Transactions of the VŠB Technical University of Ostrava, Mechanical Series. article No. 2005

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1 Tansactions of the VŠB Technical Univesity of Ostava, Mechanical Seies No. 1, 2016, vol. LXII aticle No Vladimi BULEJ *, Juaj URICEK **, Manfed EBERTH *** THE MODELLING OF MECHANISM WITH PARALLEL KINEMATIC STRUCTURE IN SOFTWARE MATLAB/SIMULINK MODELOVANIE MECHANIZMU S PARALELNOU KINEMATICKOU ŠTRUKTÚROU V SOFTVÉRI MATLAB/SIMULINK Abstact The aticle deals with the pepaation of simulation model of mechanism with paallel kinematic stuctue called hexapod as an electo-mechanical system in softwae MATLAB/Simulink. The simulation model is composed fom functional blocks epesented each pat of mechanism s kinematic stuctue with cetain popeties. The esults should be used fo futhe simulation of its behaviou as well as fo geneating of contol algoithms fo eal functional pototype. Abstakt Článok sa zaobeá modelovaním mechanizmu s paalelnou kinematickou štuktúou nazývaného hexapod ako elektomechanického systému v postedí softvéu MATLAB/Simulink. Simulačný model je zložený z funkčných blokov, ktoé pedstavujú jednotlivé časti kinematickej štuktúy mechanizmu s učitými vlastnosťami. Výsledky môžu byť použité pe budúcu simuláciu jeho spávanie, ako aj pe geneovanie iadiacich algoitmov pe eálny funkčný pototyp. Keywods Simulation model, Stewat platfom, MATLAB/Simulink, PKS, hexapod. 1 INTRODUCTION Duing the last decade, thee was fomed a goup of eseaches led by pofesso Poppeova at the Faculty of Mechanical Engineeing at Univesity of Žilina, which deals with the paallel kinematic stuctues (PKS) [9]. Thee have been poposed vaious design concepts of the mechanisms with paallel kinematic stuctue as well as diffeent kind of simulation softwae fo these mechanisms within this peiod (figue 1 - ight). One of designed pototypes is machine tool (figue 1 - left) based on mechanism Hexapod equipped with automatic tool changing and automatic pat changing system. Hexapod, also known as Stewat platfom, is multi-axis manipulato o moving platfom capable of full six degees of feedom (DOFs) motion [7]. This aticle contains the basic infomation about the fist steps in modelling and design of simulation model of mechanism mentioned above in envionment of softwae MATLAB/Simulink. Ou main aim is to ceate the model fo calculation and simulation of hexapod pototype in the softwae MATLAB/Simulink. * Ing., PhD., Depatment of Automation and Poduction Systems, Faculty of Mechanical Engineeing, Univesity of Zilina, Univezitna 8215/1, Zilina, Slovak Republic, phone. (+421) , vladimi.bulej@fstoj.uniza.sk ** Assoc.-pof., Ing., PhD., Depatment of Automation and Poduction Systems, Faculty of Mechanical Engineeing, Univesity of Zilina, Univezitna 8215/1, Zilina, Slovak Republic, phone. (+421) , juaj.uicek@fstoj.uniza.sk *** Dipl.-Ing.(FH), MSc., AUDI AG, I/EE-554, D Ingolstadt, phone. (+49) , manfed.ebeth@audi.de 1

