der sliding mode control to mechanical systems

A survey of applications of second-order sliding mode control to mechanical systemsGIORGIO BARTOLINI {*,ALESSANDRO PISANO {,ELISABETTA PUNTA {and ELIO USAI {The effective application of slidingmode control to mechanical systems is not straig htforward because of the sensitivity of these systems to chattering.Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable,so that the latter results are continuous.Generally,this approach requires the availability of a number of time derivatives of the slidingvariable,and,in the presence of noise,this requirement could be a practical limitation.A class of second-order slidingmode controllers,g uaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control,could be helpful both in reducingthe number of differentiator stag es in the controller and in dealingwith unmodelled actuator dynamics.I n this paper different second-order slidingmode controllers,previously presented in the literature,are shown to belongto the above cited class,and some challeng ingcontrol problems involvingmechanical systems are addressed and solved.Simulations and experimental results are provided throughout the paper.1.IntroductionThe control of dynamical systems in the presence of uncertainties is a common problem to deal with when consideringreal plants.Actually,the real plant behav-iour is affected by uncertainties the influence of which should be carefully taken into account when considering the system performance.For this reason,in recent years,the control of uncertain processes has attracted great interest in the research community (Corless and Leitmann 1981,Doyle et al.1994,Ho and Khalil 1997,Young et al.1999,Serrani et al.2001).Among existingmethodolog ies,the slidingmode control (SMC)technique (Slotine and Li 1991,Utkin 1992,1999)turns out to be characterized by high simplicity and robustness.The main idea at the basis of SMC techniques is that of designing a sliding surface to which the controlled system trajectories must belong.On the sliding manifold the behaviour of the system is the expected one.In order to obtain the control aim a control must be designed with an authority sufficient to dominate the uncertain-ties and the disturbances actingon the system.The con-trol promptly reacts to any deviation,however small,from the prescribed behaviour steeringthe system back to the slidingmanifold.An advantag e of this approach is that the slidingbehaviour is insensitive to model uncertainties and disturbances which do not steer the system outside from the chosen manifold.In spite of the claimed robustness properties,the real-life implementation of SMC techniques presents a major drawback,due to the finite switchingfrequency of real control devices.The high-frequency components of the control could excite parasitic resonant modes so that the system trajectories largely differ from the ideal ones.Similarly,unmodelled parasitic dynamics in measure-ment devices and actuators may lead to higher-order slidingmodes which can be either stable or unstable (Fridman and Levant 1996,Fridman 2002).Generally speaking,the system and the actuators non-ideal behav-iour can produce the so-called chatteringphenomenon,which is a high frequency motion that makes the state trajectories rapidly oscillatingabout the slidingmani-fold.Chatteringand the need for discontinuous control constitute two of the main criticisms of Variable Structure Systems (VSS)with slidingmodes,and these drawbacks are much more evident when dealingwith mechanical systems.In fact,it is meaningless to suppose that a system could be driven by a discontinuous torque or force.Furthermore,rapidly changing control actions induce fatigue in mechanical parts and the system could be damaged in a short time.In spite of the above cited drawbacks,VSS with SlidingModes (SM)have been considered often in the technical literature for the robust control of mechanical systems (Slotine and Sastry 1983,Cho and Hedrick 1991,Man Zhihong et al.1994,Rundell et al.1995,Bartolini et al.1998b,2000b,d,2002b,Allen et al.2000,Levant et al.2000,Shu et al.2000,Utkin et al.2000,Satoshi et al.2001,Xu and Cao 2001).Most of them used a straightforward approach to avoid chatter-ing:the sign function of the discontinuous control is approximated by the saturation function.As a result,the system motion is confined within a boundary layer of the slidingmanifold (Slotine and Sastry 1983,Burton and Zinober 1986,Slotine and Li 1991).Nevertheless,if the parasitic dynamics is not well modelled and takenInternational Journal of Control ISSN 0020–7179print/ISSN 1366–5820online #2003Taylor &Francis Ltd/journals DOI:10.1080/0020717031000099010INT.J.CONTROL,2003,VOL.76,NOS 9/10,875–892Received 20May 2002.Accepted 17September 2002.