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IntJAdvManufTechnol DOI10.1007/s00170-011-3613-y ORIGINAL ARTICLE Apply fuzzy interpolation method to calibrate parallel machine tools Ying Bai&Hanqi Zhuang&Dali Wang Received:7May2009/Accepted:25August2011 #Springer-VerlagLondonLimited2011 Abstract A novel and efficient fuzzy interpolation method their applications [1–5]. PMTs have the advantage of great is proposed to simplify the calibration process for parallel operational accuracy due to their high rigidity. However, machine tools (PMTs). Either inverse or forward kinematic because of the nonlinear relationship that existed between models must be used in the traditional PMT calibration the actuators and the platform, it is difficult to develop a methods to perform the identification and compensation of forward kinematic model for PMTs. Most popular methods the pose errors for PMTs, which made the calibration for kinematic analysis are iterative algorithms, which are processtime consumingandinefficient inrealapplications. inefficient and may lead to unstable solutions. Instead of using a model, the proposed method presents a DifferentcalibrationmethodsandstrategiesforPMTshave modeless technique combined with the fuzzy interpolation beenreportedrecently[6–20]. However, these methods used method to obtain high calibration accuracies when a small forward and inverse kinematic models to establish relation- workspace is adopted. This new approach can significantly ships between the pose of the end-effector and the link reducethecomplexintraditionalPMTcalibrationprocesses lengths as well as the joints on the mobile and base and greatly simplify the calibration procedures. platforms. The complication in developing such forward or inverse kinematic models makes those calibration processes Keywords Parallel machine tools.Robots calibrations. very complicated, time consuming, and inefficient. Fuzzy interpolations.Kinematic models Gengetal.developedsomeneuralnetworkalgorithmsto derive an approximate forward kinematic model of PMTs [21]. However, the algorithm is computationally intensive 1 Introduction and does not guarantee convergence. Tosimplifythecalibrationprocess,amodelesscalibration In recent years, parallel machine tools (PMTs) have drawn method with a fuzzy interpolation technique (FIM) is more and more interest in the academia and industry for developedinthispaper.Themainadvantageoftheproposed techniqueisthatitdoesnotinvolveanykinematicmodeling and identification process. The calibration process can be Y.Bai(*) significantly simplified, and calibration performance can be JohnsonC.SmithUniversity, improvedbyusingtheproposedmethodaslongasthePMTs Charlotte,NC,USA e-mail:[email protected] workinarelativelysmallworkspace. Compared with other interpolation techniques, the FIM H.Zhuang has the advantages of high calibration accuracy, shortened FloridaAtlanticUniversity, calibration process, and potentially low operational cost. BocaRaton,FL,USA e-mail:[email protected] One of the most prominent features of PMTs is that they have relatively small workspace. This makes them good D.Wang candidates for the proposed technique. ChristopherNewportUniversity, The paper is organized in the following sequence: after NewportNews,VA,USA e-mail:[email protected] the introduction section, a description of the structure and IntJAdvManufTechnol kinematics of PMTs is given. The architecture and equationoftheparallelmachinetools.Foragiven setof li, algorithm of FIM are provided in Section 3. Simulation a unique pose of the mobile platform {x y z α β γ} can be results are discussed in Section 4, and the paper ends with determined. The goals of the calibration for the parallel conclusions in Section 5. machinetools areto find the actual vector parameters ai, bi and link length offset Δli of li and to perform the compensations for the nominal poses of the mobile 2 Kinematics of parallel manipulators platform based on those identified parameters in order to improve the pose accuracy of the parallel manipulators The typical structure of a spatial six SPS parallel machine while they are in operation. tool known as Stewart platform is shown in Fig. 1. Generally, it is difficult to derive a forward kinematic Theframesofthemobileplatformandthebaseplatform model for parallel manipulators since there is no closed are defined as {A} and {B}, respectively. The spherical formsolution,andsomenumericalalgorithmsmustbeused