2 Fig. 1 Pototype of machine tools based on Hexapod mechanism and second vesion of simulation softwae designed in Delphi pogamming language [2] 2 SIMULATION MODEL OF MECHANISM HEXAPOD At the beginning it was necessay to pepae an appopiate 3D model of selected mechanism fo what we used softwae PoEnginee and SolidWoks as well. Subsequently, the 3D model was conveted to a fomat that can be pocessed by the pogam MATLAB/Simulink. Finally, the simulation model of Stewat platfom was developed with the help of Simulink libaies. The tansfomation of mechanism s 3D model into the functional schema eadable by MATLAB/Simulink can be consideed as the fist step of whole pocess. The distibution of assembly components into the individual subassemblies is needed with espect to the oganization of the whole assembly and especially with espect to minimizing the functional blocks at its tansfe into the MATLAB/Simulink. Fig. 2 Hexapod mechanism (left) and its components (ight) desciption in text below [1, 2] The size of geneated functional block diagam depends on the total numbe of fee components. It means that if we want to educe the complexity of the block diagam, it is necessay to cove the components which move togethe into the subassembly [8]. The esult of descibed components aangement is minimized stuctue composed fom seveal subassemblies without educing the functionality and any negative impact on the simulation esult. Individual pats ae aanged accoding to the figue 2 into the seveal subassemblies: 1. main fame with fixed pat of uppe univesal joints, 2. uppe univesal joint, 3. body of linea actuato with moving pat of uppe univesal joint and dive unit, 4. piston of linea actuato with one half of lowe univesal joint, 5. lowe univesal joint, 6. moving platfom with the spindle and second half of lowe univesal joints. 2

3 2.1 Expot of XML File The second step of the pocess is the data expot to an XML file and its subsequent impot into the envionment of softwae MATLAB/Simulink. Fo this opeation we need to install the fee module SimMechanics Link fo softwae Solidwoks, Autodesk Invento and PTC Ceo Paametic [11]. Registation is done though the instuction smlink_linkinv witten in MATLAB command line. Afte the egistation in softwae Solidwoks you can find in uppe toolba the SimMechanics Link icon. Fo the expoting of block diagam you can choose the option Expot>>SimMechanics Fist Geneation. Afte the expoting opeation is automatically geneated the block file hexapod.xml as well as the file of 3D model with suffix *.STL. Then we can impot these files into the envionment of softwae MATLAB. Fo this opeation can be used command mech_impot ('hexapod.xml') in Simulink window. Late will appeas the 3D model of hexapod in gaphic window. 3 SIMULATION MODEL OF HEXAPOD The figue 3 shows the final model of hexapod which we ceated on the base of the Stewat platfom demo. Model ceating, gadual adjustment and paamete settings equie extensive study of patial poblems on fee intenet foums, fee help available, o study mateials. The block model of Hexapod is divided into thee main subsystems (shown in figue 3) [1]: subsystem of tajectoy calculation, PID contol unit, and subsystem of mechanism hexapod. 3.1 Tajectoy Calculation Subsystem Fig. 3 Model of whole Hexapod [1] In the following text is a simple desciption of the individual blocks of the fist subsystem Tajectoy calculation (figue 4). The block Constant geneates a constant input paamete. In this case we will geneate constant linea motion (late it will be eplaced by anothe motion e.g. cicula tajectoy). The ecoding time is set up to 0.02s. This value was chosen with espect to the speed of simulation and quality of output data as well. The blocks X Pos, Y Pos, Z Pos define us the linea coodinates of the TCP point (placed on moving platfom with the end-effecto). Thei composition gives us the esult motion of the platfom in all thee coodinates. Blocks X Pos and Y Pos can be eplaced by othe souces of input paametes (individual coodinates). It means that instead of block Constant should be placed the block with equation of cicula motion. 3

4 Fig. 4 Model of Tajectoy calculation The blocks X Angle, Y Angle and Z Angle allow us to ente the angula settings of the moving platfom in all thee coodinates. Block EuleXYZ calculates the final otation matix (vecto of size 9x1) coesponding to the oientation of the moving platfom. These vectos will be tansfomed to the matix with dimensions 3x3. Block body_pts epesents the moving platfom with 6 theedimensional values the joints. The block Poduct will poduce the final tansfomation matix which is composed fom two input matices EuleXYZ and body_pts. Block Ceate desied position matix ceates the multidimensional aays of size 6x3. Then the matices ae poceeding by two Sum blocks. Block pos_base stoes coodinates of uppe univesal joints connected to the fame. The block Nominal Leg Length gives us the length of each leg. 3.2 PID egulato subsystem In figue 5 the block pos efes to the senso of actuato s cuent position, block _pos stoes the calculated value fom the pevious subsystem (tajectoy calculation). Matix _pos will be subtacted fom the matix pos. The esulting matix is sent to the amplifie Kp (popotional featue). Into the Ki (integation featue) will be loaded the integated value of a matix. The block vel efes to the same senso of the actuato, but in this case we measue the velocity. The block Gain invets the sign of the esult (fom positive to negative o negative to positive). The block Kd makes the diffeentiate of this value. At the end ae summed all blocks Kp, Ki, Kd and the esult is a foce intended fo the actuato. Fig. 5 Model of PID contol unit The motion eo is the diffeence of the desied o efeence length of the leg and its instantaneous o actual length [10]: E ef actual t R p p, L (1) taj, t, b, whee: 4

