- Email: [email protected]
An alternate approach to free-form surface fabrication
Journal of Materials Processing Technology 192–193 (2007) 465–469
An alternate approach to free-form surface fabrication Ka-Lok Yiu, Hon-Yuen Tam ∗ Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Ave, Kowloon Tong, Kowloon, Hong Kong
Abstract Surface milling and grinding are widely used for the fabrication of free-form surfaces for molds and dies. The continuous demand for better product quality has led to demand in higher surface accuracy. Limiting factors are the positioning accuracy of the tool and the accuracy of the manufacturing process. Computer controlled optical surfacing (CCOS) has been developed mainly for the fabrication of spherical and aspherical optics. The surface is repeatedly measured and corrected using abrasives until the target accuracy is attained. In principle, sub-micron accuracy is achievable. The accuracy is limited by surface measurement, and not by the processing equipment. The current research investigates the adoption of CCOS for the fabrication of free-form steel surfaces. This paper reports on the development of a test bed and initial experiment results. The test bed is based on a six-axis RX robot as the motion platform with a spindle attached to the wrist of the robot. Surface measurement is carried out using a Talysurf surface profiler. Surface error correction is performed by abrasion of material from the surface. The amount of material to be removed is based on the measured surface error. Following a predetermined tool path, the feed rate along the tool path for surface correction is to be varied according to the required among of material to be removed locally. A free-form surface specimen of mold steel of about 40 mm × 40 mm across is used for the initial experiment. The surface is prepared by CNC surface machining with the initial maximum error over 50 m. The positioning accuracy of the motion system is estimated to be no better than 0.1 mm. Through surface measurement and correction, the surface accuracy is significantly improved. © 2007 Elsevier B.V. All rights reserved. Keywords: CCOS; Polishing; Free-form surface; Molds and dies
1. Introduction High quality and precision surface fabrication is becoming more important in manufacturing. The needs for free-form surface fabrication are widely recognized in the aerospace, automotive and die/mold industries. However, it is difficult to produce precision free-form surfaces by conventional manufacturing processes. Such as multi-axis NC machining, sculptured surface machining (SSM) and high speed machining (HSM) [1]. Many have reported on the use of numerical controlled abrasion for surface finishing. Target applications are the finishing of surfaces for molds and dies [2] and surfaces for optics [3]. A typical setup involves a polishing tool mounted to a motion platform such as a robotics arm [4] or a machine tool. Both fixed abrasives [5] and loose abrasives [6] with felt are used. The control of the polishing force is either active (involving a force sensor [7]) or passive (with the tool mounted to a linear
∗
Corresponding author. Tel.: +852 27888436; fax: +852 27887660. E-mail address: [email protected] (H.-Y. Tam).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.04.057
slide). There were also investigations of tool path planning [8] and selection of abrasives to improve the processing efficiency. Computer controlled optical surfacing (CCOS) is a high precision finishing technique for optics manufacturing. It is an iterative process. The surface error is determined through measurement and is corrected through controlled abrasion. There are many papers on adopting CCOS for making aspheric [9] and off-axis [10] surfaces. Interferometry is the usual method for surface measurement [11]. Accurate surfacing uses loose abrasives. Surface correction can be executed through controlling the dwell time of the tool [12] or varying the tool pressure [13]. Surface accuracy of accuracy in the 0.2–0.3 m and surface roughness of 2 nm were demonstrated [14]; precision surfaces 2 m long was produced [15]. Some have worked on alternate processing methods like the use of fluid-jet abrasion [16] and magnetorheological fluid [17]. Surface accuracy of 30 nm peak-to-valley and surface roughness of 1 nm rms were achieved with the later process [17]. This paper reports an initial attempt to use CCOS to improve the form accuracy of a free-form metallic surface. A summary of CCOS is given in the next section. This is followed by a
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K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
Fig. 2. Surface abrasion by rotary tools: (a) cylindrical and (b) spherical.
Fig. 1. Flow chart of CCOS [15].
study of abrasive processes for surfacing. A general formulation of the problem to compute tool feed rates for surface correction is outlined. Then the experimental setup and initial results are presented and discussed. A conclusion is given in the end. 2. A summary of CCOS This section summarizes the use of CCOS for precision fabrication of surfaces. A flow chart of CCOS from the reference is shown in Fig. 1. Surfacing is performed iteratively to improve the surface accuracy. The input surface is obtained by precision manufacturing methods such as precision grinding or lapping. In the beginning of each iteration, the surface form is measured. The amount of material to be removed is planned based on comparing the measurement and the reference geometry. A suitable tool path is then selected. Algorithms are used to determine how the dwell time of the tool should be varied along the tool path and processing parameters selected to effect the desired material removal. The result is checked by simulation software. Dwell time calculation and simulation may be repeated until the pre-
dicted surface improvement is satisfactory. Then, a control file is generated for the execution of surface correction with CCOS [15]. 3. Abrasive processes For surface correction, one needs to know how much material is removed from the surface as the tool moves across a surface. Two types of tools are analyzed for processing of steel surfaces (Fig. 2). Experiments were conducted in which tools were moved along straight lines on planar surfaces. The spindle speed was fixed at 9000 rpm and the normal force was 5 N from a dead weight. The first set of experiments used sand paper adhered to the end of a cylindrical tool of 6.4 mm diameter. The tool feed rate was varied from 0.8 to 2.4 mm/s. The surface profile along a straight line perpendicular to the tool motion was measured before and after the experiment. The difference is termed the removal profile. The removal profile for the case of 0.8 mm/s feed rate is shown in Fig. 3. The shape is like a flat bell with depth of about 9 m and width 6 mm. The width remains approximately unchanged for various feed rates. The depth changes to 7.4 and 5 m for feed rates of 1.6 and 2.4 mm/s, respectively.
Fig. 3. The profile of material removal as the sand paper tool follows a straight line path on the surface at 0.8 mm/s.
Fig. 4. The profile of material removal as the felt tool (diamond paste) follows a straight line path on the surface at 0.8 mm/s.
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
467
The second experiment used 15 mm radius spherical felt tool. Diamond paste of 6 m grains was used as the abrasive. The removal profile for 0.8 mm/s feed rate is shown in Fig. 4. The width and depth of the profile are 2 mm and 4 m, respectively. The shape is like a V-shaped bell. The difference in the shape of the removal profile for the two types of tool is attributed to the difference in pressure and velocity distribution within the tool and surface contact. Again, the width of the profile remains stable for various feed rates; the depth is changed to 3 and 2 m when the feed rates are 1.6 and 2.4 mm/s, respectively. 4. Surface correction planning Without loss of generality, suppose zij represents the measured height of the part surface at coordinates (xij , yij ), where i = 1, . . ., m and j = 1, . . ., n. The m × n matrix E constitutes the surface error map where the (i, j)th element of E is eij = zij − f(xij , yij ) and f(x, y) is the given surface function. A portion of the tool path for surface abrasion is illustrated in Fig. 5. Let Pk for k = 1, . . ., s + 1 be points of the tool paths and the tool feed rate from Pk to Pk+1 is given by vk . Material is removed from the part surface due to abrasion as the tool moves along the tool path. Suppose material removal as the tool moves from Pk to Pk+1 to along the tool path with unity feed rate is given by the m × n matrix Ak such that the (i, j)th element corresponds to the depth of abrasion at (xij , yij ). Assume the local surface curvatures and the tool path curvatures do not change significantly so that, for the same processing conditions (i.e. abrasive type, normal force, spindle speed, etc.), material removal profile experimentally determined for a given abrasion tool travelling along a straight line path on a planar surface from the previous section can be correlated to the removal profile of the tool following a given tool path on a part surface specimen. In this way, matrix Ak can be computed when Pk and Pk+1 and vk are given (Fig. 5). For each of the locations on the surface covered by the tool as it travels from Pk to Pk+1 , the depth of removal can be derived from experimentally determined removal profiles (from last section) and according to the distance of the location from the tool path.
Fig. 6. The robotic test bed for surface correction.
Suppose the depth of removal is proportional to the time of abrasion, material removal from the surface by the tool as it follows a specified tool path can be represented by the material removal matrix where A=
s
Ak tk
(1)
k=1
where 1 tk = . vk The problem of surface correction planning may be treated as finding a way to determine the tool feed rate vk along a given path to reduce the surface error. One way to approach the solution is to formulate the problem as a quadratic optimization problem with inequality constraints. The cost function is written as m n 2 min (2) rij tk i=1 j=1 where rij is the (i, j)th element of E − A; values of tk for k = 1, . . ., s are sought to minimize the cost function subject to constraints tk > T > 0 and rij > 0; the first type of constraints is set so that feasible feed rates should be no greater than certain maximum feed rate 1/T, where T is fixed; and the second type of constraints require the material removal to be no greater than the local surface error to avoid over-correction. Alternate ways to determine the feed rates could include the use of heuristics or iterative methods [12]. 5. Experiment setup
Fig. 5. Top view of the tool sweep as the tool travels from Pk to Pk+1 .
A test bed was set up for surface correction experiments. A Staubli RX60 six-axis robot was used as the motion platform as illustrated in Fig. 6. The positioning accuracy of the motion platform was estimated to be about 0.1 mm. The end-effector was equipped with a linear slider. This allowed the spindle to slide to prevent excessive tool and surface contact force due to positioning errors. A close-up view of the spindle and the specimen is shown in Fig. 7. The specimen is a block of DF2 mold steel of about 40 mm × 40 mm across. The surface was prepared by CNC surface machining. Excluding the part of
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K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
Fig. 7. Close-up view of the spindle and the specimen.
the surface close to the edges, the surface to be used for precision forming is 19 mm × 19 mm (Fig. 8). Within this area, the maximum height of the surface is about 2 mm, corresponding to the locations of two diagonal corners of the surface. The initial error map of the surface is given in Fig. 9. The surface profile was measured using a Form Talysurf surface profiler. It was actually surprising that the initial surface error was more than 50 m as given by measurement; this error is large compared to the resolution of the machine with which the surface was fabricated. The error is large at locations where the local surface slope is large. The unexpectedly large error may be attributed to the error of the cutting process and error due to tool holding. While the surface error could possibly be reduced by selecting alternate machining parameters and tool holding methods, this fabrication experience does reveal that manufacture of precision free-form surfaces is not an easy task. Fig. 9. Surface error before correction: (a) error map and (b) error contour.
6. Initial results In view that the initial surface error is quite large. An heuristic method was adopted that focused on: (1) removing material at locations where the error was large; and (2) avoiding removal at locations where the error was very small. The purpose was to achieve significant reduction of the maximum surface error which was considered large (more than 50 m from Fig. 9) and to avoid over-correction where the error was already small.
Fig. 8. Design surface of the specimen.
Let ek be the (non-negative) surface error at tool path point Pk , feed rate vk was set according to: if ek > eH , set vk = vmin , elseif ek < eL , set vk = vmax ,
(1/e) − (1/eH ) else set vk = vmin + (vmax − vmin ) (1/eL ) − (1/eH )
(3)
An iteration of surface correction was performed using the sand paper tool described in the previous section. Based on this heuristic approach, the focus of the present iteration is to reduce the maximum error through flattening the two diagonal peaks of the error map in Fig. 9. The threshold values eH and eL were chosen so that the tool feed rate was at vmin in the neighborhood of the two error peaks and at vmax near the other two corners of the surface where the error was small. This was followed by a number of iterations using the felt tools. The surface profile was measured afterward and the error map produced was shown in Fig. 10. It is quite remarkable that the maximum surface error was significantly reduced from above 50 m to slightly over 20 m. Significant removal of material took place near the two error peaks. Very little material was removed at the opposing two corners where the original error was small. One can actually observe over-correction around the inner peak of the surface at Fig. 9; a valley was formed in the corresponding region as seen in Fig. 10. The main cause of
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
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ficient stability of the material removal process itself. Upon successful fine-tuning of the removal process, the ultimate bottleneck would likely be the accuracy of surface measurement. For measurement using Talysurf-like instruments, form accuracy for free-form surfaces is in the vicinity of 1 m. Acknowledgement The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 120105]. References
Fig. 10. Surface error after correction: (a) error map and (b) error contour.
the over-correction is attributed to insufficient stability of the material removal process. 7. Concluding remarks This paper reports initial experiments of free-form metallic surface fabrication based on CCOS. This was motivated by the fact that CCOS could be used in optics manufacturing to produce mirrors and lenses of remarkable precision. The experiment demonstrated that the maximum error of the DF2 steel specimen was reduced from over 50 m to slightly about 20 m in a small number of iterations. The amount of material removed from the surface co-related well with the planned removal. The positioning accuracy of the motion system in the present test bed is estimated to be no better than 0.1 mm. The current experiment shows that correction of surface error smaller than the precision of the motion system is feasible. The achieved accuracy is slightly over 20 m. The main bottleneck is insuf-
[1] Y.S. Lee, Computer geometry in design and manufacturing of free form surface, in: AMS-DIMACS, v67, Geometric and Algorithmic Aspects of Computer-Aided Design and Manufacturing. [2] F. Nagata, et al., Generation of normalized tool vector from 3-axis CL data and its application to a mold polishing robot, in: Proceeding of 2004 IEEE/RSJ International Conference on Intelligent Robot and Systems, 2004. [3] M. Schinhaerla, et al., Filter algorithm for influence functions in the computer controlled polishing of high-quality optical lenses, Int. J. Mach. Tools Manuf. v47 (I1) (2007) 107–111. [4] Z.W. Yang, F.F. Xi, B. Wua, A shape adaptive motion control system with application to robotic polishing, Robot. Comp. Integr. Manuf. 21 (2005) 355–367. [5] L. Zhang, H.Y. Tam, C.-M. Yuan, Z.-D. Zhou, An investigation of material removal in polishing with fixed abrasives, Proc. I Mech. E Part B, J. Eng. Manuf. 216 (1) (2002) 103–112. [6] N.B. Kirk, J.V. Wood, Glass polishing, Br. Ceram. Trans. 93 (1994). [7] E.A. Erlbacher, Force Control Basics, Ind. Robot v27 (n1) (2000) 20. [8] M.J. Tsai, J.-J. Fang, J.-L. Chang, Robotic path planning for an automatic mold polishing system, Int. J. Robot. Autom. 206 (2004). [9] H.-B. Cheng, Z.-J. Feng, K. Cheng, Y.-W. Wang, Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors, Int. J. Mach. Tools Manuf. 45 (2005) 1085–1094. [10] H.J. Juranek, et al., Off-axis telescopes—the future generation of earth observation telescopes, SPIE 3439 (1988) 104–115. [11] FISBA News Issue 10/June 2004, http://www.fisba.ch/english/ messtechnik/documents/mnews10 special.pdf. [12] Q.S. Li, Y. Cheng, F.Z. Cai, Z.J. Feng, B.P. Zhang, Algorithm of dwell time for computer-controlled optical surfacing, Opt. Technol. (n3) (1999) 56–59. [13] B.-S. Ryuh, S.M. Park, G.R. Pennock, An automatic tool changer and integrated software for a robotic die polishing station, Mech. Mach. Theory 41 (2006) 415–432. [14] D.D. Walker, et al., Recent development of precessions polishing for larger components and free-form surfaces, SPIE, Denver, August 2004. [15] R.A. Jones, W.J. Rupp, Rapid optical fabrication with computer-controlled optical surfacing, Opt. Eng. v30 (i12) (1991) 1962–1968. [16] O.W. F¨ahnle, H. Van Brug, H.J. Frankena, Fluid jet polishing of optical surfaces, Appl. Opt. v37 (n28) (1998) 6771–6773. [17] W.I. Kordonski, Adaptive structures based on magnetorheological fluids, in: Proceedings of the Third International Conference, Adaptive Structure, San Diego, CA, USA, 1992, pp. 13–17.
