UNIVERSAL COMMAND GENERATOR FOR ROBOTICS AND CNC MACHINERY

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1 UNIVERSAL COMMAND GENERATOR FOR ROBOTICS AND CNC MACHINERY A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY ARDA AKINCI IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING MAY 9

2 Approval of the thei: UNIVERSAL COMMAND GENERATOR FOR ROBOTICS AND CNC MACHINERY ubmitted by ARDA AKINCI in partial fulfillment of the requirement for the degree of Mater of Siene in Mehanial Engineering Department, Middle Eat Tehnial Univerity by, Prof. Dr. Canan Özgen Dean, Graduate Shool of Natural and Applied Siene Prof. Dr. Süha Oral Head of Department, Mehanial Engineering Ait. Prof. Dr. Melik Dölen Supervior, Mehanial Engineering Dept., METU Ait. Prof. Dr. A. Buğra Koku Co-Supervior, Mehanial Engineering Dept., METU Examining Committee Member: Prof. Dr. Ere Söylemez Mehanial Engineering Dept., METU Ait. Prof. Dr. Melik Dölen Mehanial Engineering Dept., METU Prof. Dr. M. Kemal Özgören Mehanial Engineering Dept., METU Ait. Prof. Dr. A. Buğra Koku Mehanial Engineering Dept., METU Ao. Prof. Dr. Veyel Gazi Aeropae Engineering Dept., TOBB-ETU Date:

3 I hereby delare that all information in thi doument ha been obtained and preented in aordane with aademi rule and ethial ondut. I alo delare that, a require by thee rule and ondut, I have fully ited and referened all material and reult that are not original to thi work. Name, Lat Name : Arda, Akını Signature : iii

4 ABSTRACT UNIVERSAL COMMAND GENERATOR FOR ROBOTICS AND CNC MACHINERY Akını, Arda M.S., Department of Mehanial Engineering Supervior: Ait. Prof. Dr. Melik Dölen Co-Supervior: Ait. Prof. Dr. A. Buğra Koku May 9, 67 page In thi tudy a univeral ommand generator ha been deigned for roboti and CNC mahinery. Enoding tehnique ha been utilized in order to repreent the ommand and their effiienie have been diued. The developed algorithm generate the trajetory of the end-effetor with linear and irular interpolation in an offline fahion, the orreponding joint tate and their error envelope are omputed with the utilization of a numerial invere kinemati olver with a predefined preiion. Finally, the ommand enoder employ the reulting data and produe the repreentation of poition in joint pae with uing propoed enoding tehnique depending on the error tolerane for eah joint. The enoding method onidered in thi thei are: Lole data ompreion via higher order finite differene, Huffman Coding and Arithmeti Coding tehnique, Polynomial Fitting method with Chebyhev, Legendre and Berntein Polynomial and finally Fourier and Wavelet Tranformation. The algorithm i imulated for Puma 56 and Stanford Manipulator for a trajetory in order to evaluate the performane of the above mentioned tehnique (i.e. approximation error, memory requirement, number of iv

5 ommand generated). Aording to the ae tudie, Chebyhev Polynomial ha been determined to be the mot uitable tehnique for ommand generation. Propoed method have been implemented in MATLAB environment due to it veratile toolboxe. With thi reearh the way to develop an enoding/deoding tandard for an advaned ommand generator heme for omputer numerially ontrolled (CNC) mahine in the near future ha been paved. Keyword: Univeral Command Generator, Invere Kinemati Solution, Data Compreion Tehnique, Kinemati Modeling, Enoding Method. v

