Research ArticleIntegral Sliding Mode Control for Trajectory Tracking ofWheeled Mobile Robot in Presence of Uncertainties
Aicha Bessas1 Atallah Benalia1 and Faregraves Boudjema2
1LACoSERE Laboratory Amar Telidji University Laghouat BP 37G Route de Ghardaıa 03000 Laghouat Algeria2LCP Laboratory Ecole Nationale Polytechnique 10 avenue Hassan Badi BP 182 El-Harrach Algiers Algeria
Correspondence should be addressed to Aicha Bessas aibessaslagh-univdz
Received 9 November 2015 Revised 25 April 2016 Accepted 28 April 2016
Academic Editor Yongji Wang
Copyright copy 2016 Aicha Bessas 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
Wheeled mobile robots present a typical case of complex systems with nonholonomic constraints In the past few years thedominance of these systems has been a very active research field In this paper a new method based on an integral sliding modecontrol for the trajectory tracking of wheeled mobile robots is proposed The controller is designed to solve the reaching phaseproblemwith the elimination ofmatched disturbances andminimize the unmatched oneWe distinguish two parts in the suggestedcontroller a high-level controller to stabilize the nominal system and a discontinuous controller to assess the trajectory trackingin the presence of disturbances This controller is robust during the entire motion The effectiveness of the proposed controller isdemonstrated through simulation studies for the unicycle with matched and unmatched disturbances
1 Introduction
Generally wheeled mobile robots (WMRs) are the mostwidely used classes of mobile robots This is due to theirpractical importance and theoretically interesting proper-ties These systems are a typical example of nonholonomicmechanisms where the constraints imposed on the motionsare not integrable resulting from the assumption that thereis no slipping of the wheels The main consequence of anonholonomic constraint for the WMRs is that not eachpath of the admissible configuration space corresponds toa feasible trajectory for the robot In the literature of thewheeled mobile robot control there are two fundamentalproblems posture stabilization and trajectorypath trackingThe aim of posture stabilization is to stabilize the robotto a desired point [1] while the trajectory tracking is toenforce the robot to follow a reference trajectory [2] ForWMRs it is difficult to control such system by continuoustime-invariant controller This is due to the uncontrollabilityof their linear approximation and to Brockettrsquos necessarycondition which is not satisfied for this kind of system[3] To overcome these difficulties various control strategieshave been investigated among them homogeneous and time-varying feedback [4] sinusoidal and polynomial controls
[5] backstepping approaches [6 7] and hybrid controls[8] In the real implementation it is desired to design aninherently robust control which provides fast convergenceand good robustness properties with respect to the parametervariation and the disturbances One of the robust techniquesis the discontinuous control such as sliding mode control(SMC) There are a number of references on sliding modecontrol devoted to this type of discontinuous control the bestknown one of which is by Utkin et al [9] More theoreticalanalyses and comparison study of performances for differentSMC controllers are presented in [10] The sliding modecontrol has many advantages among them its finite timeconvergence to a stable manifold and its insensitivity to thedisturbances andmodel uncertainties satisfying thematchingcondition However it has some disadvantages such as thechattering phenomena the reaching phase and sensitivity tothe unmatched perturbation To enhance the robustness ofthe slidingmode control in the whole motion it is interestingto eliminate the reaching phase andminimize the effect of theunmatched disturbances This idea can be done by applyingthe integral sliding mode design concept proposed in [9 11ndash14] The integral sliding mode control seeks to eliminatethe reaching phase by enforcing sliding mode throughout
Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 7915375 10 pageshttpdxdoiorg10115520167915375
2 Journal of Control Science and Engineering
the entire system response The basic idea of this control(ISMC) is the inclusion of an integral term to the slidingmanifoldThis integral term enables the system to start on thesliding manifold at the initial condition hence eliminatingthe reaching phase From the integral sliding manifold wedefine two controllers [9] a continuous control and a dis-continuous control The continuous controller is a nonlinearcontinuous feedback designed to stabilize the nominal systemand the discontinuous control is used to reject the matcheddisturbances and minimize the unmatched one
The main objective of this work is the design of a robustcontroller for the trajectory tracking of the unicycle subjectto state-dependent uncertainties (matched and unmatched)To attain this objective we use an integral sliding modebased controller This suggested controller combining non-linear time-varying feedback with an integral sliding modecontroller An integral sliding mode controller is constructedby incorporating an integral term in the switching manifold
The outline of this paper is as follows In Section 2 theproblem statement and the integral sliding mode controllerdesign are presented for nonlinear uncertain system Thenthe kinematic model of the unicycle-type wheeled mobilerobot is derived in Section 3 In Section 4 the design ofintegral sliding mode controller for tracking control of theunicycle is presented Then some simulation results arediscussed in Section 5 Finally Section 6 concludes this paper
2 Problem Statement
Consider the following nonlinear uncertain system
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875 (119902
119890 119905) (1)
where 119902119890isin 119876 sub IR119899 is the state of the system with initial
condition 119902119890(119905
0) = 119902
0and 119906 isin IR119898 is the control variable
The function 119891 isin IR119899 is a known vector and the matrix119892 isin IR119899times119898 is a known full rank state-dependent matrix Wesuppose also that the origin is an equilibrium point of (1)that is 119891(0 119905) = 0 forall119905 gt 0 119875(119902
119890 119905) isin IR119899 is an unknown
vector representing the modeling uncertainties and externaldisturbances The following assumption is introduced
Assumption 1 The uncertain vector 119875(119902119890 119905) is bounded
119875 (119902
119890 119905) isin Φ
Φ ≜ V isin IR119899 st V2le 119863sup
(2)
where119863sup is a known positive constantThe uncertain vector 119875(119902
119890 119905) for system (1) can always
be expressed by separating thematched disturbance119875119872(119902
119890 119905)
and the unmatched one 119875119880(119902
119890 119905) as follows
119875 (119902
119890 119905) = 119875
119872(119902
119890 119905) + 119875
119880(119902
119890 119905) (3)
119875
119872(119902
119890 119905) = 119892 (119902
119890) 119892
+
(119902
119890) 119875 (119902
119890 119905)
119875
119880(119902
119890 119905) = 119892
perp
(119902
119890) 119892
perp+
(119902
119890) 119875 (119902
119890 119905)
(4)
where 119892perp(119902119890) isin IR119899times(119899minus119898) is a matrix with indepen-
dent columns that span the null space of 119892(119902119890) that is
119890) is the left pseudoinverse of 119892(119902
119890) that is 119892+(119902
119890) =
(119892
119879
(119902
119890)119892(119902
119890))
minus1
119892
119879
(119902
119890) analogously for 119892perp+(119902
119890) This separa-
tion principle relies on proposition 1 [15] which ensures that119868
119899= 119892(119902
119890)119892
+
(119902
119890)+119892
perp
(119902
119890)119892
perp+
(119902
119890) for any full rank119892(119902
119890) being
119868
119899isin IR119899times119899 an identity matrixOur aim is to construct a robust feedback controller
