Copright IFAC 5th riennial World Congress, Barcelo, Spain RAJECORY CONROL OF CRANE LOAD POSIION AND INCLINAION Harald Aschemann, Oliver Sawodn, Eerhard P. Hofer Department of Measurement, Control and Microtechnolog Universit of Ulm, D-8969 Ulm, German E-mail: harald.aschemann@e-technik.uni-ulm.de el.: +49 73 56336, Fax: +49 73 563 Astract: A gain-scheduled trajector control is exemplar presented for two of six availale axes of an overhead crane that allows for positioning and inclining the crane load according to specified trajectories. Besides its tracking capailities, the proposed control provides an active oscillation damping. he overall control structure consists of independent axis controllers, which are adapted to measurements of varing sstem parameters. hese axis controllers take advantage of comined feedforward and feedack control as well as oserver ased disturance rejection. he achieved control performance is shown selected experimental results from an implementation on a 5 t - ridge crane. Copright @ IFAC Kewords: rajector control, gain scheduling, disturance oserver, crane control, rootics. INRODUCION Until now, most papers concerning crane control have focussed on position control of the crane load in the three dimensiol workspace ignoring the additioll three degrees of freedom as regards load orientation (Boustan and d`andréa-novel, 99; Brfors and Sluteij, 996; Delaleau and Rudolph, 998; Nguen 994). At the Universit of Ulm, a trajector control concept for an overhead crane has een developed that focuses on the load motion and allows for trajector control as well as active oscillation damping concerning all six load degrees of freedom. For this purpose, the crane has een upgraded with an orientation unit that is euipped with three additiol axes, each actuated an torue controlled DC-motor. B this, the capailities of an automated overhead crane are extended consideral and, moreover, it can e considered as a root manipulator that comines the capailit of handling heav loads with a large workspace. he overhead crane consists of six actuated axes (fig. ). he x-, - and z-axes allow for positioning of the crane load in the three dimensiol workspace. he crane load can e hoisted or lowered in z-direction means of the rope suspension. he x-axis represents the direction of the ridge motion, whereas the direction of the trolle motion on the ridge is referred to as -axis. left rail Fig.. Crane structure trolle right rail crane ridge rope suspension orientation unit and suspended crane load
he orientation unit with additioll three axes for orientating the crane load is euipped with two additiol translatiol axes (a- and -axis) and one rotatiol axis (c-axis) for the orientation around the z- axis, i.e. the rope suspension. In the seuel, oth the -axis and the according axis of the orientation unit, the -axis, are regarded exemplar. his involves the derivation of an appropriate mathematical model, feedforward and feedack controller design as well as the design of disturance oservers to compensate for non-linear friction forces acting on the drives. he implementation of the proposed control at a 5 t - ridge crane is descried in detail and measurement results taken from experiments at the ridge crane emphasise the efficienc of this control approach.. MULIBODY MODEL OF HE CRANE First, a central mathematical model for oth crane axes under consideration shall e estalished. his common model is then used to derive a decentralised design model for each of these axes, i.e. the -axis and the -axis. x z F (t) - d M m R (t) m, J, k C MC GC F (t) - d MC C C L (t) Fig. : Multiod model of the crane sstem J M, k F FC C L G F F l R s L(m L) m, J (m) L Lx L he motion of the mechanical sstem in -direction is descried a multiod sstem consisting of three rigid odies (fig. ): the trolle (mass m, drive mass moment of inertia J M, resulting gear transmission ratio k G ), the carriage of the orientation unit (mass m C can e neglected, drive mass moment of inertia J MC, resulting gear transmission ratio k GC ) and the crane load (mass m L, mass moment of inertia J Lx ). he load center of gravit C L shows an offset s L in z- direction. he rope suspension is modelled as a massless rigid link with length l R. he kinematics of the mechanical sstem in -direction can e descried four generalised coordites: the position of the trolle (t), the rope angle ϕ R (t), the displacement of the carriage C (t), and the orientation angle of the load ϕ L (t). he controlled variales are the position L (t) of the load center of gravit and the orientation angle of the load ϕ L (t). B appling Lagrange s euations, non-linear euations of motion are computed in smolic form. Under the assumption of oth small rope angles and small load angles, the following linearised euations can e stated M + D + K = fu + f e = Fu u + Fe e, () with the mass matrix () m lr = m L l R m L l R lr