CDCREG ASimpleOutputFeedbackPDControllerforNonlinearCranes B
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CDCREG ASimpleOutputFeedbackPDControllerforNonlinearCranes B

Kiss J L evine and Ph Mullhaupt Centre Automatique et Syst emes Ecole Nationale Sup erieure des Mines de Paris 35 rue SaintHonor e F77305 Fontainebleau email kisslevinemullhaupt casensmpr Abstract A simple output feedback PD controller is proposed t

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CDCREG ASimpleOutputFeedbackPDControllerforNonlinearCranes B

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CDC00-REG1133 ASimpleOutputFeedbackPDControllerforNonlinearCranes B. Kiss, J. L evine and Ph. Mullhaupt Centre Automatique et Syst` emes Ecole Nationale Sup erieure des Mines de Paris 35, rue Saint-Honor e, F-77305 Fontainebleau e-mail: kiss,levine,mullhaupt ,cas.ensmp.-r Abstract: A simple output feedback PD controller is proposed that stabilizes a nonlinear crane. Global asymptotic stability is achieved at any equilibrium point specified by the controller. The control scheme relies solely on the winches position and velocity and hence no cable angle measurement,or no

direct measurement of the load position,is needed. The controller can be ex- tended to many different kinds of existing cranes. Keywords Crane control,Output feedback,PD controller,Underactuated mechanical system. 1 Introduction Cranes constitute good examples of nonlinear os- cillating systems with challenging industrial ap- plications. Their control has been approached by various techniques,linear [(,)0],or nonlin- ear [,,-,.]. As noted by [))],the productiv- ity of harbor cranes might be significantly im- proved if one could decrease the time needed to damp the oscillations of

the load,without requir- ing the installation of fragile or complicated sen- sors. /ndeed,measurements on all configuration variables are generally not available 0especially as far as the rope angles or the load position are concerned1 due to the severe operating environ- ment. 2ad weather,dust,oil,frequent shock risk restrict the panel of efficient and reliable sensors at the designer4s disposal and in particular makes the use of sophisticated artificial vision systems uneasy. Consequently,state feedback techniques Research supported by the Nonlinear Control Net- work,

European Commissionís Training and Mobility of Re- searchers (TMR)Contract # ERB#MRX-CT970()7 cannot be directly applied. /n this paper,we pre- cisely address the question of damping the load4s oscillations to swiftly bring the load to its equilib- rium,using only sensors 0incremental encoders1 mounted on the motor axes and therefore giving only an indirect information on the load4s position. 5e propose a simple output feedback controller of the proportional derivative type that ensures global asymptotic stability under the hypothesis that the ropes are rigid. The proof of stability relies on

the application of 6a7alle invariance principle [8,9,)] and on the particular structure of the crane dynamics [-]. Un- fortunately the 6yapunov function does not pro- vide information on the rate of convergence and the gain tuning may be achieved using simulation owing to the reduced number of design parame- ters. The paper is organized as follows. 7ection . recalls basic stability definitions and main theo- rems that assess this property. /n 7ection :,we recall from [;,-] the model of the crane used in this study. Then 7ection ( gives the controller for equilibrium stabilization with

its proof of stabil- ity. 7imulations confirm the good closed loop be- haviour of the controlled crane,followed by some conclusions and open questions. 2 Stability definitions and theo- rems Consider the system ,x 0)1 001 = 0
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where 1 is lipschitz continuous and let t,x denote the unique solution of the above system with initial condition 001 = . This material is standard and can be found in [8]. Definition 1 (Stability) The equilibrium =0 of (1) is stable if for all > , there exists a δ> such that < t,x < , for all Definition 2

(Asymptoticstability) The equilibrium =0 of (1) is asymptotically stable if it is stable and if, lim t,x 1=0 A sufficient stability condition is given by the fol- lowing theorem. Theorem 1 (Lyapunovíssecondmethod) If there is a function such that 1. ∈U \{ 2. ∈U \{ where is a neighborhood of then is locally asymptotically stable. Moreover, if and is radially unbounded, i.e. as , then is globally stable. /f 1 = 0 for a set of points including the ori- gin then the stability is not guaranteed. /n order to deal with this case one needs some additional definitions.

