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ORIGINAL ARTICLEFunctional simulator of 3-axis parallel kinematicmilling machineMilos Glavonjic i 1; 2; 3. The position vector of the tool tip is defined in theframe PasPPT; xTPyTPzTPC138T, where zTPC0h. The position vectors of simulators driving axesreference points Riare defined as,BPRi; i 1; 2; 3.Joint coordinates vector: l l1l2l3C138T, l1,l2, and l3are scalar variables poweredand controlled by serial CNC machine within the rangeof lminC20 liC20 lmax, whileBaiare unit vectors,Ba1100C138T;Ba2 010C138TandBa3 00 C0 1C138T.World coordinates vector:BPT xTyTzTC138Trepresents the programmed positionvector of the tool tip, while x BPOP xpypzpC2C3Trepresents the location of the platform, i.e., the origin Opof the coordinate frame P attached to it. The relationshipbetween these two vectors is obvious since coordinateframes BandP are always parallel, i.e.,BPTBPOPPPT1Other vectors and parameters are defined as shown inFig. 6, whereBwiandBqiare unit vectors while c is fixedlength of joint parallelograms.The relationships between the simulators joint coordi-nates vector l l1l2l3C138Tand the serial machine jointcoordinates m x0MyMzMC2C3T, as shown in Fig. 6, are:x0M l3; yM l2; zMC0112On the basis of geometric relations shown in the Fig. 6,the following equations are derived:kBiwiBPOPPBPCiC0BPRi3kBiwi lBiai cBqi4By taking the square of both sides in Eq. 4 the followingrelation is derived:c2 k2i l2iC0 2liBaikBiwiC0C15By adoptingPBPCiC0BPRi 0 6in Eq. 3, kinematic modelling is very simplified. In order tofulfill this requirement, specific calibration method, i. e.,setting of reference points Rihas been developed. ByFig. 6 Geometric model of the simulatorFig. 5 Example of a simulator on the vertical CNC milling machine816 Int J Adv Manuf Technol (2009) 42:813821substituting other mechanisms parameters in Eq. 5, thesystem of the following three equations is obtainedx2p y2p z2p l21C0 2l1xpC0 c2 0x2p y2p z2p l22C0 2l2ypC0 c2 0x2p y2p z2p l23 2l3zpC0 c2 08:7from which are derived: inverse kinematic equations asl1 xpC6c2C0 y2pC0 z2pql2 ypC6c2C0 x2pC0 z2pql3C0zpC6c2C0 x2pC0 y2pq8:8as well as direct kinematic equations asypC0s6C6s26C04s5s7p2s5xp s1 s2ypzp s3 s4yp8:9where ares1l21C0 l222l1; s2l2l1; s3l22C0 l232l3; s4C0l2l3;s5 1 s22 s24; s6 2 s1s2 s3s4C0 l1s2;s7 s21 s23C0 2l1s1C0 c2 l21; lminC20 liC20 lmax; i 1; 2; 3As it was mentioned, by adjustment of simulatorsmechanism parameters, Eq. 6, the solution of inverse anddirect kinematics is greatly simplified. To satisfy theconditions from Eq. 6 six calibration struts of selectedreferent length were used, Fig. 7. With the use of inverseand direct kinematics solutions with the calibrated strutlengths, the positions of reference points Riof sliders Si,(i=1, 2, 3) are defined and fixed by calibration plain rings,Fig. 7.4.1 The analysis of inverse and direct kinematics solutionsWith the analysis of inverse kinematics variance solutions,Eq. 8, different configurations of parallel mechanism for agiven platform position may be noted: the basic configuration, Fig. 2a, when in the Eq. 8, allsigns before the square root are negative, one of alternative configurations, Fig. 2b, when inEq. 8, all signs before square root are positive, other possible mechanism configurations, when in theEq. 8, signs before the square root are combined.In a similar way, through the analysis of direct kinematicsolution, Eq. 9, different configurations of parallel mecha-nism for given positions of driving axes may be established: the basic configuration, Fig. 2a, corresponding to thecase, when in Eq. 9, there is a positive sign beforesquare root, alternative configurations, Figs. 2c and d, when inEq. 9, there is a negative sign before square root.The basic and alternative configurations shown in Fig. 2may be realized in different ways subject to the structure ofthe driving serial machine.4.2 Jacobian matrices and singularity analysisIn view of the significance of PKM singularity, thisproblem has been analyzed in detail for the mechanismvariant shown in Fig. 2a, used for the development of theFig. 7 Setting of simulators reference pointsInt J Adv Manuf Technol (2009) 42:813821 817simulator on horizontal machining center, Fig. 1. Differen-tiating Eq. 8 with respect to the time, Jacobian matrix isobtained asJ 1ypxpC0l1zpxpC0l1xpypC0l21zpypC0l2C0xpzpl3C0ypzpl3C01264375 10As the equations in Eq. 7 are implicit functions of jointand world coordinates, Jacobian matrix may be alsoobtained by their differentiation asJ JC01lC1 Jx11whereJC01l121xpC0l1001ypC0l200 C01zpl3264375 12Jx 2xpC0 l1ypzpxpypC0 l2zpxpypzp l3243513are Jacobian matrices of inverse and direct kinematics.In this way, three different types of singularities can beidentified, e.g., singularities of inverse and direct kinemat-ics as well as combined singularities.With careful analysis of Jacobian matrices determinantsdet Jxpl2l3 ypl1l3C0 zpl1l2C0 l1l2l3xpC0 l1C0C1ypC0 l2C0C1zp l3C0C1 14det JxC08 xpl2l3 ypl1l3C0 zpl1l2C0 l1l2l3C0C115det JlC08 xpC0 l1C0C1ypC0 l2C0C1zp l3C0C116the singularities of inverse and direct kinematics as well ascombined singularity may be noticed.Figure 8 shows these possible simulators singularityconfigurations with corresponding descriptions and equa-tions. As it can be seen from Fig. 8, all singularities are onthe borders of theoretically achievable