CONTROL RAPID PROTOTYPING OF REDUNDANTLY ACTUATED PARALLEL KINEMATICAL MACHINE

Transcription

1 CONROL RAPID PROOYPING OF REDUNDANLY ACUAED PARALLEL KINEMAICAL MACHINE Michael Valášek, Zbyněk Šika, Jan Zavřel, omáš Skopec, Pavel Steinbauer CU in Prague, Faculty of Mechanical Engineering Abtract Reunantly actuate parallel tructure can ubtantially improve all mechanical propertie of machine tool. hey achieve higher tiffne, eigenfrequencie an acceleration. he workpace i without ingularitie an ratio between workpace an machine overall pace i improve. Neverthele the tanar control of reunantly actuate machine rive reult into mutual fighting of rive. he moifie control trategy which remove rive conflict ha been evelope an implemente. hi ha been emontrate on the mechanical eign an control evelopment of functional moel of reunantly actuate parallel kinematical machine tool Sliing Star. he mechanical tructure evelopment tool are bae on Matlab environment an computational power, the control eign an implementation are bae on the RAD chain Matlab-Simulink-RealimeWorkhop-Space (RI an ControlDek). 1 Introuction Reunantly actuate parallel tructure can ubtantially improve all mechanical propertie of machine tool [1]. hey achieve higher tiffne, eigenfrequencie an acceleration. he workpace i without ingularitie an ratio between workpace an machine overall pace i improve. Neverthele the tanar control of reunantly actuate machine rive reult into mutual fighting of rive. he moifie control trategy [2] which remove rive conflict ha been evelope an implemente. hi ha been emontrate on the mechanical eign an control evelopment of functional moel of reunantly actuate parallel kinematical machine tool Sliing Star. he mechanical tructure evelopment tool are bae on Matlab environment an computational power, the control eign an implementation are bae on the RAD chain Matlab-Simulink-RealimeWorkhop-Space (RI an ControlDek).o cover overall mechanical an control eign tep, compact tool et of methoologie, algorithm an oftware ha been evelope. 2 Sliing Star he kinematical tructure of machine tool have been recently change from traitional erial one into parallel one [1]. he parallel kinematical tructure mean that the platform (machine tool carrier) i upene on everal link imultaneouly (Fig. 1b) in contraiction to erial kinematic with jut one link (Fig. 1a). Serial kinematic mut move the rive within it tructure, the parallel one ha all rive on the frame. he main requirement on the machine tool are the high accuracy, high prouctivity, mall pace eman an certainly the price. he manufacturer try to accomplih thee requirement by machine with high tiffne, high ynamic capabilitie, large workpace an to reuce prouction cot. he achievement of thee goal interfere with the long-term conceptual eign problem of machine tool. he frame of machine tool i loae by bening, there are large moving mae in the machine tool contruction, the backlahe an inaccuracie in erial kinematical chain are ae an the ratio of workpace an machinepace (the overall pace occupie by the machine on the hopfloor) houl be alway minimize. he parallel kinematic concept eeme to overcome all thee problem.

2 (a) (b) (c) Fig. 1 Kinematical tructure Parallel kinematic i a tru tructure an therefore the loaing of link i jut the troke/preure. he remarkable property of parallel kinematic (Fig. 1) i that all rive can be locate on the frame an therefore the moving mae are reuce. Both of that improve the ma-tiffne ratio an the reulting accuracy. Beie that the kinematical chain from the frame to the machine tool are hortene that contribute to the improve kinematical accuracy. Unfortunately the parallel kinematic tructure uffer from everal evere problem. If ome of the link interect each other then there arie o-calle ingular poition. In ingular poition an aitional egree of freeom intantaneouly occur which however reult into complete lot of tiffne in thi poition. A there are more link their mutual colliion happene often than by erial kinematic. Both thee fact reult into maller workpace an unfavourable ratio of workpace an machinepace than by erial kinematic. Beie that parallel kinematic machine have alo problem with accuracy. he tranformation between the rive coorinate an machine tool (platform) coorinate i nonlinear. he tranformation i accomplihe by the computer control ytem. However, it accuracy i epenent on the knowlege of real imenion of the machine. Even epite very accurate manufacture of machine it i not poible in cae of parallel kinematic to ue the eign imenion for the nonlinear kinematical tranformation in control ytem. It i neceary to etermine the really manufacture imenion a accurate a poible. In cae of parallel kinematic it i not poible to etermine the real imenion by irect meaurement therefore thee imenion mut be compute from ome inirect meaurement. It i ocalle calibration well known from robotic applie to machine tool. raitional calibration proceure are bae on the imultaneou meaurement of tool center poition of parallel kinematic machine by an external meaurement evice an meaurement of internal rive poition of parallel kinematic machine. he erial kinematic tructure o not have uch problem becaue the link within the erial tructure can be calibrate irectly an inepenently from each other. All thee problem an iavantage of parallel kinematic can be remove by the application of the principle of reunant actuation an the principle of reunant meaurement [1].