Definition 3 (Invariantset) Aset is said to be invariant with respect to 0)1 if, ∈I ,x t,x ∈I Definition 4 (Positivelyinvariantset) set is said to be positively invariant with respect to 0)1 if, ∈I ,x t,x ∈I Definition 5 (Approachingaset) We say that approaches a set as ,iffor each > , there is a T> such that, inf ∈M <, t>T. Theorem 2 (LaSalleinvariancetheorem) Let C⊂U be a compact set that is positively invariant with respect to (1). Let U be a continuously differentiable function such that for all ∈U .Let be

the set of all points in where 1=0 .Let be the largest invariant set in . Then every solution starting in approaches as 3 Nonlinear Crane Model 5e will consider the model of an onboard disem- barkment crane used by the U7 Navy. Aor sim- plicity of the exposition we restrict the system to evolve in a fixed vertical plane. This restriction does not impart on generality. mg Aigure )? U7 Navy crane The crane illustrated in Aigure ) consists of the following main parts? a pole making a fixed angle with respect to the vertical,equipped with two winches,one located at the top,denoted by

and chosen as the origin,and the second one located at ,at a fixed distance from
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a vertical rope of variable length ,starting from ,whose upper part makes an angle with the vertical,passing through a pulley located at the point ,the lower part of the rope making an angle with the vertical. The length of the upper part is denoted by and the one of the lower part by . 7ince the total length of the rope is ,we have a horizontal rope of variable length relat- ing the winch to the pulley a load with mass attached to the vertical rope at the point ,located at a distance from

the pulley the winches at the points and with radii and are supposed to be torque con- trolled using electric motors with incremental encoders on their axes. All friction forces are supposed to be compensated. 5e consider a reference orthonormal frame O,x,z 1with Oz oriented upwards. 6et denote the gravity acceleration and 0 x,z 1 the coordinates of the load . The masses of the ropes are ne- glected and the ropes are assumed to be unstretch- able. Also denote the modulus of the force in the rope at and the modulus of the force in the rope at The modeling of this system has been under- taken in

[;] which concludes to an implicit model. The dynamics of the load are given by sin cos 0.1 the force equilibrium at the pulley reads? sin0 1C sin0 1C sin =0 cos0 1C cos0 cos =0 0:1 and the geometric constraints are sin0 cos0 sin cos sin sin0 cos cos0 0(1 The dynamics of the winches are given by 0,1 081 Notice that the unstrechability of the ropes im- plies . Eoreover,using the equations 0:1, it is easily verified that 1091 and that =. cos The crane has three degrees of freedom and a possible choice of the generalized coordinates is =0 1 which will be used in the sequel. The only

external efforts are the torques and delivered by the motors. 6et 0 x, 1 denote the coordinates of the load at equilibrium. Then one may calculate the equilib- rium of the remaining variables using the following relations? sin sin =0 C arcsin sin sin sin sin sin0 sin cos0 sin sin =. mg cos γ, mg. 0;1 Notice finally that due to the geometry of the crane, 4PD Controller and Stability Analysis 5e wish to stabilize the crane at a given equilib- rium 0 x, 1. 5e claim that this can be achieved using the following PD controllers? dA pA 0-1 dO pO 0)01
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where the a

priori rope tensions and are determined using Fquation 0;1 and pA pO dA dO are constant gains,yet to be determined,so as to achieve satisfactory performance. The crane depicted in Aigure ) has,in the ab- sence of the controllers,kinetic and potential en- ergy due to the load and kinetic energy due to the inertia of the winches and .6et kin de- note the total kinetic energy and pg the potential gravitic energy. 5hen the controller is present, extra energy can be stored in the controller due to the constant a priori and proportional terms. This energy will be denoted by ctrl Thus,the energy