6 ÖZ ROBOTİK UYGULAMALAR VE CNC TAKIM TEZGAHLARI İÇİN EVRENSEL KOMUT ÜRETECİ Akını, Arda Yükek Lian. Makina Mühendiliği Bölümü Tez Yönetiii: Yard. Doç. Dr. Melik Dölen Ortak Tez Yönetiii: Yard. Doç. Dr. A. Buğra Koku Mayı 9, 67 ayfa Bu çalışmada, çeşitli robotik uygulamalar ve CNC Takım tezgâhları için evrenel bir komut üretei taarımı yapılmıştır. Daha onra bu komutların en verimli şekilde ifade edilmei için çeşitli kodlama metodları kullanılmıştır. Bu yordamın kabiliyetleri, önelikle uç işlemi için verilmiş NC kodunu okuyarak, doğrual ve daireel enterpolayon kullanarak verilmiş örnekleme zamanına bağlı uç işlemi konum komutlarının üretilmeidir. Ardından yinelemeli nümerik metodu ile verilmiş bir konumlama hata toleranı kullanılarak ter kinematik çözümü yapılarak eklem konumlarının ve bu eklemlerin hata bantlarının oluşturulmaı. Son olarak komut kodlayııı araılığı ile bu konumların heaplanmış hata bandı içinde kalaak şekilde kodlanıp en uygun şekilde depolanmaıdır. Bu çalışmada göz önüne alınan metotlar, yükek dereeden onlu farkların heaplanmaı, ve bu farkların Huffman ve Aritmetik kodlama yordamları ile ıkıştırılıp hataız bir şekilde aklanmaı, Chebyhev, Legendre ve Berntein Polinomları kullanarak verinin polinomlara uyarlanmaı ve on olarak Fourier ve Dalgaık dönüşümleri ile frekan-zaman tanım kümeinde tanımlanmaıdır. vi

7 Geliştirilen yordam, kodlama metotlarının verimliliğini ( yaklaştırma hataı, depolamak için gerekli yer miktarı ve kullanılan komut miktarı ininden) karşılaştırmak için, Puma 56 ve Stanford Manipülatörleri kullanılarak, belirlenen yörünge üzerinde uygulanmıştır. Sonuçlar göz önünde bulundurulduğu zaman, en az miktarda komut üreterek, en düşük aklama alanına ihtiyaç duymaı ve itenen hata miktarlarının altında bir yaklaşım ağladığından dolayı, Chebyhev polinomları en uygun metot olarak belirlenmiştir. Yordam taarımında, gelişmiş ve çok yönlü uygulama alanlarından dolayı MATLAB programı kullanılmıştır. Bu çalışma ile çeşitli robotik uygulamalar ve CNC Takım tezgâhları için kodlayıı/çözümleyii tandardı oluşturarak, ileri düzeyde komut üretim yordamlarının oluşturulmaının temelleri atılmıştır. Anahtar kelimeler: Evrenel Komut Üretei, Ter Kinematik Çözümleri, Data Sıkıştırma Teknikleri, Kinematik Modelleme. Kodlama Metotları vii

8 To My Family Günel Adnan & Alı viii

9 ACKNOWLEDGEMENTS I would like to expre my inere appreiation to my upervior At. Prof. Dr. Melik Dölen and o-upervior At. Prof. Dr. A. Buğra Koku for their guidane and upport throughout thi thei tudy. I onider myelf privileged to have a mentor with the trong work ethi, never ending knowledge, unyielding patiene and tolerane of At. Prof. Dr. Melik Dölen and At. Prof. Dr. A. Buğra Koku. I would alo like to thank to Dr. Gürel Erarlanoğlu and Tolga M. Güçyetmez for their undertanding and tolerane. I would like to thank to Alı for her upport, enouragement and many other thing. And finally I am deeply in debt to my parent Günel and Adnan Akını for onvining me to tart thi tudy, their never-ending love and piritual upport at ritial and opportune time. ix

10 TABLE OF CONTENTS ABSTRACT... iv ÖZ... vi LIST OF FIGURES... xvi LIST OF TABLES... xx CHAPTERS. INTRODUCTION.... Motivation.... Sope of the Thei....3 Organization LITERATURE SURVEY Interpolator Trajetory Generation Kinemati Modelling and Solution Method Algorithm and Toolboxe Optimization of Manipulator Data Compreion Open Reearh Area... 8 x

11 3. KINEMATIC MODELING OF ARTICULATED MECHANISMS Artiulated Mehanim Bakground Knowledge Homogenou Tranformation Denavit Hartenberg Notation Forward Kinemati Invere Kinemati Multiple Solution Solution Method Numerial Invere Kinemati Singularity Handling Cloure POSITION GENERATION IN JOINT SPACE Poition Generation NC Code Motion Type Frame (Coordinate) Tranformation Developed Algorithm Trajetory Generation... 5 xi

12 4.3. Invere Kinemati Segmentation Cae Study Poition Generation Invere Kinemati Cloure COMMAND GENERATION VIA DIRECT DATA STORAGE Data Storage Enoding and Storage Spae Diret Storage Finite Differene Finite Compoition Tehnique Simulation of Finite Differene Tehnique Finite Differene Method Data Compreion Tehnique Huffman Coding Arithmeti Coding Algorithm Simulation of Compreion Tehnique xii

