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International Journal of Machine Tools & Manufacture 43 (2003) 871877Accuracy enhancement of five-axis CNC machines through real-time error compensationW.T. Leia, Y.Y. HsubaDepartment of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROCbDepartment of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan, ROCReceived 18 January 2003; accepted 11 March 2003AbstractAlthough error modeling and compensation have given significant results for three-axis CNC machine tools, a few barriers haveprevented this promising technique from being applied in five-axis CNC machine tools. One crucial barrier is the difficulty ofmeasuring or identifying link errors in the rotary block of five-axis CNC machine tools. The error model is thus not fully known.To overcome this, the 3D probe-ball and spherical test method are successfully developed to measure and estimate these unknownlink errors. Based on the identified error model, real-time error compensation methods for the five-axis CNC machine tool areinvestigated. The proposed model-based error compensation method is simple enough to implement in real time. Problems associatedwith the error compensation in singular position of the five-axis machine tool are also discussed. Experimental results show thatthe overall position accuracy of the five-axis CNC machine tool can be improved dramatically. 2003 Elsevier Science Ltd. All rights reserved.Keywords: Accuracy enhancement; Five-axis machine tool; Probe-ball; Error compensation1. IntroductionIn the past decades, much research has focused on themachine tool accuracy of three-axis CNC machine toolsin the presence of geometric and thermally inducederrors 14. Based on the established error model, acompensation method can be developed to improve theaccuracy of the target machine tool 5,6. The error com-pensation in the three-axis machine tool delivers verygood results if the machines operating condition is well-defined and repeatable. In contrast, previous studies onthe five-axis CNC machine tool are mainly based ontheory and simulation 7,8. Due to the lack of a propermeasurement device, some dominant errors in the errormodel of the five-axis CNC machine tool are notmeasurable. In 9, a neural network model was used forthe error compensation, thus bypassing the problem.Since the error model describes the effects of individualCorresponding author. Tel.: +886-3-574-2321; fax: +886-3-572-2840.E-mail address: .tw (W.T. Lei).0890-6955/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0890-6955(03)00089-0error sources on the overall position errors exactly, it isobvious that error model-based compensation willdeliver the most effective results. This research effortfocuses on the identification of unknown components inthe error model.The components appearing in the error model of thefive-axis CNC machine tool can be classified into twocategories: motional errors and link errors. Motionalerrors are those associated with the inaccurate motion ofthe servo-driven linear or rotary axis. All motional errorsof the servo-driven linear axis can be measuredefficiently with modern laser interferometers 10.Incontrast, the motional errors of the rotary axis are onlypartly measurable, with an electronic level or multi-facemirror. Link errors are those due to the erroneous mount-ing of structural components such as column, spindle,and rotary block. Measurable link errors include thethree squareness errors between the three linear axes.Link errors in the rotary block are normally not measur-able, due to lack of accessibility.To enhance the accuracy of five-axis CNC machinetools, model-based real-time error compensation hasmany advantages. For one, nearly all identified errors872 W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 43 (2003) 871877can be fully compensated. For another, high-level pathdefinitions in the NC program, such as the spline, canbe processed by the CNCs interpolator directly andavoid the typical problem of discontinuous feedrate dur-ing five-axis machining. In a previous study 11, a newmeasurement device with corresponding methods waspresented to test the overall position errors of the five-axis CNC machine tool. Also, an estimation method wasdeveloped to identify the unknown link errors in theerror model 12. This study goes further and investi-gates model-based real-time error compensation.2. Basic conceptIn the information