Dynamic modeling and control of a new automatic corking machine for threaded plastic caps

Transcription

1 Dynamic modeling and control of a new automatic corking machine for threaded platic cap Roberto Zanai Federica Groi Univerity of Modena and Reggio Emilia Modena Italy (roberto.zanai federica.groi)@unimore.it Nicola Giuliani Schneider Electric S.p.A. Bologna Italy nicola.giuliani@it.chneider-electric.com Abtract The aim of thi work i to model a new electromechanical ytem for application in the field of automated corking machine. The paper preent the dynamic modeling of a new automatic corking machine for threaded platic cap. The model i obtained uing the energy-baed modeling technique named Power-Oriented Graph (POG). The conidered corking machine i an electromechanical ytem with two degree of freedom compoed by two electrical motor moving a ball crew/pline that realize the linear/rotary motion neceary to crew a platic cap on a bottle. The paper preent the dynamic model of the machine and the interaction between the cap and the bottle. In the paper a control algorithm i propoed. Some imulation reult are preented and compared to experimental reult.. Introduction When dealing with the modeling of dynamic phyical ytem the chooe of a modeling technique i a very important iue. Many graphical energy-baed modeling technique have been introduced in the pat year: the Bond Graph (BG) 3 the Power-Oriented Graph (POG) 4 and the Energetic Macrocopic Repreentation (EMR) 2. All thee technique are baed on the concept of energy moving within the ytem. In thi work the Power-Oriented Graph (POG) modeling technique i exploited taking advantage of it main propertie uch a compactne direct correpondence with tate pace equation poibility to tranlate block cheme directly into Simulink for imulation and poibility to tranform (reduce and/or invert) dynamical ytem. POG cheme clearly how the power flow within the ytem thu allowing a precie analyi and to keep alway the correponding phyical meaning. Becaue of the POG modular tructure complex phyical ytem can be modeled by compoing ubytem model. Thi work wa upported by Fondazione di Vignola in collaboration with Democenter-Sipe Modena Italy. The dynamic ytem conidered in thi paper i an automatic corking machine for threaded platic cap. It i an electromechanical ytem with two degree of freedom compoed by two controlled electrical motor which move a ball crew/pline that realize the linear/rotary motion neceary to crew a platic cap on a bottle. A prototype of the machine ha been realized for the part concerning the haft with linear/rotary motion while the part concerning the crewing of the platic cap on the bottle ha not been realized yet. The aim of the paper i to give the dynamic model of the whole machine. In particular the interaction between the cap and the bottle i able to reproduce the dynamic behavior of the ytem. The ue of POG cheme allow to have imple implementation of the model in Simulink and to calculate internal variable of the ytem ueful for the dimenioning of the ytem itelf. An accurate modeling of the whole ytem i very ueful in the project of the machine for the chooe of parameter the evaluation of performance that can be obtained and the building of the control. The imulation model allow the tet of different control algorithm in imulation before the implementation on the real machine. 2. Power-Oriented Graph baic feature The POG block cheme are tandard block diagram combined with a particular modular tructure eentially baed on the ue of the two block hown in Fig..a and Fig..b: the elaboration block (e.b.) tore and/or diipate energy (i.e. pring mae damper capacitie inductance reitance etc.); the connection block (c.b.) reditribute the power within the ytem without toring nor diipating energy (i.e. gear reduction tranformer etc.). The e.b. and the c.b. are uitable for repreenting both calar and vectorial ytem. In the vectorial cae G() and K are matrice: G() i alway a quare matrix compoed by poitive real tranfer function; matrix K can alo be rectangular. The circle preent in the e.b. i a ummation element and the black pot repreent a minu ign that multiplie the entering variable. The POG put in evidence the power flow within the ytem and keep a direct correpondence between the dahed ection of the graph and real power ection of the modeled y-