An alternate approach to free-form surface fabrication Ka-Lok Yiu, Hon-Yuen Tam ∗ Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Ave, Kowloon Tong, Kowloon, Hong Kong
Abstract Surface milling and grinding are widely used for the fabrication of free-form surfaces for molds and dies. The continuous demand for better product quality has led to demand in higher surface accuracy. Limiting factors are the positioning accuracy of the tool and the accuracy of the manufacturing process. Computer controlled optical surfacing (CCOS) has been developed mainly for the fabrication of spherical and aspherical optics. The surface is repeatedly measured and corrected using abrasives until the target accuracy is attained. In principle, sub-micron accuracy is achievable. The accuracy is limited by surface measurement, and not by the processing equipment. The current research investigates the adoption of CCOS for the fabrication of free-form steel surfaces. This paper reports on the development of a test bed and initial experiment results. The test bed is based on a six-axis RX robot as the motion platform with a spindle attached to the wrist of the robot. Surface measurement is carried out using a Talysurf surface profiler. Surface error correction is performed by abrasion of material from the surface. The amount of material to be removed is based on the measured surface error. Following a predetermined tool path, the feed rate along the tool path for surface correction is to be varied according to the required among of material to be removed locally. A free-form surface specimen of mold steel of about 40 mm × 40 mm across is used for the initial experiment. The surface is prepared by CNC surface machining with the initial maximum error over 50 m. The positioning accuracy of the motion system is estimated to be no better than 0.1 mm. Through surface measurement and correction, the surface accuracy is significantly improved. © 2007 Elsevier B.V. All rights reserved. Keywords: CCOS; Polishing; Free-form surface; Molds and dies
1. Introduction High quality and precision surface fabrication is becoming more important in manufacturing. The needs for free-form surface fabrication are widely recognized in the aerospace, automotive and die/mold industries. However, it is difficult to produce precision free-form surfaces by conventional manufacturing processes. Such as multi-axis NC machining, sculptured surface machining (SSM) and high speed machining (HSM) [1]. Many have reported on the use of numerical controlled abrasion for surface finishing. Target applications are the finishing of surfaces for molds and dies [2] and surfaces for optics [3]. A typical setup involves a polishing tool mounted to a motion platform such as a robotics arm [4] or a machine tool. Both fixed abrasives [5] and loose abrasives [6] with felt are used. The control of the polishing force is either active (involving a force sensor [7]) or passive (with the tool mounted to a linear
∗
Corresponding author. Tel.: +852 27888436; fax: +852 27887660. E-mail address: [email protected] (H.-Y. Tam).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.04.057
slide). There were also investigations of tool path planning [8] and selection of abrasives to improve the processing efficiency. Computer controlled optical surfacing (CCOS) is a high precision finishing technique for optics manufacturing. It is an iterative process. The surface error is determined through measurement and is corrected through controlled abrasion. There are many papers on adopting CCOS for making aspheric [9] and off-axis [10] surfaces. Interferometry is the usual method for surface measurement [11]. Accurate surfacing uses loose abrasives. Surface correction can be executed through controlling the dwell time of the tool [12] or varying the tool pressure [13]. Surface accuracy of accuracy in the 0.2–0.3 m and surface roughness of 2 nm were demonstrated [14]; precision surfaces 2 m long was produced [15]. Some have worked on alternate processing methods like the use of fluid-jet abrasion [16] and magnetorheological fluid [17]. Surface accuracy of 30 nm peak-to-valley and surface roughness of 1 nm rms were achieved with the later process [17]. This paper reports an initial attempt to use CCOS to improve the form accuracy of a free-form metallic surface. A summary of CCOS is given in the next section. This is followed by a
466
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
Fig. 2. Surface abrasion by rotary tools: (a) cylindrical and (b) spherical.
Fig. 1. Flow chart of CCOS [15].
