Mechatronics Design of Ball and Beam System; Education and Research

Transcription

1 Iteratioal Joural of Computatioal Egieerig Reearch Vol, 3 Iue, 9 Mechatroic Deig of Ball ad Beam Sytem; Educatio ad Reearch Farha A. Salem, Departmet of Mechaical Egieerig, Faculty of Egieerig, Mechatroic prog,, Taif Uiverity, 888, Taif, Saudi Arabia., Alpha Ceter for Egieerig Studie ad Techology Reearche, Amma, Jorda. ABSTRACT: The key elemet i Mechatroic deig i the cocurret yergetic itegratio, modelig, imulatio, aalyi ad optimizatio of multidicipliary kowledge through the deig proce from the very tart of the deig proce. Mechatroic egieer i expected to deig egieerig ytem with yergy ad itegratio toward cotrai like higher performace, peed, preciio, efficiecy, lower cot ad fuctioality. Thi paper propoe the coceptio ad developmet of ball ad beam ytem baed o Mechatroic deig approach. A complete overall ytem ad ubytem electio, modelig, imulatio, aalyi, ad itegratio are preeted. The propoed Mechatroic deig ad model were created ad verified uig MATLAB /Simulik oftware ad are iteded for reearch purpoe, a well a, applicatio i educatioal proce. KEYWORDS: Mechatroic, Mechatroic deig approach, Ball ad beam, modelig/imulatio. I. INTRODUCTION Mechatroic i defied a yergitic itegratio of mechaical egieerig, electric egieerig, electroic ytem, iformatio techology, itelliget cotrol ytem, computer hardware ad oftware to maage complexity, ucertaity, ad commuicatio i deig ad maufacture of product ad procee, a workig defiitio ca be; the yergitic itegratio of eor, actuator, igal coditioig, power electroic, deciio, cotrol algorithm, computer hardware ad oftware to maage complexity, ucertaity, ad commuicatio i egieered ytem. The key elemet i Mechatroic deig i the cocurret yergetic itegratio (itead of equetial), aalyi ad optimizatio of thee area through the deig proce reultig i product with more yergy [] ad a balace betwee modelig/aalyi ad experimetatio /hardware implemetatio. Baed o thi, Mechatroic egieer i expected to deig egieerig ytem with yergy ad itegratio toward cotrai like higher performace, peed, preciio, efficiecy, lower cot ad fuctioality. Referece to [-], Mechatroic ytem deig proce ca be divided ito ytematic, imple ad clear deig tep icludig; Problem tatemet; Coceptual deig ad fuctioal pecificatio; Parallel (cocurret) deig ad itegratio of ytem ad all ubytem a whole ad cocurretly icludig; electio, deig ad yergetic itegratio of mechaical, electroic, oftware, cotrol uit, cotrol algorithm ad iterface ub-ytem; Modelig ad imulatio; Prototypig, Tetig ad Optimizatio; ad fially Maufacturig, ad commercializatio, thee tep will be followed ad explaied i thi paper, were thi paper propoe the coceptio ad developmet of ball ad beam ytem baed o Mechatroic deig approach. A complete overall ytem' ad ubytem' compoet' electio, deig, itegratio, a well a, modelig, imulatio ad aalyi are to be preeted.the igificace of the ball ad beam ytem i due to it' complexity, ad at the ame time it' implicity, where it i a double itegrator ytem that i ope-loop utable ad imply, it i a poitio cotrol ytem, preet a challegig deig ad cotrol problem, it i ofte ued by may cotrol theorit ad egieer a a bech mark problem for tetig ad aalyzig the reult of may differet theoretical cotrol cheme ad ew cotroller ad oberver method. II. PRE-STUDY PROCESS-PROBLEM STATEMENT The ball ad beam ytem i motio cotrol ytem coitig of three mai part; cotrol ytem, a ervo or electric motor, free rollig back ad forth, Ball o top of log V-grooved Beam whoe icliatio ca be adjuted by maipulatig the agular poitio of the ervo. Ball ad beam ytem i ofte ued a a bech mark problem for may differet cotrol cheme ad preet a challegig deig ad cotrol problem, due to fact that it i double itegrator (two pole at the origi) ytem that i ope-loop utable, i uch ytem eve at optimal beam poitio coditio the ball will wig o Beam, up to roll off the ed of the beam. Target uer; I 5-35 September 3 Page 54

2 Mechatroic Deig Of Ball Ad Beam Reearcher ad Educator.Uer' requiremet; it i required to develop a ball ad beam ytem; liear, oedimeioal igle-variable experimetal uit, that ca be ued for tetig, aalyi, educatio, ad demotratio the effect of variou cotrol trategie, maily SISO ytem, precie, cot-efficiet, with imple, eay to udertad ad ue iterface allowig the uer to maipulate a machie, gatherig ad outputtig data, pace avig, ad with attractive deig. Sytem parameter, requiremet ad aalyi, are lited i Table. III. CONCEPTUAL DESIGN The purpoe of deig i to propoe the coceptio ad developmet of ball ad beam ytem baed o Mechatroic deig approach, it i required to propoe a deig of a ball ad beam ytem that fit uer' (Reearcher, Educator, tudet) eed ad requiremet, with overall fuctio ( iteded) to help uer to tet, viualize ad aalyze the elected ad deiged cotrol trategy to cotiually meaure ad tabilize the rollig ball poitio a log a give beam by adjutig the rotatig agle of a beam by tiltig it frotward, backward, leftward, or rightward, all thi accomplihed by maipulatig the agular poitio of a elected ervo. Sytem' fuctioal tructure block diagram i how i Figure. Morphological table, aalyi ad evaluatig the bet olutio are how i Table. Block diagram repreetatio, prelimiary tructure icludig mechaical, electroic ad ytem dyamic are how i Figure ad Figure 3 Table requiremet aalyi Requiremet type Requiremet Value/ Uit Fixed Req. Soft Req. Quatitative Req. Qualitative Req. Cotrol ytem Compact, cot effective. - Fixed Qual. Cotrol algorithm Beam Ball Simple, quick, precie, efficiet, eay to program V-grooved, Ma, m Beam legth, m Radiu Ma, M - 4 cm.5 kg 4 cm.5 m. kg Fixed Fixed Soft Qua. Actuator Oe actuator, for adjutig the beam rotatig agle, available, iexpeive, imple to iterface, ad cotrol. -4 v - Soft Qual. Seor Poitio eor, imple to iterface, available, iexpeive - Soft Qual. Gear, belt, chai.3 m Soft Tramiio Output gear radiu Overall gear ratio Height: 4 Cm - Qual. Overall Sytem dimeio Width Legth 3 Cm 5 Cm Fixed Uer Iterface Simple ad eay to ue ad udertad Qual. Machie aethetic deig Compact, attractive fit uer requiremet. Soft Qua I 5-35 September 3 Page 55

3 Cotrol ytem Mechatroic Deig Of Ball Ad Beam Table Morphological table, aalyi ad evaluatig the bet olutio electio Switchig ytem o-off Uer iterface Deired ball poitio Cotrol ytem Actuator, electric eergy Actuator motio repoe Gear,lever arm Motio tramiio Poitio eor Beam agular poitio Poitio eor Ball poitio & meaurig Figure Fuctioal Structure block diagram IV. PARALLEL (CONCURRENT) SELECTION, DESIGN AND INTEGRATION OF SUB-SYSTEMS AND OVERALL SYSTEM. Mechatroic deig i the yergitic electio, evaluatio ad itegratio of the ytem ad all it compoet a a whole ad cocurretly icludig eor, actuator, igal coditioig, power electroic, deciio, cotrol algorithm, computer hardware ad oftware. All the deig diciplie work i parallel ad collaboratively throughout the deig ad developmet proce to produce a overall optimal deig. The ball ad beam ytem i to be divided ito realizable hardware ad oftware ubytem icludig; Mechaical ubytem; eor ubytem ; actuator ubytem ; power upplie, cotroller ad Cotrol algorithm, drive, ad coditioig circuit), the optimal electio, modelig, imulatio, itegratio, optimizatio ad the exchage of iformatio betwee differet module are to be deiged cocurretly. 4. The optimal mechaical deig: The mechaical deig i the keleto of Mechatroic ytem, there are everal arragemet of cotructig ball ad beam ytem, three ytem' arragemet are how i Figure. I Figure (a), oe ed of the beam i coupled to electric motor through lever arm ad gear, the other ed i fixed, ad upported with vertical beam. I Figure (c) oe ed of the beam i coupled to electric motor through lever arm ad gear, the other ed i free to move up ad dow, the beam i upported with vertical bar at it ceter. I Figure (d) the I 5-35 September 3 Page 56

