Journal of
Manufacturing and Materials Processing
Article
Using the Segmented Iterative Learning Control Method to Generate Volumetric Error-Compensated Part Programs for Three-Axis CNC Milling Machine Tools Ying-Chen Lu 1 and Syh-Shiuh Yeh 2, * 1 2
*
ID
Institute of Mechatronic Engineering, National Taipei University of Technology, Taipei 10608, Taiwan;
[email protected] Department of Mechanical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan Correspondence:
[email protected]; Tel.: +886-2-2771-2171
Received: 17 July 2018; Accepted: 10 August 2018; Published: 13 August 2018
Abstract: This study proposes using the iterative learning control method to adjust the volumetric error-compensated tool path, where the working volume motion accuracy of three-axis computerized numerical control (CNC) milling machine tools is increased by segmented modification of the part program. As the three-axis CNC milling machine tools generally have volumetric error of working volume, this study refers to the measured and established table of volumetric errors and uses the method of the modifying part program for volumetric error compensation of machine tools. This study proposes using part-program single-block positioning segmented for volumetric error compensation, as the generated compensated part program with multiple compensated blocks can effectively compensate the volumetric error of working volume in the tool moving process. In terms of the compensated tool path computing method, this study uses the iterative learning control (ILC) method and refers to compensated tool path and volumetric errors along the compensated tool path for iterative computation. Finally, a part program with multiple blocks is modified by the converged optimal compensated tool path, in order that the modified part program has higher-precision volumetric error compensation effect. The simulation result shows that the rate of improvement of error of the volumetric error compensation method proposed in this study is 70%. The result of cutting tests shows that the average rate of improvement of the straightness error of the test workpiece is 60%, while the average rate of improvement of height error is 80%. Therefore, the results of simulation and cutting tests can prove the feasibility of using the ILC method for segmented modification of the volumetric error-compensated part programs proposed in this study. Keywords: iterative learning control; volumetric error; part program; CNC machine tools
1. Introduction Manufacturing mechanical parts with higher accuracy has been the machining target of various computerized numerical control (CNC) machine tools through the ages, as the conventional pitch error measurement and compensation methods cannot meet the requirement. Using volumetric error measurement and implementing the geometric error compensation of machine tool working volume according to this measurement result have become an important method to improve machine tool machining quality [1–5]. There are six static geometric errors resulting from the axial movement of three-axis CNC milling machine tools, including displacement errors and angular errors. The displacement errors are linear displacement errors, vertical straightness errors, and horizontal straightness errors. The angular errors are pitch angular errors, yaw angular errors, and roll angular J. Manuf. Mater. Process. 2018, 2, 53; doi:10.3390/jmmp2030053
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errors. In addition, the three-axis CNC milling machine tools have three squareness errors; therefore, for the three-axis CNC milling machine tools, there are 21 static geometric errors, leading to the volumetric error of the working volume of the machine tool [6–10]. The volumetric error has significant effect on the motion accuracy of a machine tool, and severely influences the geometric dimension accuracy of the workpiece, thus, measurement and compensation are required [11–16]. Xiang et al. [17] developed a volumetric error compensation technique for real-time and simultaneous compensation of the volumetric errors of multiple machine tools by integrating an ethernet distributed numerical control system with a generalized volumetric error kinematic model, which was also developed in this study. Due to the increased importance of integrating metrology systems to machine tools to online compensate volumetric errors of machine tools for improving machining efficiency and accuracy, Wang and Maropoulos [2] developed a real-time volumetric error compensation algorithm to improve the dynamic path accuracy of a machine tool using a laser tracker-assisted 3-axis positioning system. Holub et al. [18] identified volumetric errors of a small three-axis vertical machining center using a laser interferometer, and ballbar tests were used to validate the volumetric error compensation results. Later, Holub et al. [19] performed on-the-fly tracking experiments on the small three-axis machining center to measure the volumetric errors of the small three-axis machining center using a self-tracking interferometer, and then compared the measurement time and results with conventional trigger- or time-based measurements to show the superior performance of on-the-fly measurement. Circular tests based on ISO 230-4 were also employed for verification. Owing to the influence of thermal distortion on the manufacturing accuracy of large machine tools, Gomez–Acedo et al. [20] developed a thermal distortion compensation system using parametric state-space representation as the model architecture, and the Kalman filter as the estimator for the thermal drift of the machine tool center point for different spindle speeds and temperatures of the motor gearbox and room air, respectively. Later, Gomez–Acedo [21] developed a new methodology for the measurement of thermal distortion in large machine tools using a laser tracking interferometer to achieve low measurement uncertainty. The measurement method for the angular thermal distortion of the horizontal plane was also developed by referring to the temperatures of the machine headstock and column. López De Lacalle et al. [22] developed a methodology for the geometric accuracy estimation of five-axis machine tools using the Denavit–Hartenberg (DH) convention. The assembly errors were considered as the additional geometric parameters in DH transformation matrices in order to present the imperfect tool position in absolute reference system; therefore, the developed methodology can be applied to precisely estimate the dimensional errors of a manufactured workpiece. Based on the DH convention, Díaz–Tena et al. [23] developed a methodology for the geometric accuracy assessment of a multitasking machine consisting of a swivel tool-spindle head, two spindles, and a turret. Assembly errors were also considered as additional geometric parameters in transformation matrices in order to analyze the propagation and influence of assembly errors on the tool position and orientation in detail. While there are high-end machine tool controllers with volumetric error compensation functionalities, it is expensive, which limits the generality of volumetric error compensation. To make the conventional machine tool controller have volumetric error compensation functionalities, the software compensation methods of part-program generation and modification have become an important trend to improve the influence of the volumetric error of machine tools [24–27]. Eskandari et al. [28] utilized a neuro-fuzzy algorithm to model the geometric errors of a machine tool using a laser interferometer, and then, developed a compensation method to modify the Numerical Control (NC) part program with G-codes, which consider the developed error model and the volumetric errors calculated through kinematic analysis of the machine tool. Among numerous volumetric error software compensation methods, using CAD/CAM to generate part programs with volumetric error compensation generally has excessive complexity and difficult operation. The design of the modifying part program calculates the volumetric error and makes compensation according to the part program characterized tool moving path. While the method is simple, its volumetric error compensation effect is limited. As the volumetric errors induced by the