joints connected to the mobile platform are indicated as Ai to find the “true” forward kinematic parameters [3–7]. andtheU-jointslocatedonthebaseplatformaredefinedas Bi (i=1,2,…6). The position vectors of 6 joint points Bi in thebaseplatformwithrespectto{B}aredenotedasb ∼b , 3 Fuzzy interpolation method 1 6 and the position vectors of the joint points A ∼A in the 1 6 mobile platform with respect to {A}are denoted as a ∼a . Generally, themodelesscalibration can bedivided into two 1 6 ThepositionvectoroftheoriginO of{A}withrespectto steps: measurement and compensation. A {B} is defined as t, and the rotation matrix of {A} respect to {B} can be expressed as R that is a 3×3 rotation matrix 3.1 Measurement process with a set of Euler angles [α, β, γ]T. The translation vector t, and the rotation matrix R can be written as To begin, a measuring device is used to measure a set of selected poses that are distributed around a group of grid t ¼½xt;yt;zt(cid:2)T ð1Þ points (vertices) of all cubic cells in the PMTworkspace in R¼Rðz;gÞRðy;bÞRðx;aÞ ordertodeterminetheerrorsbetweenthedesiredposesand the nominal poses of the end-effector. All the pose errors 2 3 are stored into computer memory based on each pose’s cgcb cgsbsa(cid:3)sgca cgsbcaþsgsa ¼4sgcb sgsbsaþcgca sgsbca(cid:3)cgsa5 ð2Þ coordinates. During the calibration process, a target pose canbeinterpolatedbyusingtheFIMbasedontheerrorsof (cid:3)sb cbsa cbca neighboringposeslocatedattheverticesofenclosingcubic whereR(z,γ)isarotationofγaboutthez-axis(Roll),R(y,β) cell.Onebenefitofthistechniqueisthatitcancombinethe is a rotation of β about the y-axis (Pitch), and R(x, α) is a identificationandcompensationprocessintoone.Thisgreatly rotation of α about the x-axis (Yaw). reducesthetimeconsumedinthecompensationprocess. The kinematic equation of the parallel machine tool can When implementing the modeless method to calibrate a be expressed as: robot, it is necessary to use a measurement tool, such as a high-accuracy coordinate measuring machine or a laser li ¼jjRAaiþt(cid:3)Bbijjði¼1;2;...6Þ ð3Þ tracking system (LTS) to record pose errors as the robot whereAairepresentstheithpositionvectoraiinthemobile moves its end-effector to all grid points on pre-determined platform respecttothe{A}andBbidenotestheithposition cubic cells. Figure 2 shows an experiment setup in which vector bi in the base platform respect to {B}. li is the ith link length. Equation 3 is a typical inverse kinematic Fig.1 Stewartplatform Fig.2 Ameasurementsetupformodelessmethod IntJAdvManufTechnol an LTS is used to measure pose errors of a PMT. A similar are ex, ey, and ez. The outputs are eex, eey, eez, eeα, eeβ, arrangement can be made for measuring pose errors of a and eeγ, which are shown in Fig. 3b. robot. The relationship between the frame of the LTS {W} The control rule is shown in Fig. 3c, which is and the Base of the PMT {B} is also shown in Fig. 2. straightforward and based on the human knowledge. It is The PMTworkspace is divided into a sequence of small worth to note that each Pi should be considered as a cubic cells. Each small cell is surrounded by eight combination of three position and three orientation error neighboring grid points (vertices) as shown in Fig. 2. At components on each grid point. each grid point, the LTS is used to measure the pose error Inourcurrentstudy,thedistancebetweentheneighboring of the end-effector of the manipulator. In Fig. 2, the gridpointsofeachcellis20mminthreedirections,whichisa nominalposeofthegridpointi is Pi=[xi, yi, zi, αi, βi, γi]T, standardintervalforsmallcalibrationworkspaces. and the actual pose of the end-effector is Pia=[xia, yia, zia, In total, the workspace includes 20×20×20 cells, and it αia, βia, γia]T. Thus, the pose error for this grid point is is equivalent to a 400×400×400-mm3 space. This is a Pia−Pi. The parallel manipulator will be moved to all grid typical workspace of most parallel manipulators imple- points in its workspace, and all pose errors on these grid mented in real operations [6, 7]. The input membership points are measured and stored in the memory for future functions for x, y, and z directions, and predefined output use. membership functions are shown in Fig. 4. To measure the orientation of the end-effector, different Thepredefinedoutputmembershipfunctionsareusedas methods [3, 10, 11, 22, 23] have been reported. In this thedefaultones,andtheactualoutputmembershipfunction