2 tem: the calar product x T y of the two power vector x and y involved in each dahed line of a power-oriented graph ee Fig. ha the phyical meaning of power flowing through that particular ection. x x 2 x K x 2 G() y y a) e. b. y K T b) c. b. Figure. POG baic block and variable: a) elaboration block; b) connection block. Another important property of the POG technique i the direct correpondence between the POG cheme and the correponding tate pace dynamic equation. For example the POG cheme hown in Fig. 2 can be repreented by the tate pace equation () where the energy matrix L i ymmetric and poitive definite: L = L T >. It can be eaily hown that when D = it follow that C = B T. u B D A L - y C x y 2 { Lẋ = Ax+Bu y = Cx + Du () (x = Tz) { Lż = Az+Bu y = Cz +Du Figure 2. POG block cheme of a generic dynamic ytem. 3. Decription of the corker ytem The phyical cheme of the electromechanical corker ytem i hown in Fig. 3. The ytem i compoed by two three-phae motor ELAU ISH 7 67 two pulley-belt tranmiion and a linear-rotary unit that contain Ball Screw groove and Ball Spline groove croing with each other on a ingle haft (ball crew/pline). The two motor are tied to the pulley-belt tranmiion and through their coordinate operation the linear-rotary motion of the Tool Center Point (TCP) i obtained. In particular the upper motor i reponible for the crew motion and the lower motor i reponible for the pline motion. The motion control of the pulley i realized by the controller C4 included in the electric motor. (2) Figure 3. Picture of the corker ytem. 4. POG modeling of the ytem The POG cheme of the two DC electrical motor i hown in Fig. 4 where a vectorial notation i ued. Vector and matrice preent in the POG cheme of Fig. 4 have the following meaning: V a = Ve V r I a = Ie I r Le L a = L r Ke K m = K r J m = ω m = ωe ω r τ m = Re R a = R r Je J r B m = τme be b r τ mr. The variable and parameter of the ytem are explained in Table. V a I a L - a Inductance R a Reitance K T m Km Torque contant ω m J - m Inertia B m Friction Figure 4. POG cheme of the DC electrical motor. ω m τ m The parameter characterizing the pulley the belt and the crew/pline haft are hown in Fig.. The POG cheme of the conidered corking ytem i hown in Fig. 6 where vector and matrice have the following meaning: F c = Fe F r ω t = ωte ω tr τ t = τte τ tr

3 ω ω m R R T t 2 H H T 2 Ẋ a ż J - 2 J - a B 2 D B a K K c τ m R T R2 H T H2 F a+f g F c τ t F F a belt connection pulley connection coupling tiffne crew/pline haft 2 3 Figure 6. POG cheme of the corking ytem downtream the electric motor including the pulleybelt tranmiion the nut of the ball crew and the ball pline and the haft. 4 R 2e V e I e armature voltage and current of crew motor V r I r armature voltage and current of pline motor L e R e inductance and reitance of crew motor L r R r inductance and reitance of pline motor K e K r torque contant of crew and pline motor J e b e inertia and friction coefficient of crew motor J r b r inertia and friction coefficient of pline motor ω e ω r angular velocity of crew and pline motor F e F r tangential force of the crew and pline belt ω te ω tr angular velocity of crew and pline pulley τ te τ tr torque acting on internal pulley ẋ θ tranlational and rotational haft velocitie R e R r radii of external crew and pline pulley K ce K cr tiffne of crew and pline belt R 2e R 2r radii of internal crew and pline pulley J 2e b 2e inertia and friction of the internal crew pulley J 2r b 2r inertia and friction of the internal pline pulley r a l a radiu and length of the haft α angle between the crew groove and the haft axle K e d e tiffne and friction coefficient between crew pulley and crew groove K r d r tiffne and friction coefficient between pline pulley and pline groove m a b ma ma and linear friction coefficient of the haft J a b ja inertia and rotational friction coefficient of the haft Table. Meaning of variable and parameter of the ytem. ẋ ma g Re Ẋ a = F θ a = R = R r Kce R2e J2e K c = R K 2 = J cr R 2 = 2r J 2r b2e ma bma B 2 = J b a = B 2r J a = a b ja r a co α K e H = r a in α r a K = K r in α r a co α D = d e HT 2 = co α r a in α r a. d r J e b e R e K ce R 2r J 2e b 2e m a bma J 2r b 2r J a b ja r a K cr J r b r R r Figure. Parameter characterizing the pulley the belt and the crew/pline haft. Vector F a and F g indicate the weight and the external force acting on the haft. The power ection of the POG cheme of Fig. 6 are in direct correpondence with the power ection of the ytem: the input ection correpond to the power ection hown alo in Fig. 7 between the electric motor haft and the belt connection with the pulley while the output ection correpond to the power ection aociated to the TCP of the haft. The block between ection and 2 repreent the belt tranmiion between pulley block between ection 2 and 3 repreent the dynamic of the econdary pulley block between 3 and 4 repreent the interaction between the nut of the ball crew/pline and the haft and finally the dynamic of the haft i repreented by block between ection 4 and. Matrice H and H 2 are ued to decompoe the tangential velocitie of the pulley r a ω te and r a ω tr along the crew groove and the vertical pline groove a it i hown in Fig. 8. When the tiffne between the nut and the haft tend to infinity K a contraint in the ytem appear then it i poible to apply to the ytem a coordinate tranformation in order to reduce the ytem. Therefore the part of the POG cheme between ection 2 and can be repreented in a compact form a hown in Fig. 9 by applying the aforementioned