lrsl M lr + J MCkGC lrsl + J Lx containing m = m + J M kg +, the damping matrix D = diag( d dc ), (3) the stiffness matrix lr g K =, g g g (4) and the input matrices F u = Fe =. (5) according to the control input u = [ F M FMC ] as well as the disturance input e = [ F F FFC ], respectivel. he smolic mathematical model contains two sstem parameters with domint variations during operation: the rope length l R and the load mass m L, which are comined in the vector of varing sstem parameters p = [l R m L ]. o take these model variations into account, the gain scheduling techniue is utilised. As the varing sstem parameters are availale direct measurements, the complete control structure can e adapted. With the usual choice of the state vector x = [ ], (6) the state space representation ecomes I = M K M D + u + e M F M F u e or in areviated notation (7) x = A x + Bu u + Be e. At this, the sstem matrix A, the input matrices B u and B e of the control input u and the disturances e are introduced, respectivel. 3. DECENRALISED DESIGN MODELS he mechanical models that are used for the decentralised feedack control design of the -axis and the -axis are directl derived from the central design model appling model reduction techniues. For each axis =, the vector of generalised coordites is expressed as a linear comition of gene-
ralised coordites and given reference motions g according to = J + g, (8) where denotes the vector of generalised coordites for the design model of reduced order. he corresponding time derivatives result in = J + g, = J + g. (9) B inserting these expressions into the euations of motions, the reduced order euations of motion for the -axis are otained M + D + K = f u + fe. () he new sstem matrices are given M = J M J, J D J D =, K = J K J.() he new control input is f u = J fu, whereas the new disturance vector f = J f M g D g K g () e [ ] e consist of non-linear friction forces as well as coupling forces due to g and its first two time derivatives. 4. DECENRALISED FEEDBACK CONROL he decentralised design model for the -axis can e stated as M + D + K = fu u + Fe e. (3) he new sstem matrices are given m L + J MCkGC M =, J Lx + d = C g D, K =, (4) g g f u =, F e =. he motor force of the carriage drive represents the control input u = F MC, whereas the vector of disturances e = [ FFC S ] consists of oth the nonlinear friction force F FC acting on the carriage drive and the coupling acceleration of the rope suspension S = + lr ϕ R. he euations of motion are transformed into state space representation x = A x + uu + Bee, (5) according to the state vector x = [ ]. For control design the LQR approach ased on the time weighted performance index J = α t ( Q x + r u ) e dt x (6) is emploed, with the diagol weighting matrix Q and the scalar weighting factor r >. Using this time weighted criterion, the maximum real part s = - α of the resulting closed-loop poles can e specified. he corresponding linear feedack control law, FB u,fb = k (,i ) x, which contains the gain vector, FB k (,i ) = r u(,i ) P(, i ), (7) is calculated with the solution P(m L,i ) of the algeraic Riccati-euation (ARE) (8) A ( m ) P( m ) P( m )A ( m ) α L,i L,i + L,i α L, i P( m ) ( ) L,i u,i r u (,i )P(,i ) + Q =. Here, the matrix A α = A + α I has to e utilised according to the chosen performance index. he solution of ARE P(m L,i ) and, hence, the gain matrix k are continuous if oth the sstem matrices A FB, and u depend continuousl on the load mass m L and constant weightings are used, i.e. Q = const. and r = const. In order to allow for gain scheduling, the control design is performed for a specified numer of operating points i representing the range of possile sstem parameter variation. hen, an approximated description is derived for this set of controller gains using look-up tales in comition with linear interpolation etween the operating points. In fig. 3, the controller gain k FB, (m L ) for the first state variale C is depicted exemplar. control gain k FB.5 x 5..5..5.95.9.85 8 4 6 8 load mass [kg] Fig. 3. Feedack control gain k FB, (m L ) in dependence on the load mass m L o avoid hidden weightings, the elements of the diagol matrix Q and r are divided the suared maximum values of the according variales, respectivel. he resulting closed-loop poles due to this gain-scheduled LQR design are variale in contrast to a gain scheduling ased on a fixed pole configuration (Aschemann, et al., a). he control design for the -axis is performed alogousl and leads to the feedack control law u,fb = k, FB ( p ) x, in which the vector x = [ L L] denotes the state vector and L = + ( lr + sl ) ϕr represents the load position that is used for feedack. 5. CENRAL FEEDFORWARD CONROL he central feedforward control action consists of oth a linear part and a non-linear part. he linear feedforward control part provides the reference forces according to the linear sstem model and is calculated with the reference trajector. Aim of the feedforward control design is an perfect tracking of specified trajectories for the controlled variales: the