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tool moving process cannot be neglected, some studies segmented and compensated the tool moving path [29–31]. Zhu et al. [32] employed statistical analysis methods to extract the component errors of a machine tool, and then, a B-Spline mathematical model was further developed to represent component error functions using the least-squares fitting method. Here, the NC part program with initial G-codes was recursively subdivided and segmented according to the developed component error functions with the calculated deviation errors. In addition, in order to improve the volumetric error induced by the compensated tool path, some studies used the iteration method to compute more accurate tool moving paths for machine tools [29,30]. Based on the measurement and modeling of volumetric errors, Wang and Lin [27] developed a software-based error compensation method with an error interpolation scheme to improve machining accuracy of a machine tool by recursively compensating the cutting path and machining trajectory. Tool path deviation, as induced by the kinematic and geometric errors of a CNC machine tool, usually corrupts the dimensional accuracy of the machined mechanical parts; therefore, based on the developed error estimation model, Vahebi Nojedeh et al. [26] developed NC program editor software to modify and generate the errors the compensated NC program by recursively computing the compensated positions along the modified tool path. Lee et al. [33] developed a kind of capacitance-sensor to measure the five geometric errors of a miniaturized machine tool, and then, a volumetric error model was developed to synthesize and recursively compensate the geometric errors of the machine tool to efficiently improve its positioning accuracy. While the iterative computation of the volumetric error-compensated tool moving path can effectively improve the influence of the volumetric errors in the tool moving process of a machine tool, the unit recursive gain may fail to effectively converge to appropriate compensated tool path positions due to the machine tool working volume error distribution. Therefore, this study refers to the measured and established table of volumetric errors, as based on the part-program modification method, to segment the tool moving path, and uses the concept of ILC (iterative learning control) adjustment to design a segmented compensation method for the volumetric error of the part program, meaning the part program can be modified to have volumetric error compensation effect with higher tool moving accuracy. This study uses the Laser Doppler Displacement Meter (LDDM) laser displacement measurement system and volumetric error analysis software to establish a table of the volumetric errors of the machine tool [34,35], and then, performs path segmentation according to the part program characterized tool moving path. Finally, the segmented tool moving path of the part program is iteratively adjusted according to the concept of ILC. The ILC registers the actuation input, system output, and output error information of the control system execution process through an iterative computation process as reference for adjusting the control system actuation input of the next iteration, in order that the output error can converge to a limited range within a limited number of iterative learning. According to the segmented tool moving path of the part program, this study uses the interpolation table of volumetric errors to compute the volumetric errors induced by the tool moving path, and modifies the segmented tool moving path according to the volumetric errors. At this point, to reduce the volumetric errors induced by the execution of the segmented tool moving path after modification, the segmented tool moving path is adjusted by ILC iteratively till it converges to the optimal segmented tool moving path. The structure of this paper is described, as follows. Section 2 describes the experimental and measuring equipment used in this study. Section 3 describes the volumetric error compensation and calculation method proposed in this study, including computation for error compensation and application of the iterative learning control method. Section 4 tests volumetric error compensation and discusses the results, thus, the feasibility of the volumetric error compensation method, as proposed in this study, can be validated. Section 5 summarizes this paper. 2. Introduction to Experimental System and Architecture The three-axis CNC milling machine tool used in this study is shown in Figure 1. Its working volume is 560 × 410 × 450 mm3 , the controller is a FANUC 32i series controller, which is suitable for
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the application a high-speed CNC J. Manuf. Mater.of Process. 2018, 2, x FOR PEERmachine REVIEW tool [36]. The machine tool uses a C-shaped mechanical 4 of 15 the application a high-speed CNC machineistool [36]. The machine tool the usesballscrew a C-shaped structure; the Y-axisoffeed drive servomechanism installed on the base, and drives mechanical structure; the Y-axis feed drive servomechanism is installed on the base, and the the application of a high-speed CNC machine tool [36]. The machine tool uses a C-shaped the saddle. The X-axis feed drive servomechanism for driving the working table is installed on the ballscrew drives the saddle. The X-axis feed drive servomechanism for driving table is mechanical structure; the Y-axis feed the drive servomechanism is installed onthe the base, and the Y-axis feed drive servomechanism, and workpiece to be machined is fixed toworking the working table. installed on the Y-axis feed drive servomechanism, and the workpiece to be machined is fixed to the drives theservomechanism, saddle. The X-axis which feed drive servomechanism thetable, working table is on Theballscrew Z-axis feed drive is perpendicular to for thedriving working is installed working on table. Z-axis feed drive servomechanism, which is perpendicular to the working table, installed theThe Y-axis drive servomechanism, the workpiece to be tool machined to the is the column to control thefeed vertical movement of theand spindle. The cutting used is forfixed machining is installed on the column to control the vertical movement of the spindle. The cutting tool used for working table. The Z-axis feed drive servomechanism, which is perpendicular to the working table, installed on theis spindle. machining installed on the spindle. is installed on the column to control the vertical movement of the spindle. The cutting tool used for machining is installed on the spindle.