5 E ef actual L taj (t) p p t, p b, R motion eo [m], efeence length of leg [m], actual length of leg [m], efeence tajectoy length [m], vecto defined the distance between the cente point of base and top plate [m], vecto defined the position of attachment point with espect to the cente of top plate [m], vecto defined the position of attachment point with espect to the cente of base plate [m], otation matix. Fig. 6 Desciption of Hexapod kinematic stuctue The efeence length L taj (t) is given as a function of time by the output of the Leg Refeence Tajectoy subsystem. The vectos p, p t,, and p b, ae defined in the figue 6. The othogonal otation matix R specifies the oientation of the top plate with espect to the bottom. The Stewat platfom model use a simple PID contolle and Joint Senso blocks to measue motion. The simplest implementation of tajectoy contol is to apply foces to the actuato popotional to the motion eo. PID feedback is a common fom of linea contol. A PID contol law is a linea combination of a vaiable detected by a senso, its time integal, and its fist deivative. This Stewat platfom's PID contolle uses the leg position eos E and thei integals and velocities. The contol law fo each leg has the fom [10]: t F act, K E K E dt K de dt, (2) p i 0 The contolle applies the actuating foce F act, along the leg. d If E is positive, the leg is too shot, and F act, is positive (expansive). If E is negative, the leg is too long, and F act, is negative (compessive). If E is zeo, the leg has exactly the desied length, and F act, is zeo. The eal, nonnegative K p, K i, and K d ae, espectively, the popotional, integal, and deivative gains that modulate the feedback senso signals in the contol law: The fist tem is popotional to the instantaneous leg position eo fom efeence. The second tem is popotional to the integal of the leg position eo. The thid tem is popotional to the deivative of the leg position eo. 5

6 The esult is F act,, the actuato foce (input) applied by the contolle to the legs. The popotional, integal, and deivative tems tend to make the legs' top attachment points p t, follow the efeence tajectoies by suppessing the motion eo. 3.3 Hexapod Subsystem In figue 7 is shown the hexapod block diagam (the oientation of blocks is upside-down egading to the eal inveted configuation of the hexapod pototype). Lowe ed ectangle highlights the chain of blocks, which epesente the fixed fame of hexapod (static uppe platfom). Fig. 7 Simulation model of mechanism Hexapod ceated in MATLAB/Simulink Then the uppe ed ectangle descibes the moving platfom include the end-effecto (in eality it is placed unde the fixed fame). Between the fixed fame and the moving platfom ae located the guiding chains (extendable stuts composed fom linea actuatos). They must be linked popely. Env block must be connected to the block Root Gound due to machine settings to the envionment. It's actually a machine envionment that lets you view and change the mechanical envionment. Block Env has the possibility of enteing the gavitational signal into a thee-dimensional envionment, as we set. When you click on a block top_plate-1 a window of basic settings of moving platfom appeas. The model of each leg (moe detailed desciption of blocks Leg1-Leg6) is shown in figue 8. 6