*Author for correspondence.e-mail:giob@diee.unica.it {Department of Electrical and Electronic Engineering (DIEE),University of Cagliari,Piazza d’Armi,09123Cagliari,Italy.{Department of Communications,Computer,and System Sciences (DIST),University of Genova,Via Opera Pia,13,16141Genova,Italy.into account,the approximation of the discontinuous control could compromise the disturbance rejection properties of SMC(Young et al.1999).As chattering is due to switchingdelays and parasitic system dynamics.A different solution is to embed an asympto-tic state observer into the controller so that the discon-tinuous control is confined within a high-frequency loop by-passingthe real plant(Youngand Kwatny1982, Utkin1992).A different approach to avoid chatteringis to aug-ment the controlled system dynamics,by addinginteg ra-tors at the input side,so as to obtain a higher-order system in which the actual control signal and its deriva-tives explicitly appear.If the discontinuous signal coin-cides with the highest derivative of the actual plant control,the latter results are continuous with a smooth-ness degree depending on the considered derivative order.This procedure refers to higher order SM (Emelyanov et al.1986,Levant2001a),dynamic SM (Sira-Ramırez1993),and terminal SM(Man Zhihong et al.1994).VSC implementinghig her-order slidingmodes require the knowledge of a number of time derivatives of the slidingoutput,dependingon the specific control algorithm and on the input/output system relative degree.Even if the use of differentiators(e.g.high-gain dif-ferentiators)has been assumed to be a‘routine matter’as far as the theoretical development of output feedback is concerned(Isidori2000),the practical behaviour of such kinds of devices needs particular care in real imple-mentation,due to the measurement noise,whose nega-tive effects on the overall closed-loop performance dramatically increase with the number of differentiation stages.The research activity is devoted to develop real-time differentiators less sensitive to the noise propagation as well as control algorithms for non-linear uncertain systems with relative degree higher than one,so as to reduce the number of differentiation stages needed (Bartolini et al.2002a).As for thefirst topic,some differentiation algorithms based on second,and higher,order sliding modes have been recently presented(Levant1998,2001b,Bartolini et al.2000a),showingan interestingtrade-offbetween precision and noise-immunity.The second research line led,up to now,to control algorithms belonging to the family of2-SM controllers, that is algorithms in which the relative degree between the constraint output(zeroingwhich the system motion meets the desired performance specification)and the dis-continuous control is two.The main difficulty when dealingwith this kind of controllers for uncertain systems is due to the need to solve differential inequalities of order greater than one.The comparison principle,which permeates most of existingresults in non-linear control theory(including Lyapunov stability theorems)in general does not work in the high relative degree case,and special treatments, often non-systematic,are needed.In this paper various2-SMC algorithms are pre-sented,in a unified framework,as a way to solve classes of second-order differential inequalities.A suitable set of parameters are identified affectingsome qualitative tran-sient properties such as peaking,monotonicity and the global nature of the control results.Finally,some recently presented applications of this class of algor-ithms are surveyed.2.A class of second-order sliding mode controllersConsider an uncertain system,possibly non-linear, characterized by the dynamics_xðtÞ¼FðxðtÞ;tÞþGðxðtÞ;tÞuðtÞð1Þwhere x2X&R n is the state vector,u2U&R is the scalar control variable,F:R nþ1!R n and G:R nþ1!R n are uncertain,sufficiently smooth,vector fields.Assume that the control specifications are fulfilled by constrainingto zero a suitable output variable,i.e. the slidingvariableðtÞ¼ ðxðtÞ;tÞð2Þhavingwell-defined relative deg ree r=2with respect to the control variable u,and that a diffeomorphism C:R nÀ2ÂR2!R n exists such that the dynamics of the internal state wðtÞ2R nÀ2is BIBO stable. Therefore,system(1)can be reduced to the normal form(Isidori1995)€ ðtÞ¼’ðwðtÞ; ðtÞ;_ ðtÞ;tÞþ ðwðtÞ; ðtÞ;_ ðtÞ;tÞuðtÞ_wðtÞ¼ðwðtÞ; ðtÞ;_ ðtÞ;tÞ)ð3ÞwithxðtÞ¼CðwðtÞ; ðtÞ;_ ðtÞÞAssume that the second-order input–output dynamics € ðtÞ¼’ðwðtÞ; ðtÞ;_ ðtÞ;tÞþ ðwðtÞ; ðtÞ;_ ðtÞ;tÞuðtÞð4Þis globally bounded,and let also the sign of the control gain ðÁÞbe constant and known.Then,the second-order slidingmode control problem for system(1) entails thefinite-time stabilization of system(4),which satisfies the global boundedness conditionsj’ðw; ;_ ;tÞj F0<G m ðw; ;_ ;tÞG M)ð5Þ876G.Bartolini et al.Let the slidingvariable and the sign of its total time derivative_ be available for feedback.This stabilization problem has been addressed and solved in many ways(Emelyanov1986,Levant1993, Bartolini et al.1997,1998a,1999,2001a,b).The restric-tive condition(5)was assumed to hold locally in (Emelyanov et al.1986,Levant1993)where specific con-trol actions were devoted to keep the system within the boundedness region.Recently,the global boundedness assumption(5)has been explicitly relaxed by the authors.In Bartolini et al.(2001a)the‘sub-optimal’2-SMC algorithm was considered,and it was shown that if the modulus of the drift term’ðÁÞcan be upper bounded by a known function affine in j_ ðtÞj,then a set of con-stant controller parameters guaranteeing the solution of the problem exists.On the contrary,if j’ðÁÞj can be upper bounded only by a known function of the phase variables ðtÞand_ ðtÞ(without any assumption regard-ingthe g rowth rate w.r.t.j_ ðtÞjÞ,then the convergence to the2-SM can be assured by suitable adaptation of the controller parameters(Bartolini et al.2001b).In order to make the treatment clearer and to present the very idea of the controller design,in the following we shall consider that conditions(5)hold globally.A large number of2-SMC algorithms studied in the cited literature can be practically implemented by setting the parameters of the followingcontrol law properly uðtÞ¼À ðtÞU signð À MÞðtÞ¼1ifð À MÞ M!0 Ãifð À MÞ M<0 (2½0;1Þ9 >>>>>=>>> >>;ð6Þwhere U>0is the minimum control magnitude, Ã>1 is called the modulation factor, is the anticipation factor,and M is the last extremal value of the sliding variable (i.e.the value of at the last time instant at which a local maximum,minimum or horizontalflex point of ðtÞhas occurred). M can be initialized to ðt iÞ(t i beingthe initial time instant)and then updated either by checkingthe sig n of_ or by inspection of the past values of ðtÞ;in the latter case no information about_ is needed.U, Ãand are the controller par-ameters that must be tuned accordingto the inequalitiesU>F G mÃ2½1;þ1Þ\2Fþð1À ÞG M Uð1þ ÞG m U;þ19>>>=>>>;ð7ÞThefirst part of(7)can be referred to as the dominance condition,ensuringthat the control has sufficient author-ity to affect the sign of€ .The second part of(7)repre-sents the convergence condition,sufficient to guarantee the stability of the SM and determiningthe rate of con-vergence as well.Generally,these conditions lead to transient trajectories twistingaround the orig in of the N plane.The additional requirement of monotonic convergence to zero of the sliding variable may be ful-filled by imposinga stricter inequality than(7),i.e.U>FmÃ2½1;þ1Þ\Fþð1À ÞG M UG m U;þ19>>>=>>>;ð8ÞController(6)satisfyingeither condition(7)or(8) assures the establishment of the2-SM behaviour in a finite time T c s.t.T c t M1þmax f T c1;T c2gð9ÞT c1¼UÃG mþG MÃG m UÀFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1À Þj M1jG M UþFsÂ11Àffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij Fþ½ð1À ÞG MÀ à G m U jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG m UÀFpT c2¼UÃG mþG MÃG m UþFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1À Þj M1jG m UÀFsÂ11Àffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij FÀ½ð1À ÞG mÀ à G M U jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÃG M UþFpwhere t M1is the time instant at which thefirst extremal value of the slidingvariable, M1,occurs.The absolute value of M1defines also the maximum magnitude of the time derivative of the slidingvariable,_ ,after the time instant t M1,i.e.j_ ðtÞjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1À ÞðG M UþFÞj M1jq8t2½t M1;þ1Þð10ÞIn order to speed-up the convergence,the following con-trol law should be implemented from the initial time instant t i until thefirst extremal value is reached at t M1 uðtÞ¼ÀU i signð À ðt iÞÞU i>FG m8t2½t i;t M19>=>;ð11ÞThe initializingcontrol(11)assures thatt M1t iþj_ ðt iÞjG m U iÀFð12ÞFrom equations(9),(10)and(12),it is apparent that the larger the control magnitude U,the shorter the conver-gence time and the larger the maximal values of the time derivative of the slidingvariable.Nevertheless,once theApplications of sliding mode control to mechanical systems877the parameters U and Ãare set,accordingto proper considerations and conditions,the parameter still allows for limiting j _ ðt Þj .Therefore,by a proper choiceof U , Ãand it is possible to obtain both the mono-tonic convergence to zero of the sliding variable and the counteraction of the peakingphenomenon duringthe reachingphase of the 2-SM behaviour.Obviously,very strict requirements on j _ ðt Þj imply a very long reachingphase;at the limit,if !1then j _ ðt Þj !0and the slidingbehaviour is reached in an almost infinite time,i.e.T c !1.The proof of the above presented convergence prop-erties for the 2-SMC (11),(6),with tuningrules (7)or (8)entails algebraic computations analogous to those in Bartolini et al.(1997,2001b),and it is omitted for the sake of brevity.It should be noted that different 2-SMC algorithms presented in the literature (Levant 1993,Bartolini et al.1997,2001b)are characterized by different values of the anticipation coefficient .2.1.Twisting algorithmThe ‘twisting’algorithm (Levant 1993)was one of the first proposed algorithms belonging to the consid-ered class of 2-SMC.It is based on the knowledge of the sign of both and _u ¼À 1sign ð Þþ 2sign ð_ Þ;1> 2ð13ÞI n the ideal case,by settingparameter to zero,the control law (6)causes the system to have the same tra-jectories of the twistingalg orithm with 2¼U and1¼ ÃU .The phase trajectories of the slidingvariable twist around the origin of the two-dimensional phase plane (figure 1),and this implies that the sliding variable cannot have a monotonic behaviour;furthermore,the maximal values of j _ ðt Þj depend directly on the chosen values of U .If a large control authority is needed,because of large uncertainties,transient peaking will arise.The con-vergence conditions for this 2-SMC are easily obtained by setting ¼0in (7),i.e.U >F G mÃ!2F þG M U G m U 9>>>=>>>;ð14Þ2.2.Sub-optimal algorithmThe ‘sub-optimal’algorithm was developed by some of the authors derivingit from the time-optimal control law for a pure double integrator (Bartolini et al.1997).It singles out by the specific value of the anticipation coef-ficient ,i.e. ¼0:5.The anticipation of the switching,with respect to that defined by the ‘twisting’algorithm,implies that the trajectories in the phase plane can show both twistingand bouncingbehaviours (fig ure 2).This property allows for a monotonic convergence to zero of ðt Þby a proper choice of parameters U and Ãaccord-ingto (8).As for the ‘twisting’algorithm,the maximal values of j _ ðt Þj depend directly on the chosen values of U ;therefore,the transient peakingphenomenon will arise.The convergence conditions for the sub-optimal con-troller are easily obtained by setting ¼0:5in (7),i.e.U >F G mÃ2½1;þ1Þ\4F þG MU 3G m U;þ1 9>>>=>>>;ð15Þ878G.Bartolini et al.(t)d /d t(t )The phase trajectories for the twisting algorithmFigure 1.Twisting algorithm:phase trajectory.