4 Mechatroic Deig Of Ball Ad Beam beam at it ceter, i mouted o the output haft of a electric motor. Sytem CAD model i how i Figure (e). The whole ytem layout i how i Figure (a), ad the mechaical ytem part, material ad optimal dimeio are lited i Table 3. Prelimiary block diagram ad layout repreetatio of propoed ytem ad mai compoet are how i Figure (f)), α, Beam agle mg Mg Lever arm θ, output gear agle Motor with gear Figure (a) Oe ed of the beam i coupled to electric motor through lever arm ad gear, the other ed i fixed. Motor mg α,beam agle Figure (b) Electric motor with gear Figure (c) upport at beam' ceter Figure (d) The beam at it ceter, i mouted o the output haft of a electric motor. Figure (e) 3-D CAD model D Deired Poitio, x - + Ball cotroller α - + Actuator cotroller + + Actuator θ Ball & Beam x Uer iterface α=d/l Feedback Figure (f) ytem block diagram I 5-35 September 3 Page 57

5 Mechatroic Deig Of Ball Ad Beam Table 3 mechaical ytem part, type of joit material ad dimeio Sytem dimeio Beam Ball Tramiio Joit Support Type of joit Bae, Frame Value: Height Width Legth Shape & material Beam ma, m Beam legth, L Shape & material Radiu Ma, M Outer gear radiu Overall gear ratio Revolute joit Steel bar Revolute joit Uit 4 Cm 3 Cm 5 Cm V-grooved,teel.5 kg 4 cm Steel.5 m. kg.3 m - - 3J 4..Cotroller ad cotrol algorithm electio Due to fact that Ball ad beam ytem i double itegrator (two pole at the origi) ytem, the behavior of movig Ball o a beam i behavior of ope loop utable ytem, i order to tabilize the double itegrator ad ope-loop utable ball ad beam ytem, a pecial cotrol ytem i to be deiged. The cotrol of utable ytem i critically importat to may of mot difficult cotrol problem ad mut be tudied i laboratory, the problem i that real utable ytem are uually dagerou ad ca ot brought i laboratory, uch utable ytem iclude; the cotrol model i the rocket or aircraft topplig cotrol ytem durig vertical take-off, it i the cotrol where a feedback ytem i ued to prevet rocket or aircraft to topple out of balace durig lauch by force ad momet that could perturb the vertical motio, the agle of thruter jet or diverter mut be cotiually cotrolled to prevet the rocket tumblig or aircraft tippig [-4]. The ball ad beam ytem wa developed to reolve thi paradox, it i imple, afe mechaim ad yet it ha the importat dyamic feature of utable ytem [],. Cotrol algorithm electio i the mot critical deciio i the Mechatroic deig proce, there are umber of alterative cotrol algorithm trategie that ca be applied to the ball ad beam ytem, icludig but ot limited to ; PD cotrol, lead compeator, Fuzzy cotrol, Robut cotrol, Liear quadratic Gauia, Liear quadratic regulator. The cotrol ytem tak, i ball ad beam ytem, i to cotiually meaure ad tabilize the rollig ball poitio a log the beam by adjutig the rotatig agle of a beam ( tiltig it frotward or backward) by maipulatig the agular poitio of the ervo, i uch a way that the actual poitio of the ball reache deired poitio, thi i difficult cotrol tak, becaue eve at optimal beam poitio coditio the ball will wig o Beam with acceleratio proportioal to tilt agle, thi behavior i behavior of ope loop utable ytem, where the ytem output (ball poitio o beam) icreae without limit for fixed iput (beam agle) ad i a challegig deig ad cotrol problem. I [5] W. Yu ad F. Ortiz tudied complete oliear model of beam ad ball ytem ad tability aalyi of PD cotrol. I [6] a deiged ad implemeted both covetioal pole-placemet ad eural etwork method for beam ad ball ytem are itroduced. I [7] the robut oliear ervomechaim theory wa applied to deig a trackig cotroller for beam ad ball ytem. I [8] witchig cotrol for beam ad ball ytem i tudied. I [9] fuzzy logic cotroller ad real time implemetatio of beam ad ball balacig ytem i tudied. I thi paper will be itroduce PD cotroller with deadbeat repoe, PID, ad Lead itegral compeator. Cotroller electio :A vat variety of cotroller (phyical cotroller) are available i the market, embedded Microcotroller i optimal electio, ice it i iexpeive igle chip computer, eay to embed ito larger electroic circuit deig, alo, becaue of their veratility, Microcotroller add a lot of power, cotrol, ad optio at little cot; capable of torig ad ruig program, programmed to perform a wide rage of cotrol tak. Optimal microcotroller i PICmicro Microcotroller, upplied with 5VDC. I 5-35 September 3 Page 58

6 DC motor, gear Vertical upport Electroic circuitry Mechatroic Deig Of Ball Ad Beam 4.3 Actuator (electric machie) electio ad itegratio Actuator covert a iformatio igal from the microcotroller cotrol uit, ito eergy actig o the baic ytem. Mechatroic ytem ofte ue electric motor to drive their work load, the electric actuator mot uitable to be ued for ball ad beam i DC motor, due to imple priciple of workig, quick itataeou ad accurate torque geeratio, available, iexpeive, reverible, ad eae deigig ad implemetig cotroller to achieve optimal itataeou, precie motio cotrol performace. PMDC motor tur electrical eergy ito mechaical eergy ad produce the torque required to move (rotate) the beam to the deired poitio, θ L, Actuator placemet ad itegratio: to phyically itegrate Mechatroic ytem compoet, ad to give the overall ytem afe, compact ad attractive deig, the elected DC motor ad gear, are embedded withi mechaical deig ad to be located o adjacet ( right ide, a whole ytem bae) cuboide houig, with lot for digital uer iterface ad iput/output port(ee Figure 3). Reitive wire eor DC motor, gear Reitive wire eor Revolute joit V-grooved beam Lever arm V-grooved beam DC motor, gear Uer iterface Electroic circuitry Figure 3(a) Figure 3(b) Revolute joit Reitive wire eor V-grooved beam Lever arm Uer iterface Electroic circuitry Figure 3(c) Figure 3(a)(b)(c) Three arragemet of overall ytem deig, compoet ad itegratio 4.4 Seor electio Seor covert a tate variable of the baic ytem, ito a iformatio igal to the cotrol uit. To calculate the error betwee the actual ad deired poitio, we eed to covert the actual poitio, ito voltage, V, ad the compare thi voltage with the iput voltage; the differece betwee both i the error igal i volt. Ball ad beam ytem, provide two output meauremet ad correpodigly two eor, oe to meaure ball tralatioal poitio, x a log the beam ad other to meaure motor haft agular poitio θ, correpodigly, the optimal cotrol ytem deig require two cotrol loop deig, ier for motor agle cotrol, to make motor follow the iput agle ad other outer loop to cotrol the ball poitio o the beam, to make ball poitio follow the referece iput poitio. The two agular quatitie the beam agle, α ad DC motor output poitio agle, θ are ot the ame, but are correlated i imple relatiohip, relatig thee two quatitie reult i that oly oe eor ca be ued to meaure the poitio. The poitio of the ball o the beam ca be meaured uig a pecial eor, e.g. liear Potetiometer, ifrared, ad a liear reitive wire, the optimal electio i liear reitive wire, to ued a a poitio eor ad the coductive ball movig alog it actig a a voltage divider ( ee Figure 3). I 5-35 September 3 Page 59

7 Mechatroic Deig Of Ball Ad Beam 4.5 Output igal, Coditioig ad Iterfacig electio, deig ad itegratio A mot uitable ad imple to implemet drive for PMDC motor are H-bridge ad IC circuit e.g.l93d.the H-bridge circuit i upplied with VDC ad the four bit output of microcotroller to drive the deire coditio of electric Motor. By H-bridge four NPN Power traitor are ued a witch to cotrol the directio of curret flow to the Motor, ad correpodigly cotrol the agular poitio of beam, elected microcotroller type i upported with ADC pi, to covert the aalog iput of eor readig to digital value, fed to microcotroller. A commo carrier (ee Figure 3-4) for electroic, data outputtig ad cotrol circuitry to be located iide the cuboide houig, that will iclude lot for uer iterface ad iput/output port Figure 4 Microcotroller baed cotrol Circuitry ad iterfacig diagram V. Modelig, imulatio, aalyi ad evaluatio The key eetial characteritic of a Mechatroic egieer are a balace betwee modelig/aalyi kill ad experimetatio/hardware implemetatio kill. Modelig, imulatio, aalyi ad evaluatio procee i Mechatroic deig coit of two level, ub-ytem model ad whole ytem model with variou ub-ytem model iteractig imilar to real ituatio, all egieerig ubytem hould be icluded i whole ytem model. Baed o the pecificatio of requiremet ad deig, the ubytem model ad the whole ytem model, are to be teted ad aalyzed, pecificatio to tet ad check whether the give deig pecificatio are atified, If the pecificatio are ot atified,, modificatio ca be made, if the pecificatio are atified the model ca be Optimized. The derivatio of complete ad accurate ball ad beam ytem' dyamic i quit complicated, i thi paper, differet ytem deig ad dyamic model will be itroduced, implified ad more actual model ad correpodig cotrol ytem trategie electio ad deig. Ball ad beam dyamic are how i Figure Modelig of baic phyical ub-ytem model with o cotrol ivolved; Ball ad beam balacer Sytem Modelig 5.. Ball ad beam ytem' dyamic, baic mathematical model derivatio: lidig ball. I thi paper, differet approache for derivig the mathematical model of ball ad beam will be itroduced. Baed o [] ad o ytem arragemet how i Figure (a), dyamic are how i Figure 5. I the abece of frictio or other diturbace ad aumig the ball i lidig, the dyamic of the ball ad beam ytem ca be obtaied a follow; The ball move a log the beam with acceleratio, the force that accelerate the ball a it roll come from the compoet of gravity that act parallel to the beam. The equatio of motio for the Ball i give by the followig equatio: I 5-35 September 3 Page 6

8 Mechatroic Deig Of Ball Ad Beam α, Beam agle Mg α α, Beam agle x +x*dα/dt Figure 5 (a) Ball dyamic Figure 5 (b) Ball dyamic Subtitutig ad maipulatig, give: F M * a F M * g * i ( ) d x d x M * M * g * i ( ) g * i ( ) d t d t Thi model ca be liearized, where for mall agle, i (α) α, the model become: d x dt g * () Thi i the baic mathematical model of the ball ad beam ytem, thi imple mathematical model how that, the acceleratio of the ball i proportioal to the beam agle α ad to gravity g. Where: x : the ball poitio o the beam, M : the ball ma. to write a approximated overall trafer fuctio of the ball ad beam ytem, that relate the iput voltage ad meaured output acceleratio, we eed to replace the beam agle α, by cotrol voltage, V i, ad replace the ball poitio x, by the poitio eor output y, ad fially Combie the actuator ad eor cotat with the gravity cotat, we have a igle cotat b, Subtitutig, performig Laplace traform ad rearragig, we have the trafer fuctio that relate the iput voltage ad meaured output acceleratio d y b * V Y ( ) b * V ( ) i i dt Y ( ) b Y ( ) g V ( ) V ( ) i i Where: b: the actuator ad eor with the gravity cotat. The tate pace repreetatio of thi approach i give by: x x u x x b x y x The liear model give above i good approximatio of true ytem dyamic, all what we eed i to meaure the cotat parameter, b, it ca be obtaied by meaured the time take for the ball to accelerate from oe ed of the beam to the other at fixed agle of the beam, the law of motio ca be the ued to itegrate Eq.() ad calculate cotat b. The real ball ad beam ytem ha additioal dyamic compoet due to the motor, plu o-liear ad oie compoet that ifluece it cotrol behavior,( coulomb frictio, dead zoe aturatio i the motor iput amplifier)[4]. 5.. Ball ad beam ytem' dyamic, ecod approach: rollig ball without lippig. Baed o ytem arragemet how i Figure (a), Ball dyamic how i Figure 5(b), ad i the abece of frictio or other diturbace ad aumig the ball i rollig without lippig, the dyamic of the ball ad beam ytem ca be obtaied a follow; There are two force ifluecig the motio of the rollig ball, thee force are: F tx : the force due to tralatioal motio ad F rx : the force due to ball rotatio. The equatio of motio for the Ball i give by: F M * a M * g * i ( ) Subtitutig force ifluecig the motio of the rollig ball, give: F F F M * g * i ( ) (3) tx rx Now, to fid force ifluecig the motio of the rollig ball, ad ubtitutig i Eq.(3). Force F tx due to tralatioal motio i give by: I 5-35 September 3 Page 6 ()

9 Mechatroic Deig Of Ball Ad Beam Ftx m x (4) The force due to ball rotatio, F rx i foud a follow: The torque produced by the ball' rotatioal motio i equal to the radiu of the ball, R, multiplied by the rotatioal force, ad i give by T F R r Alo, the torque i equal to the ball' momet of iertia,j, multiplied by it agular acceleratio, dω b /dt, which the ca be writte a it momet of iertia multiplied by the double-derivative of it tralatioal motio (x b ) divided by it radiu, thi i a how ext: rx d b T F R J r rx dt d d ( v / R ) d ( x / R ) b b b J T F R J J J x r rx d t d t dt R F rx = (J *a)/r Subtitutig, the momet of iertia, J, for the ball ad give by ( J=(/5)MR ), we have: M (5) F / 5 d rx dt x Subtitutig Eq.(5) ad Eq. (4),i Eq.(3) we have the baic mathematical model of ball ad beam ytem that relate the ball diplacemet ad agular poitio of the motor haft, ad give a: F F F M * g * i ( ) tx rx d x d x M * g * i ( ) M M 5 d t d t Maipulatig ad rearragig, give: M*g*i(α) = d x/dt [ (/5) M + M ] g*i(α) = d x/dt [ (7/5) ] (5/7) *g*i(α) = d x/dt Thi model ca be liearized, ice for mall value of agle α, i(α) = α, thi approximatio i good for agle value -π/4 <α< π/4, ie of the agle i approximately the agle itelf. Takig Laplace traform ad rearragigiput diplacemet, X(), over output agular poitio, α(), of the motor haft give : X ( ) (5 / 7 ) g ( ) The tate pace repreetatio of for thi approach i give by: (5/7) g α () = X() (6) x x u x x 5 / 7 x y x The trafer fuctio ca be derived baed o ytem arragemet how i Figure (b), a follow; the beam agle cotrol acceleratio, ot poitio, from gravity the equatio of motio for the Ball i give by: F M * a M * g * i ( ) Aumig the ball i rollig without lippig, we have r*ω= dx/dt, The momet of iertia of the olid ball i give by: ( J=(/5)MR ). Give cetripetal acceleratio with rotatio of the rod a a cetripetal = xα, altogether we have: d x d d x x g i d t d t 5 d t By liearizatio, takig Laplace traform, rearragig for trafer fuctio, give: X ( ) (5 / 7 ) g ( ) (7) 5..3 The Lagragia approach to the Ball ad Beam ytem' dyamic: I the abece of frictio or other diturbace, the dyamic of the ball ad beam ytem ca be obtaied by Lagragia method baed o the eergy balace of the ytem, The Lagragia method i utilized to derive the equatio of motio for the ball ad beam ytem i the mot model baed reearch work o the ball ad beam ytem [6][-3]. I order to derive the Euler-Lagrage equatio, the firt tep i to defie the kietic ad potetial eergy for the ball ad beam. The kietic eergy of the ytem i give by: I 5-35 September 3 Page 6

10 Mechatroic Deig Of Ball Ad Beam T = T + T (8) Where T ad T are kietic eergie of the beam ad the ball, thee kietic eergie iclude radial ad circular motio. The rotatioal kietic eergy of the beam i give by: T.5 J a Where: J, i the momet of iertia of the beam ad a i the agle velocity of the frame. The ball ha kietic eergy, give by: T.5 ( M x ) * a.5 M v.5 J b Subtitutig v =Rω ad J b = (/5)*M*R, the rotatioal kietic eergy of the ball i give by:.5((/5)*m*r ), ubtitutig, the kietic eergy of the ytem i Eq.( 8) will give: T.5 ( J M x ) * a (7 / 5 ) M x The potetial eergy of the ytem i exhibited by the rollig ball aloe, ad give by: P M * g * x * i.5 m * g * L * i P M * g * x.5 * m * g * L i Where m : the ma of the beam,m: the ma of the ball, L: the beam legth. We ca ow write the Lagragia, where the differece betwee kietic ad potetial eergy i called the Lagrage fuctio, which i defied by (L) equatio ad give by: L=T -P Where : T i the kietic eergy ad P i the potetial eergy i the ytem. L.5 ( J M x ) * a (7 / 5) M v M * g * x.5 *m * g *L i The dyamic equatio repreetig the variatio effect of ytem variable i give by: d L L d t q q Q J d x M M g * i m * x * R d t Where θ: the motor output poitio agle. α: the beam agle. x: ball coordiate poitio, m: beam ma. A how i Figure (a) ad Figure 7, It hould be coidered, that the beam agle, α, ad DC motor output haft agle, θ, are ot the ame, but there i alo relatiohip betwee thee two agle that ca be ee from Figure 6, thi relatio ca be approximated a liear, where the ditace traveled by the beam at radiu equal to L, i equal to ditace traveled by gear at radiu, r, that i: r r L fo r r L () L (9) () Figure (a) Figure 7 the beam agle, α, ad DC motor output poitio agle,θ Ball ad beam ytem' dyamic i term of ball radiu R, ad r: baed o ytem arragemet how i Figure (a), dyamic how i Figure 5, ad i the abece of frictio or other diturbace, we have: x r x v x ( x ) r U m v I I b a Writig ad ubtitutig the Lagragia equatio, give: I 5-35 September 3 Page 63