7 Fig. 8 Model of one leg ceated in MATLAB/Simulink 3 FIRST SIMULATIOUN OUTPUTS Constucted model can be consequently used to cay out vide vaiety of simulations, diffeent motion analysis, simulation of diffeent egulation methods fo dive units, influence of contolle settings, o contol stability duing idle o unde load. The esults fom simulation depend on the type of input paametes as well as on the system settings. Accoding to ou goal it is necessay to pepae suitable input infomation like fo example mathematical desciption of the tajectoy, loading foces, the impact of distubance vaiables, etc. Fig. 9 Cicula tajectoy fo fist testing cycle Thee we have the ability to add vaious input blocks geneating the tajectoy of vaious shapes, such as Sine Wave Geneato Pulse, Ramp, Random Numbe, Step and othes. We chose a cicula tajectoy as input infomation fo the fist functional tests of the mechanism. Then was composed a system of blocks geneating a cicula tajectoy with a adius of 5 cm (the value of X = 0.05), along which is moving the tool end point with moving platfom and the spindle. Fom the cente, the tool will move along a line until it touches the oute cicumfeence and subsequently complete the cicle (fig. 9 and 10). Whole system is coveed by block XY Gaph [1]. Late we can simulate the eaction of the mechanism on the mateial esistance duing machining pocess. Fig. 10 Simulation esults of hexapod motion duing testing of cicula intepolation 7

8 4 CONCLUSIONS Hexapod can be consideed as a geneal paallel mechanism with complex DOFs because it poduces 6 DOF of a geneal igid body motion. Machine tools based on paallel o hybid mechanisms can be successfully applied fo high speed cutting, etc. [3, 5, 6]. At the authos wokplace was designed the pototype of machine tool with hexapod kinematic stuctue. In the aticle is descibed the modelling pocess of hexapod mechanism in MATLAB/Simulink. The ceated model is composed fom individual functional blocks mutually inteconnected by links. The model descibes the kinematics as well as the chaacteistics of its individual pats. The simulation model can be consequently used to cay out vide vaiety of simulations, diffeent motion analysis, simulation of diffeent methods fo dive units egulation, contolle settings with espect to minimizing the output positioning eo, o simulation of contol system stability duing idle o unde load. The model will be in next peiod impoved to bette descibe the chaacteistics of the eal mechanism. ACKNOWLEDGEMENT This aticle was ceated by the solution of poject - code ITMS : The Development of Paallel Kinematic Stuctue Pototypes fo Application in the Aea of Machine Tools and Robots suppoted by opeational pogam Development and eseach, financed fom Euopean foundation fo egional pogess. REFERENCES [1] DRESTO, D.: Use of Simulink fo dive contol of machine with paallel kinematic stuctue. Diploma thesis. Žilina, [2] POPPEOVÁ, V. et al.: Paallel Mechanism and its Application in Design of Machine Tool with Numeical Contol. In Applied mechanics and mateials, 2013, Vol. 282, p , ISSN [3] D. ZHANG: Paallel Robotic Machine Tools, Faculty of Engineeing and Applied Science, Univesity of Ontaio Institute of Technology,Canada,LLC 2010, ISBN [4] BŘEZINA, T., FLORIAN, Z., HOUŠKA, P., et al.: 2006, Device fo Expeimental Modeling of Biomechanical Systems Popeties. Simulation Modelling o Mechatonic Systems II, 2004, ISBN , p , pp. 276 [5] ZELENÝ, J.: Numeically Contolled Machine Tools and Accesoies. ČVUT Paha, [6] MERLET, J.-P Paallel Robots. Dodecht: Spinge Academic Publishe, 2006, ISBN [7] LUNG WEN TSAI: Robot Analysis: The Mechanics of Seial and Paallel Manipulatos. Intescience 1999, ISBN , s. 520 [8] KUMIČÁKOVÁ, D., RENGEVIČ, A.: New possibilities of obot am motion simulation. In: Communications : scientific lettes of the Univesity of Žilina. - ISSN Vol. 18, no. 1A (2016), s [9] KURIC, I. et al.: Expeimental device fo pacticing outines of machine tool pecision measuement. In: Academic jounal of manufactuing engineeing. - ISSN Vol. 13, no. 1 (2015), s [10] SimMechanics in Academia MathWoks [online]. [cit ]. Available: [11] GREPL, R. Modelování mechatonických systémů v Matlab/SimMechanics. BEN - technická liteatua, 2007, p. 152, ISBN: