A Novel Geometric Error Compensation Method for GantryMoving CNC Machine Regarding Dominant Errors


 萱军 蓟
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1 processes Article A Novel Geometric Error Compensation Method for GantryMoving CNC Machine Regarding Dominant Errors Hong Lu 1, Qian Cheng 1, Xinbao Zhang 2, *, Qi Liu 1, Yu Qiao 1 Yongquan Zhang 1 1 School of Mechanical Electronic Engineering, Wuhan University of Technology, Wuhan , China; (H.L.); (Q.C.); (Q.L.); (Y.Q.); (Y.Z.) 2 School of Mechanical Science Engineering, Huazhong University of Science Technology, Wuhan , China * Correspondence: Received: 7 July 2020; Accepted: 27 July 2020; Published: 31 July 2020 Abstract: Gantrytype computer numerical control (CNC) machines are widely used in manufacturing industry. A novel structure with moveable gantry is proposed to improve traditional gantrytype machine structure s disadvantage of taking up too much space. Geometric s have direct impacts on actual position of tool, which significantly reduces accuracy of machines. Errors of different components are always coupled have uncertain effects on total geometric. Thus, it is essential to find an effective way to identify dominant s do targeted. First, a novel identification method using value leaded global sensitivity analysis (VLGSA) is proposed to find dominant s. In VLGSA, weighting factors which show influence of range are used to modify multibody system (MBS) model. Results show that dominant s in three directions respectively contribute 80%, 86% 85% of total in ir directions. n, s identified by VLGSA are modeled by leastsquare linear fitting Bspline interpolation methods, respectively, according to feature of data. Finally, models are applied in a new realtime system developed on Beckhoff TwinCAT servo system. Experimental results from gantrytype CNC engraving milling machine show proposed method can help figure out most dominant s reduce around 90% of total. Keywords: gantrytype CNC machine; geometric ; multibody system; VLGSA; method 1. Introduction Gantrytype manufacturing equipment has already played an indispensable role in modern manufacturing industry. In traditional gantrytype machines, Xdirection feed movement is realized by motion of workbench, while gantry is fixed on machine bed. A vast room of more than twice feed stroke should be reserved to ensure processing capability, which causes waste of space. Thus, a novel structure with a moveable gantry is proposed. Different from traditional structure, workbench is fixed two ballscrew feed systems are symmetrically arranged to drive gantry. Only half space is needed comparing with traditional one. However, machines with moveable gantries require higher accuracy. Many factors are affecting accuracy of gantrytype machine tools, such as geometric, rmal, servo system so forth. rmal s widely appear in engineering machinery devices many researchers have done great jobs on optimizing heat transformation [1 4]. Geometric is anor common component usually Processes 2020, 8, 906; doi: /pr
2 Processes 2020, 8, of 21 donates most in final inaccuracy [5]. geometric s directly influence final position of tool nose s are generally related to tiny defect of machine itself, like wrong structure of feed system. re are two ways to resolve problem. One is adjusting physical structure, which is complicated expensive may be limited by skill of technical staff. or one is compensating for existing s by numerical control (NC) code or software, which is much more flexible can achieve high quality [6]. However, needs a clear understing of existing s. Error modeling is firstly required to reach goal. technique of geometric modeling has been persistently developed in recent decades. In application of robot industry, Paul, R.P. [7] Veitschegger, W.K. [8] gave methods using homogeneous transformation matrix differential transformation. precise location of robot s terminal point with existence of geometric size s kinematic s was found in ir research. se works established basis of analysis. Chen, J.X. et al. [5] took furr research based on fundamental principle of Paul Veitschegger, by analyzing differential movement differential transformation. y proposed a comprehensive method of both discovering how geometric propagates through every single axis. Zhao, Q.Q. et al. [9] established topology structure of machine tool using multibody system ory proposed two conceptions, which were relative motion matrices of axis connectors connection matrix of machine tool connectors. Based on homogeneous coordinate transformation ory, Zhou, X.T. [10] built model of classic threeaxis machine tool, decomposed final into s existing in components in whole motion chain. se works showed development application of different modeling methods, gave inspirations to furr research. Still, many of m were limited in modeling. Models based on se methods are usually complex mamatical expressions of a vast amount of items, making it a sophisticated work to compensate for all items. However, towards several specific items, which can be defined as dominant s, may provide enough accuracy improvement. Thus, more research could be done to simplify. Sensitivity analysis (SA) is a kind of method to quantify uncertainties of an unknown system using great data, which may help in finding most valuable items to compensate. For a specific model, a vast amount of different input output data can be obtained through all kinds of scientific sampling methods powerful calculation tools. Thus sensitivity analysis can help to identify weight each input has in affecting system outputs. SA consists of two main branches, one is local sensitivity analysis (LSA) or is called global sensitivity analysis (GSA). [11,12] GSA is more commonly used in manufacturing industry with its ability to find out result differences under changes of parameters. It can emphasize interaction power of parameters by using additional methods. Usually, SA is used in economic analysis. However, it is also found suitable in engineering field recently. Fan, J.W. et al. [13] established a prediction model of component tolerance proposed a sensitivity analysis based on component motion. proposed sensitivity analysis was experimentally verified several main parameters that donate to s most were found out. Yan, Y. et al. [14] established model of multiprocess assembling built sensitivity analysis model of parameters towards assembling using method of matrix differentiation. Li, J. Xue, F.G. [15] proposed general local sensitivity indices (GLSIs), general global sensitivity indices (GGSIs) general global sensitivity fluctuation indices (GGSFIs) based on traditional sensitivity analysis. A fiveaxis machine was analyzed using above indices. One common limit of above works is that sensitivity indices calculation is carried out on original model without considering impact of value range towards sensitivity analysis. Besides, sensitivity analysis of machine tools was only done for performance assessment, targeted was missing validity was not verified. Sensitivity analysis tells what to concentrate on; measurement fitting tell how to describe deal with m. With all kinds of advanced measurement equipment, precision data can be obtained. According to data, model can be established for. However, appropriate fitting methods are needed to ensure accuracy. To improve