function consists of three terms? kin pg ctrl 0))1 with kin 0 C 1C pg mgz ctrl pA pO R. 0).1 where,if we use generalized coordinates γ,L ,R sin cos sin sin0 arcsin0 /l sin sin 0 C arcsin0 /l sin cos sin sin sin0 arcsin0 /l sin 11 cos 0 C arcsin0 /l sin 0):1 Therefore using the 6agrangian kin pg ctrl 0)(1 the crane dynamics can be obtained by applying dt ∂q 0),1 where ,and is the asso- ciated generalized force,i.e. =0, dO and dA due to the derivative terms in the controllers. Notice that the proportional term and the constant a priori forces are already in the potential function and

thus absent in the gener- alized forces. Notice also that the actual choice of generalized coordinates does not lead to the most compact formulation of the dynamics but will make the derivation of a necessary lemma easy. Lemma 1 The time derivative of the energy function is dt dA dO The proof is an easy adaptation of derivations appearing in most textbooks on classical mechan- ics that prove energy conservation in purely 6a- grangian systems 0no dissipation1 [).,:]. Gere extra terms are present due to the derivative com- ponents in the controller. Gence,it remains to characterize the sets of

sys- tem traHectories such that = 0 and = 0. Note the usage of to signify that the quantity stays for all times at the value Lemma 2 The only invariant trajectory compat- ible with =0 and =0 is the equilibrium trajectory, i.e. Proof: The input torques and are respon- sible for forces in the ropes and and motion along and 0)81 0)91 2y using the control strategy proposed,i.e. ap- plying PD controllers on both winches,the torques and satisfy 0-1 and 0)01 where and are the forces corresponding to the equilibrium posi- tion and Putting these equations together and under the condition that = 0 and =

0 0since we are interested in the traHectories compatible with = 0,i.e. both stay at constant values say and 1 yields, pA 1 0);1 pO 0)-1
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Notice that whatever the traHectory of the load is we have that . cos 1= .7ince and are constant,so must be This shows that all configuration variables are constant if 0and 0. /t follows that the only traHectory compatible with = 0 is an equi- librium of the system. 6et us denote the values of the variables at this equilibria by a hat. /t remains to show that the equilibrium charac- terized by the hated variables coincides with the

desired equilibrium given by the bared variables. Airst,observe that for every equilibrium posi- tion of the load mg . Using 0)-1 we conclude that . The equalities and =I will be proved by contradiction. Aor,sup- pose that γ> . Jecall that = 0,thus 091 implies β> .7inceI γ, ] it is easily verified that sin sin sin0. sin is a strictly increasing function of its argument, thus we conclude that . Noticing that pA 0 and using 0);1 we have that 2ut then the relations =. mg cos and mg cos I imply that γ< ,a contradiction. One arrives to a similar contradiction supposing

that γ< thus we conclude that =I and and the lemma is proved. 5e can now state our main stability theorem for the nonlinear crane together with the PD con- trollers given by the equations 0--)01. Theorem 3 The crane with rigid cables equipped with P( controllers for both winches is globally asymptotically stable. Proof: Choose a sufficiently large such that, for both the initial condition and the equilibrium, W with being the function defined in 0))1. Define the set Using 6emma ),we get dO dA 7ince 0,the system4s traHectory stays in . Eoreover is bounded from below in the

set hence this latter set is positively invari- ant and compact. 6emma . characterizes the set 1=0 as being a finite set con- sisting of the equilibrium point x, . The claim follows by applying Theorem . with both previ- ously defined sets and and Remark1 )otice that the model was obtained under the hypothesis that the cables were rigid and thus could transmit positive and negative forces to the winches which is normally not the case. As long as γ< is guaranteed to be positive and the force can be transmitted. 5hen the cables are not rigid,they can get out of the pulleys due

to the negative tension that can- not be delivered. 7ome extra mechanical device should be present to prevent such an event. Al- though this does not lead to an instability as such, the set of initial conditions that are handled prop- erly by the controller is somewhat reduced as in the case of rigid cables. 5 Simulation study Note that,though this controller has been suc- cessfully experimented on our reduced-size model of crane,we can only present simulation results since we do not have sensors to measure the posi- tion of the load or the angles of the cables and to record them. 7uch