13 5.8 Cloure POLYNOMIAL BASED COMMAND GENERATION Polynomial Tehnique Chebyhev Polynomial Legendre Polynomial Berntein Polynomial Computation of Polynomial Evaluation of Error Tolerane Band Cae Study Polynomial Baed Command Generation Coeffiient Optimization Implementation of Coeffiient Cae Study Cloure COMMAND GENERATION VIA TRANSFORMATIONS Fourier Analyi Fourier Tranform Invere Fourier Tranform Fourier via Leat Square Method... 3 xiii

14 7..4 Signal Partitioning Invere Fourier Tranform via Look-up Table Wavelet Tranformation Wavelet Analyi Wavelet Familie Continuou Wavelet Tranform Multilevel -D wavelet deompoition Wavelet Reontrution Algorithm Simulation Fourier with Leat Square Method Wavelet tranformation Cloure CASE STUDIES Introdution Manipulator Trajetory and Invere Kinemati Solution Roundabout Signal Simulation... 5 xiv

15 8.4. Puma Stanford Manipulator Cloure CONCLUSIONS AND FUTURE WORK Conluion Future work REFERENCES... 5 APPENDICES A. NC CODE OF ROUNDABOUT SIGN CASE STUDY B. NC CODE OF PUMA 56 FOR ROUNDABOUT SIGN C. LIST OF FINDCENTER D. ANALYTICAL SOLUTION OF PUMA MANIPULATOR... 6 E. ANALYTICAL SOLUTION OF STANFORD MANIPULATOR... 6 F. LISTING OF M FILES... 6 xv

16 LIST OF FIGURES FIGURES Figure. Flow hart of the propoed method Figure. Command deoding heme Figure 3. Different type of manipulator [34]... Figure 3. Standard frame of a manipulator.... Figure 3.3 Bai Rotation and Tranlation Figure 3.4 Denavit Hartenberg Frame aignment [35] Figure 3.5 Hand frame aignment Figure 3.6 Shemati of forward and invere kinemati Figure 3.7 Multiple Solution []... 3 Figure 3.8 Puma 56 Manipulator Figure 3.9 Stanford Manipulator Figure 3. Orientation of end-effetor w.r.t working plane Figure 4. Linear Motion Figure 4. Cirular Motion Figure 4.3 Coordinate tranformation of omplex trajetorie Figure 4.4 Flowhart of trajetory generation Figure 4.5 Flowhart of the parer xvi

17 Figure 4.6 Flowhart of Rapid and Linear Motion Figure 4.7 Flow Chart of Cirular interpolator Figure 4.8 General Flowhart of Invere Kinemati Figure 4.9 Flowhart of preparation phae of invere kinemati Figure 4. Segmentation of the Trajetory of -D manipulator Figure 4. Deired Trajetory Figure 4. Generated Trajetory Figure 4.3 Trajetory in eah axi Figure 4.4 Joint value in degree Figure 4.5 Angular Veloitie of Eah Joint Figure 4.6 Error band of the realulated trajetory Figure 5. Bai data tranfer heme Figure 5. Joint angle and trajetory error with enoder uage Figure 5.3 Alloated Spae v. order of the finite differene Figure 5.4 Planar two link mehanim... 7 Figure 5.5 Trajetory and the joint angle Figure 5.6 Code Mapping in Arithmeti Coding Figure 5.7 Deoding by Huffman Coding Method and approximation error Figure 5.8 Deoding by Arithmeti Coding Method and approximation error.. 8 xvii

18 Figure 6. Firt few Chebyhev Polynomial in domain -<x< Figure 6. Firt few Legendre Polynomial in domain -<x< Figure 6.3 Berntein polynomial up to fourth level Figure 6.4 Error band of the tool tip Figure 6.5 Error Band of joint throughout the trajetory in Figure Figure 6.6 Polynomial pae to time domain Figure 6.7 Error of the joint by polynomial fitting Figure 6.8 Error in trajetorie generated with fitted data Figure 7. Signal Partitioning... 4 Figure 7. Partitioned ignal... 5 Figure 7.3 Reult of Direte Fourier Approximation... 6 Figure 7.4 Reult of Fourier Approximation by Linear Interpolation Figure 7.5 Contituent wavelet of different ale and poition [43]... 9 Figure 7.6 Commonly ued wavelet funtion[7].... Figure 7.7 The effet of the ignal to C.... Figure 7.8 Shifting the wavelet.... Figure 7.9 Saling of the wavelet.... Figure 7. Wavelet deompoition Figure 7. Deompoition of original Signal [43] xviii