3 Processes 2020, 8, of 21 manufacturing performance, many researchers have done great works. Fang, K.G. Yang, J.G. [16] considered rmal factors, used orthogonal polynomial method to build total model, compensated 90% of spindle geometric. Wang, W. et al. [17] used Newton interpolation method to construct rmal geometric model of computer numerical control (CNC) milling machine, established a realtime system. experiment based on Fanuc CNC machine was carried out to show system works well. Tuo, Z.Y. et al. [18] using cubic spline interpolation established rmal geometric model improved position accuracy of CNC machine. Bspline curve is commonly used in computer aided design (CAD) areas for its ability to describe continual material. It is also used in trajectory planning for its smooth character. Li, Y.H. et al. [19] used quantic Bspline curve to optimize motion path of pickplace robots. His work showed that Bspline curve is appropriate for smooth machine motion, which is also suitable for relative movement of machine tools. In this paper, multibody system ory was used to establish geometric model of gantrymoving engraving milling machine. n translation s in all three directions are extracted from matrix to analyze significant items which impact precision most. Value leaded global sensitivity analysis (VLGSA) is proposed used in above analysis process, in which measurement value of each item is considered to modify total geometric model. Bspline curve was developed into Bspline interpolation method used on result of machine measurement. A realtime system based on Beckhoff servo system, TwinCAT control software C# language was built to eliminate s. experiment verified validity of above crucial finding method. 2. Geometric Modeling Based on Multibody System multibody system (MBS) illustrates a kind of mechanical system in which a series of rigid or flexible bodies interconnect [20]; every single body may undergo translation rotation in different scales. It is a general systematic method for geometric modeling is widely used in analysis of robots, machine tools coordinate measurement. core of MBS is using loworderbody arrays to describe topology structure of system [20], using transformation matrices to calculate motion relationship. Usually, inertial coordinate system is chosen as body B 0, basic component is selected as body B 1, or elements are divided into several branches numbered along direction away from B 1, branch by branch Coordinate System Definition Topology Structure Description of GantryMoving CNC Machine In this paper, a gantrymoving engraving milling machine with threedimension movements is taken to describe ideal of presented geometric modeling method. structure of chosen machine Processes 2020, is shown 8, x FOR PEER in Figure REVIEW 1. 4 of 24 Figure 1. Figure Structure 1. Structure of of gantrymoving engraving milling milling machine. machine. B0bed, B1moveable B gantry, 0 bed, B 1 moveable B2Y B3Z direction slide, B4spindle with tool, B5workbench, B6workpiece. T01 gantry, B 2 Y direction slide, B 3 Z direction slide, B 4 spindle with tool, B 5 workbench, B 6 workpiece. T56 represent coordinate transformation between components. T 01 T 56 represent coordinate transformation between components. gantrymoving CNC machine has two servo motors in X direction. Two motors are symmetrically arranged to drive gantry through ballscrew feed systems. X1 X2 represent movement of two feed systems. synchronization between two movements is an important cause of Xdirection yaw. Since machine locates on a stable ground coordinate system, machine bed itself can be directly chosen as body B0. As orange lines show in Figure 1, engraving mill machine can be seen as a twobranch topology structure [21]. first branch (tool branch) consists of
4 Processes 2020, 8, of 21 gantrymoving CNC machine has two servo motors in X direction. Two motors are symmetrically arranged to drive gantry through ballscrew feed systems. X1 X2 represent movement of two feed systems. synchronization between two movements is an important cause of Xdirection yaw. Since machine locates on a stable ground coordinate system, machine bed itself can be directly chosen as body B 0. As orange lines show in Figure 1, engraving mill machine can be seen as a twobranch topology structure [21]. first branch (tool branch) consists of moveable gantry (B 1 ), Y direction slide (B 2 ), Z direction slide (B 3 ) spindle with tool on it (B 4 ). second branch (workpiece branch) consists of workbench (B 5 ) workpiece (B 6 ). Table 1 shows main sizes of se components. Table 1. Size of main components. Main Parameters of Components Value (mm) Size of workbench (width length) Stroke of gantry (Xdirection stroke) 2400 Stroke of Y slide (Ydirection stroke) 1200 Stroke of Z slide (Zdirection stroke) 240 final topology structure coordinate definition with position motion relationships are shown in Figure 2, where vectors between bodies show ir relations. Take B 0 B 1 as an example, T 01p means position transformation between B 0 B 1 ; T 01s means motion transformation between B 0 B 1 ; T 01pe T 01se represent of position motion transformations respectively. T 01p, T 01pe, T 01s T 01se constitute total coordinate transformation betweenprocesses B , B8, x 1 (TFOR 01 ). PEER REVIEW 5 of 24 Figure 2. Figure 2. topology topology structure coordinate definition of of machine. Orange Orange lines show lines show coordinate coordinate transformation transformation between between different bodies, which are are marked with with T01p, T01p, T01pe, T01pe, etc. etc. Table 2 shows loworderbody array of machine, in which L is loworderbody Table 2 shows loworderbody array of machine, in which L is loworderbody operator. operator. For body Bj, its nth order loworderbody number can be defined as Equation (1): For body B j, its nth order loworderbody number can be defined as Equation (1): n L(j) = i, (1) where n means that Bi is nth order loworderbody L n (j) = i of body Bj (1) Similarly, or equation can be inferred as Equations (2) (3): where n means that B i is nth order loworderbody n n 1 of body B L(j) = L(L (j)) j (2) 0 L(j) = j. (3) Table 2. Loworderbody Array of Chosen Machine. Body L1(j)