load position in -direction L as well as the load orientation ϕ L. Both controlled variales are comined in the output vector (9) L lr sl = = C x = x, ϕ L where the matrix C descries the linear dependence on the state vector x. For the central feedforward control part the following form has een chosen, where the index d denotes the decentralised parts on the diagol and the index c stands for the central off-diagol parts u k k,lf w u = = d c LF. () uϕ k k,lf w c d ϕ It represents a linear comition of all reference values and its first four time derivatives, which are comined in the reference vectors () ( ) [ ] IV ( ) [ ϕ ϕ ϕ ϕ ϕ ] IV w =, w =. ϕ Altertivel, this feedforward control law can e expressed in the freuenc domain using a transfer function matrix u LF = GLF ( s ). ϕ () B rewriting oth decentralised feedack control laws in dependence on the state vector x u,fb ufb = = K FB x u,fb (3) the command transfer function matrix ecomes (4) G s ) = C( si A + B K ) B G ( s ) ( u FB u LF ij kij ( he feedforward gain vectors k = p ), which depend on the varing sstem parameters p, can e calculated in smolic form appling the fil value theorem of the Laplace transform to the elements of G(s). Up to the fourth time derivative of the reference trajector, ideal tracking ehaviour is aimed at for the diagol elements, whereas ideal decoupling is aimed at for the off-diagol elements (Aschemann, et al., ). Fig. 4. Identified friction characteristic and nonlinear friction model of the -axis he non-linear feedforward control part is dedicated to provide an open-loop compensation for the nonlinear friction forces. he identified non-linear friction characteristic and an approximating friction model are shown in fig. 4 for the -axis, exemplar. he non-linear feedforward control part for the -axis is given the non-linear friction model u,nf = FFC = F ( FC C, ref ) evaluated with the reference velocit of the -axis. his reference velocit can e derived in smolic form from the linear feedforward control part. It results in a linear comition of the reference vectors w and w ϕ. (5) s w L J Lx C,ref = sl g m w L g ϕ he non-linear friction model FF = FF (, ref ) of the -axis is evaluated with the reference trolle velocit, ref. 6. DISURBANCE OBSERVER DESIGN As the friction characteristics depend on lots of factors and ma change during operation time, the nonlinear feedforward control part cannot achieve a perfect disturance rejection. herefore, additiol measures are necessar to cope with changeale friction characteristics. For this purpose, a reduced order disturance oserver is emploed for each axis. In the following, the design approach is presented at the example of the -axis. As disturance model for the friction force F FC an integrator model has proved advantageous, i.e. F FC = (Aschemann, et al., a, ). Since all state variales are forthcoming, the measurement vector of the oserver is identical to the state vector B = x. he input vector of the oserver ub = [ uo,lin S ] consists of the control input u O,lin = u,fb + u,lf + u, DC except the nonlinear feedforward part and the acceleration of the load suspension S. he state space representation of the reduced oserver is given (6) ρ = a ρ ρ + B + u ub, Fˆ FC = ρ + h B At this, the oserver gain vector [ h ] = 3 h, (7) the scalar sstem matrix 3 ( J Lx ) / a ρ = h + with n m J J k ( J m s a = L Lx + MC GC Lx + L L ), (8) and the input vectors ( J Lx + )h3 J Lx h3 u =, (9) h m s g J m g h ( J + m s 3 L L Lx L 3 Lx L L ) ( h3 + dc )h3 =
concerning measurement vector B and oserver input u B are introduced, respectivel. he oserver dmics can e specified pole placement. B proposing a closed-loop characteristic polnomial s +α =, the oserver gain h 3 results in α h3 =. (3) ( J Lx + ) he estimated friction force Fˆ FC can e directl applied to the carriage drive, i.e. k,dc =. Hence, the oserver ased disturance compensation results in u = k Fˆ = Fˆ. (3),DC,DC FC FC 7. CONROLLER IMPLEMENAION he trajector control scheme is shown in fig. 5 for the -axis, exemplar. he complete control law for the -axis u = u,lf + u,nf + u,fb + u,dc (3) represents the sum of linear feedforward u,lf, nonlinear feedforward u,nf, linear state feedack u,fb, and oserver ased disturance compensation u,dc. he corresponding motor torue u M is otained premultipling with the inverse gain transmission ratio, i.e. um = kgc u. Note that all susstems are adapted to the measured parameter vector p. he carriage position C is measured an encoder. he load angle ϕ L is otained as the sum of the small angular rope deflection at the trolle and the relative angular deflection etween rope and orientation unit, which oth are measured incremental encoders. w w ϕ k,nf k,o LF k, u,lf f ( ) N u,dc k,dc u,fb k,fb p ê x S u,nf u=f MC u M k GC u O,lin disturance oserver x sigl processing V L Fig. 5. rajector control scheme for the -axis 8. EXPERIMENAL RESULS overhead crane -axis he efficienc of the proposed control scheme shall e shown measurements at a 5 t ridge crane. At first, trajector tracking concerning the -axis with deactivated remaining axes is considered. he reference trajector for the -axis consists of a movement from ϕ L = to ϕ L = 6, an seconds reak, and the return movement to ϕ L =. As can e seen in the upper part of fig. 6, the feedforward disturance rejection works uite well. Moreover, a further reduction of the tracking error as L depicted in the lower part of fig. 6 can e achieved additiol use of a disturance oserver. [degree] [degree] 7 6 5 4 3 5 5 5 3 35 4 7 6 5 4 3 5 5 5 3 35 4 Fig. 6. Reference and measured load angle without (upper part) and with (lower part) oserver ased disturance compensation for m L =8 kg In the following figures, a reference trajector is tracked with the -axis, while the -axis should keep its reference value identical to zero, i.e. ϕ. During the specified motions of the trolle m, the acceleration of the rope suspension S acts as a disturance on the -axis control. Fig. 7 and 8 provide a comparison of reference values and measured values for oth controlled variales: the load position L and the load inclition ϕ L. In fig. 7, measurement results are shown for a pure decentralised feedforward control, where the -axis feedforward control is exclusivel calculated with the reference w ϕ and the -axis feedforward control is solel ased on the reference w. he tracking ehaviour concerning the load position is uite well with maximum deviations of less than 6 cm ut the load angle ϕ L shows maximum deviations of aout from its reference value zero. Stead-state accurac is aout cm for the load position and aout.3 for the load inclition. In fig. 8, measurement results are presented for the same reference motion as in fig. 7, ut now with the complete feedforward control as descried in chapter 5. It is ovious that the compensation of the coupling forces leads to a reduction of the tracking error of the -axis approximatel factor four with deviations of aout.5. he remaining oscillations with an
amplitude of approximatel. can e tolerated from a practical point of view. L [m] [degree].5 -.5.5 - reference position measured position -.5 5 5 5 3 35 4 45.5.5 -.5.5-5 5 5 3 35 4 45 Fig. 7. Comparison of reference and measured controlled variales for m L = 93 kg, l R = 4.85 m with decentralised instead of central feedforward control L [m] l [degree].5 -.5.5 - reference position measured position -.5 5 5 5 3 35 4 45.5.5 -.5.5-5 5 5 3 35 4 45 Fig. 8. Comparison of reference and measured controlled variales for m L = 93 kg, l R = 4.85 m with central feedforward control 9. CONCLUSION A trajector control approach is presented for the load position as well as the load inclition of an overhead crane, which has een upgraded with an additiol orientation unit. Based on a multiod model of the crane, feedack and feedforward controllers as well as disturance oservers are derived in smolic form. his allows for an adaptation of the complete control structure using the gain scheduling techniue with respect to the varing sstem parameters rope length and load mass. Measurements, taken at a 5 t ridge crane, show the enefits of the proposed control scheme as regards control performance and stead-state accurac. Concerning the load position the achieved tracking error remains elow 6 cm with a stead-state error less than cm. he tracking error with respect to the load inclition is aout.6 with a stead-state error of aout.3 in the case of coupling force compensation using central feedforward control. REFERENCES Aschemann H., Sawodn O., Lahres S., Hofer E.P. (a). Disturance Estimation and Compensation for rajector Control of an Overhead Crane. In: Proceedings of American Control Conference ACC, Chicago, USA, pp. 7 3 Aschemann H., Sawodn O., Hofer E.P. (). Active Damping of ilt Oscillations and rajector Control of Overhead Cranes. Proceedings of IEEE Conference on Industrial Electronics, Control, and Instrumentation IECON, Nagoa, Japan, pp. 7 75 Boustan F., d`andréa-novel B. (99). Adaptive Control of an Overhead Crane using Dmic Feedack Linearization and Estimation Design. In: Proceedings of the 99 IEEE Intertiol Conference on Rootics and Automation, Nice, France, pp. 963 968 Brfors, U., Sluteij, A. (996). Integrated Crane Control. In: Proceedings of the 6 th Conference on ransportation Sstems, Zagre, Croatia, pp. 67 7 Delaleau, E., Rudolph, J. (998). Control of flat sstems uasi-static feedack of generalized states. In: Intertiol Jourl of Control, Vol. 7, No. 5, pp. 745 765 Nguen, H.. (994). State Variale Feedack Controller for an Overhead Crane. In: Jourl of Electrical and Electronics Engineering, Australia, Vol. 4, No., pp. 75 84