A Near-Optimal Probing Strategy for Workpiece Localization - CiteSeerX
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IEEE TRANSACTIONS ON ROBOTICS, VOL. 20, NO. 4, AUGUST 2004
A Near-Optimal Probing Strategy for Workpiece Localization Zhenhua Xiong, Michael Yu Wang, Senior Member, IEEE, and Zexiang Li, Member, IEEE
Abstract—This paper addresses an optimal planning problem for workpiece localization with coordinate measurements. The fundamental issue is to find the best probing locations and a suitable sampling size, such that the uncertainty of the localization error is within a predefined limited bound. First, we introduce two sequential optimization algorithms to incrementally increase the localization accuracy, defined by the determinant of the information matrix of the measurements. Then, a reliability analysis method is incorporated for finding a sample size that is sufficient to reduce the uncertainty of the localization error to a limited bound. By combining these two analysis tools, we present a near-optimal probing strategy for finding the best probing locations and a suitable sampling size. With this strategy, given the desired translation and orientation error bounds and desired confidence limit, we can experimentally determine the least number of points needed to measure. Simulation and experimental results show the efficiency of the proposed probing strategy.
Due to the accuracy of the probing device as well as the geometric and surface inaccuracy of the workpiece, there exists certain measurement error with each probed point. Thus, the positioning errors of the coordinate data will result in positional (translational and rotational) errors of the workpiece derived from the measurement data. Given a set of measurement points sampled from the workpiece surfaces in the machine reference and the corresponding surface descriptions in a frame , the problem is computer-aided design (CAD) model frame formulated as a least-squares problem, with the objective function given by
Index Terms—D-optimization, dimension inspection, measurement synthesis, reliability analysis, workpiece localization.
is the Euclidean transformation transforming where of the workpiece to a known machine refthe CAD frame erence frame , and is the corresponding point of on the corresponding surface , which we call the home point of . Workpiece localization plays a vital role in automation of many manufacturing processes. Workpiece localization and related problems, such as geometric localization algorithms [3], [10], [13], [17], [20], performance evaluation [8], and workpiece setup [7], [30], have received much attention from many researchers in the manufacturing community and robotics. When we measure the point set in the machine reference using a computer-controlled coordinate measuring frame machine (CMM) or on-machine touch probe, there are two questions that need to be answered. 1) In order to do a reliable workpiece localization, how many points should we probe on the workpiece surfaces? 2) If a point number is specified, how should we plan the measurement points on the CAD model? For the first question, a reliability-analysis method of workpiece localization [8] can be applied to check whether the localization result can be accurate enough with the measurement point set . The second question is about measurement synthesis on the CAD model. We call it the optimal planning problem. In this paper, a near-optimal probing strategy is proposed. The strategy is a combination of reliability analysis of workpiece localization and near-optimal planning algorithms. With this probing strategy, a near-minimal number of optimally planned points, which can be computed online, will be probed, while the transformation errors of the workpiece are within given error bounds. The remainder of the paper is organized as follows. In Section II, we review existing coordinate-sampling strategies
I. INTRODUCTION
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VER the last 20 years, a great deal of effort has been dedicated to improving production efficiency in manufacturing through automating production processes, and by eliminating human interaction as much as possible. In many manufacturing processes and robotic applications, accurately localizing geometric features is very important and useful. In the manufacturing literature, workpiece localization is a problem defined as follows. Assuming a rigid workpiece is arbitrarily fixtured to a machine table, determine the position and orientation of the workpiece frame relative to a known machine frame from a set of coordinates measured on the workpiece.