11 Mechatroic Deig Of Ball Ad Beam x L m [ x ( x ) ] I I b a r d d d L L m g i d t d x d x I b I b m x m x m g i r r I b m x m g r Takig Laplace traform, Subtitutig I b ( / 5 ) M R, ad maipulatig for trafer fuctio give: X ( ) m g m g g G ( ) ( ) I b m R R m m r 5 r 5 r The trafer fuctio i term of iput gear agle, θ, to output ball poitio, X, alo ca be writte a follow: Liearizatio of Eq.() about the beam agle, α=, give J d x M M g * i m * x * R d t J d x M M * g * R d t Liearizatio of thi equatio, ubtitutig Eq.() takig Laplace traform, ad maipulatig for trafer fuctio give: J d x d X ( ) M * g * d M * * * M g R d t L ( ) J L M R The tate pace repreetatio of thi model i give below, here it i importat to otice i thee model, that by combiig the ball radiu R, ad r cotat with the gravity cotat, g, we have a igle cotat b, thi will give the baic trafer fuctio model previouly obtaied, thi i how i below compario of differet approache for model derivatio: X ( ) M * g * d X ( ) b X ( ) g X ( ) b * a d ( ) J ( ) ( ) R ( ) L M R 5 r The trafer fuctio betwee the ball poitio ad rotor agle ca be obtaied baed o baic trafer fuctio Y ( ) b g V ( ) i ad o relatio give by Eq.(), ad give by Y ( ) r g r * g V ( ) L L i 5. MATLAB/Simulik repreetatio, imulatio ad aalyi The loop ytem Simulik model for the implified model i how i Figure 8. The Simulik model of mathematical model give by Eq.(6) i the ame a Simulik model how i Figure 8, but the cotat b, ow i more accurately defied ad i give by (5/7)*g, the Simulik model of ope loop ball ad beam ytem i how i Figure 8(c), The ytem repoe uig MATLAB, ca be foud uig the followig code >> g=9.8; um=[5*g/7]; de=[ ]; G_ope=tf(um,de); prity(um,de); tep(um,de); [A,B,C,D]=tf(um,de); Figure, tep(a,b,c,d) Now, Baed o [4], the oliear Lagragia equatio of motio i directly modeled, where Eq.() give d(x)/dt a a fuctio of the tate ad iput variable, x, d(x)/dt, alpha, ad d(alpha)/dt. The referece made ue of the oliear fuctio block to expre thi fuctio; the fuctio take the iput vector u=[ x dx/dt α dα/dt ] ad retur d x/dt, where each compoet i referred to a u[], u[], etc, ad correpodigly, u[]=x, u[]=d(x)/dt, u[3]=alpha, ad u[4]=d(alpha)/dt, the fuctio i give by: I 5-35 September 3 Page 64

12 -68 Step3 Sigal Geerator3.., i(u) co(u) SiCo. g Gai Ball Acceleratio dv/dt Itegrator Ball peed. Ball poitio. Ball poitio dx/dt Itegrator3 Ball poitio,. Mechatroic Deig Of Ball Ad Beam d x d ( / ( J / R M ))( M g i M x ) d t d t Ad ca be writte i MATLAB a: (-/(J/(R^)+m))*(m*g*i(u[3])-m*u[]*(u[4])^) Thi model i how i Figure 9 (a), firt it i required, to defie ball ma m, Ball radiu, R, beam legth, L, gear radiu, r, ad ball' momet of iertia J,(e.g. m=.; R =.5; g = -9.8; L =.6; d =.3; J = 9.99e-6). Ruig all Simulik model for derived ytem' model, whe ubjected to radom iput, will reult i the ame poitio, peed ad acceleratio repoe curve how i Figure (a). I Figure (b) i how ball poitio ad acceleratio ope loop repoe whe ubjected to tep iput. ope loop ball ad beam Step3 i(u) co(u).., SiCo. Step Sigal Geerator3 Sigal Geerator.., Ball AcceleratioBall peed. Ball poitio. Ball poitio Step x' = Ax+Bu i(u) g Ball Acceleratio Ball peed y = Cx+Du dv/dt dx/dt co(u) Itegrator Itegrator3 Ball State-Space Ball poitio,. Ball Ball Gai. SiCo acceleratio peed poitio i(u) (5*g)/7 dv/dt dx/dt Sigal co(u) Itegrator Geerator Itegrator Ball poitio SiCo. Gai Figure 8(a) Simulik model of implified repreetatio Ball poitio' Ball poitio Ball poitio'. Figure 8(b) Simulik model of tate pace repreetatio -95 Ball poitio, Ball poitio, ball_beam Ball poitio. Step Icliatio agle Ball peed Sigal Geerator, Ball Acceleratio ball ad beam dyamic ubytem Ball peed Ball Speed ball_poitio To Workpace Ball Acceleratiom Ball Acceleratio ball_poitio3 To Workpace Figure 8(c) Simulik mode of ope loop ball ad beam ub-ytem dx/dt, Ball peed 3 Ball acceleratio. Theta r/l Gai alpha du/dt Derivative Mux Mux f(u) Ball-Beam Lagragia Model d/dt(x) x x, Ball poitio Itegrator Itegrator Fig, 9 (a) The oliear Lagragia equatio of ball ad beam ope loop motio ub-ytem model Ball poitio x, Ball poitio Step Theta dx/dt, Ball peed Sigal Geerator, Ball ad Beam Model Ball acceleratio Ball peed Ball acceleratio Figure 9 (b) The ball ad beam ub-ytem ope loop model ad ubytem. I 5-35 September 3 Page 65

13 Acceleratio, a Poitio x mm mm Mechatroic Deig Of Ball Ad Beam Beam agle VS time Ball poitio VS time 3 x 4 Ball poitio VS time Ball Speed VS time 5-5 Ball Accel. VS time 5 Ball Speed VS time 6 Speed 5-5 Speed (a) Repoe for radomly chagig iput (b) Repoe for tep iput Figure ope loop, ball poitio/time, peed/time ad acceleratio/time repoe curve The ext MATLAB code ca be ued to defie ytem' parameter i MATLAB ad retur trafer fuctio ad ope loop tep repoe clc, clear all M=iput('Eter ma of the ball, M = '); m=iput('eter ma of the beam, m = '); R=iput('Eter radiu of the ball, R = '); d=iput('eter lever arm offet, d = '); L=iput('Eter legth of the beam, L = '); J=iput('Eter ball'' momet of iertia, J= '); g= -9.8; % gravitatioal acceleratio = tf([ ],); G_ope = -(M*(g)*d)/(L*((J/R^)+ M)*^); tep(g_ope) % aother implified way of obtaiig the trafer fuctio K = (M*(g*d)/(L*(J/R^+M))); um = [-K]; de = [ ]; G_ope =tf(um,de); tep(g_ope); [A,B,C,D]= tf(um,de); tep(a,b,c,d); % Obtaiig cloed loop trafer fuctio with uity feedback, o gai G_cloe=feedback(G_ope,); axi([,,,]); title ('tep repoe of cloed loop ytem with H()= ') 5.3 Modelig of actuator ub-ytem dyamic The DC motor i a example of electromechaical ytem with electrical ad mechaical compoet, a implified equivalet repreetatio of DC motor' two compoet with gear attached are how i Figure (c) up. DC motor tur electrical eergy ito mechaical eergy ad produce the torque required to move the beam to the deired agular poitio, θ L, [5]. I [6] a detailed derivatio of DC motor electric ad mechaical part dyamic are itroduced. Baed o the Newto law combied with the Kirchoff law, the mathematical model i the form of differetial equatio decribig dyamic characteritic of the armature cotrolled PMDC motor ca be derived, ad correpodigly DC motor trafer fuctio ope loop trafer fuctio without ay load attached relatig the iput voltage, V i (), to the motor haft output agle, θ m (), to be give by: G a g le ( ) ( ) K / t ( ) V ( ) L R J b K K i a a m m t B K t / 3 V ( ) i L J ( R J b L ) ( R b K K ) a m a m m a a m t b The PMDC motor ope loop trafer fuctio relatig the iput voltage, V i (), to the agular velocity, ω(), give by: (3) G p eed ( ) K / t ( ) V L J R J b L R b K K ( ) i ( ( ) a m a m m a a m t b I 5-35 September 3 Page 66 ()