5 Processes 2020, 8, of 21 Similarly, or equation can be inferred as Equations (2) (3): L n (j) = L(L n 1 (j)) (2) L 0 (j) = j. (3) Table 2. Loworderbody Array of Chosen Machine. Body L1(j) L2(j) L3(j) 0 1 L4(j) Coordinate Transformation Model Establishment actual position motion between bodies can be expressed by matrix, more specifically by position matrix, motion matrix, position matrix motion matrix. After binging every single body to ir fixed coordinate, position motion transformation can be demonstrated by multiplication of a series of 4 4 homogeneous eigenmatrix [22]. For a threedimensional Cartesian coordinate system, translation can be expressed as Equation (4), in which x, y z are translation along three axes, respectively. Rotation around three axes can be expressed as Equations (5) (7) x y T = (4) z Rot(x, θ x ) = Rot(y, θ y ) = Rot(z, θ z ) = cosθ x sin θ x 0 0 sin θ x cos θ x 0 cos θ y 0 sin θ y sin θ y 0 cos θ y 0 cos θ z sin θ z 0 0 sin θ z cos θ z However, both structure motion exist in manufacturing process. Take Xdirection movement as an example (shown in Figure 3). Errors between two coordinates can be divided into six types, including three linear three angle. Linear s are position straightness along with two directions. Angle s are roll, pitch yaw. Similarly, geometric s in or directions can be defined as Table 3. (5) (6) (7)
6 sinθ cosθ 0 0 z z Rot(z, θ ) = z However, both structure motion exist in manufacturing process. Take X direction movement as an example (shown in Figure 3). Errors between two coordinates can be divided into six types, including three linear three angle. Linear s are position Processes 2020, 8, of 21 straightness along with two directions. Angle s are roll, pitch yaw. Similarly, geometric s in or directions can be defined as Table 3. (7) (a) (b) (c) (d) Figure 3. Error measurement. (a) Principle of position measurement; (b) Principle of straightness measurement; (c) Position measurement experiment; (d) Straightness measurement experiment. Figure 3. Error measurement. (a) Principle of position measurement; (b) Principle of straightness measurement; (c) Position measurement experiment; (d) Straightness measurement experiment. Table 3. Geometric s in chosen caving milling machine. Error Type Error Item Error Symbol Position x x Ydirection straightness y x Xdirection feed s Zdirection straightness z x Roll α x Pitch β x Yaw γ x Position y y Xdirection straightness x y Ydirection feed s Zdirection straightness z y Roll β y Pitch α y Yaw γ y Position z z Xdirection straightness x z Zdirection feed s Ydirection straightness y z Roll γ z Pitch β z Yaw α z XY perpendicularity γ xy Structure s XZ perpendicularity β xz YZ perpendicularity α yz According to principle of homogeneous matrix transformation, general matrix between B 0 B 1 which containing all six s can be calculated by multiplying six homogeneous eigenmatrices, as Equation (8) shows (where c for cos s for sin ). T 01e = T x x T y x T z x T θx x T θy x T θz x c θy x c θz x c θy x s θz x s θy x x x c θx x s θz x + s θx x s θy = x c θz x c θx x c θz x s θx x s θy x s θz x c θy x s θx x y x s θx x s θz x c θx x c θz x s θy x c θz x s θx x + c θx x s θy x s θz x c θx x c θy x z x (8)
7 Processes 2020, 8, of 21 In translation movement, three angle s are usually tiny, so approximation that sin θ θ cos θ 1 can be introduced. Putting approximation into Equation (8) ignoring secondorder highorder elements, equation can be simplified as Equation (9). T 01e = 1 θz x θy x x θz x 1 θx x y θy x θx x 1 z According to method above, transformation matrices of chosen engraving mill machine are listed in Table 4. Table 4. transformation matrix of chosen caring mill machine. Neighbor Bodies Ideal Position Motion Matrix Position Motion Error Matrix 1 γ xy β xz 0 γ T 01p = I 4 4 T pe = xy 1 α yz 0 β xz α yz x 1 γ x β x x x γ T 01s = T se = x 1 α x y x β x α x 1 z x (9) T 12s = T 23s = T 12p = I 4 4 T 12pe = I y T se = 1 γ y β y x y γ y 1 α y y y β y α y 1 z y T 23p = I 4 4 T 23p = I y T se = 1 γ z β z x z γ z 1 α z y z β z α z 1 z z T 34p = I 4 4 T 34pe = I 4 4 T 34s = I 4 4 T 34se = I 4 4 T 05p = I 4 4 T 05pe = I 4 4 T 05s = I 4 4 T 05se = I 4 4 T 56p = I 4 4 T 56pe = I 4 4 T 56s = I 4 4 T 56se = I 4 4 coordinate system eigenmatrix of two neighboring bodies can be written as Equation (10). Eigenmatrix with s can be written as Equation (11). T 01 = T 01p T 01pe T 01s T 01se (10) T 01e = T 01 T 01i. (11) Coordinate system eigenmatrix of any two bodies can be written as Equation (12). T SG = K=1 K=m,L m (S)=Q T K L (S)LK 1 (S) 1 J=1 J=n,L n (G)=Q T J L (G)LJ 1 (G) (12) As for topology structure of chosen engraving milling machine mentioned above, coordinate transformation eigenmatrix in both ideal situation actual situation (with s) should be calculated as Equation (13) T 64 = ( k=1 1 j=1 T L k (6)L (6)) k 1 T L j (4)L j 1 (4) k=2 j=4 (13) = (T 05 T 56 ) 1 T 01 T 12 T 23 T 34
8 Processes 2020, 8, of 21 geometric of machine can n be calculated by subtracting ideal transformation eigenmatrix actual transformation eigenmatrix between tool workpiece, like Equation (14). result T 64e is final matrix containing angle s linear s between tool coordinate system workpiece coordinate system. In planar milling, angle s represent small posture fault linear s along three axes have most significant impact on misalignment between tool nose target point on workpiece. 3. Dominant Errors Identification with VLGSA T 64e = T 64 T 64i (14) From perspective of black box, system models can always be demonstrated as a function Y = f(x), in which x is vector of several uncertain inputs of system y is one of system outputs which is chosen to be analyzed. In this paper, model targets total geometric of engraving milling machine. Y represents final X equals vector of each item like Xdirection position. By doing VLGSA to model function, effects of inputs on a particular output can be quantified into indices. In or words, impact of each item on total geometric can be described with numbers dominant s, which cause most of total, can be found Decomposition of Variance Indices Calculating In system, model function Y = f(x) is under assumption that inputs distribute independently within a unit uniform space. refore, actual input scales should be transformed into unit space [23]. function can be decomposed as Equation (15). Y = f 0 + d f i (X i ) + i=1 1 i<j d f ij (X i, X j ) f 12...d (X 1, X 2,..., X d ), (15) where f 0 is a component of constant; f i is function of X i, which represents effect X i donates alone ( geometric caused by one specific component); f ij is function with two inputs X i X j, representing effect to output given by X i X j toger. Similarly, or higher parts can be defined as above. This composition must be in condition that subcomponents are independent of each or. In or words, all components in equation are orthogonal. It can be shown in following Equation (16). 1 f i1 i 2...i s (X i1, X i2,..., X is )dx ik = 0, (k = 1, 2,..., s). (16) 0 It makes result that components of decomposition function can be defined in terms of expected values, as Equations (17) (19) show. f 0 = E(Y) (17) f i (X i ) = E(Y Xi ) f 0 (18) f ij (X i, X j ) = E(Y Xi, X j ) f 0 f i f j (19) Or components containing more inputs can be defined as equations above. f i which shows effect of change input X i can be known as primary interaction or firstorder interaction. f ij shows effect of changing both X i X j at same time. It can be known as secondorder interaction. higherorder interactions can be defined similarly.
9 Processes 2020, 8, of 21 Square integrate decomposition Equation (15), new equation is got can be organized as Equations (20) (21). 1 0 f2 (X)dX = f d 1 0 i=1 1 0 f i 2 dx i + 1 i<j d f 2 (X)dX f 0 2 = f ij 2 dx i dx j f 12...d 2 dx 1 dx 2... dx d (20) d j=1 1 i<j d f 2 i...j dx i... dx j (21) 0 It is found that left part of equation is variance of Y(X) right part is variance terms. This leads to decomposition with variance expression, shown as Equations (22) (24). Var(Y) = d V i + i=1 1 i<j d V ij V 12...d (22) V i = Var Xi (E X i (Y Xi )) (23) V ij = Var Xij (E X ij (Y X i, X j )) V i V j, (24) where X i represents whole series of variables except X i. n firstorder sensitivity index, which is direct sensitivity factor based on variance, can be defined as Equation (25). S i = V i Var(Y). (25) After estimating, component Var Xi (E X i (Y Xi )) can be written as Equation (26). Var Xi (E X i (Y X i )) 1 N N f(b) j (f(ab i ) j f(a) j ) (26) j=1 secondorder higherorder sensitivity index can be defined in same way, which represents effect of output variance causing by change of several inputs, simultaneously. sensitivity indices satisfy following principle. d S i + i=1 1 i<j d S ij S 12...d = 1 (27) For some functions which are simple in structure, firstorder sensitivity indices can be obtained by calculating integrals of decomposition function. However, more common way for most cases is estimating, which provides an easy solution to all kinds of functions it is often finished by Monte Carlo method. Monte Carlo method needs to generate a set of rom points in unit hypercube space, n use points to calculate use statistic method to estimate character of model. In actual work, rom points are usually pseudorom. most used points sequence is Latin hypercube sequence Sobol sequence. Several steps should be taken as follow to calculate sensitivity index using estimation method. 1. Generate a matrix of N 2d, in which N is number of sample points d is dimension of model (number of inputs). Points in se matrices should be romly distributed. 2. Generate matrix A using first d columns of above matrix matrix B using rest d columns. se two matrices distribute first two samples, which have n points in ddimension hypercube space.
10 Processes 2020, 8, of Generate d more matrices AB i by replacing ith column of matrix A with same column in matrix B. 4. Use above d + 2 matrices as inputs of model get d + 2 result matrices. ses matrices contain f(a) j, f(b) j f(ab i ) j mentioned in Equation (26). 5. Use matrices above to calculate sensitivity index with Equation (27) Machine Geometric Error Measurement for Value Leaded Factors Construction sample points of global sensitivity analysis distribute in unit hypercube space. However, items are different in scale usual sample method may cause inaccuracy in estimating actual impact of each item. Thus, a valueleaded factor based on measurement is proposed to modify geometric model, which normalizes s effects to same level. Keysight 5530 dualfrequency laser interferometer measurement system is used in measurement, cooperating with several kinds of mirror groups, 21 geometric s of engraving milling machine can be detected. Measurements are done with stard of ISO2301 processes are shown in Figure 3. This work includes many timeconsuming steps. For each component, measurement is done in different feed speeds, with range from 500 mm/min to 6000 mm/min interval of 500 mm/min. measurement stroke is x = 2000 mm, y = 1000 mm, z = 200 mm, which covers main processing area. start points are set in x = 200 mm, y = 100 mm z = 20 mm in machine coordinate system to leave enough space for installing measurement instruments. All measurements are done for three times. scale of each ir ratios are obtained from measurements. y are used for normalization of sample of global sensitivity analysis. measured s are shown in Table 5. Table 5. Max value in whole stroke. No. Item Max Value No. Item Max Value 1 x x mm 12 γ y mm/1000 mm 2 y x mm 13 z z mm 3 z x mm 14 x z mm 4 α x mm/1000 mm 15 y z mm 5 β x mm/1000 mm 16 γ z mm/1000 mm 6 γ x mm/1000 mm 17 β z mm/1000 mm 7 y y mm 18 α z mm/1000 mm 8 x y mm 19 γ xy mm/1000 mm 9 z y mm 20 β xz mm/1000 mm 10 β y mm/1000 mm 21 α yz mm/1000 mm 11 α y mm/1000 mm mm/1000 mm means deviation caused by α x in stroke of 1000 mm is mm. Or angle s are described in same way Indices Calculation Global Sensitivity Analysis analysis program is made in Matlab according to steps of proposed VLGAS method. geometric model of engraving milling machine built with multibody system ory is adjusted for normalization based on measurement. Putting model into analysis program setting analysis parameters, firstorder sensitivity indices in three directions are calculated listed in Table 6. angle s are at a very low level, highorder elements linking to angle s can be eliminated from MBS model. Thus, some elements sensitivity indices are zero in a particular direction. For Xdirection movement, as shown in Figure 4, with geometric s changing in a certain range, sensitivity indices of x x, x y x z are largest contribute 80% of total value toger. For Y direction, most dominant element is y y, which has highest sensitivity index value of 0.424, or two dominant elements are y x y z, se three elements contribute about
11 Processes 2020, 8, of 21 86% of total value. Similarly, three linear s z x, z y z z have largest impact on Z direction, with proportion of around 85%. Number Table 6. Firstorder sensitivity index of each item. Item FirstOrder Sensitivity XDirection YDirection ZDirection 1 x x y x z x α x β x γ x y y x y z y β y α y γ y z z x z y z γ z β z α z γ xy β xz α yz data is drawn shown in Figure 4 to demonstrate clearly. In conclusion, for chosen engraving milling machine, dominant s are mainly Processes 2020, 8, x FOR PEER REVIEW 14 of 24 linear items, specifically speaking position two straightness s from or two feed directions. feed directions. In orin words, or words, aiming at at se se three three s s can provide can provide remarkable remarkable improvement improvement to chosen to chosen machine machine with with minimum cost. (a) (b) (c) Figure 4. Figure Image4. Image of firstorder of sensitivity index. index. (a) Firstorder (a) Firstorder sensitivity of sensitivity Xdirection ; of (b) Xdirection Firstorder sensitivity of Ydirection ; (c) Firstorder sensitivity of Zdirection. ; (b) Firstorder sensitivity of Ydirection ; (c) Firstorder sensitivity of Zdirection. 4. Error Compensation In this section, data in whole stroke are listed modeled based on ir character. Different models are applied to a novel realtime system compared. linear in X direction is chosen as an example. Errors in or directions can be dealt same way. As shown in last section, main components, which have largest impact on Xdirection linear, are Xdirection position, Xdirection straightness along with Y direction movement Xdirection straightness along with Zdirection movement. y all have a close relation to feed movement, which means y can be compensated by
12 Processes 2020, 8, of Error Compensation In this section, data in whole stroke are listed modeled based on ir character. Different models are applied to a novel realtime system compared. linear in X direction is chosen as an example. Errors in or directions can be dealt same way. As shown in last section, main components, which have largest impact on Xdirection linear, are Xdirection position, Xdirection straightness along with Ydirection movement Xdirection straightness along with Zdirection movement. y all have a close relation to feed movement, which means y can be compensated by adjusting feed trajectory Position Error Model Based on LeastSquare Method Position s are measured three times on each point average values are calculated to reduce measurement. values are listed in Tables 7 9. Position (mm) Table 7. Position in X direction. Error (µm) First Second Third Average Origin of position located on x = 200 mm in machine coordinate system. Position (mm) Table 8. Position in Y direction. Error (µm) First Second Third Average Origin of position located on y = 100 mm in machine coordinate system. According to tables, position s show an obverse trend of linear distribution in whole stroke. value accumulates through feed movement maximum value appears at end of stroke. maximum values in three directions are , , respectively. Thus, least square method is chosen to establish mamatical model of position s to describe linear trend.
13 Processes 2020, 8, of 21 Position (mm) Table 9. Position in Z direction. Error (µm) First Second Third Average Processes 2020, 8, x FOR PEER REVIEW 16 of 24 Origin of position located on z = 20 mm in machine coordinate system. end of stroke. maximum values in three directions are , , respectively. Thus, least square method is chosen to establish mamatical model of leastsquare method is a commonly used fitting method, in which fitting curve not position s to describe linear trend. necessarily cross leastsquare sample points method but is a makes commonly se used points fitting as method, closein aswhich possible fitting to curve curve. not For sample points necessarily (x i, y i ), cross a series sample of equations points but can makes bese built points as used close toas create possible to function curve. For of points, sample points ( x residual sum of squares n i, y i ), a series δ 2 of equations can be built used to create function of points, is chosen to decide best coefficients of unknown function. i residual sum of squares i=1 n 2 δ is chosen to decide best coefficients of unknown function. i i= 1 fitting functions are shown as Equations (28) (30). fitting curves of position s are fitting functions are shown as Equations (28) (30). fitting curves of position s are shown in Figure 5. shown in Figure 5. y = x (28) y = x (28) y = 0.071x (29) y = 0.071x (29) y = 0.108x y = (30) (30) (a) (b) (c) Figure 5. Image Figure of 5. fitting Image of curves. fitting curves. (a) Fitting (a) Fitting curve of Xdirection position position ; ; (b) Fitting (b) curve Fitting of Ycurve of direction position ; (c) Fitting curve of Zdirection position. Ydirection position ; (c) Fitting curve of Zdirection position Straightness Error Model Based on BSpline Interpolation. straightness s are also measured three times to calculate average value. curve of s is drawn in Figure 6 to show intuitively.
14 Processes 2020, 8, of Straightness Error Model Based on BSpline Interpolation. straightness s are also measured three times to calculate average value. curve of Processes 2020, 8, x FOR PEER REVIEW 17 of 24 s is drawn in Figure 6 to show intuitively. (a) (b) Figure 6. Figure Image 6. of Image straightness of straightness. (a). Straightness (a) Straightness in Xdirection in Xdirection movement; movement; (b) Straightness (b) Straightness in Ydirection movement; (c) Straightness in Zdirection movement. in Ydirection movement; (c) Straightness in Zdirection movement. According to Figure 6, max value of straightness s is lower than position s. According to Figure 6, max value of straightness s is lower than position s. curves fluctuate with total trend of increasing first n decreasing, except Ydirection curves fluctuate along with with Xdirection totalmovement. trend of increasing maximum first values respectively n decreasing, appear in except point Ydirection of along x1 = 1600 withmm, x2 Xdirection = 1000 mm, movement. y1 = 400 mm, y2 = 500 maximum mm, z1 = values 100 mm respectively z2 = 100 mm appear for insix point of x 1 = straightness 1600 mm, xs. 2 = 1000 mm, distortion y 1 = 400 of mm, guides y 2 = during 500 mm, manufacturing z 1 = 100 mm is leading z 2 = 100 cause mmof for six straightness s. distortion of guides during manufacturing is leading cause of Bspline curve is a kind of unique curve, which is developed from Bezier curve is straightness s. constructed by linear combination of Bspline base curves. Bspline curves have many excellent properties, Bspline such curve as geometric is a kind invariance, of unique convex curve, hull, which convexity is developed retention, from variation reduction Bezier curve is constructed local support by linear so forth. combination Bspline surface of Bspline reconstruction base curves. is one of Bspline research curves hotspots have many critical excellent properties, technologies such asof geometric reverse engineering. invariance, refore, convexin hull, this paper, convexity Bspline retention, interpolation variationis reduction used to local support establish so mamatical forth. Bspline surface model of reconstruction straightness is, onewhich of can research describe hotspots continual critical technologies material of property reverse of engineering. guide. Bspline refore, base in function this paper, curve construction Bspline interpolation function are listed is used to in Appendix A. establish mamatical model of straightness, which can describe continual Exping measurement point number doing simulation with material interpolation property of result, guide. it is shown Bspline that values base function of residual curve keep construction at a low level function after are listed in Appendix interpolation A.. Interpolation results are shown in Figure 7. Exping measurement point number doing simulation with interpolation result, it is shown that values of residual keep at a low level after interpolation. Interpolation results are shown in Figure 7. (c)
15 Processes 2020, 8, of 21 Processes 2020, 8, x FOR PEER REVIEW 18 of 24 (a) (b) (c) (d) (e) (f) Figure Figure 7. Compensation 7. Compensation simulation simulation with with Bspline Bspline interpolation model. model. (a) (a) Ydirection straightness in Xdirection in Xdirection movement; movement; (b) Zdirection (b) Zdirection straightness straightness in Xdirection in Xdirection movement; movement; (c) Xdirection (c) direction straightness in Ydirection movement; (d) Zdirection straightness in Ydirection straightness in Ydirection movement; (d) Zdirection straightness in Ydirection movement; movement; (e) Xdirection straightness in Zdirection movement; (f) Ydirection straightness (e) Xdirection straightness in Zdirection movement; (f) Ydirection straightness in Zdirection movement. in Zdirection movement Compensation 4.3. Compensation System System Experiment Experiment Aiming to, many researchers in previous works try to change G code Aiming to, many researchers in previous works try to change G code according to modeling result, which performs well in reducing value [24]. according However, to this kind of modeling method need result, prework which for performs each program well in cannot reducing be realtime. With value [24]. However, advantages this kind of Beckhoff s of method open need CNC prework system, for eachmodel program can help to cannot improve bemachine realtime. accuracy With advantages in a more of Beckhoff s effective open realtime CNCway. system, can help to improve machine accuracy in a more effective realtime way. strategy combines virtual axis electronic cam technology based on TwinCAT control system of Beckhoff. core of strategy is uploading model points into cam table to help change actual tool path offsetting trajectory deviation. Firstly,
16 Processes 2020, 8, x FOR PEER REVIEW 19 of 24 strategy combines virtual axis electronic cam technology based on Processes 2020, 8, of 21 TwinCAT control system of Beckhoff. core of strategy is uploading model points into cam table to help change actual tool path offsetting trajectory deviation. Firstly, a virtual axis is established to accept motion comm. virtual axis is a kind of electronic symbol in control software. It It does not connect to any physical component only has digital motion simulation. refore, it itrepresents represents ideal ideal motion motion state. state. Secondly, Secondly, set set virtual virtual axis axis as as master master create create anor anor axis axis symbol symbol connecting connecting to to physical physical axis axis motor motor as as slave. slave. Thus, Thus, slave slave axis axis can can move move with with master master axis, axis, which whichis is virtual virtualaxis axis has hasno physical output, according to to specific specific rules. rules. Thirdly, Thirdly, set set rule rule between between two axes two as axes electronic as electronic cam, n cam, n physical axis physical can do axis superimposed can do superimposed movements movements of both ideal of both movements ideal movements camcurve camcurve movements movements based on based on models. models. cam cam table table can can be created be created in in two two ways: ways: cam cam design design tool tool assembled in TwinCAT software software user user interface interface linking linking to softto programmable soft programmable logic controller logic controller (PLC) program. (PLC) Soft program. PLC method Soft PLC has method better realtime has better property realtime property table value can table bevalue changed can in be changed feed process. in feed process. system can be described system in can Figure be described 8. hardware in Figure of 8. hardware system of is shown in Figure system 9. is shown in Figure 9. Figure 8. structure of system. Figure 9. hardware of system. Suppose that x is ideal position when feed system moves along X direction xδx is is position generated in in feed feed stroke, stroke, n n cam cam table table value value in this in this position position should should be set be as set x. as Δx Give. Give motion motion comm comm to to virtual virtual axis, axis, electronic electronic cam cam offset offset will will compensate compensate in position x. With help of Beckhoff ErCAT communication system, massive data can be transmitted in realtime. refore, coordinate information in all directions can be read cam
17 Processes 2020, 8, x FOR PEER REVIEW 20 of 24 Processes 2020, 8, x FOR PEER REVIEW 20 of 24 Processes 2020, data of 21 in 8,position x. With help of Beckhoff ErCAT communication system, massive can transmitted refore, coordinate information in all directions be read be in positioninx.realtime. With help of Beckhoff ErCAT communication system,can massive data cam table can be modified with calculated value based on models, can can be transmitted realtime. refore, coordinate information in all can be read table be modifiedinwith calculated value based on directions models, without delay. without process expressed in Figure 10.value based on models, camdelay. table can be modified with is calculated process is expressed in Figure 10. without delay. process is expressed in Figure 10. Figure 10. process of proposed method. Figure processof of proposed proposed method. Figure process method. system is composed of both hardware software. hardware platform composedof ofboth bothhardware hardware software. software. hardware platform system system is is composed hardware platform of system is provided by Beckhoff industrial personal computer (PC) ofof by Beckhoff Beckhoffindustrial industrialpersonal personal computer (PC) system system is provided provided by computer (PC) software is written in C# language. With Beckhoff ADS communication protocol, C# language. language.with With Beckhoff Beckhoff ADS communication protocol, software software is is written written in C# ADS communication protocol, data in PLC Servo controller areare easy toto bebe dealt with algorithm inin software data PLC Servo controller easy dealt with algorithm software data inin PLC Servo controller are easy to be dealt with algorithm in software n transmitted back. n ntransmitted transmittedback. back. Straightness is measured verify effect. Straightness after after is effect. Straightness after ismeasured measuredtoto toverify verify effect. models built by three or usual fitting methods (Polynomial Fitting, Newton Interpolation Cubic models built by three or usual fitting methods (Polynomial Fitting, Newton Interpolation models built by three or usual fitting methods (Polynomial Fitting, Newton Interpolation Spline are usedare to compare with with chosen method ofmethod Bspline interpolation. As for CubicInterpolation) SplineInterpolation) Interpolation) are used to chosen of