Manuscript received January 16, 2003; revised August 25, 2003. This paper was recommended for publication by Associate Editor F.-T. Cheng and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported in part by the National Science Foundation of China under Grant 50305019 and Grant 50390063, in part by the Research Grant Council (RGC) under Grant HKUST6226/02E and Grant CUHK4376/02E, and in part by the Scientific Research Foundation for Returned Overseas Chinese Scholars (SRF for ROCS), State Education Ministry (SEM). This paper was presented in part at the IEEE International Conference on Robotics and Automation, Taipei, Taiwan, R.O.C., September 2003. Z. Xiong is with the Robotics Institute, School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200030, China (e-mail: [email protected]). M. Y. Wang is with the Chinese University of Hong Kong, Shatin, NT, Hong Kong, on leave from the Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]). Z. Li is with the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TRO.2004.829474
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1042-296X/04$20.00 © 2004 IEEE
XIONG et al.: A NEAR-OPTIMAL PROBING STRATEGY FOR WORKPIECE LOCALIZATION
for reverse engineering or dimension inspection. In Section III, based on an accuracy-analysis model from fixture planning, we present two sequential optimization algorithms for near-optimal planning on workpiece surfaces. In Section IV, we present the near-optimal probing strategy for workpiece localization. Simulation and experimental results of the near-optimal planning and probing strategy are given in Section V. After some discussions in Section VI, we draw conclusions in Section VII. II. LITERATURE REVIEW Minimizing sampling time and cost while maintaining accurate measurement is a key concern in the coordinate metrology of workpiece surfaces. In [11], the authors reviewed error sources of sampling and different sampling strategies. It is noted that, in order to accurately measure part geometry, much higher sampling densities than those in the current practice must be incorporated. Minimal sampling is always desirable in industry practice for efficiency and productivity. Many efforts have been devoted to developing a rapid and accurate sampling strategy. Woo [28], [29] investigated two deterministic sequences of numbers, namely, the Hammersley sequence and the Halton–Zaremba sequence, which give a low level of discrepancy. After comparing with uniform sampling in two dimensions (2-D), it is found that the adoption of either sequence would give a rather significant improvement in the accuracy or savings in sample size, and hence, in time. Later on, in [18] and [19], it is shown that, by modeling a machined surface as a Wiener process, the root mean square (RMS) error of measurement is equivalent discrepancy of the complement of the sampling to the points. The Zaremba sequence, which is optimal in terms of discrepancy, was shown to require quadratically fewer points than does the uniform or random sequence with the same order of accuracy in measurement. In [16], a feature-based methodology which integrates the Hammersley sequence and a stratified sampling method was developed. The method can be used to derive a sampling strategy for multiple feature surfaces with multiple variances. The stratified Hammersley sampling was found to be more robust than the stratified random sampling and the stratified uniform sampling. In [15], experiments were conducted with the measurements of actual surfaces taken to evaluate different types of sampling strategies. Concerning the accuracy of flatness through the total sample size, a systematic random sampling method and the Halton–Zaremba sequence sampling method are found to yield better performance than the Hammersley sequence sampling method and the aligned systematic sampling method. It is noted that the above sampling methods are mostly used in reverse engineering or dimensional inspection applications. The purpose is to reduce the measurement error of sampled geometric features with real surfaces. For workpiece localization application, it is important to accurately recover the position and orientation of the workpiece subject to sampling errors. For this purpose, it is necessary to consider the geometric relations among the measured surfaces. It is known that a different set of sampling points will yield different transformation results
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[20]. Since sampling errors are inevitable, it becomes valuable to use a good sampling plan to ensure satisfactory recovery of Euclidean transformation . In [20], an upper bound of the transformation error was estimated with a normalized sensitivity measure. This measure serves as an index that reflects the joint effect of both the number of measurement points and the geometric attributes of measurement locations. Although such an index has been proposed for a given part geometry, a method for the synthesis of the measurement points has yet to be developed. In [5], a simple example is used to show how the probing points affect the possible displacement region of the object. A method using hitting sets and set covers was developed to obtain near-optimal probe placements for any known polygonal object, while the exact optimal solution is NP-hard. For a sculptured workpiece, it is difficult to apply this sampling method. In [4], an index was used for planning fixture locators based on the variance of the resultant localization error. Nonlinear programming was used to minimize the variance. The method can only deal with one continuous surface. When there are many surfaces in consideration, there exists a combinatorial problem on the point number assignment of different surfaces. For the application of medical image registration in [24], a 3-D model is constructed from images, using a sensor such as a computed tomographic (CT) scanner. In the synthesis procedure, a noise-amplification index is used to automatically generate near-optimal data configurations over a discrete point set [22]. Four methods were applied, namely, steepest-ascent hill climbing (SAH), near-ascent hill climbing (NAH), population-based incremental learning (PBIL), and a hybrid PBIL-hill-climbing approach. It was shown that the planning of 10–75 optimal points may need up to several hundred minutes of computational time. Such a method is suitable only for offline planning. III. SEQUENTIAL OPTIMAL PROBING In this section, we first discuss the general procedure for workpiece localization with CMM probing data. Then, we present a model for localization accuracy analysis originally developed for fixture planning. Finally, two sequential optimization algorithms are given for near-optimal planning of the probing points. A. Workpiece Localization Procedure The objective function given in (1) depends on the Euclidean transformation (Special Euclidean Group), and the , . If either set of home surface point the variables are fixed, the objective function becomes a simple function of the other set of variables. The alternating-variable method [9], [23], in which a function is minimized by changing one variable at a time, and the process continues iteratively until convergence occurs, is particularly suited for this problem. Given an initial Euclidean transformation , the first stage is to compute optimal home surface points. Once the home surface points are found, we then update the Euclidean transformation to . In this stage, there are many different methods, for example, the variational algorithm [14], the tangent algorithm [6],
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In this paper, we further restrict the feasible set of probing points in a discrete point-set domain instead of continuous surfaces, as in [26] and [27]. The advantages of point-set data are obvious. It makes the planning applicable for arbitrary geometric features. It is also noted that a fine point sample should not significantly affect the planning results. In [26], it is noticed that (4)
Fig. 1. Workpiece localization with probing data.