14 Mechatroic Deig Of Ball Ad Beam Modelig oliearitie: Coulomb frictio i a o-liear elemet i which force ted to appoe the motio of bodie i cotact i mechaical ytem, it act a diturbace torque feedback for the mechaical ytem, Coulomb frictio i coidered to be a cotat retardig force but i dicotiuou over zero croig, that i, whe a DC motor revere directio it mut come to a top at which poit Coulomb frictio drop to zero ad the oppoe the revered directio. I effect Coulomb frictio i cotat whe rotatioal velocity i ot zero. Baed o differetial equatio decribig mechaical characteritic of DC motor; the um of the torque mut equal zero, we have: T = J *α = J*d θ/dt Coulomb frictio ad dead zoe frictio, where (T load =), we have: K t *i a = T α + T ω + T load + T f Where: i a armature curret, K t : torque cotat, T f : Coulomb frictio torque, takig Laplace trafom: K t *I() - J m * θ() b m * θ() - T f = At teady tate coditio, d/dt =, give: K t *i a = - b*ω T f 5.4 Simplificatio of ope loop PMDC motor ytem trafer fuctio model. Referrig to [6] ad baed o the fact that, the PMDC motor repoe i domiated by the low mechaical time cotat, where the electric time cotat i much fater (e.g. te time) tha the mechaical time cotat, thi ca motivate u to aume that the armature iductace, L a i low compared to the armature reitace, R a. eglectig motor iductace, (L a =), will reult i the followig implified firt order form of PMDC motor trafer fuctio i term iput voltage, V i () ad output peed, ω m () give by: G ( ) ( ) p e e d V ( ) i R J a m R b a m K K t b Rearragig thi firt order equatio ito tadard firt order trafer fuctio form yield: ( ) K K R J K b t a B G ( ) / p eed V ( ) R b K K R b K K i a t b a t b A implified firt order form of PMDC motor trafer fuctio i term of output agle ca i alo be obtaied by ubtitutig (L a =), ad give by: ( ) K / K R J K t t a M V a J R m a 3 ( ) ( ) ( ) ( ) ( ) i L J R J b L R b K K R J R b K K a m a m m a a m t b a m a m t b K K t b b m by ubtitutig,(l a =), motor equatio ca be implified to firt ad ecod order ytem relatig iput voltage ad output agle, a well a equatio relatig iput voltage ad output peed, to be give by: ( ) K ( ) K t t G ( ), G ( ) a g le p e e d V ( ) ( R J K K ) V ( ) ( R J K K ) i a m t b i a m t b The trafer fuctio relatig iput voltage ad output agle ca be more implified to have the ext form: G a g le ( ) ( ) V ( ) i J R m a * K t * K b The geometry of the mechaical part determie the momet of iertia, the total equivalet iertia, J equiv ad total equivalet dampig, b equiv at the armature of the motor with gear attache, are give by Eq.(6), The iertia of the gear, eor, beam ad ball have to be icluded i the calculatio of total equivalet iertia ad dampig, for implicity are give by Eq.(7): N N b b b J J J eq u iv m L o ad eq u iv m L o ad N N ( ) / ( *M * R ) / 5 (7) lo ad b eam b all lo ad J J J J m l K t K t (4) (6) (5) 5.4. DC motor ub-ytem imulatio ad aalyi. The derived mathematical model of DC motor, implified ad more actual, ca be repreeted i Simulik, a how i Fig, with applied iput voltage ad output agular poitio, agular peed, curret ad torque; where i Figure (a) i how Simulik model of more accurate DC motor mathematical model I 5-35 September 3 Page 67

15 Mechatroic Deig Of Ball Ad Beam ad the correpodig fuctio block widow i how i ( b). The Simulik model of implified mathematical model of DC motor i how Fig (c) ad the correpodig fuctio block widow i how i Figure (d). Thee model ca be ued to aalyze ad evaluate DC motor ub-ytem performace, a well a, deiged cotrol ytem. Defiig i MATLAB DC motor parameter ued, ad ruig the Simulik model will retur agular poitio,/time, agular peed/time, curret/time ad torque/time repoe curve how i Figure (e). The followig code ca be ued to plot repoe curve:load Motor,load Motor,load Motor,load Motor3,ubplot(,,),plot( motor_agle ), xlabel(' '),ylabel(' Poitio \theta'),title(' Shaft agular poitio VS time '), grid,ubplot(,,),plot( motor_peed ), xlabel(' '),ylabel(' Speed \omega '),title(' Shaft agular Speed VS time '), grid, ubplot (,,3),plot( motor_torque ), xlabel(' '),ylabel(' Torque N.m'), title(' Torque VS time '), grid, ubplot(,,4),plot( motor_curret ), xlabel(' '),ylabel(' Curret Amp'),title(' Curret VS time '), grid, Rreitace, Ra Ra Curret Step, Vi= Kb Kb Torque Torque load um Sum /La Iductace, /La /Jm Iertia, /Jm bm Dampig, b d/dt Output peed d/dt(theta) Itegrator curret,i Itegrator Kt -K- Output agle d/dt(theta) Itegrator, Motor.mat To File Curret Torque. Motor.mat To File Agular peed Motor.mat To File alge Motor3.mat Step Iput igal Curret Torque Agular peed Agular Poitio MOTOR Subytem Motor.mat To File Torque. Motor.mat To File Agular peed Motor.mat To File Theta Motor3.mat Figure ( a) DC motor imulik model To File3 alge Figure ( b) fuctio block model Curret Torque Curret Step, Vi= Sum. /Ra Iductace, /La Curret Motor5.mat To File Kt Kt. Torque TL Sum.4 Motor6.mat To File /Jm Iertia, Itegrator.. Itegrator /Jm Motor7.mat bm Dampig,bm Agular poitio To File3 Step, Vi= Curret Torque Iput igal Agular peed Agular Poitio Motor Subytem Motor.mat Torque. Motor.mat Agular peed Motor.mat Agular.. poitio Kt Tload Motor3.mat To File7 Figure ( c) Simplified DC motor Simulik model Figure ( d) fuctio block model I 5-35 September 3 Page 68

16 Torque N.m Curret Amp Torque N.m Curret Amp Mechatroic Deig Of Ball Ad Beam Shaft agular poitio VS time. Shaft agular Speed VS time Shaft agular poitio VS time.5 Shaft agular Speed VS time.5 Poitio.5.5 Speed.5..5 Poitio..5 Speed..5 5 Torque VS time.3 5 Curret VS time Torque VS time Curret VS time Figure ( e) DC motor poitio,/time, agular peed/time, curret/time ad torque/time, both for tep ad radom repoe curve 5.4. Overall Ball ad beam ope loop ytem imulatio ad aalyi A how i Figure (a), buildig overall Ball ad beam ope loop ytem Simulik model, ca be accomplihed by coectig,i erie, both ope loop trafer fuctio of DC motor ad ball ad beam ubytem, give by Eq.(8).all three propoed DC motor Simulik model, ca be merged i oe fuctio block model how i Figure (b), that ca be coected to ball ad beam ubytem model, ad ay DC motor model ca be activated by uig Simulik maual witche, baed o all thi, the ope loop Simulik model of ball ad beam ytem will have the form how i Figure (c). Depedig o required accuracy, elected cotrol ytem type, algorithm ad deig requiremet, the iput igal ca be coected to ay of three DC motor model. Thi propoed ope loop ball ad beam model give deiger readig (umerical ad graphical) about both ub-ytem' dyamic, ball ad beam a well a, DC motor ubytem. By defiig i MATLAB, both DC motor ad ball ad ytem parameter, ad ruig the Simulik model will retur the followig repoe curve of DC motor : agular poitio,/time, agular peed/time, curret/time ad torque/time repoe curve, ad the followig repoe curve of ball ytem: ball poitio,/time, ball peed/time, ball acceleratio/time repoe, applyig radom iput igal ( to mimic ball poitio ad correpodig haft agle repoe) will reult i all thee repoe curve are how i Figure (d) (8) K / X ( ) g t G ( ) G ( ) o p e L R ( J b ) L R T K K ( ) a a e q u iv e q u iv a a b t R 5 r Derivative du/dt.,.,,., Step r/l -K- Sigal Geerator Load torque, Load torque Iput igal Curret Torque Agular peed Agular Poitio MOTOR Subytem Curret Motor.mat To File Torque. Motor.mat To File Agular peed Motor.mat To File Theta Motor3.mat alge Motor haf t agle Theta Ball ad Beam Subytem Ball poitio Ball peed Ball Acceleratio Out5 Ball poitio Ball peed Ball Acceleratio Motor haft agle Figure (a) Ope loop imulik model of ball ad beam ytem. Ball poitio, Ball peed Beam agle Ball poitio, ball_beam 3.34 Ball Speed ball_beam.964 Ball Acceleratiom Ball Acceleratio.. ball_beam haft agle ball_beam3.3 I 5-35 September 3 Page 69

17 Mechatroic Deig Of Ball Ad Beam Rreitace, Ra Ra Torque.4 VOLTAGE Iput Actual Kb Kb Sum /La Iductace, /La d/dt curret,i Itegrator5 Kt3 Kt. 4 Curret 3 Torque Load Torque Torque um Output peed d/dt(theta) /Jm Iertia, /Jm Itegrator4. Agular peed bm Dampig, b Itegrator,.3 Agular Poitio.5 3 VOLTAGE Iput SIMPLIFIED Sum. Curret /Ra Iductace, /La Kt Kt. Torque Sum.4 /Jm Iertia, Itegrator.. Itegrator /Jm bm Dampig,bm TL Tload Kt Kt 4 VOLTAGE Iput SIMPLIFIED DC motor Figure (b) three actuator model ubytem Ball poitio Agular peed Ball poitio, du/dt Derivative.,.,,., r/l -K- Load Torque VOLTAGE Iput Actual Agular peed Agular Poitio Motor.mat Agular.. poitio Motor.mat To File7 Torque Agular peed3 Agle, Theta Motor haft agle Theta Ball poitio Ball peed Ball Acceleratio Out5 Ball poitio Ball poitio, Ball peed ball_beam. Ball Speed ball_beam. Ball peed Ball Acceleratiom Ball Acceleratio VOLTAGE Iput SIMPLIFIED Torque Ball Acceleratio ball_beam Step Motor.mat Motor haft agle. Sigal Geerator, VOLTAGE Iput SIMPLIFIED Curret Curret 3 Ball ad Beam Subytem Ball poitio Beam agle haft agle ball_beam3 Subytem Motor3.mat.6.3 Figure (c) Ope loop Simulik model of ball ad beam ytem Ball poitio, I 5-35 September 3 Page 7

18 Torque N.m Mechatroic Deig Of Ball Ad Beam Poitio Curret Amp Acceleratio, a Poitio x mm Shaft agular poitio VS time Torque VS time.. -. Speed Shaft agular Speed VS time Curret VS time Beam agle VS time Ball Speed VS time 3 Speed 4 Ball poitio VS time - 5 Ball Accel. VS time Figure (d) agular poitio,/time, agular peed/time, curret/time ad torque/time repoe curve of ued DC motor ad ball poitio,/time, ball poitio,/time, ball peed/time, ball acceleratio/time repoe, both for radom iput. 5.5 Cotrol ytem electio, modelig, deig, ad imulatio Due to fact that it i double itegrator (two pole at the origi) ytem ad i ope-loop utable, i order to tabilize the ytem a pecial cotrol ytem i to be deiged. Ball ad beam ytem, provide two output meauremet ad correpodigly two eor oe to meaure ball tralatioal poitio, x, ad other to meaure motor haft agular poitio, θ. Correpodigly, the cotrol ytem deig require two cotrol loop deig, ier for motor agle cotrol, to make motor follow the iput agle ad other outer loop to cotrol the ball poitio o the beam, to make ball poitio follow the referece iput poitio. Thee cotroller ca be deiged a oe or eparately, maily if the motor ytem i required to be fater tha the beam cotroller Proportioal Derivative, PD- cotroller: The trafer fuctio of PD-cotroller i give by Eq.(9), Rearragig, we have the form give by Eq.(): G PD () =K P +K D (9) G ( ) K K K ( K / K ) K ( Z ) () P D P D D P D D P D The PD-cotroller i equivalet to the additio of a imple zero at Z PD =K P /K D. The additio of zero ha the effect of improvig the traiet repoe Sytem deig with prefilter: Prefilter i defied a a trafer fuctio G p () that filter the iput igal R() prior to calculatig the error igal. Addig a cotrol ytem to plat, will reult i the additio of pole ad/or zero, that will effect the repoe, maily the added zero, will igificatly iverely effect the repoe ad hould be cacelled by prefilter, therefore the required prefilter trafer fuctio to cacel the zero i give by (). G Pefilter () =Z o /(+ Z o ) 5.5. Ier Motor PD cotroller loop with deadbeat repoe Ofte, the goal for a cotrol ytem i to achieve a fat repoe to a tep commad with miimal overhoot. Deadbeat repoe mea the repoe that proceed rapidly to the deired level ad hold at that level with miimal overhoot. The characteritic of deadbeat repoe iclude; Zero teady tate error, Fat repoe, (hort rie time ad ettlig time), percet overhoot greater or equal to.% ad le or equal % ad miimal uderhoot, le tha ±% error bad. I cotroller deig for deadbeat repoe approach, the cotroller' coefficiet,(cotroller gai, pole ad zero), deped o the phyical parameter of the ytem. To determie the optimal coefficiet, that yield the optimal deadbeat repoe, the ytem' overall equivalet cloed-loop trafer fuctio, T() i compared with tadard, of correpodig order, ad ormalized trafer fuctio, (particularly the characteritic equatio are compared).thee coefficiet were elected to achieve deadbeat repoe ad miimize ettlig time T ad rie time, T R to % of the deired commad. The deired Stadard ecod order cloed-loop trafer fuctio for achievig deired deadbeat repoe pecificatio i give by Eq.(): () I 5-35 September 3 Page 7

19 Mechatroic Deig Of Ball Ad Beam T Baed o [8], the followig derivatio ad deig for PD cotroller with deired deadbeat repoe pecificatio i term of deired output agular poitio. Coiderig that ytem dyamic ad diturbace torque deped o applicatio hape ad dimeio (robot arm, coveyer etc), the mechaical DC motor part, will have the form: K t *i a = T α + T ω + T load + T f The coulomb frictio ca be foud at teady tate, to be: ( K t *i a - b*ω =T f. ). I the followig calculatio the diturbace torque, T, i all torque icludig coulomb frictio, ad give by: T=T load +T f. Applyig PD cotroller with deadbeat repoe deig for output deired output agular poitio, the ope-loop trafer fuctio of the PMDC, i give by: G ( ) K t / o p e L R ( J b ) a a m m L R a a T K K b t Maipulatig for forward ad cloed loop trafer fuctio, icludig diturbace torque, eor ad gear ratio, give: G G G G o p e o p e c lo ed c lo ed K K K t t ( ) L R ( J b ) L R T K K L J ( R J b L ) p o t ( R b L T ) ( R T K K ) 3 a a m m a a b t a m a m m a a m a a b t K K K K K K K K p o t t p o t t D p o t t P ( ) T J ( T ) R ( T ( R T K K ) T L J ( R J b L ) ( R b L T ) ( R T K K ) 3 3 ( ) ( ) m a a a b t a m a m m a a m a a b t 3 K K K K K K p o t t D p o t t P L J ( R J b L ) ( R b L T K K K ) R T K K K K K a m a m m a a m a p o t t D a b t p o t t P K K K K K K p o t t D p o t t P L J ( R J b L ) ( R b L T K K K ) R T K K K K K 3 a m m a a m a p o t t D L J L J L J a m a b t p o t t P a m a m a m Comparig with tadard ormalized third order ytem ad maipulatig, for K D ad K P, give by Eq.() () by: T ( ) ( R b a L T a K K K ) L J a ( R b a L T ) a m p o t t D m m K D L J K K a m p o t t R T a K K K K K 3 L J a R T a K K b t p o t t P m b t ( ) 3 K P L J K K a m p o t t For deig coideratio, ad baed o required deig accuracy we ca ue the implified ecod order model, aumig La =, ad maipulatig for cloed loop with PD cotroller, give: ( ) K R J t a a G ( ) G ( ) a g le V K K J R m a a g le 3 ( ) i L J ( R J b L ) ( R b K K ) a m a m m a a m t b t b b m ( ) K t ( ) K K K K t P t D G ( ) G ( ) a g le V V R J R b K K G c lo ed f o rw ard ( ) ( R J ) ( R b a K K ) ( ) ( ) ( ) a m m t b i i a m a m t b ( ) ( ) K K K K t P t D V ( ) ( R J ) ( R b K K K K K ) K K K i a e q u iv a e q u iv t b p o t t D p o t t P Comparig with tadard ormalized ecod order ytem ad maipulatig, for K D ad K P, give : G c lo ed ( ) K K K K t P t D ( R J ) a e q u iv R b K K K K K K K K R J ( R J ) a e q u iv a e q u iv a e q u iv t b p o t t D p o t t P R b K K K K K R J ( R b K K ) a e q u iv t b p o t t D a e q u iv a e q u iv t b K D R J K K a e q u iv p o t t K t I 5-35 September 3 Page 7