4 Ẋ a ω te τ te F a + F g TCP ω tr τ tr Figure 7. Power ection characterizing the ytem compoed by pulley and haft. ω t τ t 2 J - t Inertia Ta B t Friction T T a Link F a+f g Ẋ a Figure 9. Reduced POG cheme of the ytem pulley-crew/pline haft. Σ t K e d e α r aω te Σe Σ t d e K e r aω tr Figure 8. Decompoition of tangential velocitie of the pulley r a ω te and r a ω tr along the crew groove and the vertical pline groove. reduction rule to the cheme. Note that J t i the equivalent inertia including the inertia of both pulley and of the haft B t i the equivalent friction and T a i defined a T a = ra cot(α) r a cot(α).. POG model of the interaction between cap and bottle The ytem preented in the previou ection ha been tudied in order to evaluate it applicability a a new automated corker for threaded platic cap. A POG model of the conidered platic cap and the bottle i hown in Fig.. The conidered ytem i compoed by the following dynamic: ) elaticity of the graper; 2) ma and inertia of the cap; 3) interaction (elatic and diipative) of the thread teeth of the cap; 4) ma and inertia of the bottle; ) elaticity of the bottle upport. Vector and matrice hown in the POG cheme of Fig. have the following meaning: F g = Fg F b = ẋtp Ẋ tp = F θ tp = tp τ g Fb τ b K g = Kgt K gr Ftp τ tp Ẋ b = ẋb θ b dgt D g = d gr mtp btpt J tp = B J tp = tp b tpr bbt Kbt B b = K b b = br K br J b = D b = mb J b dbt. d br The parameter of the cap and the bottle are reported in Table 2. Function Φ( ) repreent the elatic character- K gt d gt K gr d gr m tp b tp J tp d tp m b b b J b d b K bt d bt K br d br r t h t β tranlational tiffne and friction between cap and haft rotational tiffne and friction between cap and haft ma and tranlational friction of the cap inertia and rotational friction of the cap ma and tranlational friction of the bottle inertia and rotational friction of the bottle tranlational tiffne and friction between the bottle and it upport rotational tiffne and friction between the bottle and it upport radiu and height of the thread inide the cap angle between the thread and the horizontal plane of the cap Table 2. Parameter of the cap and the bottle. itic of the interaction between cap and bottle. Vector X tb F tb and matrice K d K n and T tb have the following tructure: X tb = K n = xtb θ tb F tb = Ftb Kn T tb = τ tb K d = Kd co β rt in β inβ r t co β where r t i the radiu of the crew thread inide the cap and β i the lope of the the crew thread with repect to the horizontal plane of the cap ee Fig.. The function Ψ( ) repreent the Coulomb friction and the linear friction in the cap-bottle interaction. The Coulomb friction between the cap and the bottle when a normal force F n i preent i the phyical phenomenon which keep the cap attached to the bottle after that the crewing of the cap i