the iterative closest point (ICP) algorithm [3], Menq’s algorithm [20], and the Hong–Tan algorithm [13]. After updating the Euclidean transformation, the updated home surface points then are computed. This procedure will continue until a given precision is reached. Detailed procedures and a performance comparison of the different algorithms can be found in [6]. B. Localization Accuracy Analysis As described above, the position and orientation of the workpiece are determined from the measured coordinate data. At each probe point, there exists a certain measurement error, due to the accuracy of the probing device, as well as the geometric and surface inaccuracy of the workpiece. Therefore, the positioning errors of the coordinate data will result in positional (translational and rotational) errors of the workpiece derived from the measurement data. As shown in Fig. 1, for each measured coordinate point in the world frame , its home point may be specified in the on the workpiece. From the kinematic analysis body frame previously developed for workpiece fixturing [1], [26], we have the following linear model of workpiece localization: (2) where is the unit normal at the th home point, represents the projection of the positioning error of the th measured point defines the along the normal direction, and small perturbation of the workpiece position in the twist coordinate [21]. For a collection of measured points, we can combine the equations at all measurements to obtain (3) where and establishes the linear relation between probing errors workpiece location errors .
. This and the
C. Sequential Near-Optimal Probing Algorithms A goal of probe planning is to find the best set of positions on the workpiece for measurement among a set of virtually infinite feasible probing positions. The globally optimal solution is certainly difficult to find, even for a polygonal workpiece [5]. Therefore, any optimal planning algorithm is usually developed for suboptimal solutions.
is called the information matrix. where the matrix is used as the index of optimization. By maxiThen mizing , one can minimize the variance of the workpiece location error in (3). It is also called D-optimization [2]. One advantage of using the determinant function in designing a sequential optimization algorithm is as follows. Given a dispoints, for an initial set of points, the crete point set of information matrix is (5) Now, if we need to add (or delete) the th point from the set (or ) points, the resulting information matrix of and its inverse are given as (6) (7) where
In fact, is a diagonal element of the so-called prediction matrix , and it is easy to show that . Furthermore (8) These recursive relations lead us to a sequential deletion algorithm. Starting from a discrete point set of size , the candi, is chosen and is subtracted date point , which minimizes from the point set. This procedure is repeated till a prespecified number of measurement points remain. This algorithm represents a “top-down” approach, and its efficiency is quite limited when the initial number is large. An alternative approach of “bottom-up” [27] is a sequential addition algorithm. In an initial step, we select six points out discrete point set, such that matrix is of full rank. of the Then, an interchange step is engaged to improve the initial planning. This is achieved by, at each iteration, selecting a candidate point from the current set and exchanging it with another point from the set of candidate points, so that a maximum increase in is obtained. From (6)–(8), the interchange of a current as point and a candidate point results in a change in (9) . When is greater than 1, the interwhere . By maximizing change will increase the resulting at each iteration, we can maximize until there is no further increase.
XIONG et al.: A NEAR-OPTIMAL PROBING STRATEGY FOR WORKPIECE LOCALIZATION
After the six points, we can add an additional point from the points by maximizing using (8). One by remaining one, we will obtain a set of measurement points with the prespecified number. IV. NEAR-OPTIMAL PROBING STRATEGY With the plan of sequential sampling and a rough estimation of workpiece location, the touch probe can be programmed to collect a set of points on the workpiece surfaces. The set of points will be used to localize the workpiece in the machine with a localization algorithm. In general, reference frame for a least-squares algorithm, more points give a better result. To determine the quality of sampling and whether enough points are sampled, we first perform a reliability analysis of workpiece localization to evaluate the localization results. A. Reliability Analysis Assume that we have determined a workpiece localization algorithm to use, such as the Hong–Tan algorithm [17]. Let and , be the solution, and the value . Because of inevitable errors of the objective function at in the measurement and a finite number of points are to be probed over certain regions of the workpiece, the computed transformation will differ from the (unknown) actual (or true) transforma. Using the -test [12] in statistical analysis, the translation tional error between the computed and the actual transformations [8], by is bounded, with a confidence limit
(10) is the smallest eigenvalue of the translational error where residue matrix
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.. . is the surface normal at , and , where the critical value at the level in statistic analysis [12]. Similarly, assuming that
is
(12) for some unit vector by
, the orientation error
is bounded
(13) is the smallest eigenvalue of the orientation error where residue matrix
.. .
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and , . defined in (10) and (13) give It should be noted that upper bounds on the translation and orientation errors, based on
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and of the geometric transforthe two specific measures mation. While these measures are widely recognized in the current industrial practices, they have undesirable properties related to certain frame-dependent characteristics, as discussed recently in [25]. We will discuss implications of using these measures in the discussion section. B. Near-Optimal Probing Strategy Suppose that the desired accuracy requirement of a particular localization task has been obtained and is specified by . If we assume that the accuracy can be achieved by probing enough points on the workpiece surfaces, we then need to determine the minimal number of points so that the resulting estimaand , respectively. The impact tion errors are bounded by of the choice of measurement points on the accuracy of the computed transformation can not be underestimated. In Sections II and III, we have shown how to incrementally increase the localization accuracy (defined by the determinant of the information matrix) by adding or interchanging a probing point. The reliability analysis allows us to find a sample size that is sufficient to reduce the uncertainty of the localization error to a limited bound. By combining the above-described two tools of localization accuracy and reliability analysis, we now can define a near-optimal probing strategy for finding the best probing locations and a suitable sampling size. It should be pointed out that our probing strategy requires an initial seven points. In the sequential algorithm for optimal probing, at least six initial points are needed to make the information matrix in (5) be of full rank. Furthermore, the reliability analysis requires no less than seven points. Otherwise, the critical value in (10) and (13) for the -test will be infinity. Finally, our proposed probing strategy is described as follows. Near-optimal probing strategy: Input: sampled a) CAD model of the workpiece, with points; b) Surface finishing and sensor accuracy information. Output: a) Estimated transformation that is within given error bounds. Step 0: a) Determine acceptable transformation-error bounds and confidence limit ; b) Probe initial seven points (often manually) (set ); c) Compute the translational and orientational errors. If and , exit. Else, continue; d) Align the CAD model relative to the machine frame on the user screen, manually eliminate unaccessible regions and points of the CAD model. Let be the candidate set available for measurement. Step 1: a) Set , a new set of points to be planned; b) Use the sequential optimization algorithm to genpoints; erate points from the remaining set of . c) Set
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Simulation model for the near-optimal probing strategy.
Sampled simulation model.
Step 2: a) Cluster and sequence the previous set of points; b) Generate a path to probe these points; c) Command the touch probe to sample the points. Step 3: a) Compute the transformation using all points; b) Compute the translational and orientational errors. If they are within the error bound, exit. Else, if go to Step 1, and if , report that no satisfactory transformation can be found and exit. Note that in the last case when no satisfactory transformation can be found, either the error bounds have to be relaxed or a finer sample domain has to be generated from the CAD model. In Step 1(b), we can use the reversed sequence by the sequential deletion method or directly use the sequential addition method. V. SIMULATION AND EXPERIMENTAL RESULTS A. Simulation Results Here we show the simulation results, using a model shown in Fig. 2. Assume that we are interested in using surfaces , , and only. We first discretize the three surfaces. A sampled model is shown in Fig. 3. For this simulation, we sample surface with 877 points, surface with 365 points, and surface with 357 points. There are, in total, 1599 points, i.e., . The acceptable transformation-error bounds are set at mm and degrees, and the confidence limit is set as 95%. Since regions near surface boundaries are not desirable for probing, we leave a 5-mm margin from every boundary of the surfaces , , and . Thus, a total of 991 points remain, i.e., . These points are shown in Fig. 4. Then, the sequential deletion algorithm is applied for an optimal initial six points, as shown in Fig. 5. They are generated in 69 s on a Pentium II 400 MHz PC, using MATLAB 6. In this
Fig. 4.
Sampled simulation model for computation.
Fig. 5.
Near-optimal planning scheme with six points on the sculptured model.
case, . In the procedure, we also get a sequence of deleted points. The points that are deleted earlier are less desirable to probe than the points deleted later. and be given and Let computed Euclidean transformations, respectively. The orientation and translation errors are defined as where and . Measurement data is generated based on the 991 sampled points. A known Euclidean transformation is applied to these points. Random noise, which simulates measurement errors and dimensional errors, is added to these points. The random noise is , where the mean assumed to be of normal distribution is the measurement bias, and the variance depends on machining processes and measurement devices. Seven points with an optimal planning scheme are first used to localize the workpiece. The Hong–Tan algorithm is used in this simulation. After each localization, the translation and orientation errors are estimated, using (10) and (13). If both estimated errors are and , we terminate the within the given error bounds simulation and report success. Otherwise, several measurement points are added and the process continues. Fig. 6 shows and . The the simulation results with is set as 95%. In Fig. 6(a), translation confidence limit are shown. In error and estimated translation-error bound Fig. 6(b), rotation error and estimated rotation-error bound are shown. It turns out that when the number of points reaches 95, both error bounds are satisfied. The measurement scheme for 95 points is shown in Fig. 7. If we use the sequential addition algorithm, after several interchanges from a randomly generated six points, we then compare the six points with one generated by the sequential deletion algorithm in Fig. 8. The points planned by sequential addition are marked with triangles, while the points planned by sequential
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Fig. 6.
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Simulation results for deletion sequence with 95% confidence.
are translation error and estimated translation-error bound shown. In Fig. 9(b), rotation error and estimated rotation-error bound are shown. It turns out that when the number of points reaches 225, both error bounds are satisfied. The measurement scheme for 225 points is shown in Fig. 10. B. Experimental Results
Fig. 7. Near-optimal planning scheme with 95 points on the sculptured model.
Fig. 8.
Near-optimal planning scheme comparison on the sculptured model.
deletion are marked with asterisks. In this case, we only need, on average, 0.05 s to get the solution using MATLAB, with a result . Then it takes, on average, 0.066 of s to add an additional optimal point. From the time aspect, the sequential addition algorithm is more desirable, since we need not compute all the point sequences, as in the sequential deletion algorithm. For the point sequence generated with the sequential addition and algorithm, Fig. 9 shows the simulation results with . The confidence limit is set as 95%. In Fig. 9(a),
In the experiment, we use the workpiece shown in Fig. 11. We first discretize the model with 467, 683, and 511 points on the top, front, and side surfaces, respectively. There are, altogether, 1661 points, as shown in Fig. 11(a). For probing safety reasons, we remove the points near the surface boundaries. Since the radius of the probe tip is near 3 mm, points within 3 mm of the boundaries are removed. The candidate point set is shown in Fig. 11(b). Now the point numbers are 286, 484, and 352, which gives a total of 1122 points. is set as 95% in this experiThe confidence limit ment. We define the translation-error bound as 0.1 mm, and as 0.2 degrees. Six points are first the rotation-error bound manually probed to determine the rough transformation for auto probing [30]. Then seven optimally planned points are probed with a computer-aided setup system, which is developed on a HAAS Mini Mill center by adding a probing unit and a master computer, as shown in Fig. 12. If the two error bounds are not satisfied, the probing system probes five more points. By this procedure, finally, when the total probed points reaches 185, the error bounds are both satisfied, with the estimated translation and rotation-error bounds of 0.0900 mm and 0.1988 degree, respectively. These points are shown in Fig. 14. Fig. 13(a) and (b) shows the estimated translation-error bound and rotation-error bound during the probing procedure. VI. DISCUSSION It is known that there are two different components in sampling geometric errors of workpiece surfaces. One is a deterministic error component. The other is a random error component. In [31], two identical sculptured surfaces machined with a ball-end milling process were studied, with one using the one-way cutting
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Fig. 9.
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Simulation results for addition sequence with 95% confidence.
Fig. 10. Near-optimal planning scheme with 225 points on the sculptured model.
Fig. 12. Fig. 11. Sampled points of the experimental model. (a) Sampled model. (b) Candidate point set.
path, and the other using the zigzag cutting path. It turns out that the deterministic errors are usually much more complex than the original geometry of the surface. An iterative approach was proposed to separate the two error components. However, it is still possible in precision machining, such as high-precision datum surfaces, that the deterministic error component is negligibly small, compared with the random error component on the workpiece surfaces [31]. Our proposed near-optimal probing strategy mainly deals with this type of surface.