whichmakes system (1) asymptotically stable More preciselyfor a given known stabilizing control for the nominal systemof (1) wewant to redesign another robust stabilizing feedbackcontrol of the perturbed system (1) We can realize that wewant to robustify an existing feedback control of the nominalsystem To attempt this objective we will take into accountthe following assumption
Assumption 2 The system nominal part of (1)
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 (5)
is globally asymptotically stabilizable via a nonlinear time-varying continuous control 119906
119888(119902
119890 119905)
Since the control 119906119888(119902
119890 119905) is supposed to be not robust
with respect to dynamic (1) and to enhance the robustnesswe will add to it an integral sliding mode controller whichguarantees a good robustness during the entire motion of thestates of the obtained closed-loop system
21 Integral Sliding Mode Controller Design The enhance-ment of the robustness of the feedback control 119906
119888(119902
119890 119905) is
done by using the integral sliding mode controller to rejectthe perturbations while eliminating the reaching phase Theintegral sliding mode algorithm is designed in two designsteps [12 13] as follows
(1) The design of a suitable integral sliding manifold119904(119902
119890 119905) satisfying the control objectives on the sliding
mode
(2) The design of corresponding control input 119906 con-straining the system trajectories to evolve on theintegral sliding surface from the initial time andmakethe feedback system insensitive to the disturbances
The integral sliding function can be defined as
119904 (119902
119890 119905) = 119904
0(119902
119890 119905) + 119911
119904(119902
119890 119905) (6)
where 119904 isin IR119898 1199040isin IR119898 is designed as the linear or nonlinear
function of the system states and 119911119904isin IR119898 is an unknown
integral function of the state to be determined such that thereaching phase is eliminated The integral sliding manifold isgiven by 119904(119902
119890 119905) = 0
Differentiating 119904 in (6) yields
119904 (119902
119890 119905) = 119867 (119902
119890) [119891 (119902
119890 119905)
+ 119892 (119902
119890) (119906 + 119892
+
(119902
119890) 119875 (119902
119890 119905)) + 119875
119880(119902
119890 119905)] +
119904
(7)
where119867(119902119890) = 120597119904
0(119902
119890)120597119902
119890isin IR119898times119899
Journal of Control Science and Engineering 3
In absence of perturbation the invariance conditions ofthe integral sliding manifold with the nonlinear time varyingcontinuous control 119906
119888(119902
119890 119905) are given by
119904 (119902
119890 119905) = 0 forall119905 gt 0
119904 (119902
119890 119905) = 0 forall119905 gt 0 997904rArr
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
(8)
To satisfy this invariance condition from the initial timewe obtain from the above equations the dynamics of thevariable 119911
119904
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
119911
119904(0) = minus119904
0(119902
119890(119905
0))
(9)
According to (6) (9) we obtain
119904 = 119904
0(119902
119890 119905) minus 119904
0(119902
119890(119905
0))
minus int
119905
1199050
119867(119902
119890) [119891 (119902
119890 120591) + 119892 (119902
119890) 119906
119888] 119889120591
(10)
Note that this integral sliding manifold is analogous tothose proposed in [11] We can see that the invariance condi-tion is verified for this sliding surfacewith the control119906
119888(119902
119890 119905)
if the disturbance does not appear In order to guarantee theattractivity of the sliding manifold (10) and the robustnessagainst the perturbation we will add a discontinuous controlpart to the time varying continuous control 119906
119890 119905) is the feedback stabilizing control of the nom-
inal system (5)which guarantees the invariance of the integralsliding manifold 119906disc(119902119890 119905) is a discontinuous control actiondesigned to minimize the disturbance terms forcing thesystem state on a suitably designed integral sliding manifold119904(119902
119890 119905) = 0
Assumption 3 119867(119902119890) is such that
Rank (119867 (119902119890) 119892 (119902
119890)) = 119898 forall119902
119890isin IR119899 (12)
Take into account the reachability condition defined asfollows [9]
22 The Proposed Sliding Manifold A serval result is intro-duced for the minimization of the equivalent disturbance(18) for system (1) when the ISM control strategy is appliedhence we consider the distribution given by
Λ (119902
119890) = span 119892perp (119902
119890) (20)
That introduces the following assumption
Assumption 5 Λ(119902119890) is involutive that is
[119892
perp
119894 119892
perp
119895] =
120597119892
perp
119895
120597119902
119892
perp
119894minus
120597119892
perp
119894
120597119902
119892
perp
119895isin Λ (119902
119890) forall119894 119895 = 1 2
(21)
where [sdot sdot] is the Lie bracket of two vector fields SinceAssumption 5 is fulfilled there exists a function 119878
0(119902
119890 119905) such
that
120597
119878
0(119902
119890 119905)
120597119902
119890
=
119867(119902
119890) = 119861 (119902
119890) 119892
119879
(119902
119890)
(22)
4 Journal of Control Science and Engineering
rarrj
y
O x rarri
C = (x y)
120579
Figure 1 Kinematic model of unicycle-type mobile robot
where 119861(119902) isin IR119898times119898 is a full rank matrix Note that (22)guarantees that Assumption 3 holds
The involutivity of Λ(119902119890) is equivalent to the existence of
119898 independent functions 1198780(119902
119890 119905) such that
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
perp
(119902
119890) =
119867 (119902
119890) 119892
perp
(119902
119890) = 0
(23)
Since 119898 columns of 119867119879(119902119890) are independent they span
the orthogonal complement of Λ(119902119890) Recall that the double
orthogonal complement of a closed subspace is equal to thesubspace itself which is equivalent to
The columns of 119867119879(119902119890) and 119892(119902
119890) are basis of the same
subspace and the matrix 119861119879(119902) in (22) is simply the transfor-mation matrix relating them
3 Kinematics Model of Wheeled Mobile Robot
The kinematic model of unicycle-type wheeled mobile robotis described with consideration of the nonholonomic con-straints A complete study of the kinematics model of WMRscould be found in [16]
31 KinematicsModel of Unicycle-TypeWheeledMobile RobotA unicycle-type mobile robot is considered as depicted inFigure 1 The kinematic scheme of the robot consists ofplatform with two driving wheels mounted on the same axiswith independent actuators and one free wheel (caster) isused to keep the robot stable [16] It is assumed that thewheelsare nondeformable and roll without lateral sliding
The point 119862 at the center of the driving wheels axle isused as a reference point of the robot The configuration of
Controlled robot
Virtual reference robot
Reference trajectory
y
xO
C = (x y)
Cr = (xr yr)
ey
ex
120579
120579r
e120579
Figure 2 Tracking the reference trajectory of WMR
the robot can be described by three generalized coordinatesas
119902 = (119909 119910 120579)
119879
isin 119876 = IR2 times 1198781198741 (25)
where (119909 119910) are the coordinates of point 119862 and 120579 is the robotorientation The nonholonomic constraint that the wheelscannot slip in the lateral direction is
[minussin 120579 cos 120579 0][
[
[
[
120579
]
]
]
]
= 0 (26)
From this constraint the kinematicmodel of the unicycle canbe written as follows
= 119892 (119902) 119906 997904rArr
[
[
[
[
120579
]
]
]
]
=
[
[
[
cos 120579sin 1205790
0
0
1
]
]
]
[
V
119908
] (27)
where V and 119908 area linear velocity of the wheels and itsangular velocity respectivelyThey are taken as control inputs119906 = (V 119908)119879
The driftless nonlinear system (27) has several controlproperties most of which actually hold for the whole classof WRMs and nonholonomic mechanisms in general Thisproperty comes from the fact that any position is an equilib-rium point if the inputs are zero and hence the system has nodynamical motion
32 Posture Error Model of Mobile Robot In trajectory track-ing of wheeled mobile robots the reference trajectory 119902
119903and
the velocity 119906119903are considered where they are written as