5 Ẋ a F g K g F g Graper D g 6 Ẋ tp J - tp Cap B tp 7 Ψ( ) Φ( ) F tp Elatic Interaction Bottle 8 Ẋ b J - b B b 9 K b D b F b Bottle Support F b Figure. POG cheme of the interaction between a threaded platic cap and a bottle. Σ t K d r tω tb Figure. Decompoition of tangential velocitie of the cap along the crew thread. completed. Vector F n and matrice D tp e K tp have the following tructure: Fn dtpt Ktp F n = D tp = K d tp =. tpr The exact imulation of Coulomb friction i a very complex problem involving nonlinear dicontinuou function. In order to imulate properly and effectively the ytem cap-bottle it i neceary to modify the POG cheme of Fig. a it i hown in Fig. 2 and in Fig. 3. In Fig. 2 the elatic part of the graper (between ection and 6 ) and of the bottle upport (between ection 9 and output ection) are kept eparated from the elatic-diipative interaction between cap and bottle (between ection 6 and 9 ). Matrice T g and T b preent in Fig. 2 are tranformation needed to pa from the 2-dimenion pace of the graper and of the bottle upport to a 4-dimenion pace decribing the interaction between cap and bottle: T g = T b = Σd β. Matrix T d in Fig. 3 i ued to generate the relative peed Ẋ tp Ẋb while vector F x repreent the weight force acting on the cap and bottle: T d = F x = m tp g m b g. The elatic-diipative element repreented in Fig.3 by matrice K n K tb and D tp mut in fact take into account alo ome non linearitie therefore they mut be conidered a nonlinear element: K n generate the normal force of the cap which i different from zero only when the crewing of the cap i completed and K tb take into account the tiffne and the backlah between the thread teeth of the cap. In particular the normal force F n i different from zero only when the bottom of the cap get in contact with the top of the bottle neck that i when the firt component of vector X tpb atifie the condition x tp x b < H b where H b i the bottle height ee Fig. 4. The inertial dynamic of the interaction between cap and x b x tp Ht H b Figure 4. Height and relative poition of cap bottle and bottle upport. bottle i decribed by the following differential equation: m tp ẍ tp F tp J tp θ tp m b ẍ b = τ tp F b K tp gn θtp θ b }{{} J b θ b τ b K tp D ω }{{}}{{}}{{}}{{} J ω C D T K tp (3) where J = diag(m tp J tp m b J b ) and ω = ẋ tp θ tp ẋ b θ b T. The torque vector C and matrix D in (3) have the following tructure: C = C gb + F x F tb T T d F n + T T tb(f tk + F td ) D =.

6 Ẋ a 6 Ẋ tp D g T T g ω m T T b Ẋb Γ 9 D b F g K g F g Tg C g C b C gb T b K b F b F b Graper Cap-Bottle Bottle Support Figure 2. Modified POG cheme of the ytem compoed by the platic cap and bottle. K tp gn( ) F c θ tp θ b D D T ω m - Jtp J b ωm Td Ẋtpb Ttb Ẋde Btp B b F tb C F x F n T T d X tpb K n T T tb X de K tb D tp F td F tk Coulomb friction Fx Inertia Friction Cg C b Elatic interaction cap-bottle Friction Figure 3. Elatic-inertial interaction of the platic cap with the bottle. Vector C repreent the total torque acting on inertia J decreaed by the Coulomb friction torque K tp gn( ). The dynamic model (3) can be rewritten in a compact form a follow: J ω = C D T K gn(dω) (4) The ytem (4) i a ytem with variable dynamic dimenion ee where the model of a clutch a a variable dimenion ytem wa introduced. The term K tp gn( θ tp θ b ) i dicontinuou therefore it i reponible for high frequency commutation during the imulation. In order to avoid thi problem it i ueful to exploit the following tate pace tranformation: ω = Tz where z i the new tate vector and matrix T i defined a follow: T = J b T T 2 = J tp+j b. Jtp The phyical meaning of the component of the new tate vector z i clear if the following relation ω = Tz i inverted: z = T ω = J tp J b ẋ tp θ tp ẋ b θ b = = ẋ tp J tp θtp+j b θb ẋ b θ tp θ b = z z 2 z 3 z 4 = z z 2. The variable z z 2 and z 3 are the weighted mean velocitie decribing the main dynamic of the ytem. The variable z 4 (the relative dynamic) decribe the relative velocity of the cap with repect to the bottle. Uing tranformation ω = Tz ytem (4) i implified in the following way: J T ż = C T D T T Kgn(D T z) () where matrice J T = T T JT D T = DT and vector C T = T T C have the following tructure: JT J T = = J T2 m tp m b D T = D T2 = C t CT C T = = C t2 C T2 C t3. C t4 J tp J b