Computer-aided setup system.
We should further discuss the upper-bound estimates of and , defined in (10) and (13). In the simulation results presented in Section V-A, it is observed that these estimated upper bounds are much larger than the translation and orientation errors specified for the simulated cases. This is caused by two contributing factors. First, these upper bounds are specified based on the theory of -statistics with a conservative measure of the errors. In (10) and (13), the 3-D translational error is represented with the in (11), smallest eigenvalue of the translational-error matrix while the 3-D rotational error is represented with the smallest
XIONG et al.: A NEAR-OPTIMAL PROBING STRATEGY FOR WORKPIECE LOCALIZATION
Fig. 13.
Fig. 14.
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Estimated translation and rotation-error bound during probing.
Total probed points in the experiment.
eigenvalue of the rotational-error matrix in (14). The conservative use of the smallest eigenvalues would inevitably yield a rather loose upper-bound estimate of the errors. One may certainly use a less stringent error measure, such as the geometric mean of the eigenvalues, which should result in a tighter upper bound. There is no standard on the error measures in the current industrial practice. Another factor is related to a more fundamental issue on the measures of transformation accuracy. For the translational erin (11), it is widely realized rors represented by error matrix in the robotics literature that the eigenvalues of this matrix are frame invariant with respect to changes of the world and the workpiece reference frames [25]. They can also be explained in terms of the geometry of a level set function that is related to the total variance of the workpiece transformation [25]. Thus, they are appropriate to be used for a measure of the translational accuracy of the workpiece. However, for the rotational-error matrix in (14), it is found to be frame dependent. A measure based for the rotational errors would depend on the eigenvalues of on the choice of coordinate frames. A sampling scheme which
is optimal under one choice of the reference frame may fail to be optimal under another. It is argued in [25] that an appropriate characterization of the rotational errors should use a set of frame-invariant parameters, known as the principal rotational accuracy parameters. Their derivations and geometric interpretation are given in detail in [25]. However, these principal rotational parameters involve and the rotational-error both the translational-error matrix matrix , and their joint effect. A computation of these parameters would require an inverse of a 6 6 matrix. For our online probe planning, this computational requirement may not be practical. Fundamentally, investigations of proper characterizations of the transformation accuracy are still in their early stage. In our implementation presented here, we choose to use the conand in (10) and (13). The distribuventional measures tion pattern of the sample points, determined by the proposed procedure for the cases reported in Section V, reflects the geometric properties of these conventional measures used. See [25] for more details on the geometric interpretation. As a result of these factors, the estimated error bounds and are noticeably large, for both the simulated examples and the experimental results. It should be pointed out that these error bounds are used to serve as a reliability criterion to help determine if additional sample points are needed. Our optimal sampling algorithm remains fundamentally the same if one finds and uses a more accurate estimate of the upper bounds. Another way of making the decision on the additional point is to examine the progressive change in the values of the error bounds. When adding an additional measurement point would not result in any and , one significant decrease in the reliability measures may conclude that no further samples are necessary. This could result in a lower number of measurement points. For our example of the experimental test in Section V-B, this criterion would reduce the number of probed points to roughly 70. VII. CONCLUSION Workpiece localization has many important applications in the manufacturing process. A good planning strategy for
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measuring points on workpiece surfaces would help perform a reliable and economical workpiece localization. In this paper, we address two essential questions of what is a suitable sampling size and what are the best probing locations. Our approach combines two analysis tools. First, with a localization accuracy analysis, we propose a maximum determinant planning strategy. Two sequential optimization algorithms are then introduced, which makes online planning of measurement points possible. Then, a reliability analysis is employed to find a sample size that is sufficient to reduce the uncertainty of the localization error to a given. A combination of these analysis tools results in our proposed near-optimal probing strategy. Simulation and experimental results of different models show that, given the desired translation and orientation-error bounds and desired confidence limit, we can experimentally find the lowest number of near-optimal points to measure.
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
ACKNOWLEDGMENT
[29]
The authors would like to thank the reviewers for their useful suggestions.
[30]
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[1] H. Asada and A. B. By, “Kinematic analysis of workpiece fixturing for flexible assembly with automatically reconfigurable fixtures,” IEEE J. Robot. Automat., vol. RA-1, pp. 86–93, June 1985. [2] A. Atkinson and A. Doney, Optimal Experimental Designs. New York: Oxford Univ. Press, 1992. [3] P. Besl and N. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 239–256, Feb. 1992. [4] W. Cai, S. J. Hu, and J. X. Yuan, “A variational method of robust fixture configuration design for 3-D workpieces,” J. Manuf. Sci. Eng., vol. 119, pp. 593–602, Nov. 1997. [5] J. Canny and E. Paulos, “Optimal probing strategies,” Int. J. Robot. Res., vol. 20, no. 8, pp. 694–704, Aug. 2001. [6] Y. X. Chu, “Workpiece localization: Theory, algorithms, and implementation,” Ph.D. dissertation, Hong Kong Univ. Sci. Tech., Kowloon, Hong Kong, 1999. [7] Y. X. Chu, J. B. Gou, and Z. X. Li, “On the hybrid/envelopment problem,” Int. J. Robot. Res., vol. 18, no. 5, pp. 491–501, 1999. [8] , “Workpiece localization algorithms: Performance evaluation and reliability analysis,” J. Manuf. Syst., vol. 18, no. 2, pp. 113–126, Feb. 1999. [9] R. Fletcher, Practical Methods of Optimization. New York: Wiley, 1987. [10] J. B. Gou, Y. X. Chu, and Z. X. Li, “On the symmetric localization problem,” IEEE Trans. Robot. Automat., vol. 14, pp. 533–540, Aug. 1998. [11] R. J. Hocken, J. Raja, and U. Babu, “Sampling issues in coordinate metrology,” Manuf. Rev., vol. 6, no. 4, pp. 282–94, Dec. 1993. [12] P. G. Hoel, Introduction to Mathematical Statistics, 5th ed. New York: Wiley, 1984. [13] J. Hong and X. Tan, “Method and Apparatus for Determining Position and Orientation of Mechanical Objects,” U.S. Patent 5,208,763, May 4, 1993. [14] B. K. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Amer. A, vol. 4, no. 4, pp. 629–642, 1987. [15] W. S. Kim and S. Raman, “On the selection of flatness measurement points in coordinate measuring machine inspection,” Int. J. Mach. Tools Manuf., vol. 40, pp. 427–443, 2000. [16] G. Lee, J. Mou, and Y. Shen, “Sampling strategy design for dimensional measurement of geometric features using coordinate measuring machine,” Int. J. Mach. Tools Manuf., vol. 37, no. 7, pp. 917–934, 1997. [17] Z. X. Li, J. B. Gou, and Y. X. Chu, “Geometric algorithms for workpiece localization,” IEEE Trans. Robot. Automat., vol. 14, pp. 864–878, Dec. 1998. [18] R. Liang, T. C. Woo, and C. C. Hesieh, “Accuracy and time in surface measurement, Part 1: Mathematical foundations,” J. Manuf. Sci. Eng., vol. 120, pp. 141–149, Feb. 1998.
, “Accuracy and time in surface measurement, Part 2: Optimal sampling sequence,” J. Manuf. Sci. Eng., vol. 120, pp. 150–155, Feb. 1998. C. H. Menq, H. Yau, and G. Lai, “Automated precision measurement of surface profile in CAD-directed inspection,” IEEE Trans. Robot. Automat., vol. 8, pp. 268–278, Apr. 1992. R. Murray, Z. X. Li, and S. Sastry, A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994. A. Nahvi and J. M. Hollerbach, “The noise amplification index for optimal pose selection in robot calibration,” in Proc. IEEE Int. Conf. Robotics and Automation, vol. 1, 1996, pp. 647–656. M. J. Powell, “On search directions for minimization algorithms,” Math. Program., vol. 4, pp. 193–201, 1973. D. A. Simon, “Fast and accurate shape-based registration,” Ph.D. dissertation, Carnegie Mellon Univ., Pittsburgh, PA, 1996. M. Y. Wang, “Characterizations of localization accuracy of fixtures,” IEEE Trans. Robot Automat, vol. 18, pp. 976–981, Dec. 2002. , “An optimum design for 3-D fixture synthesis in a point-set domain,” IEEE Trans. Robot. Automat., vol. 16, pp. 939–946, Dec. 2000. M. Y. Wang and D. M. Pelinescu, “Optimizing fixture layout in a point-set domain,” IEEE Trans. Robot. Automat., vol. 17, pp. 312–323, June 2001. T. C. Woo and R. Liang, “Dimensional measurement of surfaces and their sampling,” Comput.-Aided Des., vol. 25, no. 4, pp. 233–239, 1993. T. C. Woo, R. Liang, C. C. Hsieh, and N. K. Lee, “Efficient sampling for surface measurements,” J. Manuf. Syst., vol. 14, no. 5, pp. 345–354, 1995. Z. H. Xiong, Y. X. Chu, G. F. Liu, and Z. X. Li, “Workpiece localization and computer-aided setup system,” in Proc. Int. Conf. Intelligent Robots and Systems, 2001, pp. 1141–1146. Z. C. Yan, B. D. Yang, and C. H. Menq, “Uncertainty analysis and variation reduction of three-dimensional coordinate metrology. Part 1: Geometric error decomposition,” Int. J. Mach. Tools Manufact., vol. 39, pp. 1199–1217, 1999.