of Bspline interpolation. Cubic Spline used to compare compare with chosenmethod Bspline interpolation. total effect, a space movement of x = 2000 mm, y = 1000 mm z = 200 mm carried for total total effect, effect, aa space mm, y =y 1000 mm z = z200 mmmm AsAsfor spacemovement movementofofx x= = mm, = 1000 mm =is200 out Xdirection total is measured. values are measured by absolute grating is carried out Xdirection total is measured. values are measured by absolute is carried out Xdirection total is measured. values are measured by absolute grating rulers. total isatcurbed at a low level, both modeling method rulers. Ifrulers. total is curbed a low level, both method sensitivity grating IfIf total is curbed at a low level, bothmodeling modeling method sensitivity analysis method can be verified to be effective, for only dominant components analysis method can be verified to be effective, for only dominant components analyzed by sensitivity analysis method can be verified to be effective, for only dominant components analyzed by method are compensated. Compensation resultsin are shown11. in Figure 11. method are compensated. Compensation results are shown Figure analyzed by method are compensated. Compensation results are shown in Figure 11. (a) (a) (b) Figure 11. Cont. (b)
18 Processes Processes 2020, 2020, 8, 8, 906 x FOR PEER REVIEW 21 of 1824 of 21 (c) Figure Compensation experiment result. (a) Residual Ydirection straightness in in Xdirection movement before after ; (b) Residual Zdirection straightness in in Xdirection movement before after ; (c) (c) Residual Residual Xdirection Xdirection total total before before after after. In In Figure 11a,b, straightness s along Xdirection movement are are shown. five five data data curves curves are are respectively residual residual of four of chosen four chosen models, models, including including proposed Bspline proposed model Bspline three model commonly three used commonly fitting used models fitting models without anywithout. any Three. comparisonthree models comparison are Newton models are interpolation Newton model, interpolation cubicmodel, spline interpolation cubic spline interpolation polynomial model. polynomial From image, model. From it is obvious image, that it is obvious that ability of ability Bspline interpolation of Bspline model interpolation is most significant. model is Newton most significant. interpolation Newton model interpolation has good performance model has good in performance in middle part of stroke but it also shows Runge s phenomenon, which leads middle part of stroke but it also shows Runge s phenomenon, which leads to huge residual to huge residual in beginning end of movement. polynomial model has a in beginning end of movement. polynomial model has a good ability to good ability to follow trend of curve had has a smooth residual curve. With follow trend of curve had has a smooth residual curve. With polynomial model, polynomial model, process avoids steep fluctuation of velocity acceleration. process avoids steep fluctuation of velocity acceleration. cubic spline cubic spline interpolation model has same capability of eliminating with Bspline interpolation model has same capability of eliminating with Bspline interpolation model interpolation model in most positions but still has defects in beginning end of in prediction. most positions Bspline but still interpolation has defects in method beginning is a kind of sectional end interpolation, of prediction. which means Bspline interpolation shape of method interpolation is a kindcurve of sectional only depends interpolation, on nearby which points. means Thus, shape curve of will interpolation not be curve influenced only depends by Runge s on nearby phenomenon. points. Thus, massive curve jump willof not both be velocity influenced byacceleration Runge s phenomenon. does not exist massive in jump interpolation of both velocity curve because acceleration of local doesadjustable not exist inproperty, interpolation Bspline curve interpolation because ofcan local fit irregular adjustable property, data without Bspline influence interpolation fitting canof fitor irregular flat areas. data In conclusion, without influence fitting of orbased flat areas. on In conclusion, Bspline interpolation model based improves on Bspline machine s interpolation straightness model improves accuracy to a machine s remarkable straightness level. Ydirection accuracy to a remarkable Zdirection level. straightness Ydirection s in Xdirection Zdirection straightness movement s reduce into 2.92 Xdirection μm 5.27 movement μm from reduce to μm 2.92 µm80.62 μm µm total fromstraightness µm accuracy µm. has total been straightness improved by accuracy around 95%. has been improved by around 95%. Figure Figure 11c 11c shows total total of of Xdirection position in space movement with a stroke of x = of 2000 x = 2000 mm, mm, y = 1000 y = 1000 mm mm z = 200 z = 200 mm. mm. Two Two curves curves respectively respectively show show before before after. after. max value max value before before appears appears at at end end of of stroke, stroke, with with around around 110 µm. 110 After μm., After, total total shows shows a trend a of trend fluctuating of fluctuating enlargement, enlargement, with with max of 9.88 μm. In conclusion, after aiming at dominant max of 9.88 µm. In conclusion, after aiming at dominant components components calculated by VLGSA method, total of X direction in space movement calculated by VLGSA method, total of X direction in space movement reduces around reduces around 90%. remarkable accuracy improvement of machine shows validity of 90%. remarkable accuracy improvement of machine shows validity of using VLGSA using VLGSA in finding key s. Meanwhile, considering VLGSA method is applied to in finding key s. Meanwhile, considering VLGSA method is applied to MBS model, MBS model, model accuracy is also verified. model accuracy is also verified. 5. Conclusions