flow of five-axis machining, for-ward and backward kinematic transformations are per-formed at different levels. Usually, both transformationsare based on an ideal kinematic chain, whereby geo-metric errors of the real machine are not considered.Since most five-axis CNC controllers only accept NCdata defined in machine coordinates, backward trans-formation is performed in the post-processor of theCAD/CAM system. The tool path defined in the cutterlocation data (CLDATA) file is transformed from work-piece coordinates into machine coordinates to adapt tothe input format of the target CNC controller.In this paper, transformation based on the ideal kinem-atic model is named nominal transformation. The nomi-nal backward transformation Fb,ncomputes the axis pos-ition vector u in machine coordinates from given toolpose vector v in workpiece coordinates:u H11005 Fb,n(v) (1)The tool pose vector v is defined in workpiece coordi-nates and includes the tool tip position P = xwywzwand the tool orientation Q = iwjwkw. Q is a unit direc-tional vector.The nominal forward transformation Ff,ncomputestool pose vector v in workpiece coordinates from givenaxis position vector u in machine coordinates:v H11005 Ff,n(u) (2)Note that the solution for the forward transformationis unique. For each given axis position vector u, thereis one, and only one, corresponding tool pose vector v.In contrast, the solution for the backward transformationis not unique. If the five-axis machine is not in singularposition, there are in general two solutions for the back-ward transformation. It is necessary to select a suitableone according to predefined criteria, for example mini-mal driving energy or distance. If the five-axis machineis in the singular position, the position of one rotary axisis not solvable. In the case of the XH11032YH11032ZAH11032CH11032 type five-axis milling machine, the machine is in singular positionwhen the rotary C-axis is in the vertical direction. In thiscase, the k-component of the orientation vector Q isequal to one and the other components are zero. Therotation of the C-axis does not change the tool orien-tation in workpiece coordinates.In driving the real five-axis machine with the axis pos-ition vector us, the actual tool pose vector vadeviatesfrom the setting tool pose vector vs:vaH11005 Fe(us,e) (3)where Ferepresents the error model of the real five-axismachine and e the set of geometric errors.The task of model-based error compensation is to finda necessary correction vector du for each axis positionvector ussuch that despite the existing geometric errors,the tool takes the desired pose:vsH11005 Fe(usH11001 du,e) (4)The necessary condition for finding the correctionvector du is that all errors in the error model must beknown. There are different ways to find the vector du.Since the error model is highly nonlinear, iterations arenormally necessary to get a solution with acceptable tol-erance. For real-time error compensation, the iterativeapproach is not preferred. Since the errors are small, therelationship between the differential change of tool posein workpiece coordinates and the differential change inmachine axis coordinates can be assumed to be linear.This linear relationship in matrix form is known as theJacobian matrix 13. Because the solution for the for-ward transformation is always unique, it is better to useit to compute the Jacobian matrix:J H11005Ff,n(u)u(5)The computation of correction vector du is then verysimple by using the inverse Jacobian matrix:du H11005 JH110021dv (6)where JH110021is the inverse Jacobian matrix.3. Error modeling and identificationFig. 1 shows the target five-axis milling machine. Themachine can be modeled as an open kinematic chainwith several links connected in series by prismatic androtational joints. At one end of the kinematic chain isthe tool locked with the main spindle. The spindle blockis fixed on the Z-slide. The Z-slide moves verticallyalong the column with a prismatic joint. The column onthe other hand is bolted onto the machine bed. The otherend of the kinematic chain begins with the workpiece,which is fixed onto the base surface of the C-turntable.The C-turntable is integrated with the A-tilting head,which on the other hand is mounted on the X-table. The873W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 43 (2003) 871877Fig. 1. Five-axis milling machine of the type XH11032YH11032ZAH11032CH11032.C- and A-axis