19 Figure 7. Joint approximation and trajetory error Figure 7.3 Joint approximation and trajetory error by wavelet tranform.. 8 Figure 8. Trajetory of Puma 56 for roundabout.... Figure 8. Ditributed motion in eah axi on Puma 56 for Roundabout ignal Figure 8.3 Joint value of Puma 56 for Roundabout Signal Figure 8.4 Joint value of Stanford Manipulator for Roundabout Signal Figure 8.5 Maximum error via propoed egmentation tehnique Figure 8.6 Newly added etion Figure 8.7 End-effetor deviation via Chebyhev Polynomial Figure 8.8 End-effetor deviation via Legendre Polynomial Figure 8.9 End-effetor deviation via Berntein Polynomial Figure 8. End-effetor deviation via Fourier Tranform Figure 8. End-effetor deviation via Wavelet Tranform Figure 8. End-effetor deviation via Chebyhev Polynomial Figure 8.3 End-effetor deviation via Legendre Polynomial Figure 8.4 End-effetor deviation via Berntein Polynomial Figure 8.5 End-effetor deviation via Fourier Tranform Figure 8.6 End-effetor deviation via Wavelet Tranform xix

20 LIST OF TABLES TABLES Table 3. Comparion of invere kinemati olution method Table 4. Cirular motion repreentation Table 4. Subroutine Pattern Table 4.3 Peudo ode of the invere kinemati iteration Table 4.4 Denavit Hartenberg Table Table 5. Finite differene heme... 7 Table 5. Number of bit required for eah joint variable Table 5.3 Peudo ode for Huffman Coding [57] Table 5.4 Number of byte required for Huffman Coding of n th order finite differene Table 5.5 Number of byte required for Arithmeti Coding of n th order finite differene Table 6. Number of oeffiient ued Table 6. Required pae alloation of eah joint by polynomial tehnique. Table 7. Peudo ode of Wavelet Tranformation Table 7. Fourier oeffiient found by LSM Table 7.3 Alloated torage pae with reontrut with LSM xx

21 Table 7.4 Wavelet oeffiient and their torage requirement Table 8. Denavit Hartenberg parameter of Stanford Manipulator... Table 8. Repreentation Requirement for eah Method Table 8.3 Alloated Storage Spae with eah method Table 8.4 RMS, Maximum and Minimum Error for eah axi Table 8.5 Maximum and minimum error at joint Table 8.6 Repreentation Requirement for eah Method Table 8.7 Alloated Storage Spae with eah method Table 8.8 RMS, Maximum and Minimum Error for eah axi Table 8.9 Maximum and minimum error at joint xxi

22 CHAPTER INTRODUCTION. Motivation The ue of roboti manipulator (i.e. artiulated mehanim) in the indutry ha aelerated oniderably ine 96. With the advaning tehnology, different type of manipulator have been introdued to variou etor uh a automotive, aviation/aeropae, onumer eletroni, et. Their modularity and the eae of programming make manipulator invaluable tool in bai manufaturing tak inluding welding, painting, grinding/polihing, material tranfer/handling, and aembly. Furthermore, ine roboti manipulator are apable of performing high-preiion poitioning at relatively high peed, the need for highly killed worker ould be dramatially redued, whih in turn lead to a ignifiant inreae in the quality and the quantity of the manufatured good. The appliation of a roboti manipulator to the above mentioned field i relatively eay: One the trajetory of the manipulator (i.e. tool or end-effetor) i planned for a peifi tak at hand, the orreponding angular poition of the atuator at eah joint are alulated uing invere kinemati model of the manipulator in an offline fahion. Hene, the motion ontroller of the mahine i programmed uing thee data (alo known a -a.k.a.- the deired joint poition) to ontrol the angular joint poition aurately. It i ritial to note that indutrial motion ontroller ard (like Delta-Tau PMAC and Galil DMC), whih are ommonly ued to ontrol uh mahinery, employ vetor data table to repreent a omplex trajetory in term of (hort)

23 linear pathe, provided that the tool deviation from the ideal path i within the aeptable limit (a defined by the tak at hand). Thee ard an then perform a linear interpolation between the two oneutive (table) entrie in real-time to produe the relevant referene ignal for the poition ervo-ontrol loop. It i obviou that if the manipulator need to follow a omplex (and relatively long) trajetory, the length of the linear pathe ould be too mall to abolih the effiay of linear interpolation. Furthermore, the number of required entrie for the vetor table might well exeed the available reoure on the ytem. For thoe ae, advaned ontroller unit (like Siemen Sinumerik 84DI or Fanu i erie), whih oftentime have the apability to arry out Spline- or NURBS interpolation, ould be utilized at inreaed hardware ot. However, ine the omputational burden aoiated with uh interpolation heme i extremely high, the ue of uh ytem may no longer be (tehnially/eonomially) feaible when the number of joint (axe) to be ontroller i relatively high (>5). In today tehnology, memory devie (SDRAM, SD Card, et) with large apaity ( GB++) a well a multi-ore RISC proeor running at high lok frequenie ( GHz++) are widely available in the market at relatively low ot. Conequently, there i a potential for deviing imple yet very effetive ommand generator for omputer numerially ontrolled (CNC) mahinery that benefit fully from the propertie of thee advaned devie. Suh a heme may overome the diffiultie enountered in the afore-mentioned ytem. Hene, the entral motivation of thi tudy i to look deeper into thi apet that ha not been fully explored in the indutry (or the orreponding tehnial literature per e).. Sope of the Thei The main objetive of thi tudy i to develop a general ommand generation paradigm whih an be employed for all kind of mehanim. The flow hart of the propoed tehnique i illutrated in Figure..

24 Figure. Flow hart of the propoed method. In thi method, the uer firt need to define the required trajetory for the tool (or apparatu) attahed onto the mahine (e.g. manipulator or mahine tool) by mean of an enhaned NC ode whih loely follow RS-74B onvention. Jut like onventional approah, thi NC ode repreent the trajetory in term of linear and irular egment in a loal oordinate frame ( work oordinate ytem ). Thi loal frame may be onveniently ituated inide a global (fixed) referene frame by mean of peifying the Carteian oordinate of it origin a well a it orientation. The propoed method, whih require a areful offline path planning, interpret thi NC ode to generate the poe of the tool in time (a.k.a. tool loation data ). That i, depending on the ampling time peified by the uer, three Carteian oordinate (of the tool) are alulated at equal time interval along the omplete trajetory. One the poition data are produed, the orreponding joint tate (a.k.a. joint tate data or imply JSD ) are omputed with the utilization of a numerial invere kinemati olver. Note that thi olver make good ue of the Denavit- Hartenberg parameter table that deribe the geometri propertie of the mahine ytem at hand. Finally, depending on the enoding tehnique and the error tolerane for eah joint, the ommand enoder employ the reulting data to produe the effiient repreentation of poition (and it higher order derivative 3

25 in time) in joint tate pae with minimum redundany. The following enoding method are onidered within the ontext of thi thei: Lole data ompreion of higher-order finite differene of JSD Polynomial (Chebyhev, Legendre, Berntein) repreentation of JSD Fourier and Wavelet tranform of JSD Note that in thi tudy, the performane of the above mentioned tehnique (i.e. approximation error, memory requirement, omputational omplexity, eae of deoding, et.) are omparatively evaluated for the purpoe of determining the mot uitable tehnique for ommand generation. The propoed method i implemented in MATLAB environment (via MATLAB ripting language). MATLAB, whih ha dramatially evolved over the year in addition, ha wide popularity in ientifi ommunity due to it veratile toolboxe. Hene, the tudy take full advantage of it feature to fulfill the objetive being et forth. It i ritial to note that one of the primary goal of thi reearh i to pave the way to develop an enoding/deoding tandard for an advaned ommand generator heme for omputer numerially ontrolled (CNC) mahine in the near future. A illutrated in Figure., one the enoded joint tate file i reated effiiently, the reulting file ould be uploaded to the ommand deoder (ard) whih i expeted to deode the data in real-time. Hene, the deoded joint tate (poition, veloity, aeleration) would then be fed to the (entralized or ditributed) joint-axi motion ontroller a the referene ignal. Due to the broad range of thi thei (a it i), the ommand deoding a well a it (hardware) implementation i exempted from thi work. Figure. Command deoding heme. 4

26 .3 Organization Thi thei i divided into nine hapter. The eond hapter give detailed information about the tudie relevant to the ope of thi thei. The literature urvey i onduted in variou area uh a advaned ommand generation for roboti, CNC interpolator, kinemati modeling of artiulated mehanim, data ompreion. Likewie, the third hapter deal with the kinemati modeling of artiulated mehanim. The bai information about manipulator and their kinemati are alo elaborated in that hapter. In addition, the generalized Denavit Hartenberg notation, forward kinemati, and orreponding olution method for invere kinemati are explained in detail. The fourth hapter deal with the poition generation in joint pae. An algorithm for the interpretation of the NC ode a well a the invere kinemati of artiulated mehanim are diued in thi hapter for the purpoe of generating the tool trajetory for a peifi mahine/manipulator. The hapter i onluded with an example on produing the joint poition by invere kinemati algorithm for a pre-defined error tolerane. In hapter five, the ommand generation via diret data torage method i tudied. The main idea of thi hapter i to tore the generated ommand for eah joint in the mot effiient way. Lole data ompreion method uh a Huffman Coding and Arithmeti Coding (Shannon-Fano) have been invetigated and their memory requirement have been elaborated. In the following hapter, polynomial (fitting) method uh a Chebyhev, Legendre, and Berntein polynomial ha been tudied while the relation between Chebyhev polynomial and Fourier tranformation ha been explained. Finally, an (ommand traking) error alulation algorithm for determining error tolerane band in joint pae ha been introdued in thi hapter. In Chapter 7, the Fourier- and Wavelet tranformation are invetigated o that the JSD i tranformed into another domain and inignifiant data i negleted for the purpoe of repreenting the original data effiiently. Chapter 8 evaluate the performane of the preented method on variou ae. In the lat hapter, the thei i onluded by 5

27 ummarizing the key reult of thi reearh. In addition, poible future work are preented in thi hapter a well. 6

28 CHAPTER LITERATURE SURVEY Thi hapter i dediated to a detailed literature urvey in the field relevant to ommand generation inluding kinemati modeling of manipulator, CNC interpolator, advaned ommand generation, and method for data ompreion.. Interpolator The tudy tart out with detailed invetigation about the interpolation method and the ue of interpolation tehnique in CNC appliation. By the tudy on interpolator, bakground knowledge of interpolator ha been obtained. In, Yang and Hong [4] developed a 3-dimenional (3D) Interpolator whih i baed on interetion riteria. They developed a real-time referene-pule 3D linear- and irular interpolator whih i apable of ynhronized imultaneou 3D mahining. Cheng [6] ued NURBS and offered a ommon mathematial form for repreenting both tandard analytial hape and free-form urfae. The interpolation with NURBS i high-peed and highly aurate but large data onume o muh memory and too many hort egment were lowing down the utting peed. Bahr, Xiao, Krihnan [8] implemented pline interpolator inide a CNC Controller. The main aim wa to ue finite forward differening algorithm for fat evaluation of point on a ubi parametri urve in order to prevent the aumulative error in the alulation of one piee of urve to propagate to the whole urve. Bahr ued forward differening method beaue of it effiieny for 7

29 evaluation of point. In addition to the prevention of error aumulation pline interpolation allow rendering urve point uing integer arithmeti. Following that Omirou [9] ued pae urve interpolation for CNC mahine. He propoed an effiient and aurate method for developing a la of preie interpolation algorithm whih an drive the utter of a CNC mahine along three dimenional trajetorie. Parametri programming, mathematial alulation with do-loop ubroutine, maro-apabilitie and ophitiated anned yle were ued during thi tudy.. Trajetory Generation After interpolator, a omprehenive reearh ha been done for the tudie about trajetory generation. The fundamental of the NC Code parer and tool path generation algorithm ha been founded by the information gained from here. In, Lartigue, Thiebaut, Maekawa [7] developed tool path planning algorithm for mooth free-form urfae in term of planar ubi B-pline urve. The algorithm i baed on interpolating the break point by omputing the offet urfae - driving plane interetion urve. Thi method aept urve oeffiient diretly and it i muh more aurate and require le memory. Similarly, Farouki and Tai [] ued Taylor erie oeffiient for variable feedrate CNC urve interpolator. They examined the ituation where the feedrate depend on elaped time, urve ar length and loal path urvature. In addition they preented the derivation of ompat reurive formulae. Yeung, Altinta, Erkorkmaz [] preented a omprehenive virtual imulation model of a realiti and modular CNC ytem. They implemented a trajetory generation mehanim in the Virtual CNC. The tart and end oordinate of the toolpath, the type of the tool movement and the feedrate are reognized and tored into a buffer. By exeuting the buffer blok by blok, the deription for eah tool path egment are obtained and then paed to the trajetory generation proe equentially. 8