19 Processes 2020, 8, of Conclusions Structural motional geometric s widely exist in CNC machines. y are most common factors reducing precision machine accuracy, affecting quality of products. refore, characterizing trends, finding most dominant items finally decreasing total geometric play an essential role in development of modern industry. However, considering complexity of precision CNC machine tools, process of targeted can be sophisticated. following works have been done in this paper to achieve goal of improving machine accuracy. 1. A geometric model was established with MBS method. By building topological structure relationship, sophisticated transmitting was decoupled into calculation of several matrices, which can be measured separately. 2. Based on MBS model, items dominant most to total geometric were identified by value leaded global sensitivity analysis (VLGSA), in which actual range of s was considered in process of sensitivity analysis. interaction of different parameters was analyzed in proposed analysis method. 3. Appropriate fitting methods were applied to measured data. data was measured by a highprecision dualfrequency laser interferometer. Considering its excellent ability to describe continual material significant usage in CAD, Bspline interpolation method developed on Bspline curve was used to establish straightness model. 4. A realtime online system was built based on Beckhoff TwinCAT servo system. A experiment was carried out using designed system. straightness in Xdirection movement was compensated with four models to verify ability of Bspline interpolation model. Xdirection in a space movement of x = 2000 mm, y = 1000 mm z = 200 mm was compensated as a sample. remarkable reduction of geometric in Xdirection showed above research was effective. However, this research is based on CNC machine operating at a single speed in cold start temperature. working condition is more sophisticated in actual manufacturing process. Thus, furr works need to concentrate more on relationship of geometric s working condition, such as acceleration changing workforce. Author Contributions: Conceptualization, H.L. Q.C.; methodology, X.Z. Q.C.; software, Q.C. Y.Q.; validation data curation, Q.C. Q.L.; writingoriginal draft preparation, Q.C.; writingreview editing, Q.C. Y.Z. All authors have read agreed to published version of manuscript. Funding: This research was funded by National Natural Science Foundation of China (Grant no ), Hubei Province Intellectual Property Guidance Development Project (Grant no ) Excellent Dissertation Cultivation Funds of Wuhan University of Technology (Grant no YS035). Conflicts of Interest: authors declare no conflict of interest. Appendix A For n+1 control points P i (i = 0, 1,..., n) a knot vector U = {u 0, u 1,..., u m }, a feature polygon can be formed by connecting control points. A k + 1 order Bspline curve can be expressed as follow (2 k n + 1): n C k (u) = N i,k (u)p i (A1) i=0 where N i,k (u) is k order Bspline base function, whose recursive function is shown in Equation (A1). N i,0 (u) = N i,k (u) = (u u i)n i,k 1 (u) u i+k u i (A2) { 1, ui u u i+1 0, else + (u i+k+1 u)n i+1,k 1 (u) u i+k+1 u, u i+1 k < u < u k+1
20 Processes 2020, 8, of 21 References 1. Sarafraz, M.M.; Peyghambarzadeh, S.M. Experimental study on subcooled flow boiling heat transfer to water diethylene glycol mixtures as a coolant inside a vertical annulus. Exp. rm. Fluid Science 2013, 50, [CrossRef] 2. Goodarzi, M.; D Orazio, A.; Keshavarzi, A. Develop nano scale method of lattice Boltzmann to predict fluid flow heat transfer of air in inclined lid driven cavity with a large heat source inside, Two case studies: Pure natural convection & mixed convection. Phys. Stat. Mech. Appl. 2018, 509, Sarafraz, M.M. Experimental investigation on pool boiling heat transfer to formic acid, propanol 2butanol pure liquids under atmospheric pressure. J. Appl. Fluid Mech. 2013, 6, Goodarzi, M.; Amiri, A.; Goodarzi, M.S. Investigation of heat transfer pressure drop of a counter flow corrugated plate heat exchanger using MWCNT based nanofluids. Int. Commun. Heat Mass Transf. 2015, 66, [CrossRef] 5. Chen, J.X.; Lin, S.W.; Zhou, X.L. A comprehensive analysis method for geometric of multiaxis machine tool. Int. J. Mach. Tools Manuf. 2016, 106, [CrossRef] 6. Bai, H.; Shen, J.X. Synsis modeling of CNC machine tool based on multibody system ory. Aeronaut. Manuf. Technol. 2017, 1, Paul, R.P.; Zhang, H. Computationally efficient kinematics for manipulators with spherical wrists based on homogeneous transformation representation. Int. J. Robot. Res. 1986, 5, [CrossRef] 8. Veitschegger, W.K.; Wu, C.H. Robot accuracy analysis based on kinematics. IEEE J. Robot. Autom. 1986, 2, [CrossRef] 9. Zhao, Q.Q.; Jun, H.; Liu, Z.G. Modeling method on motive axes transfer chain for machine tool of arbitrary topological structure. J. Mech. Eng. 2016, 52, [CrossRef] 10. Zhou, X.T. Research on Machining Accuracy Prediction Based on Machine Tool Error Modeling. Master s sis, Shong University, Shong, China, 20 March Chhatre, S.; Francis, R.; Zhou, Y.H.; Titchener, H.N.; King, J.; Keshavarz, M.E. Global sensitivity analysis for determination of parameter importance in biomanufacturing processes. Biotechnol. Appl. Biochem. 2008, 51, [CrossRef] 12. Cossarini, G.; Solidoro, C. Global sensitivity analysis of a trophodynamic model of Gulf of Trieste. Ecol. Model. 2008, 212, [CrossRef] 13. Fan, J.W.; Tao, H.H.; Wu, C.J.; Tang, Y.H. Method of sensitivity analysis of machine tools. J. Beijing Univ. Technol. 2019, 45, Yan, Y.; Wang, G.; Zhang, F.P. Precision assembly geometric sensitivity analysis based on transformation model for precision assembly. Trans. Beijing Inst. Technol. 2017, 37, Li, J.; Xue, F.G.; Liu, X.J. Geometric modeling sensitivity analysis of a fiveaxis machine tool. Int. J. Adv. Manuf. Technol. 2015, 82, [CrossRef] 16. Fan, K.G.; Yang, J.G.; Yang, L.Y. Orthogonal polynomialsbased rmally induced spindle geometric modeling. Int. J. Adv. Manuf. Technol. 2013, 65, [CrossRef] 17. Wang, W.; Zhang, Y.; Yang, J.G. Geometric rmal for CNC milling machines based on Newton interpolation method. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2012, 227, [CrossRef] 18. Tuo, Z.Y.; Huang, Y.Q.; Shen, M.W. Modeling of geometric rmal complex position of CNC machine tools based on cubic spline interpolation. J. Shanghai Jiao Tong Univ. 2016, 50, Li, Y.H.; Huang, T.; Derek, G.C. An approach for smooth trajectory planning of highspeed pickplace parallel robots using quantic Bsplines. Mech. Mach. ory 2018, 126, [CrossRef] 20. Guo, C.; Yang, L.; Li, Q.Y. Modeling of volumetric s for NC machine tools on multibody system ories. Mach. Des. Manuf. 2005, 3, Liu, Z.F.; Li, D.D.; Liu, Z.Z. Gantry machining tool assembly method predictive optimization based on multibody system. Comput. Integr. Manuf. Syst. 2014, 20, Fan, J.W.; Tang, Y.H.; Wang, H.L. Error modeling identification technology for a CNC camshaft grinder. J. Vib. Shock 2017, 36,
21 Processes 2020, 8, of Yu, D.J.; Li, R. Application of Sobol method to sensitivity analysis of a nonlinear passive vibration isolators. J. Vib. Eng. 2004, 17, Gao, X.; Tong, H.; Zhou, L. Geometric of CNC machine tools based on Gcode modification. Manuf. Technol. Mach. Tool 2015, 1, by authors. Licensee MDPI, Basel, Switzerl. This article is an open access article distributed under terms conditions of Creative Commons Attribution (CC BY) license (
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