study, a similar method to [23] is used. A special retro- will be obtained by shifting the default one based on the reflectorisinstalledontheend-effectorofthePMT,andthe actual error value on the grid points. For each cell, eight orientation information of the end-effector can be obtained output membership functions are implemented and each from a charged coupled device effectively [23]. oneisassociatedwiththeerroratonegridpoint.InFig.4b, only four orientation output membership functions are 3.2 Compensation process shown because of space limitation. In a real application, total eight orientation and eight position membership During the compensation stage, as the manipulator moves functions should be utilized. to a certain target point in its workspace, an interpolation The Gaussian-bell waveforms are selected as the shape technique is used to estimate the target pose error based on of the membership functions for three inputs. As shown in the errors of the neighboring grid points around the target Fig. 4a, the ranges of inputs are between −10 and 10 mm point. Finally, this error is added into the target pose to (20-mm interval on grid points). The reason for this obtainthecompensatedposewhichthePMTiscommanded selection is that the Gaussian-bell waveform has a smooth moving to. curve and can improve the accuracy. In Fig. 4, Wand E A dynamic online fuzzy interpolation algorithm is used represent the inputs located at different areas in the x for the error interpolation process of this parallel manipu- direction, N and S represent those in the y direction, and L lator’s calibration. Unlike traditional offline fuzzy inference and U represent those in the z direction. Unlike the algorithms, in which both input and output membership traditionalfuzzyinferencesystem,inwhichallmembership functionsaswellasthelookuptablehavebeendefinedinthe functionsshouldbedeterminedtoproducethelookuptable designstagebasedontheinputandoutputranges,theoutput prior to the implementation of the fuzzy system, in this membership functions and lookup table implemented in this study, the output membership functions will not be defined applicationaredefinedbasedontherealposeerrorsmeasured until the implementation of the fuzzy error mapping to during the real measurement process. The identification and compensate the pose errors. Thus, the output membership compensation accuracy of the parallel manipulator can be functions will be determined during the application of the significantly improved after using this algorithm since this fuzzy inference system online or dynamically. Figure 4b algorithmusesthereal-timeposeerrorasinputstotheFIM. shows an example of the output membership functions, The definition of this dynamic online fuzzy inference which are related to the simulated random errors at algorithm is shown in Fig. 3. Each small cube, which is neighboring grid points. Each Pxi, Pyi, and Pzi corresponds surroundedbyeightneighboringgridpoints,isdefinedasa to the pose error at the ith grid point, respectively. During cell. Furthermore, this cell is divided equally into eight thedesignstage,alloutputmembershipfunctionsshouldbe smaller cells, which are shown in Fig. 3a. initialized to a Gaussian waveform with a mean of 0 and a TheposeerrorateachgridpointisdefinedasP ,P ,P , range that is closed to the actual possible output range 1 2 3 P , P , P , P , and P . For the fuzzy inference system, the which can be estimated based on the different parallel 4 5 6 7 8 interpolation method is divided into the three dimensions manipulators for the different applications. These output separately, so the inputs to the fuzzy inference system membership functions will be determined online based on IntJAdvManufTechnol Fig.3 a–cDefinitionofthefuzzyinterpolationinferencesystem the errors of the neighboring grid points around the target The control rules shown in Fig. 3c can be interpreted as point in the workspace during the compensation process. followsaftertheoutputmembershipfunctionsaredetermined: Ifex is W;eyis N;andez is U;theneexis Px1;eeyis Py1;and eezis Pz1; andeea isa1;eebis b1;andeeg is g1:ðP1Þ Ifexis W;eyis N;and ezis L; then eexis Px3;eeyis Py3;andeez is Pz3;andeeaisa3;eeb is b3;andeegis g3:ðP3Þ Ifex is W;eyis S;andezis U;theneexis Px5;eey is Py5;andeezis Pz5;andeea isa5;eeb isb5;andeegisg5:ðP5Þ Ifex is W;ey is S;andez is L;theneexisPx7;eeyis Py7;andeezisPz7;andeea isa7;eebisb7 andeeg isg7:ðP7Þ ð4Þ Ifex