119902
119903= (119909
119903(119905) 119910
119903(119905) 120579
119903(119905))
119879
119906 = (V119903(119905) 119908
119903(119905))
119879
(28)
The mobile robot moves from posture 119902 to posture 119902119903 as
shown in Figure 2 The posture error is given by
(119890
119909 119890
119910 119890
120579)
119879
= (119909 minus 119909
119903(119905) 119910 minus 119910
119903(119905) 120579 minus 120579
119903(119905))
119879
(29)
Journal of Control Science and Engineering 5
Unicycle
controllerISM
++minus controllerNonlinear
+
yx
q =120579
u =u1u2
02468
101214161820
0 1 2 3 4 5 6 7 8 9 10
Referencetrajectory
Figure 3 Closed-loop control diagram
According to coordinate transformation the posture errorequation of the mobile robot is described as
119902
119890=
[
[
[
119909
119890
119910
119890
120579
119890
]
]
]
=
[
[
[
cos 120579119903
sin 1205791199030
minussin 120579119903cos 1205791199030
0 0 1
]
]
]
[
[
[
119890
119909
119890
119910
119890
120579
]
]
]
(30)
Thederivative of the posture error given in (30) can bewrittenas
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903+ cos 120579
119890V
minus119909
119890119908
119903+ sin 120579
119890V
119908 minus 119908
119903
]
]
]
(31)
For unicycle it is assumed that |120579119890| lt 1205872 which means
that the vehicle orientation must not be perpendicular to thedesired trajectory We can write (31) in the form affine in thecontrol as follows
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 997904rArr
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
(32)
The model of the unicycle usually has external distur-bance or unmodeled uncertainty so the behavior of (32) canbe quite different fromexpected If there is disturbances actingon the system the dynamic of the posture error is given by
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
+
[
[
[
119901
1cos 120579119890minus 119901
2sin 120579119890
119901
1sin 120579119890+ 119901
2cos 120579119890
119901
3
]
]
]
(33)
From (3) and (4) we have
119892
+
(119902
119890) = [
1 0 0
0 0 1
]
119892
perp
(119902
119890) =
[
[
[
minussin 120579cos 1205790
]
]
]
119892
perp+
(119902) = [
0 0 0
0 1 0
]
(34)
System (4) is written as
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875
119872(119902
119890 119905) + 119875
119906(119902
119890 119905)
997904rArr
119890=
[
[
[
119890
119890
120579
119890
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V119908
]
+
[
[
[
119901
1cos 120579119890
119901
1sin 120579119890
119901
3
]
]
]
+
[
[
[
minus119901
1sin 120579119890
119901
1cos 120579119890
0
]
]
]
(35)
If we assume that each component of the vector 119875 is inabsolute value smaller than a constant 119875
1 1198752 1198753 respec-
tively we obtain119863sup = radic1198752
1+ 119875
2
2+ 119875
2
3as required in (2)
In the following we applied the above control design (11)to this obtained model for the trajectory tracking
4 Integral Sliding Mode Controller DesignApplied to the Unicycle
In this section the integral sliding mode controller (ISMC) isapplied to a unicycle Firstly we mention the nonlinear time-varying continuous feedback based technique proposed byNonami et al [16] for using it as a nominal controller Thenwe determine the ISMC to assess the trajectory tracking inthe presence of matched and unmatched perturbations fromthe initial condition (Figure 3)
6 Journal of Control Science and Engineering
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
the entire system response The basic idea of this control(ISMC) is the inclusion of an integral term to the slidingmanifoldThis integral term enables the system to start on thesliding manifold at the initial condition hence eliminatingthe reaching phase From the integral sliding manifold wedefine two controllers [9] a continuous control and a dis-continuous control The continuous controller is a nonlinearcontinuous feedback designed to stabilize the nominal systemand the discontinuous control is used to reject the matcheddisturbances and minimize the unmatched one
The main objective of this work is the design of a robustcontroller for the trajectory tracking of the unicycle subjectto state-dependent uncertainties (matched and unmatched)To attain this objective we use an integral sliding modebased controller This suggested controller combining non-linear time-varying feedback with an integral sliding modecontroller An integral sliding mode controller is constructedby incorporating an integral term in the switching manifold
The outline of this paper is as follows In Section 2 theproblem statement and the integral sliding mode controllerdesign are presented for nonlinear uncertain system Thenthe kinematic model of the unicycle-type wheeled mobilerobot is derived in Section 3 In Section 4 the design ofintegral sliding mode controller for tracking control of theunicycle is presented Then some simulation results arediscussed in Section 5 Finally Section 6 concludes this paper
2 Problem Statement
Consider the following nonlinear uncertain system
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875 (119902
119890 119905) (1)
where 119902119890isin 119876 sub IR119899 is the state of the system with initial
condition 119902119890(119905
0) = 119902
0and 119906 isin IR119898 is the control variable
The function 119891 isin IR119899 is a known vector and the matrix119892 isin IR119899times119898 is a known full rank state-dependent matrix Wesuppose also that the origin is an equilibrium point of (1)that is 119891(0 119905) = 0 forall119905 gt 0 119875(119902
119890 119905) isin IR119899 is an unknown
vector representing the modeling uncertainties and externaldisturbances The following assumption is introduced
Assumption 1 The uncertain vector 119875(119902119890 119905) is bounded
119875 (119902
119890 119905) isin Φ
Φ ≜ V isin IR119899 st V2le 119863sup
(2)
where119863sup is a known positive constantThe uncertain vector 119875(119902
119890 119905) for system (1) can always
be expressed by separating thematched disturbance119875119872(119902
119890 119905)
and the unmatched one 119875119880(119902
119890 119905) as follows
119875 (119902
119890 119905) = 119875
119872(119902
119890 119905) + 119875
119880(119902
119890 119905) (3)
119875
119872(119902
119890 119905) = 119892 (119902
119890) 119892
+
(119902
119890) 119875 (119902
119890 119905)
119875
119880(119902
119890 119905) = 119892
perp
(119902
119890) 119892
perp+
(119902
119890) 119875 (119902
119890 119905)
(4)
where 119892perp(119902119890) isin IR119899times(119899minus119898) is a matrix with indepen-
dent columns that span the null space of 119892(119902119890) that is
119890) is the left pseudoinverse of 119892(119902
119890) that is 119892+(119902
119890) =
(119892
119879
(119902
119890)119892(119902
119890))
minus1
119892
119879
(119902
119890) analogously for 119892perp+(119902
119890) This separa-
tion principle relies on proposition 1 [15] which ensures that119868
119899= 119892(119902
119890)119892
+
(119902
119890)+119892
perp
(119902
119890)119892
perp+
(119902
119890) for any full rank119892(119902
119890) being
119868
119899isin IR119899times119899 an identity matrixOur aim is to construct a robust feedback controller
whichmakes system (1) asymptotically stable More preciselyfor a given known stabilizing control for the nominal systemof (1) wewant to redesign another robust stabilizing feedbackcontrol of the perturbed system (1) We can realize that wewant to robustify an existing feedback control of the nominalsystem To attempt this objective we will take into accountthe following assumption
Assumption 2 The system nominal part of (1)
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 (5)
is globally asymptotically stabilizable via a nonlinear time-varying continuous control 119906
119888(119902
119890 119905)
Since the control 119906119888(119902
119890 119905) is supposed to be not robust
with respect to dynamic (1) and to enhance the robustnesswe will add to it an integral sliding mode controller whichguarantees a good robustness during the entire motion of thestates of the obtained closed-loop system
21 Integral Sliding Mode Controller Design The enhance-ment of the robustness of the feedback control 119906
119888(119902
119890 119905) is
done by using the integral sliding mode controller to rejectthe perturbations while eliminating the reaching phase Theintegral sliding mode algorithm is designed in two designsteps [12 13] as follows
(1) The design of a suitable integral sliding manifold119904(119902
119890 119905) satisfying the control objectives on the sliding
mode
(2) The design of corresponding control input 119906 con-straining the system trajectories to evolve on theintegral sliding surface from the initial time andmakethe feedback system insensitive to the disturbances
The integral sliding function can be defined as
119904 (119902
119890 119905) = 119904
0(119902
119890 119905) + 119911
119904(119902
119890 119905) (6)
where 119904 isin IR119898 1199040isin IR119898 is designed as the linear or nonlinear
function of the system states and 119911119904isin IR119898 is an unknown
integral function of the state to be determined such that thereaching phase is eliminated The integral sliding manifold isgiven by 119904(119902
119890 119905) = 0
Differentiating 119904 in (6) yields
119904 (119902
119890 119905) = 119867 (119902
119890) [119891 (119902
119890 119905)
+ 119892 (119902
119890) (119906 + 119892
+
(119902
119890) 119875 (119902
119890 119905)) + 119875
119880(119902
119890 119905)] +
119904
(7)
where119867(119902119890) = 120597119904
0(119902
119890)120597119902
119890isin IR119898times119899
Journal of Control Science and Engineering 3
In absence of perturbation the invariance conditions ofthe integral sliding manifold with the nonlinear time varyingcontinuous control 119906
119888(119902
119890 119905) are given by
119904 (119902
119890 119905) = 0 forall119905 gt 0
119904 (119902
119890 119905) = 0 forall119905 gt 0 997904rArr
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
(8)
To satisfy this invariance condition from the initial timewe obtain from the above equations the dynamics of thevariable 119911
119904
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
119911
119904(0) = minus119904
0(119902
119890(119905
0))
(9)
According to (6) (9) we obtain
119904 = 119904
0(119902
119890 119905) minus 119904
0(119902
119890(119905
0))
minus int
119905
1199050
119867(119902
119890) [119891 (119902
119890 120591) + 119892 (119902
119890) 119906
119888] 119889120591
(10)
Note that this integral sliding manifold is analogous tothose proposed in [11] We can see that the invariance condi-tion is verified for this sliding surfacewith the control119906
119888(119902
119890 119905)
if the disturbance does not appear In order to guarantee theattractivity of the sliding manifold (10) and the robustnessagainst the perturbation we will add a discontinuous controlpart to the time varying continuous control 119906
119890 119905) is the feedback stabilizing control of the nom-
inal system (5)which guarantees the invariance of the integralsliding manifold 119906disc(119902119890 119905) is a discontinuous control actiondesigned to minimize the disturbance terms forcing thesystem state on a suitably designed integral sliding manifold119904(119902
119890 119905) = 0
Assumption 3 119867(119902119890) is such that
Rank (119867 (119902119890) 119892 (119902
119890)) = 119898 forall119902
119890isin IR119899 (12)
Take into account the reachability condition defined asfollows [9]
22 The Proposed Sliding Manifold A serval result is intro-duced for the minimization of the equivalent disturbance(18) for system (1) when the ISM control strategy is appliedhence we consider the distribution given by
Λ (119902
119890) = span 119892perp (119902
119890) (20)
That introduces the following assumption
Assumption 5 Λ(119902119890) is involutive that is
[119892
perp
119894 119892
perp
119895] =
120597119892
perp
119895
120597119902
119892
perp
119894minus
120597119892
perp
119894
120597119902
119892
perp
119895isin Λ (119902
119890) forall119894 119895 = 1 2
(21)
where [sdot sdot] is the Lie bracket of two vector fields SinceAssumption 5 is fulfilled there exists a function 119878
0(119902
119890 119905) such
that
120597
119878
0(119902
119890 119905)
120597119902
119890
=
119867(119902
119890) = 119861 (119902
119890) 119892
119879
(119902
119890)
(22)
4 Journal of Control Science and Engineering
rarrj
y
O x rarri
C = (x y)
120579
Figure 1 Kinematic model of unicycle-type mobile robot
where 119861(119902) isin IR119898times119898 is a full rank matrix Note that (22)guarantees that Assumption 3 holds
The involutivity of Λ(119902119890) is equivalent to the existence of
119898 independent functions 1198780(119902
119890 119905) such that
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
perp
(119902
119890) =
119867 (119902
119890) 119892
perp
(119902
119890) = 0
(23)
Since 119898 columns of 119867119879(119902119890) are independent they span
the orthogonal complement of Λ(119902119890) Recall that the double
orthogonal complement of a closed subspace is equal to thesubspace itself which is equivalent to
The columns of 119867119879(119902119890) and 119892(119902
119890) are basis of the same
subspace and the matrix 119861119879(119902) in (22) is simply the transfor-mation matrix relating them
3 Kinematics Model of Wheeled Mobile Robot
The kinematic model of unicycle-type wheeled mobile robotis described with consideration of the nonholonomic con-straints A complete study of the kinematics model of WMRscould be found in [16]
31 KinematicsModel of Unicycle-TypeWheeledMobile RobotA unicycle-type mobile robot is considered as depicted inFigure 1 The kinematic scheme of the robot consists ofplatform with two driving wheels mounted on the same axiswith independent actuators and one free wheel (caster) isused to keep the robot stable [16] It is assumed that thewheelsare nondeformable and roll without lateral sliding
The point 119862 at the center of the driving wheels axle isused as a reference point of the robot The configuration of
Controlled robot
Virtual reference robot
Reference trajectory
y
xO
C = (x y)
Cr = (xr yr)
ey
ex
120579
120579r
e120579
Figure 2 Tracking the reference trajectory of WMR
the robot can be described by three generalized coordinatesas
119902 = (119909 119910 120579)
119879
isin 119876 = IR2 times 1198781198741 (25)
where (119909 119910) are the coordinates of point 119862 and 120579 is the robotorientation The nonholonomic constraint that the wheelscannot slip in the lateral direction is
[minussin 120579 cos 120579 0][
[
[
[
120579
]
]
]
]
= 0 (26)
From this constraint the kinematicmodel of the unicycle canbe written as follows
= 119892 (119902) 119906 997904rArr
[
[
[
[
120579
]
]
]
]
=
[
[
[
cos 120579sin 1205790
0
0
1
]
]
]
[
V
119908
] (27)
where V and 119908 area linear velocity of the wheels and itsangular velocity respectivelyThey are taken as control inputs119906 = (V 119908)119879
The driftless nonlinear system (27) has several controlproperties most of which actually hold for the whole classof WRMs and nonholonomic mechanisms in general Thisproperty comes from the fact that any position is an equilib-rium point if the inputs are zero and hence the system has nodynamical motion
32 Posture Error Model of Mobile Robot In trajectory track-ing of wheeled mobile robots the reference trajectory 119902
119903and
the velocity 119906119903are considered where they are written as
119902
119903= (119909
119903(119905) 119910
119903(119905) 120579
119903(119905))
119879
119906 = (V119903(119905) 119908
119903(119905))
119879
(28)
The mobile robot moves from posture 119902 to posture 119902119903 as
shown in Figure 2 The posture error is given by
(119890
119909 119890
119910 119890
120579)
119879
= (119909 minus 119909
119903(119905) 119910 minus 119910
119903(119905) 120579 minus 120579
119903(119905))
119879
(29)
Journal of Control Science and Engineering 5
Unicycle
controllerISM
++minus controllerNonlinear
+
yx
q =120579
u =u1u2
02468
101214161820
0 1 2 3 4 5 6 7 8 9 10
Referencetrajectory
Figure 3 Closed-loop control diagram
According to coordinate transformation the posture errorequation of the mobile robot is described as
119902
119890=
[
[
[
119909
119890
119910
119890
120579
119890
]
]
]
=
[
[
[
cos 120579119903
sin 1205791199030
minussin 120579119903cos 1205791199030
0 0 1
]
]
]
[
[
[
119890
119909
119890
119910
119890
120579
]
]
]
(30)
Thederivative of the posture error given in (30) can bewrittenas
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903+ cos 120579
119890V
minus119909
119890119908
119903+ sin 120579
119890V
119908 minus 119908
119903
]
]
]
(31)
For unicycle it is assumed that |120579119890| lt 1205872 which means
that the vehicle orientation must not be perpendicular to thedesired trajectory We can write (31) in the form affine in thecontrol as follows
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 997904rArr
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
(32)
The model of the unicycle usually has external distur-bance or unmodeled uncertainty so the behavior of (32) canbe quite different fromexpected If there is disturbances actingon the system the dynamic of the posture error is given by
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
+
[
[
[
119901
1cos 120579119890minus 119901
2sin 120579119890
119901
1sin 120579119890+ 119901
2cos 120579119890
119901
3
]
]
]
(33)
From (3) and (4) we have
119892
+
(119902
119890) = [
1 0 0
0 0 1
]
119892
perp
(119902
119890) =
[
[
[
minussin 120579cos 1205790
]
]
]
119892
perp+
(119902) = [
0 0 0
0 1 0
]
(34)
System (4) is written as
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875
119872(119902
119890 119905) + 119875
119906(119902
119890 119905)
997904rArr
119890=
[
[
[
119890
119890
120579
119890
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V119908
]
+
[
[
[
119901
1cos 120579119890
119901
1sin 120579119890
119901
3
]
]
]
+
[
[
[
minus119901
1sin 120579119890
119901
1cos 120579119890
0
]
]
]
(35)
If we assume that each component of the vector 119875 is inabsolute value smaller than a constant 119875
1 1198752 1198753 respec-
tively we obtain119863sup = radic1198752
1+ 119875
2
2+ 119875
2
3as required in (2)
In the following we applied the above control design (11)to this obtained model for the trajectory tracking
4 Integral Sliding Mode Controller DesignApplied to the Unicycle
In this section the integral sliding mode controller (ISMC) isapplied to a unicycle Firstly we mention the nonlinear time-varying continuous feedback based technique proposed byNonami et al [16] for using it as a nominal controller Thenwe determine the ISMC to assess the trajectory tracking inthe presence of matched and unmatched perturbations fromthe initial condition (Figure 3)
6 Journal of Control Science and Engineering
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
In absence of perturbation the invariance conditions ofthe integral sliding manifold with the nonlinear time varyingcontinuous control 119906
119888(119902
119890 119905) are given by
119904 (119902
119890 119905) = 0 forall119905 gt 0
119904 (119902
119890 119905) = 0 forall119905 gt 0 997904rArr
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
(8)
To satisfy this invariance condition from the initial timewe obtain from the above equations the dynamics of thevariable 119911
119904
119904= minus119867 (119902
119890) [119891 (119902
119890 119905) + 119892 (119902
119890) 119906
119888]
119911
119904(0) = minus119904
0(119902
119890(119905
0))
(9)
According to (6) (9) we obtain
119904 = 119904
0(119902
119890 119905) minus 119904
0(119902
119890(119905
0))
minus int
119905
1199050
119867(119902
119890) [119891 (119902
119890 120591) + 119892 (119902
119890) 119906
119888] 119889120591
(10)
Note that this integral sliding manifold is analogous tothose proposed in [11] We can see that the invariance condi-tion is verified for this sliding surfacewith the control119906
119888(119902
119890 119905)
if the disturbance does not appear In order to guarantee theattractivity of the sliding manifold (10) and the robustnessagainst the perturbation we will add a discontinuous controlpart to the time varying continuous control 119906
119890 119905) is the feedback stabilizing control of the nom-
inal system (5)which guarantees the invariance of the integralsliding manifold 119906disc(119902119890 119905) is a discontinuous control actiondesigned to minimize the disturbance terms forcing thesystem state on a suitably designed integral sliding manifold119904(119902
119890 119905) = 0
Assumption 3 119867(119902119890) is such that
Rank (119867 (119902119890) 119892 (119902
119890)) = 119898 forall119902
119890isin IR119899 (12)
Take into account the reachability condition defined asfollows [9]
22 The Proposed Sliding Manifold A serval result is intro-duced for the minimization of the equivalent disturbance(18) for system (1) when the ISM control strategy is appliedhence we consider the distribution given by
Λ (119902
119890) = span 119892perp (119902
119890) (20)
That introduces the following assumption
Assumption 5 Λ(119902119890) is involutive that is
[119892
perp
119894 119892
perp
119895] =
120597119892
perp
119895
120597119902
119892
perp
119894minus
120597119892
perp
119894
120597119902
119892
perp
119895isin Λ (119902
119890) forall119894 119895 = 1 2
(21)
where [sdot sdot] is the Lie bracket of two vector fields SinceAssumption 5 is fulfilled there exists a function 119878
0(119902
119890 119905) such
that
120597
119878
0(119902
119890 119905)
120597119902
119890
=
119867(119902
119890) = 119861 (119902
119890) 119892
119879
(119902
119890)
(22)
4 Journal of Control Science and Engineering
rarrj
y
O x rarri
C = (x y)
120579
Figure 1 Kinematic model of unicycle-type mobile robot
where 119861(119902) isin IR119898times119898 is a full rank matrix Note that (22)guarantees that Assumption 3 holds
The involutivity of Λ(119902119890) is equivalent to the existence of
119898 independent functions 1198780(119902
119890 119905) such that
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
perp
(119902
119890) =
119867 (119902
119890) 119892
perp
(119902
119890) = 0
(23)
Since 119898 columns of 119867119879(119902119890) are independent they span
the orthogonal complement of Λ(119902119890) Recall that the double
orthogonal complement of a closed subspace is equal to thesubspace itself which is equivalent to
The columns of 119867119879(119902119890) and 119892(119902
119890) are basis of the same
subspace and the matrix 119861119879(119902) in (22) is simply the transfor-mation matrix relating them
3 Kinematics Model of Wheeled Mobile Robot
The kinematic model of unicycle-type wheeled mobile robotis described with consideration of the nonholonomic con-straints A complete study of the kinematics model of WMRscould be found in [16]
31 KinematicsModel of Unicycle-TypeWheeledMobile RobotA unicycle-type mobile robot is considered as depicted inFigure 1 The kinematic scheme of the robot consists ofplatform with two driving wheels mounted on the same axiswith independent actuators and one free wheel (caster) isused to keep the robot stable [16] It is assumed that thewheelsare nondeformable and roll without lateral sliding
The point 119862 at the center of the driving wheels axle isused as a reference point of the robot The configuration of
Controlled robot
Virtual reference robot
Reference trajectory
y
xO
C = (x y)
Cr = (xr yr)
ey
ex
120579
120579r
e120579
Figure 2 Tracking the reference trajectory of WMR
the robot can be described by three generalized coordinatesas
119902 = (119909 119910 120579)
119879
isin 119876 = IR2 times 1198781198741 (25)
where (119909 119910) are the coordinates of point 119862 and 120579 is the robotorientation The nonholonomic constraint that the wheelscannot slip in the lateral direction is
[minussin 120579 cos 120579 0][
[
[
[
120579
]
]
]
]
= 0 (26)
From this constraint the kinematicmodel of the unicycle canbe written as follows
= 119892 (119902) 119906 997904rArr
[
[
[
[
120579
]
]
]
]
=
[
[
[
cos 120579sin 1205790
0
0
1
]
]
]
[
V
119908
] (27)
where V and 119908 area linear velocity of the wheels and itsangular velocity respectivelyThey are taken as control inputs119906 = (V 119908)119879
The driftless nonlinear system (27) has several controlproperties most of which actually hold for the whole classof WRMs and nonholonomic mechanisms in general Thisproperty comes from the fact that any position is an equilib-rium point if the inputs are zero and hence the system has nodynamical motion
32 Posture Error Model of Mobile Robot In trajectory track-ing of wheeled mobile robots the reference trajectory 119902
119903and
the velocity 119906119903are considered where they are written as
119902
119903= (119909
119903(119905) 119910
119903(119905) 120579
119903(119905))
119879
119906 = (V119903(119905) 119908
119903(119905))
119879
(28)
The mobile robot moves from posture 119902 to posture 119902119903 as
shown in Figure 2 The posture error is given by
(119890
119909 119890
119910 119890
120579)
119879
= (119909 minus 119909
119903(119905) 119910 minus 119910
119903(119905) 120579 minus 120579
119903(119905))
119879
(29)
Journal of Control Science and Engineering 5
Unicycle
controllerISM
++minus controllerNonlinear
+
yx
q =120579
u =u1u2
02468
101214161820
0 1 2 3 4 5 6 7 8 9 10
Referencetrajectory
Figure 3 Closed-loop control diagram
According to coordinate transformation the posture errorequation of the mobile robot is described as
119902
119890=
[
[
[
119909
119890
119910
119890
120579
119890
]
]
]
=
[
[
[
cos 120579119903
sin 1205791199030
minussin 120579119903cos 1205791199030
0 0 1
]
]
]
[
[
[
119890
119909
119890
119910
119890
120579
]
]
]
(30)
Thederivative of the posture error given in (30) can bewrittenas
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903+ cos 120579
119890V
minus119909
119890119908
119903+ sin 120579
119890V
119908 minus 119908
119903
]
]
]
(31)
For unicycle it is assumed that |120579119890| lt 1205872 which means
that the vehicle orientation must not be perpendicular to thedesired trajectory We can write (31) in the form affine in thecontrol as follows
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 997904rArr
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
(32)
The model of the unicycle usually has external distur-bance or unmodeled uncertainty so the behavior of (32) canbe quite different fromexpected If there is disturbances actingon the system the dynamic of the posture error is given by
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
+
[
[
[
119901
1cos 120579119890minus 119901
2sin 120579119890
119901
1sin 120579119890+ 119901
2cos 120579119890
119901
3
]
]
]
(33)
From (3) and (4) we have
119892
+
(119902
119890) = [
1 0 0
0 0 1
]
119892
perp
(119902
119890) =
[
[
[
minussin 120579cos 1205790
]
]
]
119892
perp+
(119902) = [
0 0 0
0 1 0
]
(34)
System (4) is written as
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875
119872(119902
119890 119905) + 119875
119906(119902
119890 119905)
997904rArr
119890=
[
[
[
119890
119890
120579
119890
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V119908
]
+
[
[
[
119901
1cos 120579119890
119901
1sin 120579119890
119901
3
]
]
]
+
[
[
[
minus119901
1sin 120579119890
119901
1cos 120579119890
0
]
]
]
(35)
If we assume that each component of the vector 119875 is inabsolute value smaller than a constant 119875
1 1198752 1198753 respec-
tively we obtain119863sup = radic1198752
1+ 119875
2
2+ 119875
2
3as required in (2)
In the following we applied the above control design (11)to this obtained model for the trajectory tracking
4 Integral Sliding Mode Controller DesignApplied to the Unicycle
In this section the integral sliding mode controller (ISMC) isapplied to a unicycle Firstly we mention the nonlinear time-varying continuous feedback based technique proposed byNonami et al [16] for using it as a nominal controller Thenwe determine the ISMC to assess the trajectory tracking inthe presence of matched and unmatched perturbations fromthe initial condition (Figure 3)
6 Journal of Control Science and Engineering
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
The columns of 119867119879(119902119890) and 119892(119902
119890) are basis of the same
subspace and the matrix 119861119879(119902) in (22) is simply the transfor-mation matrix relating them
3 Kinematics Model of Wheeled Mobile Robot
The kinematic model of unicycle-type wheeled mobile robotis described with consideration of the nonholonomic con-straints A complete study of the kinematics model of WMRscould be found in [16]
31 KinematicsModel of Unicycle-TypeWheeledMobile RobotA unicycle-type mobile robot is considered as depicted inFigure 1 The kinematic scheme of the robot consists ofplatform with two driving wheels mounted on the same axiswith independent actuators and one free wheel (caster) isused to keep the robot stable [16] It is assumed that thewheelsare nondeformable and roll without lateral sliding
The point 119862 at the center of the driving wheels axle isused as a reference point of the robot The configuration of
Controlled robot
Virtual reference robot
Reference trajectory
y
xO
C = (x y)
Cr = (xr yr)
ey
ex
120579
120579r
e120579
Figure 2 Tracking the reference trajectory of WMR
the robot can be described by three generalized coordinatesas
119902 = (119909 119910 120579)
119879
isin 119876 = IR2 times 1198781198741 (25)
where (119909 119910) are the coordinates of point 119862 and 120579 is the robotorientation The nonholonomic constraint that the wheelscannot slip in the lateral direction is
[minussin 120579 cos 120579 0][
[
[
[
120579
]
]
]
]
= 0 (26)
From this constraint the kinematicmodel of the unicycle canbe written as follows
= 119892 (119902) 119906 997904rArr
[
[
[
[
120579
]
]
]
]
=
[
[
[
cos 120579sin 1205790
0
0
1
]
]
]
[
V
119908
] (27)
where V and 119908 area linear velocity of the wheels and itsangular velocity respectivelyThey are taken as control inputs119906 = (V 119908)119879
The driftless nonlinear system (27) has several controlproperties most of which actually hold for the whole classof WRMs and nonholonomic mechanisms in general Thisproperty comes from the fact that any position is an equilib-rium point if the inputs are zero and hence the system has nodynamical motion
32 Posture Error Model of Mobile Robot In trajectory track-ing of wheeled mobile robots the reference trajectory 119902
119903and
the velocity 119906119903are considered where they are written as
119902
119903= (119909
119903(119905) 119910
119903(119905) 120579
119903(119905))
119879
119906 = (V119903(119905) 119908
119903(119905))
119879
(28)
The mobile robot moves from posture 119902 to posture 119902119903 as
shown in Figure 2 The posture error is given by
(119890
119909 119890
119910 119890
120579)
119879
= (119909 minus 119909
119903(119905) 119910 minus 119910
119903(119905) 120579 minus 120579
119903(119905))
119879
(29)
Journal of Control Science and Engineering 5
Unicycle
controllerISM
++minus controllerNonlinear
+
yx
q =120579
u =u1u2
02468
101214161820
0 1 2 3 4 5 6 7 8 9 10
Referencetrajectory
Figure 3 Closed-loop control diagram
According to coordinate transformation the posture errorequation of the mobile robot is described as
119902
119890=
[
[
[
119909
119890
119910
119890
120579
119890
]
]
]
=
[
[
[
cos 120579119903
sin 1205791199030
minussin 120579119903cos 1205791199030
0 0 1
]
]
]
[
[
[
119890
119909
119890
119910
119890
120579
]
]
]
(30)
Thederivative of the posture error given in (30) can bewrittenas
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903+ cos 120579
119890V
minus119909
119890119908
119903+ sin 120579
119890V
119908 minus 119908
119903
]
]
]
(31)
For unicycle it is assumed that |120579119890| lt 1205872 which means
that the vehicle orientation must not be perpendicular to thedesired trajectory We can write (31) in the form affine in thecontrol as follows
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 997904rArr
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
(32)
The model of the unicycle usually has external distur-bance or unmodeled uncertainty so the behavior of (32) canbe quite different fromexpected If there is disturbances actingon the system the dynamic of the posture error is given by
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
+
[
[
[
119901
1cos 120579119890minus 119901
2sin 120579119890
119901
1sin 120579119890+ 119901
2cos 120579119890
119901
3
]
]
]
(33)
From (3) and (4) we have
119892
+
(119902
119890) = [
1 0 0
0 0 1
]
119892
perp
(119902
119890) =
[
[
[
minussin 120579cos 1205790
]
]
]
119892
perp+
(119902) = [
0 0 0
0 1 0
]
(34)
System (4) is written as
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875
119872(119902
119890 119905) + 119875
119906(119902
119890 119905)
997904rArr
119890=
[
[
[
119890
119890
120579
119890
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V119908
]
+
[
[
[
119901
1cos 120579119890
119901
1sin 120579119890
119901
3
]
]
]
+
[
[
[
minus119901
1sin 120579119890
119901
1cos 120579119890
0
]
]
]
(35)
If we assume that each component of the vector 119875 is inabsolute value smaller than a constant 119875
1 1198752 1198753 respec-
tively we obtain119863sup = radic1198752
1+ 119875
2
2+ 119875
2
3as required in (2)
In the following we applied the above control design (11)to this obtained model for the trajectory tracking
4 Integral Sliding Mode Controller DesignApplied to the Unicycle
In this section the integral sliding mode controller (ISMC) isapplied to a unicycle Firstly we mention the nonlinear time-varying continuous feedback based technique proposed byNonami et al [16] for using it as a nominal controller Thenwe determine the ISMC to assess the trajectory tracking inthe presence of matched and unmatched perturbations fromthe initial condition (Figure 3)
6 Journal of Control Science and Engineering
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
According to coordinate transformation the posture errorequation of the mobile robot is described as
119902
119890=
[
[
[
119909
119890
119910
119890
120579
119890
]
]
]
=
[
[
[
cos 120579119903
sin 1205791199030
minussin 120579119903cos 1205791199030
0 0 1
]
]
]
[
[
[
119890
119909
119890
119910
119890
120579
]
]
]
(30)
Thederivative of the posture error given in (30) can bewrittenas
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903+ cos 120579
119890V
minus119909
119890119908
119903+ sin 120579
119890V
119908 minus 119908
119903
]
]
]
(31)
For unicycle it is assumed that |120579119890| lt 1205872 which means
that the vehicle orientation must not be perpendicular to thedesired trajectory We can write (31) in the form affine in thecontrol as follows
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 997904rArr
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
(32)
The model of the unicycle usually has external distur-bance or unmodeled uncertainty so the behavior of (32) canbe quite different fromexpected If there is disturbances actingon the system the dynamic of the posture error is given by
119890=
[
[
[
[
119890
119890
120579
119890
]
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V
119908
]
+
[
[
[
119901
1cos 120579119890minus 119901
2sin 120579119890
119901
1sin 120579119890+ 119901
2cos 120579119890
119901
3
]
]
]
(33)
From (3) and (4) we have
119892
+
(119902
119890) = [
1 0 0
0 0 1
]
119892
perp
(119902
119890) =
[
[
[
minussin 120579cos 1205790
]
]
]
119892
perp+
(119902) = [
0 0 0
0 1 0
]
(34)
System (4) is written as
119890= 119891 (119902
119890 119905) + 119892 (119902
119890) 119906 + 119875
119872(119902
119890 119905) + 119875
119906(119902
119890 119905)
997904rArr
119890=
[
[
[
119890
119890
120579
119890
]
]
]
=
[
[
[
119910
119890119908
119903minus V119903
minus119909
119890119908
119903
minus119908
119903
]
]
]
+
[
[
[
cos 120579119890
sin 120579119890
0
0
0
1
]
]
]
[
V119908
]
+
[
[
[
119901
1cos 120579119890
119901
1sin 120579119890
119901
3
]
]
]
+
[
[
[
minus119901
1sin 120579119890
119901
1cos 120579119890
0
]
]
]
(35)
If we assume that each component of the vector 119875 is inabsolute value smaller than a constant 119875
1 1198752 1198753 respec-
tively we obtain119863sup = radic1198752
1+ 119875
2
2+ 119875
2
3as required in (2)
In the following we applied the above control design (11)to this obtained model for the trajectory tracking
4 Integral Sliding Mode Controller DesignApplied to the Unicycle
In this section the integral sliding mode controller (ISMC) isapplied to a unicycle Firstly we mention the nonlinear time-varying continuous feedback based technique proposed byNonami et al [16] for using it as a nominal controller Thenwe determine the ISMC to assess the trajectory tracking inthe presence of matched and unmatched perturbations fromthe initial condition (Figure 3)
6 Journal of Control Science and Engineering
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
41 Nonlinear Time-Varying Feedback Nonami et al [16]proposed a nonlinear controller which is globally sta-ble around a reference trajectory with the same nominalsystem structure described as in Figure 2 The NonlinearState Tracking Control law developed in [16] can be givenas
119906
119888(119902
119890 119905) = [
V119888
119908
119888
]
= [
V119903minus 119896
119909
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
119909
119890
119908
119903minus 119896
119910V119903sdot 119910
119890+ 119896
120579
1003816
1003816
1003816
1003816
V119903
1003816
1003816
1003816
1003816
tan (120579119890)
]
(36)
where 119896119909 119896119910 and 119896
120579are positive constantthat can be cal-
culated by eigenvalue placement and the first term in eachvelocity is a feedforward part
This controller makes the origin of system (32) globallyasymptotically stable if V
119903is a bounded differentiable function
with bounded derivative and does not tend to zero when ittends to infinity Readers are referred to [16] for detaileddescription From (36) we note that V
119903can entirely pass
through zero when it changes their sign
42 Integral Sliding Mode Controller Design The integralsliding mode control can be used to eliminate the effectof the matched disturbances and minimize the unmatchedone From the above design procedure we define the controlfeedback as
V (119902119890 119905) = V
119888(119902
119890 119905) + Vdisc (119902119890 119905)
119908 (119902
119890 119905) = 119908
119888(119902
119890 119905) + 119908disc (119902119890 119905)
(37)
To check if Assumption 5 is fulfilled we consider thedistribution
Λ (119902
119890) = span 119892perp = span
[
[
[
minussin 120579cos 1205790
]
]
]
(38)
The distributionΛ(119902119890) verifies Assumption 5 As a conse-
quence all the assumptions are fulfilled and the minimiza-tion of the disturbance terms can be performed Now we cancalculate the sliding manifold by solving the equations in theform
120597
119878
0(119902
119890 119905)
120597119902
119890
119892
119879
(119902
119890) = 0
(39)
where 1198780= [119878
01119878
02]
119879 and from (39) we have
[
[
[
[
[
120597119878
01
120597119909
120597119878
01
120597119910
120597119878
01
120597120579
120597119878
02
120597119909
120597119878
02
120597119910
120597119878
02
120597120579
]
]
]
]
]
[
[
[
minussin 120579cos 1205790
]
]
]
= 0 (40)
minus
120597119878
01
120597119909
sin 120579 +120597119878
01
120597119910
cos 120579 = 0 (41)
minus
120597119878
02
120597119909
sin 120579 +120597119878
02
120597119910
cos 120579 = 0 (42)
Clearly we can choose 11987801= 120579 119878
02= 119909 cos 120579 + 119910 sin 120579 as
an exact solution for (41) and (42) The sliding manifold is asfollows
119878
0(119902
119890 119905) = [
120579
119909 cos 120579 + 119910 sin 120579] (43)
The partial derivative of 1198780(119902
119890 119905) with respect to 119902
119890is
119867(119902
119890) = [
0
cos 1205790
sin 1205791
minus119909 sin 120579 + 119910 cos 120579] (44)
From this choice of the integral sliding manifold thesystem will reach the integral sliding manifold infinite timeand remain on it in the presence of disturbances On the otherhand from (9) we can see the zero initial value of 119904(119902
119890 119905) at
119905 = 0 Therefore the sliding mode exists forall119905 ge 0
5 Simulation Results
To assess the effectiveness of the proposed controller com-puter simulations using MATLABSIMULINK are imple-mentedThe simulations are performed by tracking a circulartrajectory in which the desired position orientation andvelocities of the unicycle are specified This trajectory isdescribed as
119909
119903(119905) = 119877 sin (120596119905)
119910
119903(119905) = 119877 (1 minus cos (120596119905))
120579
119903(119905) = 120596119905
V119903(119905) = 119877120596
119908
119903(119905) = 120596
(45)
Journal of Control Science and Engineering 7
minus3 minus2 minus1 0 1 2 3minus1
0
1
2
3
4
5
Position x(t)
Posit
ion y(t)
y = f(x)
yr = f(xr)
+Uc
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
Car path with matched and unmatched disturbances + Uc
(a)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched disturbances + ISMC
y = f(x)
yr = f(xr)
(b)
minus3 minus2 minus1 0 1 2 3
0
2
4
Position x(t)
Posit
ion y(t)
Car path with matched and unmatched disturbances + ISMC
y = f(x)
yr = f(xr)
(c)
Figure 4 The path of the unicycle x-y with initialization error and disturbances
With initial configuration equal to (1199090 119910
0 120579
0) = (0m 0m 0∘)
and 119877 = 2m 120596 = 1 radsTo show the robustness of our controller the sim-
ulation was run with initialization error (1199090 119910
0 120579
0) =
(2m 05m 20∘) and then with disturbances 119875(119902119890 119905) such as
119875 (119902
119890 119905) =
[
[
[
2 sin (20119905)sin (20119905)08 sin (8119905)
]
]
]
(46)
Leading to Φ asymp 26 in Assumption 1 the gain is chosenby 119863sup asymp 26 Moreover in order to reduce the so-calledchattering effect the well-known equivalent control methodis used applying a linear low-pass filter to the obtaineddiscontinuous control variable (Table 1)
First of all we show (Figure 4(a)) the path of the unicyclein the 119909-119910 plane in case there is matched and unmatcheddisturbance and the high-level controller only is used As
Table 1 The parameter values of the control law used in thesimulations
expected after a transient (since the initial condition is takenon purpose different from the reference) the car trajectory(solid line) does not settle on the desired one (dashed line)and the high level controller has a poor performance sinceit is not designed to work in presence of disturbancesUsing the proposed ISM strategy the bound on the matcheddisturbances is eliminated (Figures 4(b) and 5(a)) and theunmatched ones are not amplified (Figures 4(c) and 5(b))the performance of the overall control law is improving Theunicycle with ISMC indicates a higher robustness globalstability and higher tracking precision
8 Journal of Control Science and Engineering
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
10
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(a)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion x(t) (
m)
0 2 4 6minus5
0
5
Time (sec)
Posit
ion y(t) (
m)
0 2 4 60
5
Time (sec)
Ang
le 120579(t) (
rad)
x(t)xr(t)
y(t)yr(t)
120579(t)
120579r(t)
(b)
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
Figure 5 Tracking the reference trajectories 119909(119905) 119910(119905) and 120579(119905) respectively with initialization error (a) In case of matched disturbanceswith ISMC and (b) in case of matched and unmatched disturbances with ISMC
In this last case we show also (Figure 7) the time evolu-tion of the control variables 119906
1and 119906
2 The time evolution of
the components of the sliding manifold 119904 is steered to zero asshown in Figure 6 since the sliding mode is enforced fromthe initial instance
6 Conclusion
In automatic control the sliding mode control improvessystem performance by allowing the successful completion ofa task even in the presence of perturbations In this work theuncertain system trajectories are asymptotically regulated to
zero inspite while a sliding mode is enforced in finite timealong an integral manifold from the initial condition Theuse of the integral sliding manifold allows one to subdividethe control design procedure into two steps First high-levelcontrol and then a discontinuous control component areadded so as to cope with the uncertainty presenceThe designprocedure is relying on the definition of a suitable slidingmanifold and the generation of slidingmodes as it guaranteestheminimization of the effect of the disturbance terms whichtakes place when the matched disturbances are completelyrejected and the unmatched ones are not amplified Theperformance of closed-loop system has been verified using
Journal of Control Science and Engineering 9
0 2 4 6minus1
0
1
Time (sec)
0 2 4 6minus01
0
01
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(a)
0 2 4 6minus01
0
01
Time (sec)
0 2 4 6minus05
0
05
Time (sec)
Surfa
ces 1(t)
Surfa
ces 2(t)
(b)
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
Figure 6 Time evolution of the components of sliding manifold 1199041and 119904
2 (a) in case of matched disturbances and (b) in case of matched
and unmatched disturbances
0 2 4 6minus5
0
5
Time (sec)
Con
trol w
(t)
0 2 4 6minus10
0
10
Time (sec)
Con
trol
(t)
Figure 7 The control inputs 1199061and 119906
2of the unicycle using integral sliding mode controller with matched and unmatched disturbances
simulation results for the unicycle In a future work we willtry to enhance this technique and to verify the obtainedsimulation results on the real test bench
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Amar and S Mohamed ldquoStabilized feedback control ofunicycle mobile robotsrdquo International Journal of AdvancedRobotic Systems vol 10 article 187 2013
[2] A Zdesar I Skrjanc and G Klancar ldquoVisual trajectory-tracking model-based control for mobile robotsrdquo InternationalJournal of Advanced Robotic Systems vol 10 article 323 2013
[3] R Brockett ldquoAsymptotic stability and feedback stabilizationrdquo inDifferential Geometric Control Theory pp 181ndash195 BirkhauserBoston Mass USA 1983
[4] PMorin andC Samson ldquoControl of nonlinear chained systemsfrom the Routh-Hurwitz stability criterion to time-varyingexponential stabilizersrdquo IEEE Transactions on Automatic Con-trol vol 45 no 1 pp 141ndash146 2000
[5] J Ostrowski ldquoSteering for a class of dynamic nonholonomicsystemsrdquo Transactions on Automatic Control vol 45 no 8 pp1492ndash1498 2000
[6] E-J Hwang H-S Kang Ch-H Hyun and M Park ldquoRobustbackstepping control based on a lyapunov redesign for
10 Journal of Control Science and Engineering
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
skid-steered wheeled mobile robotsrdquo International Journal ofAdvanced Robotic Systems vol 10 article 26 2013
[7] F Zouari K Ben Saad and M Benrejeb ldquoRobust adaptivecontrol for a class of nonlinear systems using the backsteppingmethodrdquo International Journal of Advanced Robotic Systems vol10 article 166 2013
[8] W-J Ever and H Nijmeijer ldquoPractical stabilization of a mobilerobot using saturated controlrdquo in Proceedings of the 45th IEEEConference on Decision and Control (CDC rsquo06) pp 2394ndash2399San Diego Calif USA December 2006
[9] V Utkin J Guldner and J Shi SlidingModes in Electromechani-cal Systems Taylor and Francis London UK 2nd edition 2009
[10] J Huang Z-H Guan T Matsuno T Fukuda and K SekiyamaldquoSliding-mode velocity control of mobile-wheeled inverted-pendulum systemsrdquo IEEE Transactions on Robotics vol 26 no4 pp 750ndash758 2010
[11] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[12] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[13] MDefoort T Floquet A Kokosy andW Perruquetti ldquoIntegralsliding mode control for trajectory tracking of a unicycle typemobile robotrdquo Integrated Computer-Aided Engineering vol 13no 3 pp 277ndash288 2006
[14] Ch-Ch Feng ldquoIntegral sliding-based robust controlrdquo in RecentAdvances in Robust Control Novel Approaches and DesignMethods A Mueller Ed pp 978ndash953 InTech 2011
[15] M Defoort J Palos A Kokosy T Floquet and W PerruquettildquoPerformance-based reactive navigation for non-holonomicmobile robotsrdquo Robotica vol 27 no 2 pp 381ndash390 2009
[16] KNonamiMKartidjo K Yoon andA BudiyonoAutonomousControl Systems and Vehicle Springer Tokyo Japan 2013
![Fuzzy Gains-Scheduling of an Integral Sliding Mode …...implementation of an integral action on the controller based on sliding modes for a conventional helicopter. In [27], an adaptive](https://static.documents.pub/doc/80x56/5f085cdd7e708231d421a328/fuzzy-gains-scheduling-of-an-integral-sliding-mode-implementation-of-an-integral.jpg)