Figure 1. Experimental CNC machine tool.
Figure 1. Experimental CNC machine tool. Figure 1. Experimental CNC machine tool.
The LDDM™ laser displacement measurement system used in this study is shown in Figure 2, TheThe LDDM™ laser displacement system used this study is shown in Figure which measures the volumetric errormeasurement of machine tool working volume and generates the forward LDDMTM laser displacement measurement system used inin this study is shown in Figure 2, 2, and reverse table of volumetric errors on each motion axis. The laser Doppler displacement meter, which measures the volumetric error of machine tool working volume and generates the forward which measures the volumetric error of machine tool working volume and generates the forward LDDM™, istable anof instrument system with (1 axis. ppm) displacement measurement accuracy. andand reverse table volumetric errors on each axis.The The laser Doppler displacement meter, reverse of volumetric errors on1/1,000,000 each motion motion laser Doppler displacement meter, The maximum measurement range is1/1,000,000 15 m, the (1standard resolution is 0.01 ìm, theaccuracy. highest TM LDDM is an instrument system with 1/1,000,000 measurement LDDM™, is, an instrument system with (1ppm) ppm)displacement displacement measurement accuracy. measurement speed is 3600 mm/s, range is 15.5 °C to °C.highest In study, all maximum measurement range isthe 15 m, the standard resolution is32the 0.01 ìm,this the highest The The maximum measurement range is and 15 m, thetemperature standard resolution is 0.01 ìm, measurement measurementsspeed andis compensation processes were ◦ performed under similar temperature ◦ measurement 3600 mm/sec, and the temperature range is 15.5 °C to 32 °C. In this study, all speed is 3600 mm/s, and the temperature range is 15.5 C to 32 C. In this study, all measurements (approximately and 20 °C) and air humidity (relative humidity approximately 50%)temperature for ◦ a fair compensation processes under similar and measurements compensation processes were performed underwere similarperformed temperature (approximately 20 C) and air comparison. The measurement instrument for the cutting test is the NVC322 fully automatic (approximately 20 °C) and air humidity (relative humidity approximately 50%) for avision fair humidity (relative humidity approximately 50%) for a fair comparison. The measurement instrument measuring machine produced by 3DFAMILY Technology for workpiece straightness error comparison. The measurement instrument for the cutting test is the NVC322 fully automatic vision for the cutting testThe is the NVC322 fully automatic vision measuring machine by 3DFAMILY measurement. TESA MICRO-HITE Plus M350 height gauge for produced by produced Chiyeung, is used for measuring machine produced by 3DFAMILY Technology workpiece straightness error Technology for workpiece straightness error measurement. The TESA MICRO-HITE Plus M350 height height error measurement. measurement. The TESA MICRO-HITE Plus M350 height gauge produced by Chiyeung, is used for
gauge produced by Chiyeung, is used for height error measurement. height error measurement.
(a)
(b)
(a) Figure 2. LDDM laser displacement measurement system for this(b) study. (a) LDDM light alignment with close distance between source mirror and reflection mirror; (b) LDDM alignment with far Figure 2. LDDM laser displacement measurement system for this study. (a)light LDDM light alignment Figure 2. LDDM laser displacement measurement system for this study. (a) LDDM light alignment distance between source mirror and reflection mirror. with close distance between source mirror and reflection mirror; (b) LDDM light alignment with far
with close distance between source mirror and reflection mirror; (b) LDDM light alignment with far distance between source mirror and reflection mirror. distance between source mirror and reflection mirror.
3. Volumetric Error Calculation and Compensation
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3. Volumetric Error Calculation and Compensation As CNC machine tools have many coordinate systems, the part-program coordinate system used in part program, as well as the machine coordinate system used in machine tool motion, are described in this study. Generally speaking, for convenient workpiece machining, the operator usually uses the part-program coordinate system for programming the part program according to the engineering drawing. The original setting of the coordinate system varies with the tool moving process or machining process. The machine coordinate system is the referential coordinate system for mechanical system motion and is a real coordinate system. The origin of the coordinate system (machine origin) is the reference point of the machine tool, which is usually determined by the machine tool manufacturer. In the workpiece machining process, the part-program coordinate system is sometimes different from the machine coordinate system. As the machine tool volumetric error measurement refers to the machine coordinate system, the measured values of errors are represented by their positions in the machine coordinate system, thus, this study performs segmented compensation of volumetric errors by the part-program modification method. The part program characterized tool moving path must be transformed from the part program coordinate system to the machine coordinate system, and then, the machine coordinate system refers to the measured and established table of volumetric errors to compute the appropriate volumetric error compensation positions and values. The general transformation of coordinate systems is applicable to the tool moving path transformation between part-program coordinate system and machine coordinate system [37]. When the part program characterized tool moving path is transformed from the part program coordinate system to the machine coordinate system, the calculation and compensation of volumetric error in the tool moving process are performed. As shown in Figure 3, if the tool is to move from →
current position R (machine coordinate system represented R) to target position T (machine coordinate →
system represented T), due to the influence of the volumetric error of the machine tool, the cutting →
→
tool moves to actual position P (machine coordinate system represented P ), which leads to PT tool moving position error. As the table of the volumetric errors of the machine tool has been measured →
and established, the tool moving position error PT can be calculated by the table of volumetric errors, which can be used as the reference for the calculation of volumetric error-compensated position Q →
(machine coordinate system represented Q). The volumetric error-compensated position Q can be calculated according to the following steps: →
(S1) The tool moving direction from position R to position T (direction of RT vector) determines the load of the table of the forward volumetric errors or table of the reverse volumetric errors. → → (S2) The volumetric error vector E (=PT) of the tool moves from position R to position T and is calculated by the loaded table of the volumetric errors. →
(S3) The tool moving T position and the calculated volumetric error vector E are added up to generate the volumetric error-compensated position Q. →
The volumetric error vector E is calculated by interpolation according to the table of volumetric →
→
errors and RT vector. For example, as shown in Figure 4, if the RT vector means the X-axis moves from the origin position to the A position in the machine coordinate system, and it is 0 mm to 15 mm in the table of volumetric errors on X-axis, the X-axis movement results in the volumetric error on X-axis direction Dxx(A), expressed as Equation (1): Dxx(15) − Dxx(0) Dxx(15) − Dxx(A) = 15 − 0 15 − A
(1)
Dxx(15) represents the volumetric error on the X-axis direction when the X-axis moves to 15 mm in the table of volumetric errors. Dxx(0) represents the volumetric error on the X-axis direction when X-axis is at 0 mm in the table of volumetric errors. Therefore, when volumetric error vector
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coordinate system system represented represented Q which is is equal equal to to position position T T (machine (machine coordinate coordinate system system Q ),), which coordinate → represented plus volumetric error vector vector E T ))the E :: position Q can be calculated (machine coordinate E is calculated, volumetric error-compensated represented plus volumetric error T → → system represented Q), which is equal to position (machine coordinate system represented T) plus (2) Q = T + E (2) Q = T + E → volumetric error vector E : → position → → Q can be calculated by Equation (2), there While the the volumetric volumetric error-compensated error-compensated While position Q= T + E Q can be calculated by Equation (2), there (2) are two two principal principal problems problems in in practical practical application: application: are While the volumetric error-compensated position Q can be calculated by Equation (2), there are ● The The compensation compensation and and calculation calculation of of Equation Equation (2) (2) only only consider consider the the error error compensation compensation of of ● two principal problems in practical application: target position position T, T, while while the the position position error error compensation compensation of of aa tool tool moving moving from from R R to to T T is is not not target • considered. The compensation and calculation of Equation (2) only consider the error compensation of target considered. In other words, the position accuracy of the tool moving process cannot be In other words, the position accuracy of the tool moving process cannot be position T, while the position error compensation of a tool moving from R to T is not considered. guaranteed. guaranteed. In other words, the and position accuracy the tool (2) moving process guaranteed. ● The The compensation and calculation ofof Equation (2) disregard thecannot effect be of the the volumetric volumetric error error ● compensation calculation of Equation disregard the effect of of the the machine tool when the tool moves from position R to position . In other words, the • of The compensation and calculation of Equation (2) disregard the effect of the volumetric error of Q machine tool when the tool moves from position R to position Q . In other words, the the machine toolof the tool moves positionand R tothe position In otherto the actual actual position ofwhen the tool tool cannot be from guaranteed, and the targetQ. position towords, be reached reached has actual position the cannot be guaranteed, target position be has position of the tool cannot be guaranteed, and the target position to be reached has position error. position error. error. position
Figure 3. 3. The The influence of volumetric volumetric errors errors in in the the tool tool moving moving process. process. The influence influence of Figure
To overcomethe theabove aboveproblems, problems,the the original tool moving path is segmented. segmented. Figure shows To overcome the above problems, the original tool moving path is Figure 33 shows To overcome original tool moving path is segmented. Figure 3 shows that that target position T becomes the intermediate target position on the segmented tool moving path, that target position T becomes the intermediate target position on the segmented tool path, moving path, target position T becomes the intermediate target position on the segmented tool moving meaning meaning theaccuracy positionofaccuracy accuracy of the the original tool moving moving process canbybe be improved byerror the meaning the position of tool can by the the position the original tool original moving process can beprocess improved theimproved volumetric volumetric error compensation of themoving segmented tool moving path. Totool make the actual actual tool position position volumetric error the segmented moving To make the compensation of compensation the segmentedof tool path.tool To make thepath. actual position and tool intermediate and intermediate target position on the segmented tool moving path have small position error, and target positiontool onmoving the segmented moving path have smallfrom position error, targetintermediate position on the segmented path havetool small position error, differing the present differing from the the present present iterative computation method, this study uses uses the ILC ILC concept to differing from iterative computation method, study the to iterative computation method, this study uses the ILC conceptthis to iteratively compute the concept volumetric iteratively compute the volumetric error-compensated position. iteratively compute the volumetric error-compensated position. error-compensated position.
Figure 4. Interpolation computing method (Dxx). Figure Figure 4. 4. Interpolation Interpolation computing computing method method (Dxx). (Dxx).
ILC was was proposed proposed by by Uchiyama Uchiyama [38], [38], which which adjusts adjusts or or modifies modifies the the actuation actuation input input of of aa ILC ILC was proposed by Uchiyama [38], which adjusts or modifies the actuation input of a controlled controlled plant by the iterative approaching control method, in order that the actual output of the controlled plant by the iterative approaching control method, in order that the actual output of the plant by the iterative approaching control method, in order that the actual output of the controlled controlled plant plant can can converge converge to to the the referenced referenced output output within within limited limited numbers numbers of of iterative, iterative, thus, thus, it it controlled plant can converge to the referenced output within limited numbers of iterative, thus, it can be can be implemented in the manipulators and machine tools for executing repeated machine motions can be implemented in the manipulators and machine tools for executing repeated machine motions implemented in the manipulators and machine tools for executing repeated machine motions [39,40]. [39,40]. The The development development of of the the ILC ILC theory theory receives receives the the interest interest and and attention attention of of control control field field [39,40]. The development of the ILC theory receives the interest and attention of control field scholars; as scholars; as as its its control control structure structure is is simple, simple, it it is is applicable applicable to to both both linear linear systems systems and and nonlinear nonlinear scholars; its control structure is simple, it is applicable to both linear systems and nonlinear systems and can systems and can provide effective and practical solutions for repetitive tracking control and systems and can provide effective and practical solutions for repetitive tracking control and provide effective and practical solutions for repetitive tracking control and repetitive disturbance repetitive disturbance rejection problems [41–43]. ILC uses the current actuation input of repetitive disturbance rejection problems [41–43]. ILC uses the current actuation input of aa rejection problems [41–43]. ILC uses the current actuation input of a controlled plant and the execution controlled plant plant and and the the execution execution output output error error as as the the calculation calculation reference reference for for next next actuation actuation input input controlled output error as the calculation reference for next actuation input of the controlled plant, thus, there of the controlled plant, thus, there is appropriate control system execution output error correction of the controlled plant, thus, there is appropriate control system execution output error correction
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is appropriate control system execution output error correction ability, and the error decreases as the numbers of iterative learning increases. When the numbers of iterative learning approaches to infinity, the theoretical error approaches to zero. In this study, ILC is used for the iterative computation of the volumetric error-compensated tool position for the part program characterized tool moving path. When the calculated tool moving position error converges, the high-accuracy volumetric error-compensated tool position can be obtained, to modify the tool moving path of the part program. By referring to the volumetric error-compensated position computing equation, as shown in Equation (2), the iterative computation of the ILC volumetric error-compensated position is designed and expressed as Equation (3). →
→
→
Qi+1 = Qi + γ · E i , i = 0, 1, . . . , n
(3)
→
where position Qi (machine coordinate system represented Qi ) represents No. (i) calculated volumetric →
error-compensated position, position Qi+1 (machine coordinate system represented Qi+1 ) represents →
No. (i + 1) calculated volumetric error-compensated position, E i represents the calculated volumetric error of a tool moving from the current position to No. (i) volumetric error-compensated position Qi , →
γ represents the learning factor, 0 < γ < 1. Q0 equals target position T (machine coordinate system →
→
represented T), and E 0 represents the calculated volumetric error of the tool moving from the current →
position to target position T (machine coordinate system represented T). The setting of learning factor γ affects the convergence of volumetric error-compensated position calculation. When the learning factor value is large, the volumetric error-compensated position calculation converges rapidly, and there is fluctuation, thus, it fails to achieve stable calculation results. When the learning factor value is small, while the volumetric error-compensated position calculation result is stable, the convergence rate is low. When learning factor γ is larger than 1, the volumetric error-compensated position calculation may fail to converge. As this study uses the offline computing method of Equation (3), a small learning factor value is used to guarantee stable volumetric error-compensated position calculation results. The calculation convergence of Equation (3) is judged based on whether the Euclidean norm of No. (i) →
calculated volumetric error E i decreases as the number of iterative computation increases. In other words, this study uses Equation (3) for the iterative computation of volumetric error-compensated →
→
position Qi , and checks whether the Euclidean norm (k E i k) of calculated volumetric error E i decreases as the number of iterative computation (i) increases. When Equation (3) calculates convergence iteratively, the converged volumetric error-compensated position is used as the tool moving path of the modified part program. 4. CNC Volumetric Error Compensation Experiment Results and Discussion The machine tool is often provided with compensation parameters for general pitch error compensation; however, this pitch error compensation only compensates the displacement error induced by axial motion and neglects the displacement errors induced by other motion axes. Therefore, this study discusses the displacement error due to axial motion Exx , Eyy , Ezz , and considers the displacement errors induced by other motion axes Exy , Exz , Eyx , Eyz , Ezx , Ezy . Where Eij represents the displacement error on j-axis during i-axis movement. This study considers the aforesaid displacement errors and generates the error-compensated part program. The table of the volumetric errors of the experimental machine tool is measured by the LDDM laser Doppler displacement meter. The error distributions of the forward and reverse motions on each motion axis are shown in Figure 5.
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(a)
(b)
(c)
(d)
(e)
(f)
Figure 5. Volumetric errors of the experimental machine tool measured by LDDM. (a) X-axis forward Figure 5. Volumetric errors of the experimental machine tool measured by LDDM. (a) X-axis forward error; (b) X-axis X-axisreverse reverseerror; error;(c)(c) Y-axis forward error; Y-axis reverse (e) Z-axis forward error; (b) Y-axis forward error; (d) (d) Y-axis reverse error;error; (e) Z-axis forward error; error; (f) Z-axis reverse error. (f) Z-axis reverse error.
4.1. Simulation Results and Discussion 4.1. Simulation Results and Discussion This study designs two different tool moving path simulations for observation, segmentation, This study designs two different tool moving path simulations for observation, segmentation, and compensation for the circular path, and segmentation and compensation for the spiral path. and compensation for the circular path, and segmentation and compensation for the spiral path. Simulation 1 sets the tool movement path as a circle with a diameter of 20 mm, which is divided into Simulation 1 sets the tool movement path as a circle with a diameter of 20 mm, which is divided into 20 segment and 30 segment moving paths, as shown in Figures 6 and 7, respectively. Simulation 2 20 segment and 30 segment moving paths, as shown in Figures 6 and 7, respectively. Simulation 2 sets the path as a spiral path with a diameter of 20 mm and a pitch of 1 mm, which is divided into 60 sets the path as a spiral path with a diameter of 20 mm and a pitch of 1 mm, which is divided into segments of moving paths, as shown in Figure 8. Table 1 shows the numerical analysis results of 60 segments of moving paths, as shown in Figure 8. Table 1 shows the numerical analysis results of simulation. It is obvious that the segmented tool moving path position error root-mean-square value simulation. It is obvious that the segmented tool moving path position error root-mean-square value is is improved after ILC compensation, where the rate of improvement is higher than 70%, improved after ILC compensation, where the rate of improvement is higher than 70%, demonstrating demonstrating the feasibility and effect of the ILC volumetric error compensation computation the feasibility and effect of the ILC volumetric error compensation computation method, as designed in method, as designed in this study, on the volumetric error compensation of a three-axis CNC milling this study, on the volumetric error compensation of a three-axis CNC milling machine tool. In addition, machine tool. In addition, according to the analysis of the number of segments on the tool moving according to the analysis of the number of segments on the tool moving path, the larger the number of path, the larger the number of segments, the better the volumetric error compensation effect. segments, the better the volumetric error compensation effect. However, many segments have limited However, many segments have limited compensation effect on the volumetric error, as it prolongs compensation effect on the volumetric error, as it prolongs the computation time of compensation the computation time of compensation method on the contrary. Therefore, the position of a single method on the contrary. Therefore, the position of a single block of the part program shall not be block of the part program shall not be excessively segmented in practical application. excessively segmented in practical application.
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(a) (b) (a) (b) (a) (b) Figure6.6.20 20segments segmentsofoftool toolmoving movingpath pathsimulation simulationresults. results.(a) (a)3D 3Ddiagram diagramofof20 20segments segmentscircular circular Figure Figure Figure 6. 6. 20 20 segments segments of of tool tool moving moving path path simulation simulation results. results. (a) (a) 3D 3Ddiagram diagramof of 20 20 segments segments circular circular toolmoving movingpath; path;(b) (b)Euclidean Euclideannorm norm ee ofofposition positionerror errorofofeach eachsegment. segment.(cmd: (cmd:tool toolmoving moving tool tool of position position error error of of each each segment. segment. (cmd: tool tool moving moving path; path; (b) (b) Euclidean Euclidean norm norm ke ek of tool moving moving pathcommand; command;cmd_realpt: cmd_realpt:simulated simulatedtool toolmoving movingpath pathunder underthe theinfluence influenceof ofvolumetric volumetricerror; error; path command; cmd_realpt: simulated tool moving path under the influence of volumetric error; path path command; cmd_realpt: simulated tool moving path under the influence of volumetric error; final_comp:simulated simulatedtool toolmoving movingpath pathafter afterILC ILCcompensation). compensation). final_comp: simulated tool moving path after ILC compensation). final_comp: final_comp: simulated tool moving path after ILC compensation).
(a) (b) (a) (b) (a) (b) Figure7.7.30 30segments segmentsofoftool toolmoving movingpath pathsimulation simulationresults. results.(a) (a)3D 3Ddiagram diagramofof30 30segments segmentscircular circular Figure Figure Figure 7. 7. 30 30 segments segments of of tool tool moving moving path path simulation simulation results. results. (a) (a) 3D 3Ddiagram diagramof of 30 30 segments segments circular circular toolmoving movingpath; path;(b) (b)Euclidean Euclideannorm norm ke positionerror errorof eachsegment. segment.(cmd: (cmd:tool toolmoving moving tool ofofposition position error ofofeach each segment. tool moving path; (b) Euclidean norm tool moving eeek of tool moving path; (b) Euclidean norm of position error of each segment. (cmd: tool moving path command; cmd_realpt: simulated tool moving path under the influence of volumetric error; pathcommand; command;cmd_realpt: cmd_realpt:simulated simulatedtool toolmoving movingpath pathunder underthe theinfluence influenceof ofvolumetric volumetricerror; error; path path command; cmd_realpt: simulated tool moving path under the influence of volumetric error; final_comp: simulated tool moving path after ILC compensation). final_comp:simulated simulatedtool toolmoving movingpath pathafter afterILC ILCcompensation). compensation). final_comp: final_comp: simulated tool moving path after ILC compensation).
(a) (b) (a) (b) (a) (b) Figure8.8.60 60segments segmentsof ofspiral spiraltool toolmoving movingpath pathsimulation simulationresults. results.(a) (a)3D 3Ddiagram diagramof of60 60segments segments Figure Figure 60 segments of spiral moving path simulation results. (a) 3D diagram of 60 segments Figure 8. 60 segments of spiral tool moving path simulation results. (a) 3D diagram of 60 segments spiraltool toolmoving moving path; (b) Euclidean norm position eachsegment. segment. (cmd: tool spiral path; (b)(b) Euclidean norm kek of error error oferror eachofof segment. (cmd: tool moving spiral tool moving path; Euclidean norm ofofposition position each (cmd: tool eeeposition spiral tool moving path; (b) Euclidean norm of error of each segment. (cmd: tool path command; cmd_realpt: simulated tool moving path under the influence of volumetric error; movingpath pathcommand; command;cmd_realpt: cmd_realpt:simulated simulatedtool toolmoving movingpath pathunder underthe theinfluence influenceofofvolumetric volumetric moving moving pathsimulated command;tool cmd_realpt: simulated tool moving path under the influence of volumetric final_comp: moving path after ILC compensation). error; final_comp: simulated tool moving path after ILC compensation). error; final_comp: final_comp: simulated simulated tool tool moving moving path path after after ILC ILC compensation). compensation). error;
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Table 1. Comparison of simulation results. Circular Circular Spiral 20 Segments 30 Segments 60 Segments Circular Spiral Circular Tool Moving Path Euclidean norm root-mean-square value of position errors of the 30 segmented tool moving path (µm) 20 Segments Segments 60 Segments Euclidean norm root-mean-square value of position errors of the segmented tool moving path (μm) Volumetric error 3.39 3.41 3.31 (before ILC compensation) Volumetric error 3.39 3.41 3.31 (before ILC compensation) ILC compensation 0.77 0.72 0.71 ILC compensation (after ILC compensation) 0.77 0.72 0.71 (after ILC compensation) Rate of improvement (%) 77.28 78.89 78.55 Rate of improvement (%) 77.28 78.89 78.55 Tool Moving Path
4.2. Discussion Discussion of of the the Results Results of of Cutting Cutting Tests Tests 4.2. This study milling machine tooltool slotslot machining to detect the straightness error This studyuses usesthree-axis three-axisCNC CNC milling machine machining to detect the straightness and height error of a workpiece to develop technical identification. The machining results are shown in error and height error of a workpiece to develop technical identification. The machining results are Figure 9inand Table9 2. TheTable cutting tool cutting for experimentation is a ø10 mm end two flutes, andtwo the shown Figure and 2. The tool for experimentation is a mill ø10 with mm end mill with workpiece material is aluminum alloy A6061. According to the data plot of the machining results, the flutes, and the workpiece material is aluminum alloy A6061. According to the data plot of the degree of error convergence is of known, the relationship between therelationship table of volumetric and machining results, the degree error and convergence is known, and the betweenerrors the table the actual machining results can be observed. Figure 10 shows the side edge straightness measurement of volumetric errors and the actual machining results can be observed. Figure 10 shows the side edge data distribution of three machined slots. Figure shows the height measurement distribution of straightness measurement data distribution of11 three machined slots. Figure 11data shows the height the bottom surface three machined slots. The experimental results show that the The compensated tool measurement data ofdistribution of the bottom surface of three machined slots. experimental moving path has better machining results, the optimum rate of improvement of the straightness error results show that the compensated tool moving path has better machining results, the optimum rate is 79.03%, the minimum of improvement 62.57%,the andminimum the average rate improvementisis62.57%, 68.87%. of improvement of the rate straightness error is is79.03%, rate ofofimprovement The optimum rate of improvement of height error is 83.62%, the minimum rate of improvement is and the average rate of improvement is 68.87%. The optimum rate of improvement of height error is 80.21%, and the average rate of improvement is 81.90%. In addition, the workpiece measurement 83.62%, the minimum rate of improvement is 80.21%, and the average rate of improvement is result shows that the position error ofmeasurement straight cutting increases gradually. after volumetric 81.90%. In addition, the workpiece result shows that the However, position error of straight error compensation, the distribution of position errors diminishes obviously, and fluctuates nearby cutting increases gradually. However, after volumetric error compensation, the distribution of the zero-error value in the smaller error range. Therefore, the volumetric error compensation method position errors diminishes obviously, and fluctuates nearby the zero-error value in the smaller error designed in this study can effectively the volumetric errors in theintool of aeffectively three-axis range. Therefore, the volumetric errorreduce compensation method designed thismoving study can CNC milling machine tool. reduce the volumetric errors in the tool moving of a three-axis CNC milling machine tool.
Figure 9. Machining results. Figure 9. Machining results.
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Table 2. Comparison of results of cutting tests. Slot Number
1
2
3
Straightness error root-mean-square value comparison (unit: mm) Without compensation 0.0179 0.0186 0.0160 With ILC compensation 0.0067 0.0039 0.0056 Rate of improvement (%) 62.57 79.03 65.00 Height error root-mean-square value comparison (unit: Without compensation 0.0177 0.0187 With ILC compensation 0.0029 0.0037 J. Manuf. Mater. Process. 2018, 2, x FOR PEER REVIEW Rate of improvement (%) 83.62 80.21
mm) 0.0193 0.0035 81.87
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0.01 0.005 0 Error (mm)
-0.005
1 2 3 4 5
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.01
No Comp
-0.015
Comp
-0.02 -0.025 -0.03 -0.035 -0.04 Measurement point
(a) 0.01 0.005 0 Error (mm)
-0.005
1 2 3 4 5
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.01
No Comp Comp
-0.015 -0.02 -0.025 -0.03 -0.035 Measurement point
(b) 0.015 0.01 0.005 Error (mm)
0 -0.005
1 2 3 4 5
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
No Comp Comp
-0.01 -0.015 -0.02 -0.025 -0.03 Measurement point
(c) Figure 10. 10.Straightness Straightness error measurement data distribution. Slot 1 error straightness error Figure error measurement data distribution. (a) Slot 1 (a) straightness measurement; measurement; (b) Sloterror 2 straightness error (c) measurement; (c) Sloterror 3 straightness error measurement. (b) Slot 2 straightness measurement; Slot 3 straightness measurement.
0.01 0.005
Error (mm)
0 -0.005 -0.01 -0.015
1 2
3 4 5
6 7 8
9 10 11 12 13 14 15 16 17 18 19 20 No Comp Comp
-0.03 Measurement point
(c) Figure 10. Straightness error measurement data distribution. (a) Slot 1 straightness error measurement; Slot2, 253straightness error measurement; (c) Slot 3 straightness error measurement. 12 of 15 J. Manuf. Mater. Process.(b) 2018,
0.01 0.005
Error (mm)
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1 2
3 4 5
6 7 8
9 10 11 12 13 14 15 16 17 18 19 20 No Comp Comp
-0.01 -0.015 -0.02 -0.025 -0.03 Measurement point
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(a) 0.005 0
Error (mm)
-0.005
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.01 No Comp Comp
-0.015 -0.02 -0.025 -0.03 -0.035 Measurement point
(b) 0.01 0.005 0 Error (mm)
-0.005
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.01
No Comp Comp
-0.015 -0.02 -0.025 -0.03 -0.035 Measurement point
(c) Figure 11. Height error measurement data distribution. (a) Slot 1 height error measurement; (b) Slot 2 Figure 11. Height error measurement data distribution. (a) Slot 1 height error measurement; (b) Slot 2 height error error measurement; measurement; (c) (c) Slot Slot 33 height height error error measurement. measurement. height
5. Conclusions
Table 2. Comparison of results of cutting tests. Slot Number
1
2
3
This study proposed the use of ILC to adjust the compensated tool path for the 21 volumetric Straightness error root-mean-square value comparison (unit: mm) errors in three-axis CNC milling machine tools, to increase the error compensation accuracy of the Without compensation 0.0179 0.0186 0.0160 modified part program. The main conclusions of this study are as follows.0.0056 With ILC compensation 0.0067 0.0039
•
of working improvement (%) of the three-axis 62.57 79.03 65.00 machine tool strongly The volumetric error Rate of the volume CNC milling Heightdimension error root-mean-square value comparison (unit: mm) influences the geometric accuracy of the workpiece. The volumetric error compensation compensation 0.0177 parts 0.0187manufacturers 0.0193 therefore becomes an Without issue that high-precision mechanical must consider. With ILC compensation 0.0029 0.0037 0.0035 • Existing part-program modification methods only consider the compensation of the position of a Rate of improvement (%) 83.62 80.21 81.87 single block of a part program but neglect the movement bias induced by volumetric error in the tool moving process, which leads to limited volumetric error compensation effect. 5. Conclusions • In this study, the position of a single block of a part program was segmented, and volumetric error This study proposed the use of ILC to adjust the compensated tool path for the 21 volumetric compensation was implemented for the tool moving position in each segment to generate a part errors in three-axis CNC milling machine tools, to increase the error compensation accuracy of the modified part program. The main conclusions of this study are as follows.
●
The volumetric error of the working volume of the three-axis CNC milling machine tool strongly influences the geometric dimension accuracy of the workpiece. The volumetric error compensation therefore becomes an issue that high-precision mechanical parts manufacturers
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•
•
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program with multiple compensated blocks. The proposed compensation method can effectively compensate for volumetric errors in the tool moving process. In addition, this study used the ILC concept and referred to the present compensated tool path and volumetric error induced by the compensated tool path to calculate the next compensated tool path. Therefore, the optimum compensated tool path can be obtained by the iterative computation of the compensated tool path. To validate the effect of the proposed segmented volumetric error compensation, the measured and established table of volumetric errors was used for path simulation and cutting tests. Both the simulation and experimental results prove the feasibility of using the ILC method for the segmented modification of the volumetric error-compensated tool path. Several cutting tests were performed on a three-axis CNC milling machine tool. The experimental results showed that the average rate of improvement of the straightness error of the proposed volumetric error compensation method is higher than 60% while the rate of improvement of the height error is higher than 80%.
Author Contributions: Investigation, Y.-C.L.; Investigation, Supervision, S.-S.Y. Funding: This research was funded in part by the Ministry of Science and Technology, Taiwan, under Contract MOST104-2221-E-027-132 and MOST103-2218-E-009-027-MY2. Acknowledgments: The authors would like to thank representatives from the 3DFAMILY Technology for their beneficial discussions with the project team. The authors would also like to thank Chen-Chou Chung (Ford Lio Ho Motor Company) for his valuable comments and suggestions on volumetric error compensation of CNC milling machine tools. Conflicts of Interest: The authors declare no conflict of interest.
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