workspace so that itwould be possible to avoid them easily with adequatedesign solutions and/or mechanical constrains. This meansthat the achievable simulators workspace is smaller thantheoretical workspace. The boundaries of theoretical work-space are on cylinders of radius c whose axes are XB, YB,ZBderived from inverse kinematic Eq. 8 and sphere ofradius c centered in OB, Fig. 8.5 The examples of simulatorsAs it is known in addition to selecting appropriatekinematic topology, the selection of the right geometricdimensions is very important since the performance ishighly influenced by PKM geometric dimensions 1, 8.To select the right dimensions with respect to a givenapplication is a difficult task, and the development ofdesign tools for PKM is still open research 1.The design parameters of simulators shown in Figs. 1, 4,and 5 were adjusted in order to get more adequate shapesand workspace dimensions on the basis of performances ofavailable CNC machines for which simulators wereplanned. The procedure is essentially iterative because indetermination of the basic design parameters the attention ispaid to the possible interferences of structural elements andthe values of det(J) and det(J1) determinants, Eqs. 14, 15,and 16.Fig. 8 Singularity types818 Int J Adv Manuf Technol (2009) 42:813821In the geometric model of simulator variant from Fig. 6,it can be seen that workspace dimensions are primarilyaffected by parallelograms length c, as well as to theadequacy of the distance of the mechanism from D3, D3I2,and D3I1 singularities shown in Fig. 8.For available CNC machine for which the simulator wasplanned, parallelograms length c and values joint ofcoordinates l1,2,3minand l1,2,3maxwere analyzed in iterativeprocedure. In each iteration, attention was paid to thepotential design limitations, interferences, as well as to thevalues of det(J) and det(J1), i.e., to the distances fromsingularities.The parameters determined in this way have beenslightly corrected in detailed design of the simulatorprototype shown in Fig. 9. Shape, volume, and position ofachievable workspace for parallelograms length c=850 mmand l1,2,3min=200 mm and l1,2,3max=550 mm are shown inFig. 2a.On the basis of the adopted concepts and designparameters, the first two simulators have been built (Figs. 9and 10).6 Simulator programming and testingThe simulator programming system has been developed inastandardCADCAM environment on PC platform(Fig. 11). It is possible to exchange geometric workpiecemodels with other systems and simulate the tool path.Linear interpolated tool path is taken from the standard CLfile. The tool path may also be generated in some other wayselected by the simulator user. The basic part of the systemconsists of developed and implemented postprocessor,without the use of postprocessor generator. The postpro-cessor contains inverse and direct kinematics, simulatordesign parameters, and algorithm for simulators tool pathlinearization (Fig. 12). Simulators tool path linearization isessential because CNC machines linear interpolation isused as simulators joint coordinates interpolation. In thisway, simulators tool path remains within the tolerance tubeof predefined radius between points Tj1and Tjtaken fromCL file. The long program for CNC machine obtained inFig. 9 Completed simulator from Fig. 1Fig. 10 Completed simulator from Fig. 4Fig. 11 Simulator programming systemInt J Adv Manuf Technol (2009) 42:813821 819this way is transferred to CNC machine and can beverified during idle running of the simulator. The motionrange of driving axes has been already checked in thepostprocessor.The testing of the simulator in this phase included:& verification of the system for programming and com-munication, and& cutting tests by machining various test pieces (Fig. 13).7 ConclusionIn order to contribute towards the acquisition of practicalexperiences in modelling, design, control, programming,and the use of PKM, a low cost but functional simulator of3-Axis parallel kinematic milling machine is proposed. Thedeveloped functional simulator of the 3D parallel kine-matic milling machine integrates, as a hybrid system, theexisting technological equipment (CNC machine tools,CADCAM hardware and software) and parallel kinemat-ic mechanism into a comprehensive and sophisticateddidactic facility. The idea about the functional simulatorwas verified by successful making of some standardizedtest pieces out of soft materials, made under fulloperational conditions. Its capabilities and characteristicshave shown that the simulator itself was an interesting andvaluable R&D topic. This idea may be further used formaking of ones own simulators.Fig. 13 Test pieces made of foamFig. 12 Uniform linearization of simulators tool path820 Int J Adv Manuf Technol (2009) 42:813821Acknowledgements The presented work was part of Eureka projectE!3239 supported by the Ministry of Science of Serbia.References1. Weck M, Staimer D (2002) Parallel kinematic machine toolscurrent state and future potentials. CIRP AnnalsManufacturingTechnology 51(2):671683 doi:10.1016/S0007-8506(07)61706-52. Covic N (2000) The development of the conceptual design ofclass of flexible manufacturing systems. University of Belgrade,Belgrade Faculty of Mechanical Engineering, Dissertation, inSerbian3. Chablat D, Wenger P (2003) Architecture optimization of a 3-doftranslati