(t)d /d t (t )The phase trajectories for the sub -optimal algorithmFigure 2.Sub-optimal algorithm:phase trajectory.and the monotonic convergence conditions are coinci-dent to(14).If the control magnitude,U,is set as a function ofthefirst extremal value M1,then the convergence prop-erty of the‘sub-optimal’2-SMC can be extended to manage a broader class of systems in which the upper bound of the magnitude of the drift term,j’ðÁÞj,is state-dependent but grows affinely with respect to j_ ðtÞj (Bartolini et al.1998a,2001a).2.3.Global algorithmThe‘global’2-SMC has been recently proposed by the authors(Bartolini et al.2001b),and it is based on constrainingthe trajectory of the slidingvariable within a pre-specified region of the phase plane,so that a local bound for the uncertainty can be evaluated and never break through because of the control action.It should be used when the modulus of the uncertain drift term is upper bounded only by a known convex function ofand_ ,i.e.j’ðwðtÞ; ðtÞ;_ ðtÞ;tÞj~Fðj ðtÞj;j_ ðtÞj0<G m ðwðtÞ; ðtÞ;_ ðtÞ;tÞG M )ð16ÞIn this case the global convergence property for the2-SMC(6)can be assured only if the anticipation factor is properly set and thefirst extremal point is reached by means of(11)in which the control amplitude is defined by the knowledge of an upper bound for the initial mag-nitude of the slidingvariable derivative.In particular,in Bartolini et al.(2001b)the modula-tion coefficient is not used(i.e. ü1),and both the anticipation coefficient and the control magnitude U are dynamically assigned according to the following control lawuðtÞ¼À1G mð~Fðj ðtÞj;_ ðt iÞÞþ 2Þsignð ðtÞÀ ðt iÞÞt i t<t M1ð17ÞuðtÞ¼ÀU k signð ðtÞÀ k ðt MkÞÞU k>1G mð~F kþ132Þk¼max12;1À22ð~F kþG M U kÞ()t M k t<t M kþ19 >>> >>> >=>>> >>> >;ð18Þwhere_ ðt iÞ>_ ðt iÞ, ¼0is a design parameter affectingthe transient time,t Mk (k¼1;2;...)is the sequence ofsubsequent time instants at which the condition_ ðt Mk Þ¼0is fulfilled(i.e. ðt MkÞis a singular point forthe function ðtÞÞ, >0is a design parameter that defines the upper limit for j_ j after thefirst extremalpoint for is reached,and~F k¼~Fðj ðt Mk Þj,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij ðt MkÞjp)is an overestimate of the upper bound of the uncertain drift term magnitude,j’ðÁÞj,between twosubsequent singular points.It can be noted that the controller(18)could be implemented with constant parameters U1and 1since this choice still assures that the slidingvariabletrajectories are confined within the region O1¼fð ;_ Þ:j ðtÞj j ðt M1Þj;j_ ðtÞjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij ðt M1Þjpg,8t!t M1.3.Applications of2-SMC to mechanical systemsThe use of second-order slidingmodes appears to be strongly motivated in the control of uncertain mechan-ical systems.In this context,indeed,it is usuallyassumed that the control signals are mechanical quanti-ties like forces or torques(kinematic control schemesemploy control actions like the speed of the vehicle,itssteeringang le,etc.)and such control actions cannot bemade discontinuous for physical reasons,and are notsuitable to approximate the sign function by means,for instance,of satðÁÞor tanhðÁÞterms,due to resultingsystem stresses and oscillations.Let us consider robotic manipulators,which repre-sent a paradigm of strongly coupled multivariable non-linear mechanical dynamics.As far as the theoreticalinvestigation of the control of this kind of system isconcerned,the basic ideal control is the so-called‘com-puted torque’,which is the continuous control thatcauses the trajectory of the system to track,with arbi-trarily fast vanishingerror,a pre-specified set of refer-ence trajectories.This control can be parameterized as the product between a matrix of constant parameters,of suitabledimension,and a regressor vector whose componentsare non-linear functions of the state vector and of thedesired trajectories,and the relevant complexity rapidlyincreases with the number of degree of freedom(dof).The situation is even worse if the mechanism is con-strained to some surface,since the representation of theconstrained motion is derived through rather complexand constraint-dependent projection techniques,in amuch more involved way than modellingunconstrainedsystems.On the other side,at least in simulation,exploitingthe positive definiteness of the inertia matrix(Lewis et al.1993),the use of the slidingmode approachallows better performance to be achieved by means of adiscontinuous multivariable controller whose complex-ity is not dependent on the number of dof.These advantages have not been recognized to be effective in the