measurements should be made possible in the future by adding a camera. The crane model is simulated using the fol- lowing parameters? =0 .[kg], .,)0 [kgKm ], =0 :, [m], =0 ((, [rad]. These parameters correspond to a )K:0 small-scale model of a real U7-navy crane at disposal at the authors lab. The equilibrium position is set to be [m] and ,[ ]. The gains have been set to = .0, pA = )0, =)0and dA = .0. The tuning of the gains has been done in simu- lation. Note that the global stability of the regulator is not sensitive to the values of the design parameters as shown by Theorem :. 6 Conclusion

Crane control is addressed using a simple output feedback PD controller,using only angular sensors
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-0.1 -0.05 -0.55 -0.5 -0.45 -0.4 trajectory of the load 0.51 0.515 0.52 0.525 0.53 0.535 R (length of the vertical rope) 0.02 0.03 0.04 0.05 0.06 0.07 0.08 (length of the horizontal rope) -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 angle sec sec sec rad Aigure .? Closed-loop behaviour under PD control placed at the winches. 5e show that it globally asymptotically stabilizes any equilibrium position under the hypothesis that the cables are rigid. Eoreover,it is easy to implement and

efficient if the crane model is accurate enough,or more pre- cisely,if the frictions are satisfactorily compen- sated. Note that we have not used in this work the flatness property of the crane model 0see [-]1 since we are only interested in equilibrium points. Gow- ever,flatness might play an important role to ex- tend this controller design in the context of track- ing of traHectories that bring the load to an idle position,a question that still remains open. References [)] F.A. 2arbashin and N.N. Mrasowski. On the stability of the motion in the large. (oklady Akad. )auk

SSS, ,;8?(,:N(,8,)-,.. [.] E. Aliess,O. 6 evine,and P. Jouchon. A generalised state variable representation for a simplified crane description. Int. -. of .on- trol ,,;?.99N.;:,)--:. [:] D. T. Greenwood. .lassical (ynamics Prentice-Gall,Fnglewood Cliffs,N.O.,)-99. [(] T. Gustafsson. On the design and implemen- tation of a rotary crane controller. European -. .ontrol ,.0:1?)88N)9,,Earch )--8. [,] M.7 Gong,O.G. Mim,and M./ 6ee. Control of a container crane? Aast traversing,and resid- ual sway control from the perspective of con- trolling an underactuated system. /n Proceed- ings of

the A.. ,pages ).-(N).-;,Philadel- phia,PA,Oune )--;. [8] G. M. Mhalil. )onlinear Systems . Prentice- Gall,Fnglewood Cliffs,N.O.,second edition, )--8. [9] O. 6a 7alle and 7. 6efschetz. Stability by Li- apunovís (irect Method With Applications Eathematics in 7cience and Fngineering. Academic Press,New York,6ondon,)-8). [;] O. 6 evine. Are there new industrial perspec- tives in the control of mechanical systems R /n Paul E. Arank,editor, Advances in .on- trol ,pages )-,N..8. 7pringer-Serlag,6ondon, )---. [-] O. 6 evine,P. Jouchon,G. Yuan,C. Gre- bogi,2.J. Gunt,F. Mostelich,F. Ott,and O.

Yorke. On the control of us navy cranes. /n Proceedings of the European .ontrol .on- ference ,pages NN.)9,2russels,2elgium,Ouly )--9. [)0] A. Earttinen,O. Sirkkunen,and J.T. 7almi- nen. Control study with a pilot crane. IEEE Trans. Edu. ,::?.-;N:0,,)--0. [))] J.G. Overton. Anti-sway control system for cantilever cranes. 1nates States Patent 0,,,.8,-(81, Oune )--8. [).] F. T. 5hittaker. A treatise on the analytical dynamics of particles and rigid bodies . Cam- bridge University Press,Cambridge,)-(9.