30 Lately, Liu, Guo, Li, Yamazaki, Kahihara and Fujihima [] developed an intelligent NC program proeor for CNC Sytem of mahine tool. They invetigated the bai tandard of NC program: RS74D (USA), ISO698 (ISO) and DIN665 (Europe). In addition, they propoed a new truture whih adjut the CNC ytem to adopt variou NC program format by only updating a NC peifiation ditionary. In, Erkorkmaz and Altinta [3], publihed a paper about generating trajetorie not only deribing the deired tool path aurately, but alo having mooth kinemati profile in order to maintain high traking auray, and avoid exiting the natural mode of the mehanial truture or ervo ontrol ytem. In addition they preented a quanti pline trajetory generation algorithm that produe ontinuou poition, veloity and aeleration profile. Apragatho [3], preented two tehnique for generating an approximation of a given robot hand trajetory under bounded poition deviation whih i peified by the operator aording to the auray requirement of the robot appliation. The firt tehnique wa baed on bietion pattern whih determine enough knot point on a given Carteian urve wherea the eond one wa baed on rater anning whih find a minimal et of knot point on a given Carteian urve and pline interpolation i done between two ueive knot..3 Kinemati Modelling and Solution Method After ompleting the tudy on CNC interpolator and tool path generation, a wide reearh on kinemati modeling of manipulator ha been tarted. By thi reearh, different appliation of manipulator have been examined, the truture of the manipulator have been undertood and olution method have been invetigated. In 956, Denavit [3], made an important ontribution by baing the mathemati model of manipulator into logial, ytemati and effiient ytemati. He repreented all kinemati pair a axial joint. Link oordinate ytem are defined and the relative plaement of the ytem wa made by four independent 9

31 parameter. Wang, Baron and Cloutier [4], publihed a paper on topology of manipulator. They haraterized the manipulator by geometri ontraint, propoed a omprehenive topologial diagram whih enable the kinemati ompoition to be deribed preiely. In addition they ued graph truture whih make it poible to implement omputer algorithm in order to perform ytemati enumeration, omparion and laifiation of manipulator. Likewie, Lee, Go, Kim [] developed a uer friendly automati polihing ytem ompoed of a three-axi mahining enter and a two-axi polihing robot. Their robot wa able to keep the tool normal to the die urfae. In addition, they ompared ontrol mode to redue the traking error. Beide a geometri modeler wa developed in thi reearh in whih internal urve and urfae are repreented a a non uniform rational power-bai polynomial (NURP). In 5, Ho, Komura, Lau [6], propoed a linear programming baed invere kinemati (LPIK) olver for interative ontrol of arbitrary multi body truture. The advantage of uing LPIK are handling the inequality ontraint whih make eaier to handle with the range of the DOF and olliion of the body with other obtale and the performane of LPIK i omparable or ometime better than the IK method baed on Lagrange multiplier. In addition they mentioned that the omputation time by LPIK inreae only linearly proportional to the number of ontraint or DOF. Hene, LPIK i a uitable approah for ontrolling artiulated ytem with large DOF and ontraint for real-time appliation. On the other hand, Tabazynki [5] tudied and ompared Jaobian baed olution of invere kinemati problem namely, peudo invere, trunated peudo-invere, tranpoe, and damped leat quare (DLS). He howed with experimental reult that DLS i better with it mooth motion and immunity to ingularitie and unreahable target i the bet all around olution, but ould be too low if high onvergene auray and interative peed i required. Erdman [3] edited a book about the hitory and the development of the kinemati. He ummarized the diret and invere kinemati approahe throughout the hitory. Denavit Hartenberg (DH) notation ha been told to be the mot ommon ued notation, in ombination with homogenou tranformation

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