literature,due to the basic reason thatdiscontinuous torques or forces are physically meaning-less,and,when approximated in the standard way bymeans of satðÁÞor tanhðÁÞfunctions,the high-frequencycommutingmechanical input may damag e the con-trolled system due to the presence of backlash.Applications of sliding mode control to mechanical systems879Second-order slidingmodes can cope with such kind of problems by considering,as actual control signal,the electrical input of the torque/force actuators. Furthermore,switchingelectrical sig nals represent the standard operation mode in electrical power drives (Pulse Width Modulation(PWM)control).Indeed,the actuator dynamicsfilters out the high frequency com-ponents of the electrical input,and the generated force/ torque is relatively smooth,the higher the switching fre-quency the lower the force/torque ripple.These advan-tages incur the cost of a slightly increased complexity of the controller with respect to thefirst-order SMC.Various examples of the application of the2-SMC methodology are reported as a demonstration of its effectiveness in solvingpractical control problems invol-vingmechanical plants.Ubiquitous to any of the follow-ingcase studies is the counteraction of the chattering phenomenon by usingas discontinuous control action the time derivative of the mechanical input.A second aspect pointed out by means of most of the considered examples is the achievement of specific tran-sient behaviours by a suitable choice of the control law parameters.A promisingapproach in order to extend the effec-tiveness of the SM procedure consists in couplingSM, through a suitable choice of the sliding manifold,with other techniques(e.g.H1or standard methods).In this way it is possible to cope even with non-matchinguncer-tainties(Edwards and Spurgeon1998).Work is in pro-gress on this topic.3.1.Overhead cranesConsider a container crane of the type infigure3 (where the laboratory-size prototype used for our experiments is shown).By takingthe trolley position x t,the rope length l and the swingang le’(and their time derivatives)as the state variables,assumingthat the load can be reg arded as a material point and that the rope is always stretched (so that the swingang le can be uniquely defined),the equations of the motion can be derived,for instance,by the Eulero–Lagrange formalism(Sakawa and Shindo 1982)A1€x tþB1€l sin’þB1l€’cosþB1ð2_l_’cos’Àl_’2sin’ÞþB2_x t¼t1ð19ÞB3€x t sin’þA2€lÀB3l_’2ÀB3g cos’þB4_l¼t2ð20Þ€x t cos’þl€’þ2_l_’þg sin’¼0ð21Þwhere t1and t2are the applied trolley and hoistingtor-ques,respectively,g is the gravity constant,and A1;A2;B1;...;B4are positive constants(the viscous friction coefficients B2and B4were neglected in Sakawa and Shindo(1982)).The horizontal and vertical load coordinates,which will be referred to as x and y, respectively,can be straightforwardly expressed as func-tions of the trolley position,rope length and swing anglex¼x tþl sin’y¼l cos’)ð22ÞThe actuators dynamics(permanent magnet DC motors)are linear and time-invariant,and can be expressed in compact form asP1_tþP2t¼vÀP3_rð23Þwhere v¼½v1v2 T is the vector of the motor supply voltages,t¼½t1t2 T,r¼½x t l T and P i,i¼1;2;3,are suitable positive-definite diagonal matrices.Let the crane and motor parameters be uncertain but belongto a known compact set P.The goal is to move the load from the initial positionðx i;y iÞto afinal, desired,locationðx f;y fÞalonga pre-specified path, while keepingthe load oscillation as small as possible.The control problem is particularly involved since there are three degrees of freedom(dof)to be controlled (the load coordinates and the swingang le)but only two control actions(the trolley and hoistingmotor voltag es), i.e.the system is under-actuated.A suitable couple of slidingoutputs,that involve the three controlled dof, must be found such that the associated zero-dynamics is stable and meets the control objective.The dynamics of the force actuators can be explicitly dealt with by our approach.We propose the followingcouple of slidingoutputss1¼_x tÀ_x d tþc1ðx tÀx d tÞÀk’s2¼_lÀ_l dþc2ðlÀl dÞ9=;ð24Þwhere k is a positive constant and the superscript d represents the target trajectory.880G.Bartolini etal.Figure3.The overhead crane model:a front-view.。