20 Mechatroic Deig Of Ball Ad Beam K K K R J p o t t P a e q u iv K P ( R J ) K K a e q u iv p o t t K P Z PD K D Referrig to [8] The cotroller gai K P ad K D deped o the phyical parameter of the actuator drive, to determie gai that yield optimal deadbeat repoe, the overall cloed loop trafer fuctio T() i compared with tadard ecod order trafer fuctio give by (), ad kowig that parameter α ad ω are kow coefficiet of ytem with deadbeat repoe give by [3], α =.8 ad ω T = 4.8, T = ad give the followig: ω T = 4.8, ω = 4.8/=.4. I [8] to implify ad accelerate Mechatroic deig proce, a ew MATLAB built-i fuctio with pecific purpoe, which i deig ad verificatio for deired deadbeat repoe pecificatio, particularly for deired ettlig time. The preeted built-i fuctio i amed deadbeat(a), the iput argumet a, ca have the value of or. The iput argumet for cotrollig electric motor ytem output agular diplacemet with PD-cotroller deig for deadbeat repoe with deired ettlig time, thi ew built i fuctio determie the coefficiet (gai pole ad zero) that yield the optimal deadbeat repoe for deired output peed or agle cotrol, a well a performace pecificatio i term of Mp, E, T, fial output value Ier Motor PD Poitio cotrol loop with poitio ad rate feedback Coiderig the output poitio cotrol, applyig poitio ad rate feedback,( ee Figure 3(b)), to have the motor follow the deired output agle, the iput voltage to DC motor will be give by Eq.(3). Rearragig, we have Eq.(4): V K K V K K (3) i V P i o D i P i o D o K K K i P i P o D o (4) V K P i K K P D i o Fidig the overall equivalet cloed loop trafer fuctio, ad comparig with correpodig tadard trafer fuctio ( ecod or third order), ad by compario, we ca fid the gai K P ad K D depedig o motor parameter ad deired dampig ratio, ζ ad ettlig time, T. Stadard ecod order ytem, i term of ζ ad ω, i give by Eq.(5): T ( ) out d eired Aumig the cotrai for thi ytem are to be, deired dampig ratio of.77, or overhoot ad Peak time of T P =. ecod, we fid udamped atural frequecy ω, ad correpodigly K P, K D l (% OS / ), l (% OS / ) T P From, the DC motor trafer fuctio, depedig o deired accuracy, ad correpodigly applied mathematical model, ad by ubtitutig the value of V i, we have: J R m a V * K ( ) i t * K b V i J R m a K K K ( ) * K K ( ) K K ( ) P i P D o t P i P D o * K b J R m a ( ) * K K K ( ) K ( ) t P D o P i * K b J R m a ( ) * K K K K ( ) t P D P i * K b ( ) ( ) i J R m a K K K K b t D P By compario with tadard ecod order ytem, we fid PD cotroller gai K P, K D Ier Motor PD Poitio cotrol with oly poitio feedback I 5-35 September 3 Page 73 (5)

21 Mechatroic Deig Of Ball Ad Beam Coiderig the output poitio cotrol, applyig oly poitio feedback with proportioal ad derivative term, a how i Fig 3(a). The iput voltage to DC motor will give by V K P i o K i D o To fid the gai K P ad K D, apply the ame procedure, for deired dampig ratio, ζ ad ettlig time, T Ier Motor PID Poitio cotrol loop with oly poitio feedback Coiderig the output poitio cotrol, applyig oly poitio feedback with proportioal, derivative Speed ad itegral term, a how i Fig 3(b). The iput voltage to motor will give by: Agle Deired Vi Omega Theta Step Agle um() K I Kp i o K K de(), V i P i o D o, Motor.,. Itegrator Shaft agular The gai K P, K D ad K I, are foud by applyig the ame procedure, for deired dampig ratio, ζ poitio ad ettlig Sigal time, T. Geerator Kd du/dt Step Sigal Geerator, Deired Agle Kp, Vi Kd um() de() Motor.,. du/dt Speed Omega Itegrator Agle Theta Agle Shaft agular poitio Figure 3(a) Motor PD Poitio cotrol with poitio ad/or rate feedback ( to witch ue maual witch) Agle Step Sigal Geerator, Deired Agle Error Actual Agle,. du/dt '. Ki.. Kp Kd ',' Vi um() de() Motor actual output Agle Speed Omega Itegrator Agle Theta Figure 3 (b) Motor PID Poitio cotrol with poitio feedback Shaft agular poitio Outer Ball ad beam PD, PID Poitio cotrol with oly poitio feedback The ame procedure ca be applied, to calculate cotroller gai, for cotrollig beam agle, whe applyig PD or PID, thi i how i Fig 3(c) Outer ball ad beam PD Poitio cotrol loop with poitio ad rate feedback The iput beam agle, α, to ball ad beam ytem, applyig PD cotroller, will be give by: dx o K P x x i o K D dt Takig Laplace traform, K X ( ) X ( ) K ( ) P i o D Subtitutig α, ball ad beam trafer fuctio, a how below, we have X ( ) (5 / 7 ) g 7 ( ) X ( ) ( ) X ( ) ( ) (5 )(9.8 ) 7 X ( ) K X ( ) X ( ) K ( ) P i o D 7 X ( ) 7 K P X ( ) 7K 7K i D P Thi i the overall equivalet cloed loop trafer fuctio of ball ad beam, comparig it with tadard ecod order trafer fuctio, ad by compario, we ca fid the gai K P ad K D depedig o plat parameter ad deired dampig ratio, ζ ad ettlig time, T. a how ext: X ( ) 7 K P T ( ) X ( ) 7 K 7 K i D P I 5-35 September 3 Page 74

22 Step Sigal Geerator, Deired Agle Error Actual Agle K P K D 7 7 actual output Agle Mechatroic Deig Of Ball Ad Beam Aumig the cotrai for thi ytem are to be, deired dampig ratio of.77, or overhoot ad Peak time of T P = ecod, we fid udamped atural frequecy ω, ad correpodigly K P, K D l (% OS / ), l (% OS / ) Solvig we have: ω = 4.44, K.8 9 K ,. du/dt '. P Ki.. Kp Kd ',' Vi um() de() Motor D Speed Omega Itegrator T Agle Theta P Shaft agular poitio Step3 Sigal Geerator3,3. du/dt '., Kp k,' Kd,',' Actual Ball poitio Error,. du/dt '. Ki.. Kp,' Kd ' Vi Speed Omega um() de() Motor. Actual haft output Agle Shaft agular poitio (Theta) um() de() Itegrator Ball ad beam Actual ball poitio Figure 3 (c) Ball ad beam poitio cotrol with poitio feedback Seor ub-ytem modelig There i a relatiohip betwee the beam agle, α, ad DC motor output poitio agle, θ, where the ditace traveled by the beam at radiu equal to L, i equal to ditace traveled by gear at radiu, r, that i: r L ( r / L ), Where: The beam agle ( r / L ), ad the motor agle (L /r).the trafer fuctio betwee the ball poitio ad rotor agle ca be obtaied baed o baic trafer fuctio give by Eq.() ad o relatio give by (), ad give a how ext: Y ( ) b g Y ( ) r g r * g V ( ) V ( ) L L i i The poitio of the ball o the beam ca be meaured uig pecial eor, e.g. liear Potetiometer, i [9] a liear reitive wire i ued a a poitio eor ad the coductive ball movig alog it actig a a voltage divider Liear reitive wire ha a relative large reitace of about.5 ohm/m. A erie reitor i added to the circuit i order to divide the voltage to avoid uwated heat. Accordig to the poitio (X) of the ball, the voltage will liearly vary from. V to.74 V. Schematic of the reitive wire circuit i how i Figure 4. Figure 4 Schematic of the liear reitive wire circuit "poitio eor [9] For voltage rage of volt; K / 4.5 e ( V oltage chage) K. 6 e ( P oitio chage) L Electroic ad iterface imulatio ad tetig: imulatio i Proteu; To tet ad evaluate the electio ad itegratio of circuit deig, programmig ad iterface compoet, imulatio uig ISIS- Profeioal Proteu i ued. The cotrol program writte i C laguage, with the help of MikroC program i coverted to Hex. File ad dowloaded o the imulated PIC-microcotroller ad circuit, the imulatio i how i Figure 8, after tetig, evaluatig ad optimizig variou apect, the fial imulatio reult how the correcte of writte program, iterface, ad microcotroller, all thee ca be ued to build the phyical circuit. 5.7 Ball ad beam overall ytem imulatio ad aalyi. Baed o the pecificatio of requiremet ad deig, the ubytem model ad the whole ytem model, are to be teted ad aalyzed,. If the pecificatio are ot atified,, modificatio ca be made, if the pecificatio are atified the model ca be Optimized. The primary baic Simulik model, with PID cotrol algorithm for both loop i how i Figure 5, thee two cotroller will be replaced with correpodig propoed cotrol algorithm to achieve optimal I 5-35 September 3 Page 75

23 Acceleratio, a Acceleratio, a Poitio x mm Poitio x mm Mechatroic Deig Of Ball Ad Beam repoe. Switchig model, uig maual witch, to oliear PMDC motor model, applyig two Simulik built-i PD cotroller for ier ad outer loop, ubjectig ytem to tep iput, ad applyig Simulik block tuig; will reult i ball poitio,/time, peed/time, ball acceleratio/time,beam agle/time repoe curve how i Figure 6(a)), whe ytem i ubjected to radom iput, will reult i ball ad beam repoe curve how i Figure 6(b). Switchig model to implified PMDC motor model, ad ubjectig the ytem to tep iput, will reult i repoe curve how i Figure 6(c)), comparig repoe curve, how that the overhoot uig implified motor model i le, a well a ettig time, but geerally ytem take ecod repoe to reach deired poitio which i large for uch ytem.applyig Outer Ball ad beam PD Poitio cotrol with oly poitio feedback, Figure 7(a) Switchig model to implified PMDC motor model, defiig parameter, ruig the model, will reult i repoe ball poitio,/time, ball peed/time, ball acceleratio/time ad beam agle/time repoe curve repoe curve how i 7(b). Refiig the PD Poitio cotrol deig, for both loop, will reult i more acceptable repoe curve repoe curve how i 7(c), the ytem reache deired output at.3 ecod. Curret Curret Motor.mat Curret Torque Motor6.mat Torque.6 Motor4.mat To File Torque. Motor.mat To File -.58 Ball Iput igal Agular peed Agular 3peed -. Curret Agular peed Agular peed 3 Motor theta Bal Agular Poitio MOTOR Subytem SIMPLIFIED Agular peed3 Motor5.mat Agular.. poitio.6678 Agle, Motor7.mat To File7 Torque Iput igal Agular peed Agular Poitio Motor.mat To File Motor3.mat alge.787 Agle, Ball poitio Ball poitio,. Ball poitio, ball_beam Ball poitio Theta 6 Ball Acce MOTOR Subytem Ball peed Ball Acceleratio Ball peed Ball Speed ball_beam Step PD() PD() Speed eor, PID Cotroller PID Cotroller -K-.7.5 Theta Motor haft agle Theta Out5 Ball poitio Ball Acceleratiom Ball Acceleratio Sigal Geerator Alpha r/l.4 Ball ad Beam Subytem Ball peed Ball Acceleratio Motor haft agle Beam agle ball_beam haft agle ball_beam3 Ball poitio.3 Figure 5 The primary Simulik model, with PID cotroller Beam agle VS time Ball Speed VS time 4 Speed Ball poitio VS time 5 5 Ball Accel. VS time Speed Beam agle VS time Ball Speed VS time Ball poitio VS time Ball Accel. VS time Figure 6 (a) Figure 6 (b) Figure 7Ball ad beam ytem tep ad radom iput repoe, uig actual PMDC model. I 5-35 September 3 Page 76

24 Acceleratio, a Poitio x mm Mechatroic Deig Of Ball Ad Beam.6 Beam agle VS time 5 Ball poitio VS time Ball Speed VS time Ball Accel. VS time 4 Speed Figure 6 (c)) Ball ad beam ytem tep repoe, uig implified PMDC model Ball poitio Agular peed,., r/l Derivative du/dt Load Torque Agular peed Motor.mat Agular.. poitio Agular peed3.,., -K- VOLTAGE Iput Actual Agular Poitio Motor.mat Agle, To File7 Torque VOLTAGE Iput SIMPLIFIED Torque 7.48 Motor.mat Ball poitio, Curret 3 Ball poitio, ball_beam VOLTAGE Iput SIMPLIFIED Curret Ball poitio..6 Subytem Motor3.mat.6 Ball peed Ball Acceleratio Ball peed Ball Speed ball_beam Step 6. Theta Motor haft agle Theta Out5 Ball poitio Sigal Geerator, Alpha Derivative du/dt Speed eor (r/l) Gai Ball ad Beam Subytem Ball peed Ball Acceleratio Motor haft agle Ball poitio Ball Acceleratiom Ball Acceleratio ball_beam haft agle Beam agle ball_beam3 Ball poitio.3 Derivative3 Gai Derivative.5349 du/dt.37 du/dt -K- Gai4 Ball poitio,.3968 Gai Figure 6 (a) Ball ad beam PD Poitio cotrol with oly poitio feedback I 5-35 September 3 Page 77

25 Acceleratio, a Acceleratio, a Poitio x mm Poitio x mm Mechatroic Deig Of Ball Ad Beam Speed 3 Beam agle VS time - 5 Ball Speed VS time Ball poitio VS time 5 Ball Accel. VS time 3-5 Figure 7(b) Ball ad beam ytem tep repoe Speed.3.. Beam agle VS time Ball Speed VS time Ball poitio VS time 4 6 Ball Accel. VS time Figure 7 (c) Ball ad beam ytem tep repoe. VI. PROTOTYPING, TESTING, EVALUATION AND OPTIMIZATION. There i o igle model which ca ever flawlely reproduce reality, there will alway be error called a umodeled error betwee behavior of a product model ad the actual product. I order to take ito accout the umodeled error ad ehace preciio, performace i the deig proce, the Mechatroic deig approach iclude prototypig phae. Prototypig developmet may be carried out i the followig two form; Virtual Prototype ad Phyical Prototype, i thi paper, oly virtual prototype i built, Virtual Prototype i computer model of a product preeted i a virtual eviromet with, ideally, all iformatio ad propertie icluded, to viualize text, aalyze ad evaluate machie movemet ad logical operatio. Differet ource itroduce differet virtual prototype for evaluatig the overall ytem electio, itegratio ad deig, i [], itroduced model how i Figure 8(a), the oftware ued i MATLAB/Simulik. D N437A C pf X 8Mhz C pf Eablig ad PWM Eablig ad PWM U? 3 33 OSC/CLKIN RB/INT 4 34 OSC/CLKOUT RB 35 RB 36 RA/AN RB3/PGM 3 37 RA/AN RB RA/AN/VREF-/CVREF RB RA3/AN3/VREF+ RB6/PGC 6 4 RA4/TCKI/COUT RB7/PGD 7 RA5/AN4/SS/COUT 5 RC/TOSO/TCKI 8 6 RE/AN5/RD RC/TOSI/CCP 9 7 RE/AN6/WR RC/CCP 8 RE/AN7/CS RC3/SCK/SCL 3 RC4/SDI/SDA 4 MCLR/Vpp/THV RC5/SDO 5 RC6/TX/CK 6 RC7/RX/DT 9 RD/PSP RD/PSP RD/PSP RD3/PSP3 7 RD4/PSP4 8 RD5/PSP5 B 9 RD6/PSP6 5V 3 RD7/PSP7 PIC6F877A U MODFILE=DCMOTOR 3 IN VSS VS OUT 7 6 IN OUT EN 9 EN IN3 OUT3 5 4 IN4 GND GND OUT4 L93D B V Figure 8(a) overall ytem virtual prototype[] Figure 8 Electroic imulatio i Proteu oftware VII. CONCLUSION & FUTURE WORK Thi paper propoe the coceptio ad developmet of ball ad beam ytem baed o Mechatroic deig approach, where realizatio of deig i achieved applyig imultaeou, parallel, deig approach ad coideratio. A complete ytem' compoet' electio, deig ad itegratio, a well a, modelig, imulatio ad aalyi are preeted. The imulatio reult for PD cotroller deig are how. Depedig o deired accuracy, differet mathematical ad Simulik model form of the DC motor ad plat have bee developed. The propoed Mechatroic deig ad model are iteded for reearch purpoe, a well a, for applicatio i educatioal proce. The propoed model were created ad verified uig MATLAB Simulik, Proteu oftware. Thi paper i part (I) of author' reearch tudy o Mechatroic deig of ball ad beam ytem, a future work of thi reearch, author ited to built ad tet phyical prototype of thi deig, ad to propoe a geeral MATLAB/Simulik model with it fuctio block parameter widow, that ca be ued to cotrol ball ad beam ytem uig differet cotrol trategie, a well a to implify ad accelerate the electio, deig ad verificatio procee I 5-35 September 3 Page 78