together contribute to the tilt motion ofthe workpiece. The X-table moves horizontally on the Y-table with a prismatic joint. The Y-table moves on themachine bed with a prismatic joint too.Fig. 2 illustrates the coordinate frames with constantoffset parameters Z0, Z1, Z2and Z3between coordinatesystems. The parameter Ltis tool length. After definingthe homogenous transformation matrix (HTM) 14 foreach kinematic component, the spatial relationshipbetween the workpiece coordinate frame and the toolcoordinate frame can then be expressed aswTtH11005wTbbTccTaaTxxTyyTzzTssThhTt(7)where the index t represents the coordinate frame of thetool, h the tool holder, s the spindle, x, y, z the threelinear axes, a and c the two rotary axes, b the base sur-face of the turntable and w the workpiece. Note that linkerrors in the rotary blockbTc,cTaandaTxare dominant,unknown and need to be estimated.The positioning with axis position vector u gives toolpose v. The tool tips position P = xwywzw is obtainedas follows:P 1TH11005wTt,i0001T(8)wherewTt,idescribes the ideal relationship between theworkpiece coordinate frame and the tool coordinateFig. 2. The coordinate frames of the five-axis milling machine.frame and can be obtained by setting all errors to zero.The vector 0 0 0 1 represents the origin point of thetool coordinate frame. Note that the workpiece coordi-nate frame is fixed on the turntable and rotates with therotary axes C and A. Therefore, the tool orientation vec-tor, represented by unit directional vector Q =iwjwkw, is determined only by the two rotary axesand can be obtained by transforming the unit vector0 0 1 in the tool coordinate frame into the workpiececoordinate frame:Q 0TH11005wTt,i0 0 0 1T(9)The nominal forward transformation of the target five-874 W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 43 (2003) 871877axis milling machine can be derived easily by using thematrixwTt,iand can be explicitly expressed as follows:xwH11005 zmsin(qc)sin(qa)H11002ymsin(qc)cos(qa) (10)H11002xmcos(qc)H11002Xw0ywH11005 zmcos(qc)sin(qa)H11002ymcos(qc)cos(qa) (11)H11001 xmsin(qc)H11002Yw0zwH11005 zmcos(qa) H11001 ymsin(qa)H11002Z3H11002Zw0(12)iwH11005 sin(qa)sin(qc) (13)jwH11005 sin(qa)cos(qc) (14)kwH11005 cos(qa) (15)where xm, ym, zm, qaand qcare the setting position ofthe servo-controlled X-axis, Y-axis, Z-axis, A-axis and C-axis, respectively. The axis position vector is u =xmymzmqaqc. Xw0, Yw0and Zw0are offsets betweenthe workpiece coordinate frame and the base surfacecoordinate frame.Explicit expressions of the overall position errors canbe obtained after carrying out matrix multiplications andsimplifying the equations by neglecting the second- andhigher-order terms. The tool pose error dv includes thetool tips position error dP = dxwdywdzw and theorientation error dQ = diwdjwdkw, both are definedin workpiece coordinates. The errors dP and dQ can beexpressed asdP 1TH11005wTt,i0 0 0 1TH11002wTt0001T(16)dQ 0TH11005wTt,i0 0 0 1TH11002wTt0 0 0 1T(17)The error model represented by Eqs. (16) and (17) isdifferent from that presented in 11. Eq. (16) describesthe overall position errors of the tool in the workpiececoordinate frame. In contrast, the error model in 11describes the overall position errors measured by theprobe sensors. Although these two models are different,most error components are the same.In the past, the error model was used for simulationsince major error components were not known. Today,the spherical test 11 with the 3D probe-ball providesa new approach. With the measured overall positionerrors, the unknown link errors can be identified by usingthe least square estimation (LSE) method.4. Compensation algorithmNote that the tool pose error vector dv is defined inworkpiece coordinates and the correction vector du isdefined in machine coordinates. A critical step in model-based error compensation is to derive the correction vec-tor du from the error vector dv. As mentioned above,the nominal forward transformation functions are usedto compute the Jacobian matrix describing the linearrelationship between differential changes in the work-piece coordinates and the machine coordinates. FromEqs. (10)(15), this linear relationship can be explicitlyexpressed asdxmH11005 (H11002dxwCcSaH11001 dywScSaH11001 diwCcSazm(18)H11002diwCcCaymH11002djwScSazmH11001 djwScCaym)/SadymH11005 (H11002dxwScC2aH11002dywCcC2aH11001 dzwCaSaH11001 diwS2aSczmH11001 diwCcSaC2axmH11002djwScSaC2axm(19)H11001 djwCcS2azm)/CadzmH11005 (dxwSaScCaH11001 dywCcSaCaH11001 dzwC2aH11002diwScymH11002diwCcCaxmH11001 djwScCaxm(20)H11002djwCcym)/CadqaH11005 (diwScH11001 djwCc)/Ca(21)dqcH11005 (diwCcH11002djwSc)/Sa(22)where Cc, Ca, Scand Saare the simplified operators ofcos(qc), cos(qa), sin(qc) and sin(qa), respectively. In Eqs.