7 z T J - T ω T 2 z 2 J - T2 Kgn(z 2) C T Main dynamic T T T T 2 C T2 C Relative dynamic Figure. Interaction between cap and bottle: main and relative dynamic. Inverting matrix J T ytem () can be rewritten in the following way: ż = J T C T D T T Kgn(D Tz) then in the form: ż = J T C T (6) ż 2 = J T2 C T2 D T T2 Kgn(D Tz) which can be explicited a: ż m ż 2 tp C t = C t2 ż 3 C }{{} m t3 }{{ b }}{{} ż J C T. T ż 4 }{{} ż 2 = J tp J b C t4 K tp gnz 4 }{{}}{{} C T2 D T T2 Kgn(DTz) J T2 The propoed tranformation decompoe the original ytem in two parallel independent ytem: variable z z 2 and z 3 do not influence and are not influenced by variable z 4. A POG repreentation of ytem (6) i hown in Fig.. 6. Simulation The POG cheme of Fig and have been implemented in Simulink to compoe the model of the whole ytem. 6.. Experimental and imulation reult of the linearrotary motion Some experiment on the prototype (ee Fig. 6) have been made in order to identify the main parameter of the part of the ytem compoed by the pulley-belt tranmiion the nut of the ball crew/pline and the haft. The model of the electrical motor ued in imulation i given in the POG cheme of Fig. 4: two controlled DC motor ha been conidered a equivalent model of the real threephae motor ued in the real ytem. Fig. 7 how the Figure 6. Picture of the prototype. haft vertical poition and haft angular poition: imulated in blue and meaured in red. The maximum error on haft vertical poition i about mm and the maximum error on haft angular poition i about 3 deg. x mm θ deg Shaft vertical poition Shaft angular poition Time Figure 7. Shaft vertical poition and haft angular poition: imulated (b) and meaured (r) Simulation reult of the corking The imulation reult hown in thi ection are about the crewing of the cap on the bottle. For thi ytem a control algorithm ha been implemented which allow to guarantee the complete eal of the bottle with a certain level of train that i independent of the initial angular poition of the cap. The motion of the TCP i controlled according to the main tep of the corking cycle that can be ummarized a in Table 3. Fig. 8 how the vertical poition of the haft x cap x tp and bottle neck x bh and correponding angular poition θ θ tp e θ bh. Note that at about t = 2.3 the crewing of the cap i completed and

8 Time Operation -.3 decent to grap the cap riing 3 -. approaching to the bottle neck crewing of the cap releae of the cap and moving away Table 3. Step of a corking cycle then at t = 2. the graper releae the cap which remain attached to the bottle while the haft move away from the cap. Some imulation of a corking cycle lating xa xtp xbh cm Poition of the haft (r) cap (b) and bottle neck (k) θtp turn Fn kgf Angular poition of the cap Time Normal force between cap and bottle Time Figure 9. Angular poition of the cap and normal force between cap and bottle for different initial angular poition of the cap θa θtp turn θbh deg Time Figure 8. Vertical and angular poition of the haft cap and bottle neck. for T = 3. for different initial angular poition of the cap θ tp = { }2π rad have been done. Fig. 9 how the angular poition of the cap and normal force between cap and bottle. Note that depending on the initial poition of the cap the crewing can end in different final poition (differing each other of one turn) while the ealing normal force i alway the ame and do not depend on the initial poition of the cap. 7. Concluion The modeling approach baed on the Power-Oriented Graph technique ha been applied to an electromechanical ytem compoed by a crew/pline haft moved by two electrical motor and contituting the prototype for an automatic corking machine for threaded platic cap. The model parameter have been identified uing data collected on a real prototype. A model for the interaction between the platic cap and the bottle ha been propoed. The imulation reult how the effectivene of the realized model and allow to analyze the performance of the ytem and to build a control algorithm. 8. Acknowledgement The author would like to thank Paolo Zanon and Maimo Garuti (Democenter Sipe Scarl) Giovanni Zanai (Preident of Fondazione di Vignola) and Stefano Camatti (Phema Srl) for having upported the project. Reference D.C.Karnopp D.L.Margoli and R.C.Roemberg. Sytem dynamic - Modeling and Simulation of Mechatronic Sytem. Wiley Intercience 3rd ed J. C. Mercieca J. N. Verhille and A. Boucayrol. Energetic macrocopic repreentation of a ubway traction ytem for a imulation model. Proceeding of IEEE-ISIE 24 page 9 24 May H. Paynter. Analyi and Deign of Engineering Sytem. MIT-pre Camb. MA R. Zanai. The power-oriented graph technique: Sytem modeling and baic propertie. Proceeding of 2 IEEE Vehicle Power and Propulion Conference (VPPC) page 6 ept. 2. R. Zanai G. Sandoni and R. Morelli. Simulation of variable dynamic dimenion ytem: the clutch example. Proceeding of the European Control Conference 2 page