

(2) designing a control allocation method 119891CA(sdot) thatmaps the command control torque vector M

119888to actuator

deflection commands 120575119888[39]

120575119888= 119891CA (M

119888) (7)

The actual torque produced by control allocation maynot exactly equal the torque command Assume that ΔM

119888119886

is the bounded disturbance caused by the process of controlallocation and the torque vector produced by actuators canbe expressed as M

119888+ ΔM

119888119886 Hence the bounded uncertain

term of (5) can be rewritten as


ΔI120596 + ΔM + ΔM119888119886] (8)

Guidance system

Attitude controller

Control allocation

6-DOF reentry vehicle

Disturbance

120572c120573c120590c

120572 120573 120590

Ml

MmMn

δA

δRCS

Figure 1 Control architecture for reentry vehicle

3 Feedback Linearization

By the selection of control input as control torque vectorM and the output as attitude angle vector Ω the nonlinearattitude equations (3) can be expressed as [40]

x = f (x) + g (x) u + d

y = h (x) (9)

where x = [120572 120573 120590 119901 119902 119903]119879 is the state vector y = [120572 120573 120590]

119879

is the output vector u = [119872119897119872119898119872119899]119879 is the control vector

and d = [Δf119879 Δd119879]119879 stands for the system uncertain termf(x) and g(x) can be obtained by (10) and (11) respectively

f (x) =

[[[[[[[

[

1198911(x)

1198912(x)

1198913(x)

1198914(x)

1198915(x)

1198916(x)

]]]]]]]

]

=

[[[[[[[[[[[[[[[[

[

minus119901 cos120572 tan120573 + 119902 minus 119903 sin120572 tan120573

119901 sin120572 minus 119903 cos120572minus119901 cos120572 cos120573 minus 119902 sin120573 minus 119903 sin120572 cos120573

119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682119911minus 1198682119911119909

) 119902119903

119868119909119868119911minus 1198682119911119909

(119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

(minus119868119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]]]]

]

(10)

g (x) = [1198921(x) 119892

2(x) 119892

3(x)]

=

[[[[[[[[[[[[[

[

0 0 0

0 0 0

0 0 0119868119911

119868119909119868119911minus 1198682119911119909

0119868119911119909

119868119909119868119911minus 1198682119911119909

01

119868119910

0

119868119911119909

119868119909119868119911minus 1198682119911119909

0119868119909

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]

]

(11)


The vector relative degree of system (9) is (2 2 2) Afterdifferentiating output vector y twice the control input vector

u appearsy = Ku + B + RIminus1Δd + FΔf (12)

where K B and F are given by

K =[[

[

1198711198921119871119891ℎ1(x) 119871

1198922119871119891ℎ1(x) 119871

1198923119871119891ℎ1(x)

1198711198921119871119891ℎ2(x) 119871

1198922119871119891ℎ2(x) 119871

1198923119871119891ℎ2(x)

1198711198921119871119891ℎ3(x) 119871

1198922119871119891ℎ3(x) 119871

1198923119871119891ℎ3(x)

]]

]

B = [1198712

119891ℎ1(x) 1198712

119891ℎ2(x) 1198712

119891ℎ3(x)]119879

F =[[

[

119901 sin120572 tan120573 minus 119903 cos120572 tan120573 + 1 minus119901 cos120572sec2120573 minus 119903 sin120572sec2120573 0

119901 cos120572 + 119903 cos120572 1 0

119901 sin120572 cos120573 minus 119903 cos120572 cos120573 119901 cos120572 sin120573 minus 119902 cos120573 + 119903 sin120572 sin120573 1

]]

]

(13)

According to (12) the total relative degree of systemequals the order of the system Furthermore since the sideslipangle 120573 asymp 0 during reentry

det (K) =cos120573 minus sin120573 tan120573

119868119909119868119911minus 1198682119911119909

asymp1

119868119909119868119911minus 1198682119911119909

= 0 (14)

Thus the system (9) can be linearized completely withoutzero dynamics by using the following feedback control law

u = Kminus1 (minusB + k) (15)

where k = [V1 V2 V3]119879 is selected as the new control input in

this paperDefine the bounded uncertainty terms Δk = RIminus1Δd +

FΔf as the lumped uncertainty and substitute (15) into (12)and the basic model for attitude controller design can beobtained by

y = k + Δk (16)

4 Sliding Mode Attitude Controller Design

This section develops two TVSMC attitude controllers tosolve the finite-time control problem by incorporating distur-bance observer The first controller is BTVSMCDO and thesecond controller is SOTVSMCDO This part elaborates onthe design of sliding surface and reaching law of the con-trollers Moreover the design of disturbance observer is pre-sented too

41 Time-Varying Sliding Surface Design The time-varyingsliding surface is selected as [41]

s = e + ce + aeminus120582119905 (17)

where s = [1199041 1199042 1199043]119879 the tracking error vector e = y minus yc =

[1198901 1198902 1198903]119879

c = diag(1198881 1198882 1198883) is the sliding function gain

matrix and the element 119888119894

gt 0 119894 = 1 2 3 120582 isin R+ a =

[1198861 1198862 1198863]119879 is the coefficient vector to guarantee the existence

of sliding mode from the beginning of motion Hence a isdefined as

a = minuse (0) minus ce (0) (18)

Lemma 2 If the sliding mode s(119905) = 03times1

is satisfied thesystem (9) is globally asymptotically stable

Proof According to (17) s(119905) = 03times1

can be rewritten in scalarform

119890119894+ 119888119894119890119894+ 119886119894119890minus120582119905

= 0 119894 = 1 2 3 (19)

If 119888119894

= 120582 the differential equations can be solved as

119890119894(119905) =

119888119894119890119894(0)

119888119894minus 120582

119890minus120582119905

minus120582119890119894(0)

119888119894minus 120582

119890minus119888119894119905

le10038161003816100381610038161003816100381610038161003816

119888119894119890119894(0)

119888119894minus 120582

minus120582119890119894(0)

119888119894minus 120582

10038161003816100381610038161003816100381610038161003816119890minus120589119905

(20)

where 120589 = min(119888119894 120582) and thus the system (9) is globally expo-

nentially stableIf 119888119894= 120582 the differential equation can be solved as

119890119894(119905) = 119890

minus119888119894119905 (119888119894119905 + 1) 119890

119894(0) (21)

Because lim119905rarrinfin

119890119894(119905) rarr 0 the system (9) is globally asymp-

totically stableIn conclusion the asymptotic stability of the system (9)

is guaranteed when sliding mode s(119905) = 03times1

is satisfied Thiscompletes the proof

Remark 3 To simplify the selection procedure the fourparameters 119888

1 1198882 1198883 and 120582 in (17) are set equal so that

each of them is able to determine the sliding surface Asthe parameters become larger the rate of tracking error isfaster and the control input is required to be larger Howevercontrol input in real situation could not always be biggeras a faster convergence rate requires As a result a trade-off between control input and convergence rate is necessarywhich can be achieved by trial-and-error method


42 Disturbance Observer Design The first order derivativeof the sliding surface s is

s = e + ce minus 120582aeminus120582119905

= minusyc + y + ce minus 120582aeminus120582119905

= minusb + k + Δk

(22)

where b = yc minus ce + 120582aeminus120582119905Hence the control vector k can be expressed as

k = b minus Δk + s (23)

The sliding variable dynamics (22) is sensitive to theunknown bounded term Δk However the detailed infor-mation of Δk in (23) is unavailable To estimate thelumped uncertainty the robust differentiator technique [34]is employed

Assumption 4 1199041 1199042 1199043are measured by Lebesgue-measur-

able noise bounded 1205761

gt 0 1205762

gt 0 1205763

gt 0 respectivelyFurthermore k b are assumed to be bounded and Lebesguemeasurable respectively and the lumped uncertainty Δk is 2times differentiable and bounded

Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and

z2= [11991121 11991122 11991123]119879 as the estimated values of state variables

and the observer can be expressed as [34]

z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]

times sgn (z0minus s) + z

1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]

times sgn (z1minus ^0) + z2

z2= minus1205822L sgn (z

2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and

L = diag(1198971 1198972 1198973) are the parameters to be selected

Lemma 5 (see [34]) Suppose Assumption 4 is satisfied DO(24) is finite-time stable The following inequalities can beestablished in finite time

1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)

where 1205830119894 1205831119894 1205832119894 119894 = 1 2 3 are positive constants

Remark 6 The proof of Lemma 5 is similar to the studies ofShtessel et al [34] and is not presented in this paper Theparameters 120582

0 1205821 1205822can be chosen recursively and the sim-

ulation-checked set 8 5 3 is suitable for the observer design[34 42]

In absence of measurement noise the exact equalities canbe established in a finite time

z0 = s z1 = Δk z2 = Δk (26)

After DO is constructed the control vector k can bemodified as

k = b minus z1+ s (27)

43 Reaching Law Design Before giving the reaching lawdesign three lemmas to be used are presented

Lemma 7 (see [43]) Consider the system of differential equa-tions

(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630

rarr R119899 is continuous on an openneighborhood 119863

0containing the origin 119891(0) = 0

Suppose there exists a continuous positive definite function119881 119863

0rarr R In addition there exist real numbers 119896 gt 0

120572 isin (0 1) and an open neighborhood of the origin 1198631

sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)

Then the origin is a finite-time stable equilibrium of system(28) The settling time 119905 is depended on the initial value 119909

0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)

Furthermore if 119863 = 1198630

= R119899 the origin is a globally finite-time stable equilibrium of system (28)

Lemma8 (see [44]) Suppose there exists a continuous positivedefinite function 119881 119863

0rarr R In addition there exist real

numbers 119896 119897 gt 0 120572 isin (0 1) and an open neighborhood of theorigin 119863

1sube 1198630satisfies

+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)



Lemma 9 (see [45]) For 119909119894isin R 119894 = 1 119899 0 lt 119901 le 1 is a

real number and the inequality holds

(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)

Consider the reaching law with saturation function

s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904

3)]119879 stands for the saturation function

that is used to attenuate the chattering problem and sat(119904119894)

119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)

where h = diag(ℎ1 ℎ2 ℎ3) and ℎ

119894is the boundary layer thick-

nessSubstitute (34) into (27) the control algorithm of BTVS-

MCDO can be expressed as

v = b minus z1minus 120578 sat (s) (36)

Theorem 10 Based on Assumption 4 the attitude controlproblem described in (9) can be solved by BTVSMCDO (36)Furthermore the attitude tracking error e is asymptoticallystable if the exact estimate of Δk is available through the DO

Proof Consider the Lyapunov function candidate

1198811=

1

2s119879s (37)

According to (37) and (22) the time derivative of 1198811is

1= s119879 s

= s119879 (minusb + v + Δv) (38)

Substituting (36) into (38) gives

1= s119879 (minus120578 sat (119904) minus z

1+ Δv) (39)

According to Lemma 5 DO (24) is finite-time stablehence we suppose there exists a moment 119905 = 119905ob whichsatisfies z

1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)

In view of (35) consider the following two cases

(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|

Hence it is obvious that

1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816

le minus120594radic1198811

(41)

where 120594 = min(1205781 1205782 1205783) according to Lemma 7 the trajec-

tory of system will be driven into the related sliding surfaces = 03times1

in a finite time 119905119903

119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)

where 1198811(119905ob) is the value of 1198811 at 119905 = 119905ob

According to Lemma 2 once the slide mode s = 03times1

isestablished the system (9) is globally asymptotically stableThis completes the proof

Generally a thicker boundary layer (larger values of ℎ119894

119894 = 1 2 3) contributes to smaller chattering however thestatic error inside the boundary layer may be large Since theboundary layer method may result in the erosion of robust-ness and precision a novel second-order SMC is pro-posedin this paper

Consider the reaching law

s = minus k1sig(119898minus1)119898 (s) sgn (s) minus k

2s + 120591

= k3sig(119898minus2)119898 (s) sgn (s) minus k

4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894

isin R+ 119894 = 1 2 3 And sig119898(s) is defined as

sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)

Substitute (43) into (27) the related control algorithm ofSOTVSMCDO is given by

v = b minus z1minus k1sig(119898minus1)119898 (s) sgn (s) minus k

2s

+ int (k3sig(119898minus2)119898 (s) sgn (s) minus k

4s)

(45)

Remark 11 In view of (43) when the systemrsquos initial state isclose to equilibrium point (s

119894= 0 119894 = 1 2 3) the conver-

gence speed mainly depends on the nonlinear terms Other-wise when the systemrsquos initial state is far from equilibrium


point the convergence speed mainly depends on the linearterms Hence the system can hold a fast convergence speedwhether the initial state is close to equilibrium point or not

Theorem 12 Suppose that the following inequality holds

1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)

Based on Assumption 4 the attitude control problem describedin (9) can be solved by the SOTVSMCDO (45) Furthermorethe attitude tracking error e is asymptotically stable if the exactestimate of Δk is available through the DO

Proof The Lyapunov function can be expressed as

1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)

According to [46] 1198812119894is defined as

1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)

Construct the vector 120585119894= [|119904119894|(119898minus1)119898 sgn(119904

119894) 119904119894 120591119894]119879

then1198812119894and the derivative

2119894can be expressed as

1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)

1198812119894is continuous positive function

120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)

where sdot denotes the Euclidean norm 120582minsdot and 120582maxsdotdenote the minimum and maximum eigenvalues of therelated matrix respectively

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)

According to (46)Γ1119894andΓ2119894are positive definitematrix-

es hence 2119894is negative

2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172

le minus(120582min Λ

119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894

= minus1205941119894119881(2119898minus3)(2119898minus2)

2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1

(minus1205941119894119881(2119898minus3)(2119898minus2)

2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423

Since 119898 gt 2 0 lt (2119898 minus 3)(2119898 minus 2) lt 1 According toLemma 9

1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)

Similar to the proof of Theorem 10 according toLemma 8 the trajectory of system will be driven into therelated sliding surface s = 0

3times1in a finite time 119905

119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)

where 119905ob is the moment which satisfies z1

= Δv 119905 ge 119905ob1198812(119905ob) is the value of 1198812 at 119905 = 119905obAccording to Lemma 2 once the slide mode s = 0

3times1is

established the system (9) is globally asymptotically stableThis completes the proof


5 Control Allocation

With low atmosphere density reentry vehicle suffers pooraerodynamic maneuverability at high altitude In such casethe combination of RCS jets and aerodynamic surfaces isconsidered to meet the control performances To ensure thatthe command control torque M

119888can be produced jointly

by the actuators input 120575119888 a control allocation approach is

designed in this paperThe core of the control allocation problem is to solve the

nondeterministic system equations with typical constraintsSuppose that the number of aerodynamic surface is 119899

1 and

the number of RCS jets is 1198992

M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes

the vector of aerodynamic surface deflection and 120575RCS =

[120575RCS1 120575RCS2 120575RCS1198992] stands for the vector of RCS thrusterstates The matrixD(sdot) can be expressed as

D (sdot) = [D119860(sdot) DRCS (sdot)] (58)

where D119860(sdot) and DRCS(sdot) stand for aerodynamic torque

matrix and RCS torque matrix respectivelyThe typical constraints for the control allocation problem

are commonly defined as

120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575

119860119894max are the lower boundary and upperboundary of aerodynamic surface 120575

119860119894 respectively 120575

119889-119860119894minand 120575119889-119860119894max are the lower boundary and upper boundary of

deflection rate 120575119860119894 respectively

The optimizationmethod can be used to solve the nonde-terministic system equations The primary object of the con-trol allocation is to minimize the difference between com-mand control torque M

119888and the torque produced by actua-

tors [39] Moreover another objective is to minimize the useof RCS jets Hence the cost function can be expressed as

min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W

2isin R1198992 are the weights to be designed

Therefore the control allocation problem is transformedinto optimization problem tominimize the cost function (60)subject to (59)

In practical on-off RCS jets can only provide the maxi-mum torque or zero torque Thus 120575RCS should be defined asbinary variables This paper employs the PWPF modulatorto convert the continuous signal into on-off RCS commandsAs shown in Figure 2 PWPFmodulator consists of a low passfiler and a Schmitt trigger inside a feedback loop 119870

119898and 119879

119898

are the low pass filer gain 119906on and 119906off are the on-value andoff-value of Schmitt trigger

6 Numerical SimulationResults and Assessment

In order to verify the effectiveness of proposed controlmethods the comparisons between proposed control meth-ods and two conventional methods are presented The twoconventional methods are FBL and boundary layer methodbased time-varying sliding mode control (BTVSMC)

The control algorithm of FBL can be expressed as [40]

v = yc minus 119896119901e minus 119896119889e (61)

where the parameters 119896119901and 119896119889should be a positive value

As stated earlier bound layer method is a conventionalmethod to alleviate chattering phenomenon of SMC SinceDO is not employed in the control method the controlalgorithm of BTVSMC is given by [17]

k = b minus 120578 sat (s) (62)

Thenumerical tests in this paper employ a reentry vehiclewhose moments of inertia are 119868

119909= 588791 kgsdotm2 119868

119910=

1303212 kgsdotm2 and 119868119911= 1534164 kgsdotm2 and the products of

inertia are 119868119911119909

= 119868119909119911

= 24242 kgsdotm2 In addition the vehiclehas a lifting-body configuration with 8 aerodynamic surfacesand 10 RCS jets The aerodynamic surfaces include left outerelevon 120575elo right outer elevon 120575ero left inner elevon 120575eli rightinner elevon 120575eri left flap 120575fl and right flap 120575fr left rudder120575rl and right rudder 120575rr [47] The constraints of the vector120575119860

= [120575elo 120575ero 120575eli 120575eti 120575fl 120575fr 120575rl 120575rr]119879 are given by

120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)

where 120575119860max and 120575119860min are measured in degree and 120575

119889-119860maxand 120575

119889-119860119894min are measured in degree per secondEach RCS jet of reentry vehicle can produce 3559N of

thrust In addition the RCS torque matrix DRCS(sdot) is definedas [48]

DRCS (sdot) =[[

[

0 minus2048 11625 minus6912 0 2054 minus11623 6912 0 0

minus498 0 minus9466 10944 minus498 0 minus9465 11798 minus498 minus498

19897 minus15723 minus9465 minus11798 minus19897 15723 9465 11798 minus19897 19897

]]

]

(64)


Table 1 Sliding mode control parameters

Parametercontroller BTVSMC BTVSMCDO SOTVSMCDO

Sliding surface parameters c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2

Control parameters 120578 = diag(01 01 01)h = diag(0015 0015 0015)

120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms

Figure 2 PWPF modulator

The initial conditions for reentry vehicle are taken asfollows the altitude ℎ = 550 km Mach number Ma = 98Ω0

= [320∘

20∘

580∘

]119879 and w

0= [00

∘

s 00∘s 00∘s]119879And attitude angle commands are set to be Ω

119888= [300

∘

00∘

600∘]119879The reentry vehicle suffers high structural stresses for

high Mach number and bad aerodynamics for large AOAand hence the rudders are not allowed to be used [10]

Furthermore additional constraints for control allocationproblem should be satisfied

120575rl = 120575rr = 0 (65)

The planet uncertainties are set in consideration of 5percent bias conditions for moments of inertia and productsof inertia 10 percent bias conditions for aerodynamic coeffi-cients and 10 percent bias conditions for atmospheric densityIn addition the external disturbance torque vector takes theform of

ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]

times 104N sdot m (66)

To validate the robustness and the chattering reductionof the proposed methods numerical simulations of FBL (61)BTVSMC (62) BTVSMCDO (36) and SOTVSMCDO (45)are presentedThe overall attitude control system architecturefor reentry is showed in Figure 1 And the integration step is

Table 2 Control allocation parameters

Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015

specified as 001 seconds In FBL (61) the control parametersare selected as 119896

119901= 119896119889

= 3 In addition the control param-eters of the sliding mode controllers are specified in Table 1After the command control torque vector M

119888 is produced

by these controllers the actuator command is obtained bycontrol allocation algorithm as presented in Section 5 Theparameters of control allocation problem are as shown inTable 2

The variations of the attitude angles including AOAsideslip angle and bank angle under FBL BTVSMC BTVS-MCDO and SOTVSMCDO are shown in Figure 3 It isobvious that SMC has significant robustness performance inthe presence of uncertainties and disturbances Since FBLrelies on the knowledge of the exact model dynamics thetracking errors under FBL do not converge to zero Thesecond row of Figure 3 shows the attitude angle evolutionsin steady-state region Compared with FBL and BTVSMCBTVSMCDO and SOTVSMCDO achieve the goals oftracking with higher accuracy

To further evaluate the tracking performance betweenBTVSMCDO and SOTVSMCDO the local view of attitudeangle evolutions via BTVSMCDO and SOTVSMCDO isshown in Figure 4 The results suggest that all the attitudeangles converge to the desired values within 4 seconds inthe transient region for both methods As shown in the localview of attitude angles in the steady-state region the trackingerrors under SOTVSMCDO are much smaller than thoseunder BTVSMCDO


0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s) Time (s) Time (s)

Time (s)Time (s)Time (s)

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

Figure 3 Comparison of attitude angle evolutions via FBL BTVSMC BTVSMCDO and SOTVSMCDO

0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

Figure 4 Local view of attitude angle evolutions via BTVSMCDO and SOTVSMCDO

The sliding surface responses via BTVSMC BTVSMCDO and SOTVSMCDO are shown in Figure 5 It is obviousthat the sliding surface under BTVSMC does not converge tozero because DO is not introduced The system trajectories

under BTVSMCDO and SOTVSMCDO move apart fromzero after the start point and move back to zero around4 seconds later The second row of Figure 5 shows thelocal view of steady-state region under BTVSMCDO and


0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus3 times10minus3times10minus4

times10minus5times10minus6 times10minus6

BTVSMCBTVSMCDOSOTVSMCDO



s 1s 1

s 2s 2

s 3s 3

Figure 5 Comparison of sliding surface responses via BTVSMC BTVSMCDO and SOTVSMCDO

SOTVSMCDO As boundary layer method is applied thesliding surface stays inside of the boundary layer howeverthe tracking accuracy within the boundary layer is notguaranteed As a result the sliding surface under SOTVSMCDO converges to zero more precisely

The estimations of the sliding surface by BTVSMCDOand SOTVSMCDO are shown in Figures 6(a) and 6(b)respectively In addition Figure 7 depicts the estimations ofuncertainty terms Clearly all of sliding surfaces and lumpeduncertainty can be estimated by DO effectively

Figure 8 illustrates the comparison of attitude angularrate evolutions via FBL BTVSMC BTVSMCDO and SOT-VSMCDO And the produced command control torquesare showed in Figure 9 Control chattering is undesirable inpractice because it involves high control activity and mayexcite high frequency unmolded dynamics As is shown inFigure 9 the problem of chattering phenomenon for SMC iseliminated in the cases of BTVSMC BTVSMCDO and SOT-VSMCDO

This paper formulates all the numerical simulations inconsideration of the constraints for aerodynamic deflectionsand RCS thrusts both of which are actuator command andcan be obtained by control allocation algorithm presentedin Section 5 The aerodynamic deflections via BTVSMCDOand SOTVSMCDO are shown in Figures 10(a) and 11(a)respectively The rudders are kept around zero All theactuator limits in (59) and (65) are satisfied The RCS thrust

commands via BTVSMCDO and SOTVSMCDO are shownin Figures 10(b) and 11(b) respectivelyThe RCS thrusts com-pensate the torque errors caused by aerodynamic deflectionsaturation with high accuracy

7 Conclusion

This paper investigates the finite-time control problem ofreentry vehicle with aerodynamic surfaces and RCS jets andseeks for more reliable attitude controller design and thecontrol allocation design The numerical simulation resultsprove the tracking accuracy and robustness of the proposedattitude controller and control allocation method

For attitude controller design two kinds of robust finite-time TVSMC controllers are proposed Time-varying slidingsurface is employed to eliminate the reaching phase of transitdynamics and thus the global robustness is guaranteed ADO is introduced to enhance the robustness against planetuncertainness and external disturbances To alleviate thechattering boundary layer method and second-order SMCmethod are employed respectively Both of the proposedmethods can avoid chattering phenomenon effectivelyMore-over the tracking error under SOTVSMCDO converges tozero more precisely

A control allocation approach is introduced to gener-ate the actuator commands including aerodynamic surfacedeflections and on-off RCS thrusts The nondeterministic


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5

RealEstimatedEstimation error

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4

times10minus4 times10minus4

times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3



(a) Comparison between s and z0under BTVSMCDO

0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2




times10minus4times10minus4

times10minus4 times10minus4 times10minus4

times10minus3



s 1s 1

s 2s 2 s 3

s 3

(b) Comparison between s and z0under SOTVSMCDO

Figure 6 The estimations of sliding surface by BTVSMCDO and SOTVSMCDO


0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3

(a) Comparison between Δv and z1under BTVSMCDO

0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3

(b) Comparison between Δv and z1under SOTVSMCDO

Figure 7 The estimations of the uncertainty terms by BTVSMCDO and SOTVSMCDO


0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

Figure 8 Comparison of attitude angular rate evolutions via FBL BTVSMC BTVSMCDO and SOTVSMCDO

0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)

times104 times104 times104

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

Figure 9 Comparison of command control torque via FBL BTVSMC BTVSMCDO and SOTVSMCDO

system equations are solved by optimization methods andthe proposedweighted cost function is capable ofminimizingboth the utilization of RCS jets and the differences betweenthe desired torque and the torque produced by actuators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5

(a) Elevon flap and rudder commands

01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands

Figure 10 Aerodynamic surface commands and RCS commands via BTVSMCDO


minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands

Figure 11 Aerodynamic surface commands and RCS commands via SOTVSMCDO


Acknowledgments

The authors would like to appreciate the editor and all theanonymous reviewers for their comments which helped toimprove the quality of this paperThis studywas supported byNational Natural Science Foundation of China (11202024)

References

[1] Z Shen and P Lu ldquoOnboard generation of three-dimensionalconstrained entry trajectoriesrdquo Journal of Guidance Controland Dynamics vol 26 no 1 pp 111ndash121 2003

[2] R Smith and A Ahmed ldquoRobust parametrically varying atti-tude controller designs for the X-33 vehiclerdquo inAIAAGuidanceNavigation and Control Conference and Exhibit Denver ColoUSA 2000 AIAA-2000-4158

[3] W J Rugh and J S Shamma ldquoResearch on gain schedulingrdquoAutomatica vol 36 no 10 pp 1401ndash1425 2000

[4] D J Leith and W E Leithead ldquoSurvey of gain-schedulinganalysis and designrdquo International Journal of Control vol 73 no11 pp 1001ndash1025 2000

[5] R M Agustin R S Mangoubi R M Hain and N J AdamsldquoRobust failure detection for reentry vehicle attitude controlsystemsrdquo Journal of Guidance Control and Dynamics vol 22no 6 pp 839ndash845 1999

[6] P Lu ldquoRegulation about time-varying trajectories precisionentry guidance illustratedrdquo Journal of Guidance Control andDynamics vol 22 no 6 pp 784ndash790 1999

[7] A Rahideh A H Bajodah and M H Shaheed ldquoReal timeadaptive nonlinear model inversion control of a twin rotorMIMO systemusing neural networksrdquo Engineering Applicationsof Artificial Intelligence vol 25 no 6 pp 1289ndash1297 2012

[8] W R van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[9] H Xu M D Mirmirani and P A Ioannou ldquoAdaptive slidingmode control design for a hypersonic flight vehiclerdquo Journal ofGuidance Control and Dynamics vol 27 no 5 pp 829ndash8382004

[10] R R Da Costa Q P Chu and J A Mulder ldquoReentry flightcontroller design using nonlinear dynamic inversionrdquo Journalof Spacecraft and Rockets vol 40 no 1 pp 64ndash71 2003

[11] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999

[12] A Pisano and E Usai ldquoSliding mode control a survey withapplications in mathrdquo Mathematics and Computers in Simula-tion vol 81 no 5 pp 954ndash979 2011

[13] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[14] O Barambones Caramazana P Alkorta Egiguren and J MGonzalez de Durana Garcıa ldquoSliding mode position control forreal-time control of induction motorsrdquo International Journal ofInnovative Computing Information andControl vol 9 pp 2741ndash2754 2013

[15] L Wu X Su and P Shi ldquoSliding mode control with boundedL2gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012

[16] Y Shtessel C Tournes and D Krupp ldquoReusable launch vehiclecontrol in sliding modesrdquo in Proceedings of the Guidance Nav-igation and Control Conference AIAA-1997-3533 AmericanInstitute of Aeronautics and Astronautics 1997

[17] Y Shtessel J McDuffie M Jackson et al ldquoSliding modecontrol of the X-33 vehicle in launch and re-entry modesinrdquo in Proceedings of the Guidance Navigation and ControlConference and Exhibit AIAA-1998-4414 American Institute ofAeronautics and Astronautics 1998

[18] Y Shtessel C Hall and M Jackson ldquoReusable launch vehiclecontrol in multiple-time-scale sliding modesrdquo Journal of Guid-ance Control and Dynamics vol 23 no 6 pp 1013ndash1020 2000

[19] Y Shtessel J Zhu andDDan ldquoReusable launch vehicle attitudecontrol using time-varying sliding modesrdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit AIAA-2002-4779 American Institute of Aeronauticsand Astronautics 2002

[20] M Defoort T Floquet A Kokosy andW Perruquetti ldquoA novelhigher order sliding mode control schemerdquo Systems amp ControlLetters vol 58 no 2 pp 102ndash108 2009

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[22] H Sun S Li and C Sun ldquoFinite time integral sliding modecontrol of hypersonic vehiclesrdquoNonlinear Dynamics vol 73 no1-2 pp 229ndash244 2013

[23] MR Soltanpour B ZolfaghariM Soltani andMHKhoobanldquoFuzzy sliding mode control design for a class of nonlin-ear systems with structured and unstructured uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 7 pp 2713ndash2726 2013

[24] X Su P Shi L Wu and M V Basin ldquoReliable filtering withstrict dissipativity for T-S fuzzy time-delay systemsrdquo IEEETransactions on Cybernetics 2014

[25] H Lee and V I Utkin ldquoChattering suppression methods insliding mode control systemsrdquo Annual Reviews in Control vol31 no 2 pp 179ndash188 2007

[26] H Lee E Kim H-J Kang and M Park ldquoA new sliding-modecontrol with fuzzy boundary layerrdquo Fuzzy Sets and Systems vol120 no 1 pp 135ndash143 2001

[27] A Levant Higher order sliding modes and their application forcontrolling uncertain processes [PhD thesis] Institute for SystemStudies of the USSR Academy of Science Moscow Russia 1987

[28] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[29] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[30] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997

[31] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998

[32] K Ohishi M Nakao K Ohnishi and K Miyachi ldquoMicropro-cessor-controlled DC motor for load-insensitive position servosystemrdquo IEEE Transactions on Industrial Electronics vol IE-34no 1 pp 44ndash49 1985

[33] C E Hall and Y B Shtessel ldquoSliding mode disturbanceobserver-based control for a reusable launch vehiclerdquo Journal ofGuidance Control and Dynamics vol 29 no 6 pp 1315ndash13282006


[34] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[35] S N DrsquoSouza and N Sarigul-Klijn ldquoSurvey of planetary entryguidance algorithmsrdquo Progress in Aerospace Sciences vol 68 pp64ndash74 2014

[36] P Lu ldquoEntry guidance a unified methodrdquo Journal of GuidanceControl and Dynamics vol 37 no 3 pp 713ndash728 2014

[37] B Tian Q Zong J Wang and F Wang ldquoQuasi-continuoushigh-order sliding mode controller design for reusable launchvehicles in reentry phaserdquo Aerospace Science and Technologyvol 28 no 1 pp 198ndash207 2013

[38] N X Vinh A Busemann and R D Culp Hypersonic andPlanetrary Entry Flight Mechanics The University of MichiganPress 1980

[39] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[40] A J Krener ldquoFeedback linearizationrdquo in Mathematical ControlTheory J Baillieul and J C Willems Eds pp 66ndash98 SpringerNew York NY USA 1999

[41] C Binglong L Xiangdong and C Zhen ldquoExponential time-varying sliding mode control for large angle attitude eigenaxismaneuver of rigid spacecraftrdquo Chinese Journal of Aeronauticsvol 23 no 4 pp 447ndash453 2010

[42] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[43] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[44] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems Sciencevol 43 no 1 pp 73ndash78 2012

[45] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1952

[46] J A Moreno and M Osorio ldquoA Lyapunov approach to second-order sliding mode controllers and observersrdquo in Proceedings ofthe 47th IEEE Conference on Decision and Control (CDC rsquo08)pp 2856ndash2861 December 2008

[47] K P Bollino High-Fidelity Real-Time Trajectory Optimizaitonfor Reusable Launch Vehicles Naval Postgraduate School 2006

[48] D B Doman B J Gamble and A D Ngo ldquoQuantized controlallocation of reaction control jets and aerodynamic controlsurfacesrdquo Journal of Guidance Control and Dynamics vol 32no 1 pp 13ndash24 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


incorporated neural network (NN) based compensation inthe FBL design Van Soest et al [8] combined FBL with con-strained linear model predictive control (MPC) method Xuet al [9] utilized the combination of FBL and adaptive slidingmode control (SMC) method

Among the various upgraded nonlinear control methodsSMC outstands with many advantages such as simplicityof implementation fast dynamic response good transientbehavior exponential stability insensitivity to parameter var-iations and robustness to plant uncertainties and externaldisturbances [11ndash14] Therefore SMC has been successfullyapplied to a variety of complex engineering systems [15]Barambones Caramazana et al [14] develop a sliding modeposition control incorporating a flux estimator for high-performance real-time applications of induction motors Wuet al [15] investigate the key problems of SMC of Markovianjump singular time-delay systems Shtessel et al [16ndash19] stud-ied the application of SMCmethod to reusable launch vehicle(RLV) in launch and reentrymode and amultiple-time-scaleSMC strategy is proposed in [18] for RLV in ascent phase

Generally SMC design consists of two steps [20 21] (1)select a sliding surface as a function of the system statesso that the system trajectories along the surface meet thedesired performance such as stability and tracking capability(2) design a suitable control law to drive the states onto thepredefined sliding surface in finite timeWhen it comes to thedesign of conventional SMC there are two major problemsconcerned One is its unguaranteed global robustness andthe other is chattering phenomenonThe conventional slidingsurfaces [16ndash18] employ linear function of tracking errorswhich results in the fact that the transient dynamics of SMCconsists of reaching phase and sliding phase However theSMC method can only ensure the robustness against planetuncertainties and external disturbances in sliding phaseTherefore the conventional sliding surfaces do not possessthe property of global robustness Several studies are dedi-cated to global robustness of SMC Sun et al [22] introducedan integral sliding mode control (ISMC) method to solvethe longitudinal control problem of air-breathing hypersonicvehicle (AHV) Shtessel et al [19] proposed a two-loop con-troller that utilized a time-varying sliding mode control(TVSMC) method to achieve fault tolerance for RLV attitudecontrol With the elimination of reaching phase both ISMCand TVSMC can keep the system states on the sliding surfacefrom the initial time so that global robustness against planetuncertainties and external disturbances is guaranteed

As to the chattering phenomenon it is assumed that thecontrol can be switched from one structure to anotherinfinitely fast in the design of SMC [23] However it is impos-sible to achieve high-speed switching control because of theinevitable switching delay computation and the limitation ofthe physical actuatorsThe existence of time delay introducesinstability oscillation and poor performance [24] High con-trol gains of SMC lead to high frequency oscillations knownas chattering phenomenon This harmful phenomenon mayerode the performance to gain robustness decrease the con-trol accuracy and damage the actuators [25]There are essen-tially two ways to alleviate the chattering phenomenon [23]one way is boundary layermethod [17ndash19 26] and the other is

higher order sliding mode control (HOSMC) The boundarylayer method replaces the sign function (discontinuous con-trol) with smooth approximations such as high-gain satura-tion function or sigmoid function Nevertheless this methodno longer drives the system state to the origin and cannotguarantee the robustness and accuracy within the boundarylayer [23] HOSMC was proposed by Levant [27] Instead ofinfluencing the first order time derivative the discontinuouscontrol acts on sliding variablersquos higher order derivative Asa special case of HOSMC second-order sliding mode control(SOSMC) is themost popular approach in engineeringThereare many kinds of SOSMC such as twisting algorithm [28]super-twisting algorithm [29] suboptimal algorithm [30 31]and prescribed convergence law algorithm [28]

Disturbance observer (DO) is an effective way to enhancesystem robustness The disturbance estimation is used forcompensation DO was first proposed by Ohishi et al [32]Hall and Shtessel [33] combined SMC and sliding modedisturbance observer (SMDO) which estimates the boundeduncertainties and disturbances effectively to improve RLVattitude control Shtessel et al [34] proposed a homogeneousDO based on the standard robust exact differentiator to solvethe missile guidance problem

Inspired by previous work this paper proposes twoTVSMC methods to solve the finite-time attitude controlproblem by incorporating the disturbance observer One isBTVSMCDO which is the abbreviation for boundary layermethod based time-varying sliding mode controller withdisturbance observer and the other is SOTVSMCDO whichmeans the second-order time-varying sliding mode con-troller with disturbance observer With the same dedicationto systematically enhance robustness and suppress controlchattering the two methods adopt different ways to alleviatechattering The former is designed based on boundary layertechnique and the latter utilizes a novel SOSMC The maincontributions of this paper are summarized as follows

(1) This paper incorporates a novel reaching law basedon SOSMCwith the time-varying sliding function Inorder to enhance the robustness of the method a DObased on the standard robust exact differentiator isemployed to estimate the systemrsquos uncertainties anddisturbances in finite time In addition the finite-time convergence of time-varying sliding function forresulted method is proved via Lyapunov theory andconsequently the asymptotical stability of the closed-loop nonlinear system is proved according to thedefinition of the time-varying sliding function

(2) Since RVs deploy both aerodynamic surfaces andRCS jets this paper introduces a control allocationapproach to assign control responsibility amongstredundant actuators The nonlinear programmingproblem is established and solved by optimiza-tion method and the pulse-width-pulse-frequency(PWPF) is employed tomodulate the on-off thrusters

(3) The proposed control methods are applied to finite-time attitude control problem for RVs Numericalsimulation results confirm the validity and superiorperformance of the proposed control methods by




2 Preliminary



119888 and bank


119888



119889120572


+sin120590





119889120573

119889119905= 119901 sin120572 minus 119903 cos120572




119889120590




(1)




119909119910= 119868119910119911



d119901d119905

=119868119911119872119897+ 119868119911119909

119872119899+ 119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682

119911minus 1198682

119911119909) 119902119903

119868119909119868119911minus 1198682119911119909

d119902d119905

=119872119898

+ (119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

d119903d119905

=119868119911119909

119872119897+ 119868119909119872119899+ (minus119868

119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

(2)

where 119872119897 119872119898 and 119872



119911




119890= 0


Ω = R120596 + Δf


(3)



119879



1 Δ1198912 Δ1198913 ]119879



R =[[

[



]]

]

120596times

=[[

[

0 minus119903 119902

119903 0 minus119901

119902 119901 0

]]

]

I =[[

[

119868119909

0 minus119868119911119909

0 119868119910

0

minus119868119911119909

0 119868119911

]]

]

(4)


ΔI120596 + ΔM] (5)




= [119872119897119872119898



lim119905gt119905119865

e = lim119905gt119905119865

1003817100381710038171003817Ω minusΩ119888

1003817100381710038171003817 = 0 (6)



119888to actuator


120575119888= 119891CA (M

119888) (7)


119888119886


119888+ ΔM




ΔI120596 + ΔM + ΔM119888119886] (8)

Guidance system

Attitude controller

Control allocation


Disturbance

120572c120573c120590c

120572 120573 120590

Ml

MmMn

δA

δRCS




x = f (x) + g (x) u + d

y = h (x) (9)


119879



f (x) =

[[[[[[[

[

1198911(x)

1198912(x)

1198913(x)

1198914(x)

1198915(x)

1198916(x)

]]]]]]]

]

=

[[[[[[[[[[[[[[[[

[



119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682119911minus 1198682119911119909

) 119902119903

119868119909119868119911minus 1198682119911119909

(119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

(minus119868119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]]]]

]

(10)

g (x) = [1198921(x) 119892

2(x) 119892

3(x)]

=

[[[[[[[[[[[[[

[

0 0 0

0 0 0

0 0 0119868119911

119868119909119868119911minus 1198682119911119909

0119868119911119909

119868119909119868119911minus 1198682119911119909

01

119868119910

0

119868119911119909

119868119909119868119911minus 1198682119911119909

0119868119909

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]

]

(11)





K =[[

[

1198711198921119871119891ℎ1(x) 119871

1198922119871119891ℎ1(x) 119871

1198923119871119891ℎ1(x)

1198711198921119871119891ℎ2(x) 119871

1198922119871119891ℎ2(x) 119871

1198923119871119891ℎ2(x)

1198711198921119871119891ℎ3(x) 119871

1198922119871119891ℎ3(x) 119871

1198923119871119891ℎ3(x)

]]

]

B = [1198712

119891ℎ1(x) 1198712

119891ℎ2(x) 1198712

119891ℎ3(x)]119879

F =[[

[


119901 cos120572 + 119903 cos120572 1 0


]]

]

(13)



119868119909119868119911minus 1198682119911119909

asymp1

119868119909119868119911minus 1198682119911119909

= 0 (14)






y = k + Δk (16)




s = e + ce + aeminus120582119905 (17)


[1198901 1198902 1198903]119879



gt 0 119894 = 1 2 3 120582 isin R+ a =








119890119894+ 119888119894119890119894+ 119886119894119890minus120582119905

= 0 119894 = 1 2 3 (19)

If 119888119894


119890119894(119905) =

119888119894119890119894(0)

119888119894minus 120582

119890minus120582119905

minus120582119890119894(0)

119888119894minus 120582

119890minus119888119894119905

le10038161003816100381610038161003816100381610038161003816

119888119894119890119894(0)

119888119894minus 120582

minus120582119890119894(0)

119888119894minus 120582

10038161003816100381610038161003816100381610038161003816119890minus120589119905

(20)



119890119894(119905) = 119890

minus119888119894119905 (119888119894119905 + 1) 119890

119894(0) (21)













= minusb + k + Δk

(22)






gt 0 1205762

gt 0 1205763


Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and



z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]


1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]



2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and



1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)






z0 = s z1 = Δk z2 = Δk (26)





(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630






sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)


0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)







+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 Preliminary



119888 and bank


119888



119889120572


+sin120590





119889120573

119889119905= 119901 sin120572 minus 119903 cos120572




119889120590




(1)




119909119910= 119868119910119911



d119901d119905

=119868119911119872119897+ 119868119911119909

119872119899+ 119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682

119911minus 1198682

119911119909) 119902119903

119868119909119868119911minus 1198682119911119909

d119902d119905

=119872119898

+ (119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

d119903d119905

=119868119911119909

119872119897+ 119868119909119872119899+ (minus119868

119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

(2)

where 119872119897 119872119898 and 119872



119911




119890= 0


Ω = R120596 + Δf


(3)



119879



1 Δ1198912 Δ1198913 ]119879



R =[[

[



]]

]

120596times

=[[

[

0 minus119903 119902

119903 0 minus119901

119902 119901 0

]]

]

I =[[

[

119868119909

0 minus119868119911119909

0 119868119910

0

minus119868119911119909

0 119868119911

]]

]

(4)


ΔI120596 + ΔM] (5)




= [119872119897119872119898



lim119905gt119905119865

e = lim119905gt119905119865

1003817100381710038171003817Ω minusΩ119888

1003817100381710038171003817 = 0 (6)



119888to actuator


120575119888= 119891CA (M

119888) (7)


119888119886


119888+ ΔM




ΔI120596 + ΔM + ΔM119888119886] (8)

Guidance system

Attitude controller

Control allocation


Disturbance

120572c120573c120590c

120572 120573 120590

Ml

MmMn

δA

δRCS




x = f (x) + g (x) u + d

y = h (x) (9)


119879



f (x) =

[[[[[[[

[

1198911(x)

1198912(x)

1198913(x)

1198914(x)

1198915(x)

1198916(x)

]]]]]]]

]

=

[[[[[[[[[[[[[[[[

[



119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682119911minus 1198682119911119909

) 119902119903

119868119909119868119911minus 1198682119911119909

(119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

(minus119868119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]]]]

]

(10)

g (x) = [1198921(x) 119892

2(x) 119892

3(x)]

=

[[[[[[[[[[[[[

[

0 0 0

0 0 0

0 0 0119868119911

119868119909119868119911minus 1198682119911119909

0119868119911119909

119868119909119868119911minus 1198682119911119909

01

119868119910

0

119868119911119909

119868119909119868119911minus 1198682119911119909

0119868119909

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]

]

(11)





K =[[

[

1198711198921119871119891ℎ1(x) 119871

1198922119871119891ℎ1(x) 119871

1198923119871119891ℎ1(x)

1198711198921119871119891ℎ2(x) 119871

1198922119871119891ℎ2(x) 119871

1198923119871119891ℎ2(x)

1198711198921119871119891ℎ3(x) 119871

1198922119871119891ℎ3(x) 119871

1198923119871119891ℎ3(x)

]]

]

B = [1198712

119891ℎ1(x) 1198712

119891ℎ2(x) 1198712

119891ℎ3(x)]119879

F =[[

[


119901 cos120572 + 119903 cos120572 1 0


]]

]

(13)



119868119909119868119911minus 1198682119911119909

asymp1

119868119909119868119911minus 1198682119911119909

= 0 (14)






y = k + Δk (16)




s = e + ce + aeminus120582119905 (17)


[1198901 1198902 1198903]119879



gt 0 119894 = 1 2 3 120582 isin R+ a =








119890119894+ 119888119894119890119894+ 119886119894119890minus120582119905

= 0 119894 = 1 2 3 (19)

If 119888119894


119890119894(119905) =

119888119894119890119894(0)

119888119894minus 120582

119890minus120582119905

minus120582119890119894(0)

119888119894minus 120582

119890minus119888119894119905

le10038161003816100381610038161003816100381610038161003816

119888119894119890119894(0)

119888119894minus 120582

minus120582119890119894(0)

119888119894minus 120582

10038161003816100381610038161003816100381610038161003816119890minus120589119905

(20)



119890119894(119905) = 119890

minus119888119894119905 (119888119894119905 + 1) 119890

119894(0) (21)













= minusb + k + Δk

(22)






gt 0 1205762

gt 0 1205763


Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and



z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]


1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]



2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and



1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)






z0 = s z1 = Δk z2 = Δk (26)





(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630






sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)


0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)







+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879



1 Δ1198912 Δ1198913 ]119879



R =[[

[



]]

]

120596times

=[[

[

0 minus119903 119902

119903 0 minus119901

119902 119901 0

]]

]

I =[[

[

119868119909

0 minus119868119911119909

0 119868119910

0

minus119868119911119909

0 119868119911

]]

]

(4)


ΔI120596 + ΔM] (5)




= [119872119897119872119898



lim119905gt119905119865

e = lim119905gt119905119865

1003817100381710038171003817Ω minusΩ119888

1003817100381710038171003817 = 0 (6)



119888to actuator


120575119888= 119891CA (M

119888) (7)


119888119886


119888+ ΔM




ΔI120596 + ΔM + ΔM119888119886] (8)

Guidance system

Attitude controller

Control allocation


Disturbance

120572c120573c120590c

120572 120573 120590

Ml

MmMn

δA

δRCS




x = f (x) + g (x) u + d

y = h (x) (9)


119879



f (x) =

[[[[[[[

[

1198911(x)

1198912(x)

1198913(x)

1198914(x)

1198915(x)

1198916(x)

]]]]]]]

]

=

[[[[[[[[[[[[[[[[

[



119868119911119909

(119868119911+ 119868119909minus 119868119910) 119901119902 + (119868

119910119868119911minus 1198682119911minus 1198682119911119909

) 119902119903

119868119909119868119911minus 1198682119911119909

(119868119911minus 119868119909) 119901119903 + 119868

119911119909(1199032 minus 1199012)

119868119910

(minus119868119909119868119910+ 1198682119909+ 1198682119911119909

) 119901119902 + 119868119911119909

(minus119868119911minus 119868119909+ 119868119910) 119902119903

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]]]]

]

(10)

g (x) = [1198921(x) 119892

2(x) 119892

3(x)]

=

[[[[[[[[[[[[[

[

0 0 0

0 0 0

0 0 0119868119911

119868119909119868119911minus 1198682119911119909

0119868119911119909

119868119909119868119911minus 1198682119911119909

01

119868119910

0

119868119911119909

119868119909119868119911minus 1198682119911119909

0119868119909

119868119909119868119911minus 1198682119911119909

]]]]]]]]]]]]]

]

(11)





K =[[

[

1198711198921119871119891ℎ1(x) 119871

1198922119871119891ℎ1(x) 119871

1198923119871119891ℎ1(x)

1198711198921119871119891ℎ2(x) 119871

1198922119871119891ℎ2(x) 119871

1198923119871119891ℎ2(x)

1198711198921119871119891ℎ3(x) 119871

1198922119871119891ℎ3(x) 119871

1198923119871119891ℎ3(x)

]]

]

B = [1198712

119891ℎ1(x) 1198712

119891ℎ2(x) 1198712

119891ℎ3(x)]119879

F =[[

[


119901 cos120572 + 119903 cos120572 1 0


]]

]

(13)



119868119909119868119911minus 1198682119911119909

asymp1

119868119909119868119911minus 1198682119911119909

= 0 (14)






y = k + Δk (16)




s = e + ce + aeminus120582119905 (17)


[1198901 1198902 1198903]119879



gt 0 119894 = 1 2 3 120582 isin R+ a =








119890119894+ 119888119894119890119894+ 119886119894119890minus120582119905

= 0 119894 = 1 2 3 (19)

If 119888119894


119890119894(119905) =

119888119894119890119894(0)

119888119894minus 120582

119890minus120582119905

minus120582119890119894(0)

119888119894minus 120582

119890minus119888119894119905

le10038161003816100381610038161003816100381610038161003816

119888119894119890119894(0)

119888119894minus 120582

minus120582119890119894(0)

119888119894minus 120582

10038161003816100381610038161003816100381610038161003816119890minus120589119905

(20)



119890119894(119905) = 119890

minus119888119894119905 (119888119894119905 + 1) 119890

119894(0) (21)













= minusb + k + Δk

(22)






gt 0 1205762

gt 0 1205763


Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and



z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]


1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]



2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and



1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)






z0 = s z1 = Δk z2 = Δk (26)





(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630






sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)


0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)







+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













K =[[

[

1198711198921119871119891ℎ1(x) 119871

1198922119871119891ℎ1(x) 119871

1198923119871119891ℎ1(x)

1198711198921119871119891ℎ2(x) 119871

1198922119871119891ℎ2(x) 119871

1198923119871119891ℎ2(x)

1198711198921119871119891ℎ3(x) 119871

1198922119871119891ℎ3(x) 119871

1198923119871119891ℎ3(x)

]]

]

B = [1198712

119891ℎ1(x) 1198712

119891ℎ2(x) 1198712

119891ℎ3(x)]119879

F =[[

[


119901 cos120572 + 119903 cos120572 1 0


]]

]

(13)



119868119909119868119911minus 1198682119911119909

asymp1

119868119909119868119911minus 1198682119911119909

= 0 (14)






y = k + Δk (16)




s = e + ce + aeminus120582119905 (17)


[1198901 1198902 1198903]119879



gt 0 119894 = 1 2 3 120582 isin R+ a =








119890119894+ 119888119894119890119894+ 119886119894119890minus120582119905

= 0 119894 = 1 2 3 (19)

If 119888119894


119890119894(119905) =

119888119894119890119894(0)

119888119894minus 120582

119890minus120582119905

minus120582119890119894(0)

119888119894minus 120582

119890minus119888119894119905

le10038161003816100381610038161003816100381610038161003816

119888119894119890119894(0)

119888119894minus 120582

minus120582119890119894(0)

119888119894minus 120582

10038161003816100381610038161003816100381610038161003816119890minus120589119905

(20)



119890119894(119905) = 119890

minus119888119894119905 (119888119894119905 + 1) 119890

119894(0) (21)













= minusb + k + Δk

(22)






gt 0 1205762

gt 0 1205763


Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and



z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]


1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]



2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and



1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)






z0 = s z1 = Δk z2 = Δk (26)





(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630






sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)


0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)







+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













= minusb + k + Δk

(22)






gt 0 1205762

gt 0 1205763


Consider z0

= [11991101 11991102 11991103]119879 z1

= [11991111 11991112 11991113]119879 and



z0= v minus b + ^

0

^0= minus1205820L13 [[

[

100381610038161003816100381611991101 minus 1199041

100381610038161003816100381623

0 0

0100381610038161003816100381611991102 minus 119904

2

100381610038161003816100381623

0

0 0100381610038161003816100381611991103 minus 119904

3

100381610038161003816100381623

]]

]


1

z1= ^1

^1= minus1205821L12 [[

[

100381610038161003816100381611991111 minus ]01

100381610038161003816100381612

0 0

0100381610038161003816100381611991112 minus ]

02

100381610038161003816100381612

0

0 0100381610038161003816100381611991113 minus ]

03

100381610038161003816100381612

]]

]



2minus ^1)

(24)

where ^0= []01 ]02 ]03]119879 ^0= []01 ]02 ]03]119879 1205820 1205821 1205822and



1199110119894

minus 119904119894le 1205830119894120576119894

1199111119894

minus ΔV119894le 120583111989412057623

119894

1199112119894

minus ΔV119894le 120583211989412057613

119894

119894 = 1 2 3

(25)






z0 = s z1 = Δk z2 = Δk (26)





(119905) = 119891 (119909 (119905)) (28)

where 119909 isin R119899 119891 1198630






sube 1198630

satisfies

+ 119896119881120572

le 0 119909 isin 1198631 0 (29)


0

119905 (1199090) le

119881(1199090)1minus120572

119896 (1 minus 120572) (30)







+ 119896119881120572

+ 119897119881 le 0 119909 isin 1198631 0 (31)



0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

119905 (1199090) le

ln (1 + (119897119896) 119881(1199090)1minus120572

)

119896 (1 minus 120572) (32)





(10038161003816100381610038161199091

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816)119901

le10038161003816100381610038161199091

1003816100381610038161003816119901

+ sdot sdot sdot +1003816100381610038161003816119909119899

1003816100381610038161003816119901

(33)


s = minus120578 sat (s) (34)

where 120578 = diag(1205781 1205782 1205783) 120578119894

isin R+ 119894 = 1 2 3 sat(s) =

[sat(1199041) sat(119904

2) sat(119904



119894 = 1 2 3 is defined as

sat (119904119894) =

ℎminus1119894

119904119894

10038161003816100381610038161199041198941003816100381610038161003816 le ℎ119894

sgn (119904119894)

10038161003816100381610038161199041198941003816100381610038161003816 gt ℎ119894

119894 = 1 2 3 (35)








1198811=

1

2s119879s (37)


1= s119879 s

= s119879 (minusb + v + Δv) (38)



1+ Δv) (39)


1= Δv 119905 ge 119905ob

When 119905 ge 119905ob

1= s119879 (minus120578 sat (s))

=

3

sum119894=1

120578119894119904119894sat (119904119894)

(40)


(1) If |119904119894| gt ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894sgn(119904119894) = 120578119894|119904119894|

(2) If |119904119894| le ℎ

119894 119894 = 1 2 3 we can get 120578

119894119904119894sat(119904119894) =

120578119894119904119894ℎminus1119894

119904119894= 120578119894|119904119894|2

ℎminus1119894

le 120578119894|119904119894|


1le

3

sum119894=1

120578119894

10038161003816100381610038161199041198941003816100381610038161003816


(41)




119905119903le

2radic1198811(119905ob)

120594+ 119905ob

(42)








2s + 120591


4s

(43)

where 119898 gt 2 k1= diag(119896

11 11989612 11989613) k2= diag(119896

21 11989622 11989623)

k3

= diag(11989631 11989632 11989633) and k

4= diag(119896

41 11989642 11989643) with 119896

1119894

1198962119894 1198963119894 1198964119894


sig119898 (s) = [sgn (1199041)10038161003816100381610038161199041

1003816100381610038161003816119898

sgn (1199042)10038161003816100381610038161199042

1003816100381610038161003816119898

sgn (1199043)10038161003816100381610038161199043

1003816100381610038161003816119898

]119879

(44)



2s


4s)

(45)


119894= 0 119894 = 1 2 3) the conver-





1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198982

11989631198941198964119894

minus (1198983

119898 minus 11198963119894

+ (41198982

minus 4119898 + 1) 1198962

1119894)1198962

2119894gt 0

119894 = 1 2 3

(46)



1198812=

3

sum119894=1

1198812119894 119894 = 1 2 3 (47)


1198812119894

=1

2(minus1198961119894

10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898 sgn (119904

119894) minus 1198962119894119904119894+ 120591119894)2

+1198963119894119898

119898 minus 1

100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 11989641198941199042

119894+

1

21205912

119894

(48)


119894) 119904119894 120591119894]119879



1198812119894

= 120585119879

119894Λ119894120585119894

2119894

= minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120585119879

119894Γ1119894120585119894+ 120585119879

119894Γ2119894120585119894

(49)

where

Λ119894=

1

2[

[

(2119898 (119898 minus 1)) 1198963119894

+ 11989621119894

11989611198941198962119894

minus1198961119894

11989611198941198962119894

21198964119894

+ 11989622119894

minus1198962119894

minus1198961119894

minus1198962119894

2

]

]

Γ1119894

=1198961119894

119898

times [

[

1198981198963119894

+ (119898 minus 1) 1198962

11198940 minus (119898 minus 1) 119896

1119894

0 1198981198964119894

+ (3119898 minus 1) 1198962

2119894minus (2119898 minus 1) 119896

2119894

minus (119898 minus 1) 1198961119894

minus (2119898 minus 1) 1198962119894

119898 minus 1

]

]

Γ2119894

= 1198962119894

[

[

1198963119894

+ ((3119898 minus 2) 119898) 1198962

11198940 0

0 1198964119894

+ 1198962

2119894minus1198962119894

0 minus1198962119894

1

]

]

(50)


120582min Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

le 1198812119894

le 120582max Λ1198941003817100381710038171003817120585119894

10038171003817100381710038172

(51)


10038161003816100381610038161199041198941003816100381610038161003816(119898minus1)119898

le radic100381610038161003816100381611990411989410038161003816100381610038162(119898minus1)119898

+ 1199042119894+ 1205912119894=

10038171003817100381710038171205851198941003817100381710038171003817 le radic

1198812119894

120582min Λ119894

100381610038161003816100381611990411989410038161003816100381610038161119898

le (1198812119894

120582min Λ119894)

12(119898minus1)

(52)



2119894

le minus1003816100381610038161003816119904119894

1003816100381610038161003816minus1119898

120582min Γ11198941003817100381710038171003817120585119894

10038171003817100381710038172

minus 120582min Γ21198941003817100381710038171003817120585119894

10038171003817100381710038172


119894

1198812119894

)

12(119898minus1)

times 120582min Γ1119894

1198812119894

120582max Λ119894minus 120582min Γ

2119894

1198812119894

120582max Λ119894


2119894minus 12059421198941198812119894

(53)

where 1205941119894

= (120582minΛ119894)12(119898minus1)

(120582minΓ1119894120582maxΛ119894) 1205942119894

=

(120582minΓ2119894120582maxΛ119894)

1198812=

3

sum119894=1


2119894minus 12059421198941198812119894)

le minus1205941

3

sum119894=1

119881(2119898minus3)(2119898minus2)

2119894minus 12059421198812

(54)

where 1205941= max120594

11 12059412 12059413 1205942= max120594

21 12059422 12059423


1198812+ 1205941119881(2119898minus3)(2119898minus2)

2+ 12059421198812le 0 (55)



119903

119905119903le

2 (119898 minus 1) ln (1 + (12059421205941) 1198812(119905ob)12(119898minus1)

)

1205941

+ 119905ob(56)



3times1is









1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















1 and


M119888= D (sdot) 120575

119888 (57)

where 120575119888

= [120575119860 120575RCS] 120575119860 = [120575

1198601 1205751198602

1205751198601198991

]119879 denotes







120575119860119894min le 120575

119860119894le 120575119860119894max

120575119889-119860119894min le 120575

119860119894le 120575119889-119860119894max 119894 = 1 119899

1

0 le 120575RCS119895 le 1 119895 = 1 1198992

(59)

where 120575119860119894min and 120575


119860119894 respectively 120575






min 1198691= W1

1003817100381710038171003817M119888 minus D (sdot) 120575119888

1003817100381710038171003817 + W2120575RCS (60)

whereW1isin R3W




119898and 119879

119898








k = b minus 120578 sat (s) (62)


119909= 588791 kgsdotm2 119868

119910=


inertia are 119868119911119909

= 119868119909119911



120575119860max = [0 0 0 0 30 30 30 30]

119879

120575119860min = minus [25 25 25 25 10 10 30 30]

119879

120575119889-119860max = minus 120575

119889-119860119894min = [10 10 10 10 10 10 10 10]119879

(63)


119889-119860maxand 120575



DRCS (sdot) =[[

[




]]

]

(64)





c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













c = diag(2 2 2)120582 = 2

c = diag(2 2 2)120582 = 2


120578 = diag(01 01 01)h = diag (0015 0015 0015)

k1= diag(01 01 01)

k2= diag(02 02 02)

k3= diag(01 01 01)

k4= diag(01 01 01)

119898 = 3

DO parameters 1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)1205741= 82 120574

2= 41 120574

3= 20

L = diag(0005 0005 001)

A in

uoff uon

Aout

minus1

1Km

1 + Tms



= [320∘

20∘

580∘

]119879 and w

0= [00

∘


119888= [300

∘

00∘




120575rl = 120575rr = 0 (65)


ΔM =[[

[

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

05 + sin (01119905) + sin (119905)

]]

]




Parameter ValueW1

[1 1 1]

W2

[01 01]

119870119898

45

119879119898

015

119906on 045

119906off 015


119901= 119896119889


119888 is produced





0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15295

30

305

31

315

32

0 5 10 15minus05

0

05

1

15

2

0 5 10 1557

58

59

60

61

5 10 15298

30

302

304

306

308

5 10 15minus005

0

005

01

015

02

5 10 15585

59

595

60

605

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)



FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 1 2 3 4 530

305

31

315

32

0 1 2 3 4 50

05

1

15

2

0 1 2 3 4 5575

58

585

59

595

60

5 10 15299995

30

300005

30001

300015

5 10 15minus5

0

5

10

15

5 10 15599985

59999

599995

60

600005

60001

120572(d

eg)

120572(d

eg)

120573(d

eg)

120573(d

eg)

120590(d

eg)

120590(d

eg)

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

times10minus4

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO

BTVSMCDOSOTVSMCDO





0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15minus1

0

1

2

3

0 5 10 15minus5

0

5

10

15

0 5 10 15minus8

minus6

minus4

minus2

0

2

5 10 15minus5

0

5

10

5 10 15minus2

minus1

0

1

2

3

5 10 15minus15

minus1

minus05

0

05

1

Time (s)

Time (s) Time (s)

Time (s)

Time (s)

Time (s)






s 1s 1

s 2s 2

s 3s 3







7 Conclusion





0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus10

minus5

0

5


0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2times10minus4


times10minus4

times10minus4

times10minus4



s 1s 1

s 2s 2

s 3s 3




0 5 10 15minus5

0

5

10

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

minus1

0

1

2

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus2

0

2

4

6

8

0 5 10 15minus8

minus6

minus4

minus2

0

2






times10minus3



s 1s 1

s 2s 2 s 3

s 3




0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15minus005

0

005

01

015

0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus008

minus006

minus004

minus002

0

002

0 5 10 15minus005

0

005

01

015



0 5 10 15minus004

minus002

0

002

004

006

0 5 10 15minus01

minus005

0

005

01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3


0 5 10 15minus005

0

005

01

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus01

minus005

0

005

0 5 10 15minus005

0

005

015

0 5 10 15minus005

0

005

015

0 5 10 15minus015

minus005

0

005

Time (s)


Time (s) Time (s)

01 01

01

minus01




Δ 1

Δ 1

Δ 2

Δ 2

Δ 3

Δ 3




0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15minus2

minus15

minus1

minus05

0

05

minus15

minus05

05

0 5 10 15minus2

minus1

0

0 5 10 15minus02

0

02

04

06

5 10 15minus3

minus2

minus1

0

1

5 10 15minus20

minus15

minus10

minus5

0

5

5 10 15minus5

0

5

10

15



p(d

egs

)p

(deg

s)

q(d

egs

)q

(deg

s)

r(d

egs

)r

(deg

s)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO


0 5 10 15minus12

minus10

minus8

minus6

minus4

minus2

0

2

0 5 10 15minus20

minus15

minus10

minus5

0

5

0 5 10 15minus4

minus2

0

2

4

6

8


Ml

(Nmiddotm

)

Mm

(Nmiddotm

)

Mn

(Nmiddotm

)


FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO

FBLBTVSMC

BTVSMCDOSOTVSMCDO






minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0

10

10

0

20

minus5

0

5

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus2

0

2

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus8

times10minus13

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

5


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R10

120575R9

(b) RCS commands



minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus20minus10

0

minus20minus10

0

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

minus20minus10

0

0 5 10 15

0 5 10 15

0 5 10 15

0 5 10 15

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

times10minus10

times10minus12

120575el

o120575

ero

120575el

i120575

eri

120575fl

120575fr

120575rl

120575rr

0

10

20

minus1

0

1

minus050

05

0

20

40


01

01

01

01

01

0

01

1

01

01

0 5 10 1501

Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15Time (s)

0 5 10 15

0 5 10 15Time (s)

Time (s)

0 5 10 15Time (s)

120575R1

120575R2

120575R3

120575R4

120575R5

120575R6

120575R7

120575R8

120575R9

120575R10

(b) RCS commands



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article finite-time reentry attitude...

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