(18)(22), cos(qa) and sin(qa) appear in the denominatorand may cause problems.4.1. Case 1: qa= 0oIn this case, the C-turntable is in the horizontal pos-ition. The tool orientation vector is 0 0 1 in the work-piece coordinates and the five-axis machine is in itssingular position. The nominal backward transformationis not solvable because the C-axis may take any positionvalue. A small orientation deviation out of the singularposition may cause the C-axis to adapt abruptly. A simi-lar effect exists for five-axis machine tools of other kine-matic types. In path planning, the singular position mustbe avoided through tilt mounting of the workpiece. Inpractice, the singular problem may occur after 3D work-piece mounting correction. For the sake of safety, correc-tion with the C-axis must be suppressed. Only the orien-tation deviation in the YZ-plane is compensated with theA-axis. The corrections are as follows:dxmH11005H11002CcdxwH11001 ScdywH11002ymdjw(23)dymH11005H11002ScdxwH11002CcdywH11001 xmdjw(24)dzmH11005 dzw(25)dqaH11005 diw(26)4.2. Case 2: qa= 90oIn this case, the base surface of the C-turntable is ver-tical and the component kwis zero. The position of theC-axis determines the tool orientation in the XY-plane ofthe workpiece coordinate frame. The corrections are sim-plified as follows:875W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 43 (2003) 871877dxmH11005H11002CcdxwH11001 ScdywH11001 zmdjw(27)dymH11005 dzw(28)dzmH11005 ScdxwH11001 CcdywH11002xmdjw(29)dqaH11005 diw(30)dqcH11005 djw(31)5. Experimental resultsTo test the effectiveness of the proposed compen-sation method, the compensation function is developedand integrated in the CNC controller of the target five-axis milling machine. The self-developed CNC control-ler is an industrial PC with Pentium III microprocessorrunning at 330 MHz. The interface between the control-Fig. 3. The functional structure of real-time geometric error compen-sation.ler and the AC drivers is a motion card, which has five16-bit D/A converters for the output of velocity com-mands and five 24-bit counters for the encoder inputs.The motion control software has been developed inC/C+ language and runs under the real-time NT-RTXoperating system. The sampling time for the positioncontrol is 2 ms.The functional structure of the CNC software for thereal-time error compensation is shown in Fig. 3. Thereal-time kernel program performs the path and orien-tation interpolation. The NC path input can be definedin either machine coordinates or workpiece coordinates.If interpolation is performed in the workpiece coordi-nates, backward transformation based on the ideal kine-matic model follows to calculate the setting position inmachine coordinates. The error compensation is acti-vated in each real-time position control loop. The errormodel computes tool pose errors in workpiece coordi-nates for each setting axis position. From the tool poseerrors, the correction vector in machine coordinates isthen calculated.The measurement of overall position error and theestimation of unknown link errors are already describedin previous papers. Simulations show that errors in therotary axes block are dominant, contributing up to 80%of the overall position errors. In contrast, the linear axessystem contributes less than 20%.Fig. 4 shows some test paths for the real-time errorcompensation. Fig. 5 shows the measured overall pos-ition errors before and after the real-time error compen-sation, whereby path F is used for both estimation andaccuracy test. The accuracy enhancement is veryimpressive. The overall positioning accuracy can beimproved by a factor of 7.8 in the X direction, 3.2 in theFig. 4. The test path.876 W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 43 (2003) 871877Fig. 5. Results of error compensation for path F.Y direction and 8.2 in the Z direction. This high improve-ment rate is already predicted in simulation and justifiednow with real machine tests. These excellent resultscome from the fact that the non-measurable link errorsin the rotary block are dominant, repeatable and can beidentified accurately. The