is E;ey is N; andez is U;theneex is Px2;eey isPy2;andeezis Pz2;andeeaisa2;eebisb2;andeegisg2:ðP2Þ Ifexis E; ey is N; andezis L;theneexis Px4;eeyis Py4; andeezis Pz4;andeea isa4;eeb isb4;andeeg isg4:ðP4Þ Ifex is E;eyis S;andez is U;theneex is Px6;eeyis Py6;andeez is Pz6;andeeaisa6;eebisb6;andeegisg6:ðP6Þ Ifexis E;eyis S;andezis L; theneexis Px8;eeyis Py8; andeezis Pz8;andeeaisa8;eebisb8andeegisg8:ðP8Þ Thecontrolrulesarestraightforward,andtheyarebased The input error variables can be expressed as a label set on the human knowledge [24]. The error on P grid point L, with E being a linguistic input variable: 1 should carry larger weight if the target position (input) is LðEÞ¼fNWU;NWL;NEU;NEL;SWU;SWL;SEU;SELg located inside the NWU area on a cell. Similar consider- ation should be given for errors on all other grid points. ð5Þ Fig.4 a–cInputandoutputmembershipfunctions IntJAdvManufTechnol Assume that ui is the membership function, Ui the This shortcoming becomes of little importance as the universe of discourse, and m the number of contributions, availability of high-speed CPUs for the controllers. the traditional output of the fuzzy inference system can be represented as: 4 Simulation results Pm ðui(cid:4)UiÞ u¼i¼1 Pm ð6Þ To compare the performance of modeless and model-based u calibration methods, a simulation study is provided hereaf- i i¼1 ter.AStewartplatformdevelopedintheRoboticsCenterat where u is the current crisp output of the fuzzy inference Florida Atlantic University (FAU) is utilized to illustrate this implementation. The simulation study is divided into system and (6) is obtained by using the center-of-gravity method. In this study, both ui and Ui in the output twoparts,measurement andcompensation process,aslisted below. An LTS developed in the Robotics Center at FAU membership functions are randomly distributed variables, was used to perform the measurement task. and the actual values of these variables depend upon the A single-beam LTS is used for the tracking and position errors of eight neighboring grid points around the measurement purpose. The target is a retro-reflector target position. These relationships can be expressed as: installed on the end-effector of the PMT. The retro- uxi ¼FxiðPx1;Px2;Px3;Px4;Px5;Px6;Px7;Px8Þ ð7Þ reflector is built with three prisms that can reflect both the position and the orientation of the target [23]. Uxi ¼QxiðPx1;Px2;Px3;Px4;Px5;Px6;Px7;Px8Þ ð8Þ 4.1 Measurement Process Simulation Steps where Fxi is the membership function of the input position in the x direction, and it is a predetermined membership 1. For a set of given actual poses, using the inverse function as shown in Fig. 4a. Qxi is the real error output kinematics (3) with identified kinematic parameters membership function, which is a randomly distributed ra ¼½xai; yai; zai; xbi; ybi; zbi; Δli(cid:2)T[6] to obtain the function, and it gives the error output contributions in the actual leg length vector l. These given poses can be x direction.This membershipfunctionisdetermined bythe considered as the actual poses (identified poses) of the realpositionerrorsatthe8-neighboringgridpointsinthex StewartplatformXa ¼½xyzabg(cid:2)Tateachgridpointof direction, Px1∼Px8. This membership function determines the 3D workspace. the degree to which the current position input belongs to 2. Add those actual poses given in step 1 with a uniform- each different real error output based on the eight control distributed noise XaþArandðÞ to obtain the “actual” rulesdefinedin(4)inthexdirection,anditisequivalentto poses Xm ¼½xyzabg(cid:2)T. These poses can be consid- the universe of discourse or a weighing factor. Substituting ered as the measured poses of the Stewart platform at (7) and (8) into (6), one obtains: each grid point of the 3D workspace. 