3 Reunantly actuate parallel kinematical tructure have the potential for ignificant increae of all mechanical propertie of machine tool. During the evelopment of rijoint 900H (Fig. 2) there have been propoe an invetigate about 0 tructural variant with everal hunre thouan parametric variant. he important eign variant are on Fig. 3. he variant on Fig. 3a,b, are reunant one. he variant on Fig. 3c i nonreunant one correponing to rijoint. here were foun intereting propertie of thee tructure. If the level of ynamic capabilitie i the ame then the global tiffne of thee tructure (Fig. 3a : Fig. 3b : Fig. 3c : Fig. 3) i approximately in the ratio 0.7 : 2. : 2.0 : 3.0 where 100% correpon to tiffne of goo traitional machine tool (about 60 N/µm). he tiffne of the variant on Fig. 3a i urpriingly low an compare to it the tiffne of the variant on Fig. 3 i high [2]. Figure 2 rijoint 900H (a) OBR.1 (b) (c) () Figure 3 Deign variant from rijoint evelopment

4 he tiffne of the variant on Fig. 3 i ignificantly high, but it workpace i limite. he further tructural moification reulte into the variant of Sliing Star on Fig. [2]. It intereting potential i a horizontal machine tool for long profile with tiffne an ynamic of machine tool with maller workpace or a portal machine tool for complex large workpiece (Fig. ). 3 3 φ B 3 B 2 φ φ 1 B B φ 2 Figure Kinematic cheme an kinematical tructure of Sliing Star Figure Potential of Sliing Star a horizontal machine center for long profile or portal machine center for large workpiece he imenion of Sliing Star have been intenively optimize. he reulting functional moel i on Fig. 6. It i reunantly actuate parallel kinematical tructure, i.e. it ha more actuaror than DOF. It ha 3 DOF an actuator on two lier. he four carriage 1, 2, 3, are actuate by linear rive an the platform B 1 B 2 B 3 B i move to the eire poition. Figure 6 3D moel of Sliing Star

5 he challenge of machine tool with g, 200 N/ m, 10 Hz in whole workpace 1 m3 with on-line meaurement an compenation of thermal eformation uring machine operation. Such parameter are 3- time higher than the parameter of current machine tool. It concept maybe imilar to Sliing Star. 3 Control of reunantly actuate tructure One of the critical problem of reunantly actuate kinematic i the control. here i a anger of mutual fighting of reunant actuator ue to alway exiting ifference between reality an it kinematical moel (e.g. Fig. 7). here have been evelope moifie control approache that eliminate thee problem [2]. Compenate Mutual fighting Figure 7 Conflict of reunant actuator an it compenation he ynamic of parallel kinematic tructure i conveniently ecribe by Lagrange equation of mixe type M && Φ λ = g + u f(( t)) = 0 (1) where M i a ma matrix, i a vector of phyical coorinate (their number i higher than the number of DOF), which are contraint by kinematic contraint f(( t )) = 0, Φ i Jacobian of thee contraint, λ i a vector of correponing Lagrange multiplier, g i a vector of ynamic an applie force, matrix tranform the input u (n torque) into n rive. he phyical coorinate conit of the inepenent coorinate x (conveniently Carteian an orientation coorinate of the platform), rive (actuator ) coorinate ϕ an other auxiliary geometrical coorinate q. hee equation of motion can be tranforme into inepenent coorinate x uing the null pace R of the Jacobian Φ which ecribe the relation between phyical an inepenent coorinate & = Rx& R MRx & + R MRx & & = R g + R u (2) Among the coniere coorinate there are important the coorinate of the rive ϕ an the inepenent coorinate of the platform x. here i the mapping from inepenent coorinate to the coorinate of rive

6 ϕ = ϕ(x) (3) ϕ & ϕ = x& = Jx& x () he force interaction of the platform of kinematic tructure with the environment can be ecribe by the force F acting on the platform (they are acting on the inepenent coorinate x) an by the rive torque n (they are acting on the coorinate of rive ϕ). heir mutual relation i F = J n () If the traitional approach of cacae PID controller on poition, velocity an current level i applie there i the eriou problem of mutual fighting of reunant actuator. It i ue to the fact that the kinematic moel of the reunant parallel tructure i never perfect. Due to the reunancy of rive there i no one-to-one mapping from coorinate of rive ϕ to inepenent coorinate x (invere of (3)). he epenence of reunant coorinate of rive on inepenent coorinate repreent a contraint among the rive coorinate (3). hi mean that the rive coorinate ϕ are epenent. hi contraint i in reality not atifie ue to the imperfection of kinematic moel an PID controller trie to achieve zero error for all epenent rive coorinate ϕ. It i not poible an the reult i the increae of rive torque up to the aturation. Such behavior wa achieve by both imulation an laboratory experiment on laboratory moel of reunant parallel tructure. 3.1 Decentralize (Local) Control In orer to olve thi problem there have been evelope the following moification of traitional cacae rive control. In fact it i ecentralize (local) control of rive. he propoe control cheme i on Fig. 8. he only moification i the block of tranformation. It iea i that the local ecentralize controller compute the eire rive torque n. From them the eire reulting force F acting on the platform can be compute F = J n (6) Now the applie rive torque n are to be etermine from uneretermine ytem of linear equation J n = F (7) FK x IK ϕ e m u x poition PID velocity PID m u x tranf e m u x velocity current rive ROBO ϕ Figure 8 Control cheme of moifie ecentralize control (IK=invere kinematic, FK=forwar kinematic)

7 he olution of thi problem which minimize the value of n i n 1 = J(J J) J n (8) he imulation of thi cheme ha prove that the problem of reunant rive fighting i remove. Neverthele the cheme ha been implemente in implifie form on the laboratory moel of Sliing Star only on the integral component of the controller (Fig. 9). he practical experiment have hown full applicability of thi approach. A pecial evelopment wa evote for the on-line olution of forwar kinematic of reunant parallel kinematic tructure. he approach of ifferential kinematic being on-line integrate wa uccefully applie. x IK ϕ e m u x poition PD current rive ROBO ϕ I tranf FK Figure 9 Implemente control cheme of moifie ecentralize control 3.2 Centralize (Global) Control he problem of control of reunant parallel kinematic tructure i the interaction among part an reunant actuator of the parallel kinematic which are not inepenent. he general olution of thi problem i to control the tructure from the centralize (global) point of view where the interaction can be irectly taken into account. he implet control approach i to apply PID controller however on the global level. he correponing control cheme i on Fig. 10. he approach i bae on the control of the inepenent Carteian an orientation coorinate x of the platform on level of poition an velocitie. he reult of thee block i a fictitiou control force F acting irectly on the platform. hi force mut be tranforme into local rive torque n accoring to the olution of (6). aking into account the olution with minimum value the reult of the tranformation i n = J(J J) 1 F (9) x poition velocity PID PID tranf e m u x current rive ROBO ϕ FK Figure 10 Control cheme of centralize control (FK=forwar kinematic)

8 Control Implementation he objective of the reearch on thi functional moel of Sliing Star with propertie of real machine tool an inutrial control ytem ha been to emontrate the ability to control reunantly actuate (overetermine) parallel kinematical tructure an to verify the new poibilitie of calibration coming from reunant meaurement. raitional control fail for reunant actuation an non-reunant calibration require external artefact-etalon an oe not enable the calibration uring machine operation. he ultimate objective i to implement the control of reunantly actuate parallel kinematical tructure on the inutrial control ytem Siemen 80D uing OA Packet enabling to moify the original control oftware. It ha not been poible within the time perio until IM 2006 exhibition. herefore the intermeiate way of implementation ha been electe. It conit of velocity controller of all rive in Siemen control ytem Simorive for local rive control an of poition/trajectory generator an poition controller in DSP 03 of Space. Figure Control cheme in Simulink Complex control oftware ha been evelope uing control oftware rapi evelopment chain Matlab-Simulink-RealimeWorkhop-Space. It wa neceary to evelop many moule. hey have covere the invere an irect kinematical tranformation, the generation of eire trajectory, poition controller with compenation from Fig. 9, velocity controller with compenation equivalent to Fig. 8, controller for elementary Carteian motion, controller of generate trajectorie, protecting of emergency tate, etc. he part of control cheme in Simulink i on Fig.. he reult wa a lit of oftware moule:

9 Spee PI, Poition PI, Ramp Movement generator, Forwar Kinematic (newton), Invere Kinematic, rajectory Generation, rajectory Execution, rajectory Repeater, rajectory Spee, ize Ajutment, Decentralize Spee Control, Decentralize Poition Control, Communication with Siemen Simorive, Incremental Senor Communication, GUI, 3-axi incremental movement of working platform. he evelopment of uch emaning control oftware package ha been heavily upporte an accelerate by the capabilitie of control oftware rapi evelopment ytem Matlab Space. he interface Control Dek of Space DSP 03 ha enable to create a cutomize interface for eay operation of control ytem. It GUI i on Fig. 12. he leon learne i again the fact that without the control oftware rapi evelopment ytem Matlab Space the tak cannot be complete within the hort time perio of 2 month. Reult Figure 12 Sliing Star Control GUI he real ize functional moel Sliing Star incluing control ytem bae on Siemen 80D ha been built an uccefully hown at IM 2006 exhibition in Brno 2006 (Fig. 13). he functional moel ha emontrate the capability to evelop an epecially to control the reunantly actuate parallel kinematical tructure. he uage of thi technology i reay for inutrial application. 6 Concluion he paper ha ecribe the evelopment of control ytem of functional moel of reunantly actuate parallel kinematical tructure Sliing Star. he evelopment ha been enable by the application of control oftware rapi evelopment ytem Matlab Space. Reference [1] Valaek, M. et al.: he Innovative Potential of Reunantly Actuate PKM, In: Proc. of Parallel Kinematic Seminar PKS 0, IWU FhG, Chemnitz 200, pp [2] Valášek, M. et al.: Deign-by-Optimization an Control of Reunantly Actuate Parallel Kinematic Sliing Star. Multiboy Sytem Dynamic 200, vol. 1, no. 3-, p

10 Michael Valášek Figure 13 Running Sliing Star at IM 2006 Exhibition