study of abrasive processes for surfacing. A general formulation of the problem to compute tool feed rates for surface correction is outlined. Then the experimental setup and initial results are presented and discussed. A conclusion is given in the end. 2. A summary of CCOS This section summarizes the use of CCOS for precision fabrication of surfaces. A flow chart of CCOS from the reference is shown in Fig. 1. Surfacing is performed iteratively to improve the surface accuracy. The input surface is obtained by precision manufacturing methods such as precision grinding or lapping. In the beginning of each iteration, the surface form is measured. The amount of material to be removed is planned based on comparing the measurement and the reference geometry. A suitable tool path is then selected. Algorithms are used to determine how the dwell time of the tool should be varied along the tool path and processing parameters selected to effect the desired material removal. The result is checked by simulation software. Dwell time calculation and simulation may be repeated until the pre-
dicted surface improvement is satisfactory. Then, a control file is generated for the execution of surface correction with CCOS [15]. 3. Abrasive processes For surface correction, one needs to know how much material is removed from the surface as the tool moves across a surface. Two types of tools are analyzed for processing of steel surfaces (Fig. 2). Experiments were conducted in which tools were moved along straight lines on planar surfaces. The spindle speed was fixed at 9000 rpm and the normal force was 5 N from a dead weight. The first set of experiments used sand paper adhered to the end of a cylindrical tool of 6.4 mm diameter. The tool feed rate was varied from 0.8 to 2.4 mm/s. The surface profile along a straight line perpendicular to the tool motion was measured before and after the experiment. The difference is termed the removal profile. The removal profile for the case of 0.8 mm/s feed rate is shown in Fig. 3. The shape is like a flat bell with depth of about 9 m and width 6 mm. The width remains approximately unchanged for various feed rates. The depth changes to 7.4 and 5 m for feed rates of 1.6 and 2.4 mm/s, respectively.
Fig. 3. The profile of material removal as the sand paper tool follows a straight line path on the surface at 0.8 mm/s.
Fig. 4. The profile of material removal as the felt tool (diamond paste) follows a straight line path on the surface at 0.8 mm/s.
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
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The second experiment used 15 mm radius spherical felt tool. Diamond paste of 6 m grains was used as the abrasive. The removal profile for 0.8 mm/s feed rate is shown in Fig. 4. The width and depth of the profile are 2 mm and 4 m, respectively. The shape is like a V-shaped bell. The difference in the shape of the removal profile for the two types of tool is attributed to the difference in pressure and velocity distribution within the tool and surface contact. Again, the width of the profile remains stable for various feed rates; the depth is changed to 3 and 2 m when the feed rates are 1.6 and 2.4 mm/s, respectively. 4. Surface correction planning Without loss of generality, suppose zij represents the measured height of the part surface at coordinates (xij , yij ), where i = 1, . . ., m and j = 1, . . ., n. The m × n matrix E constitutes the surface error map where the (i, j)th element of E is eij = zij − f(xij , yij ) and f(x, y) is the given surface function. A portion of the tool path for surface abrasion is illustrated in Fig. 5. Let Pk for k = 1, . . ., s + 1 be points of the tool paths and the tool feed rate from Pk to Pk+1 is given by vk . Material is removed from the part surface due to abrasion as the tool moves along the tool path. Suppose material removal as the tool moves from Pk to Pk+1 to along the tool path with unity feed rate is given by the m × n matrix Ak such that the (i, j)th element corresponds to the depth of abrasion at (xij , yij ). Assume the local surface curvatures and the tool path curvatures do not change significantly so that, for the same processing conditions (i.e. abrasive type, normal force, spindle speed, etc.), material removal profile experimentally determined for a given abrasion tool travelling along a straight line path on a planar surface from the previous section can be correlated to the removal profile of the tool following a given tool path on a part surface specimen. In this way, matrix Ak can be computed when Pk and Pk+1 and vk are given (Fig. 5). For each of the locations on the surface covered by the tool as it travels from Pk to Pk+1 , the depth of removal can be derived from experimentally determined removal profiles (from last section) and according to the distance of the location from the tool path.
Fig. 6. The robotic test bed for surface correction.
Suppose the depth of removal is proportional to the time of abrasion, material removal from the surface by the tool as it follows a specified tool path can be represented by the material removal matrix where A=
s
Ak tk
(1)
k=1
where 1 tk = . vk The problem of surface correction planning may be treated as finding a way to determine the tool feed rate vk along a given path to reduce the surface error. One way to approach the solution is to formulate the problem as a quadratic optimization problem with inequality constraints. The cost function is written as m n 2 min (2) rij tk i=1 j=1 where rij is the (i, j)th element of E − A; values of tk for k = 1, . . ., s are sought to minimize the cost function subject to constraints tk > T > 0 and rij > 0; the first type of constraints is set so that feasible feed rates should be no greater than certain maximum feed rate 1/T, where T is fixed; and the second type of constraints require the material removal to be no greater than the local surface error to avoid over-correction. Alternate ways to determine the feed rates could include the use of heuristics or iterative methods [12]. 5. Experiment setup
Fig. 5. Top view of the tool sweep as the tool travels from Pk to Pk+1 .
A test bed was set up for surface correction experiments. A Staubli RX60 six-axis robot was used as the motion platform as illustrated in Fig. 6. The positioning accuracy of the motion platform was estimated to be about 0.1 mm. The end-effector was equipped with a linear slider. This allowed the spindle to slide to prevent excessive tool and surface contact force due to positioning errors. A close-up view of the spindle and the specimen is shown in Fig. 7. The specimen is a block of DF2 mold steel of about 40 mm × 40 mm across. The surface was prepared by CNC surface machining. Excluding the part of
468
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
Fig. 7. Close-up view of the spindle and the specimen.
the surface close to the edges, the surface to be used for precision forming is 19 mm × 19 mm (Fig. 8). Within this area, the maximum height of the surface is about 2 mm, corresponding to the locations of two diagonal corners of the surface. The initial error map of the surface is given in Fig. 9. The surface profile was measured using a Form Talysurf surface profiler. It was actually surprising that the initial surface error was more than 50 m as given by measurement; this error is large compared to the resolution of the machine with which the surface was fabricated. The error is large at locations where the local surface slope is large. The unexpectedly large error may be attributed to the error of the cutting process and error due to tool holding. While the surface error could possibly be reduced by selecting alternate machining parameters and tool holding methods, this fabrication experience does reveal that manufacture of precision free-form surfaces is not an easy task. Fig. 9. Surface error before correction: (a) error map and (b) error contour.
6. Initial results In view that the initial surface error is quite large. An heuristic method was adopted that focused on: (1) removing material at locations where the error was large; and (2) avoiding removal at locations where the error was very small. The purpose was to achieve significant reduction of the maximum surface error which was considered large (more than 50 m from Fig. 9) and to avoid over-correction where the error was already small.
Fig. 8. Design surface of the specimen.
Let ek be the (non-negative) surface error at tool path point Pk , feed rate vk was set according to: if ek > eH , set vk = vmin , elseif ek < eL , set vk = vmax ,
(1/e) − (1/eH ) else set vk = vmin + (vmax − vmin ) (1/eL ) − (1/eH )
(3)
An iteration of surface correction was performed using the sand paper tool described in the previous section. Based on this heuristic approach, the focus of the present iteration is to reduce the maximum error through flattening the two diagonal peaks of the error map in Fig. 9. The threshold values eH and eL were chosen so that the tool feed rate was at vmin in the neighborhood of the two error peaks and at vmax near the other two corners of the surface where the error was small. This was followed by a number of iterations using the felt tools. The surface profile was measured afterward and the error map produced was shown in Fig. 10. It is quite remarkable that the maximum surface error was significantly reduced from above 50 m to slightly over 20 m. Significant removal of material took place near the two error peaks. Very little material was removed at the opposing two corners where the original error was small. One can actually observe over-correction around the inner peak of the surface at Fig. 9; a valley was formed in the corresponding region as seen in Fig. 10. The main cause of
K.-L. Yiu, H.-Y. Tam / Journal of Materials Processing Technology 192–193 (2007) 465–469
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ficient stability of the material removal process itself. Upon successful fine-tuning of the removal process, the ultimate bottleneck would likely be the accuracy of surface measurement. For measurement using Talysurf-like instruments, form accuracy for free-form surfaces is in the vicinity of 1 m. Acknowledgement The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 120105]. References
Fig. 10. Surface error after correction: (a) error map and (b) error contour.
the over-correction is attributed to insufficient stability of the material removal process. 7. Concluding remarks This paper reports initial experiments of free-form metallic surface fabrication based on CCOS. This was motivated by the fact that CCOS could be used in optics manufacturing to produce mirrors and lenses of remarkable precision. The experiment demonstrated that the maximum error of the DF2 steel specimen was reduced from over 50 m to slightly about 20 m in a small number of iterations. The amount of material removed from the surface co-related well with the planned removal. The positioning accuracy of the motion system in the present test bed is estimated to be no better than 0.1 mm. The current experiment shows that correction of surface error smaller than the precision of the motion system is feasible. The achieved accuracy is slightly over 20 m. The main bottleneck is insuf-
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