3. Obtain the nominal poses Xn using the forward Pm FxiðPx1; Px2; :::::Px8Þ (cid:4)QxiðPx1; Px2;:::Px8Þ kinem(cid:2)aticmodel[1] with nomina(cid:3)l kinematic parameters ui ¼ i¼1Pm ð9Þ rn ¼ xnai; ynai; znai; xnbi; ynbi; znbi; lin Tand the leg length FxiðPx1; Px2; Px3; Px4;Px5;Px6;Px7;Px8Þ l obtained from step 1 as the input at each grid point i¼1 of the 3D workspace. Here, ux represents the final error output of the fuzzy 4. CalculatetheposeerrorsΔX ¼Xm(cid:3)Xn.Theseerrors interpolationmethodinthexdirection.In(9),Qxiwillnotbe can be considered as the ones at grid points for each determined until the fuzzy error interpolation technique is cell and should be stored in the memory to be used appliedinanactualcompensationprocess,whichmeansthat during the compensation process. this fuzzy inference system is an online process. The final crisp output of the fuzzy error interpolation system is 4.2 Compensation Process Simulation Steps determined by the neighboring pose errors of 8 grid points. Similarcalculationscanbeimplementedfortheerroroutputs 5. Follow a step similar to step 1 to select some actual in the y and z directions as well as in three orientations. poses Xa. The only difference is that those poses must The advantage of using the online fuzzy inference not be located at the grid points. We can obtain the system is its real-time control ability; the drawback is that actualleglengthvector lbyusingthose selectedactual this method may result in a relative longer response time poses and the inverse kinematic model (3) combined because of the calculation in the fuzzy inference system. with identified kinematic parameters IntJAdvManufTechnol Fig.5 Thesimulationresults 3 model based position errors m) modeless position errors m s ( 2 or err n sitio 1 o P 0 1 2 3 4 5 6 7 8 9 10 0.015 d) model based orientation errors a modeless orientation errors s (r 0.01 or err n o ati 0.005 nt e Ori 0 1 2 3 4 5 6 7 8 9 10 The number of poses calibrated ra ¼½xai; yai; zai; xbi; ybi; zbi; Δli(cid:2)T: 4.3 Details of Steps 3 through 7 6. Follow a step similar to step 2 above to obtain the The presentation of forward kinematic models in parallel measured or true poses Xm ¼½xyzabg(cid:2)Tby adding machine tools is a very challenge topic since no closed those actual poses with a uniform-distributed noise form solution exists. A more detailed description about our XaþArandðÞ: handling for forward kinematics of the parallel machine 7. Obtain the nominal poses Xn using the forward tools is provided in this section. kinematicmodel[1] with nominal kinematic parameters (cid:2) (cid:3) The relation between the pose and the kinematic rn ¼ xnai;ynai;znai;xnbi;ynbi;znbi;lin T and the leg length parameters is: l obtained from step 5 as the input at each non-grid point of the 3D workspace. 8. UposseetehrerofruszΔzyX ifnotrertphoolsaetionnomminetahlopdosteoscXanlc,ualnatde atdhde ðliþΔliÞl2i ¼¼ðjRRaaiiþþtt(cid:3)(cid:3)bbiijÞTðRaiþt(cid:3)biÞ calculated pose errors to those nominal poses Xn to get the calibrated poses using the fuzzy interpolation method, XF=Xn+ΔX. Table1 Modelessandmodel-basedmethodscomparison 9. Calculate the fuzzy interpolation pose errors by Cubiccellsize Error Modeless Model-based calculating the difference ΔXF=Xm−XF. Where Xm is type (mm) (mm) obtained from step 6. 10. ComparethefuzzyinterpolatedposeerrorsΔXFthatcan 20×20×20mm3 Mean 0.0108 0.0092 beconsideredasthemodelesserrorswiththeposeerrors Max 0.0912 0.0875 ΔXobtainedinstep4byusingthemode-basedmethod. STD 0.0087 0.0152 11. Display the comparison between the two calibration 10×10×10mm3 Mean 0.0011 0.0012 errors. Max 0.0495 0.0513 STD 0.0020 0.0157 Steps 3 and 7 in the above simulation process need to 5×5×5mm3 Mean 0.0001 0.0010 use the forward kinematic model of the PMT to derive the Max 0.0475 0.0500 poses based on the input leg lengths. Some details for the STD 0.0002 0.0101 forward kinematic model are provided below. IntJAdvManufTechnol The cost function is using the proposed modeless method is better than that of using a model-based method when workspaces are small. X6 (cid:2) (cid:3) The model-based method can be replaced by the modeless QðxÞ¼ lia(cid:3)jfiðx; uÞj 2 method as the PMTworks in a relative small workspace. i¼1 Additional advantage of using a modeless method to where fi(x, u) is the ith joint-link train inverse kinematic calibrate PMTs is that the kinematic modeling and solutionforli,uisthekinematicparameters(ai,bi),andxis identification process can be removed, and furthermore theposeofthePMT[25].Proceduretoperformtheforward the compensation process can be significantly simplified. kinematics can be expressed by the following sequence: The modeless method is suitable for applications requiring higher accuracy PMTs calibration. 1. Make an initial estimation of X for a given input, the actual link length la 2. Perform inverse kinematics for each joint-link train to References get an estimation for the link length vector at the kth iteration, lk 3. Compute the cost function Qk ¼jla(cid:3)lkj. 1. 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