Zhenhua Xiong received the B.E. and M.E. degrees from the Department of Aircraft Design, Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 1995 and 1998, respectively, and the Ph.D. degree from the Hong Kong University of Science and Technology, Kowloon, in 2002. He is currently a Faculty Member at the Robotics Institute, Shanghai Jiaotong University, Shanghai, China. His research interests include workpiece localization, dimensioning inspection, optimal design, and electronics manufacturing. Michael Y. Wang (SM’01) received the B.S. degree in mechanical and manufacturing engineering from Xi’an Jiaotong University, Xi’an City, China, in 1982, the M.S. degree in engineering mechanics from Pennsylvania State University, State College, PA, in 1985, and the Ph.D. degree in mechanical engineering from Carnegie Mellon University, Pittsburgh, PA, in 1989. He is currently with the Chinese University of Hong Kong, Shatin, Hong Kong, on leave from the Department of Mechanical Engineering, University of Maryland, College Park, where he is an Associate Professor. His research interests include precision fixturing and grasping, particle vibration damping for electronics manufacturing, computational design and modeling, and development of 3-D microstructures and microchannels. Zexiang Li (M’83) received the B.S. degree in electrical engineering and economics (with honors) from Carnegie Mellon University, Pittsburgh, PA, in 1983 and the M.A. degree in mathematics and Ph.D. degree in electrical engineering and computer science, both from the University of California, Berkeley, in 1985 and 1989, respectively. He is an Associate Professor with the Electrical and Electronic Engineering Department, Hong Kong University of Science and Technology, Kowloon, Hong Kong. His research interests include robotics, nonlinear system theory, and manufacturing.
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A Near-Optimal Probing Strategy for Workpiece Localization Zhenhua Xiong, Michael Yu Wang, Senior Member, IEEE, and Zexiang Li, Member, IEEE
Abstract—This paper addresses an optimal planning problem for workpiece localization with coordinate measurements. The fundamental issue is to find the best probing locations and a suitable sampling size, such that the uncertainty of the localization error is within a predefined limited bound. First, we introduce two sequential optimization algorithms to incrementally increase the localization accuracy, defined by the determinant of the information matrix of the measurements. Then, a reliability analysis method is incorporated for finding a sample size that is sufficient to reduce the uncertainty of the localization error to a limited bound. By combining these two analysis tools, we present a near-optimal probing strategy for finding the best probing locations and a suitable sampling size. With this strategy, given the desired translation and orientation error bounds and desired confidence limit, we can experimentally determine the least number of points needed to measure. Simulation and experimental results show the efficiency of the proposed probing strategy.
Due to the accuracy of the probing device as well as the geometric and surface inaccuracy of the workpiece, there exists certain measurement error with each probed point. Thus, the positioning errors of the coordinate data will result in positional (translational and rotational) errors of the workpiece derived from the measurement data. Given a set of measurement points sampled from the workpiece surfaces in the machine reference and the corresponding surface descriptions in a frame , the problem is computer-aided design (CAD) model frame formulated as a least-squares problem, with the objective function given by
Index Terms—D-optimization, dimension inspection, measurement synthesis, reliability analysis, workpiece localization.
is the Euclidean transformation transforming where of the workpiece to a known machine refthe CAD frame erence frame , and is the corresponding point of on the corresponding surface , which we call the home point of . Workpiece localization plays a vital role in automation of many manufacturing processes. Workpiece localization and related problems, such as geometric localization algorithms [3], [10], [13], [17], [20], performance evaluation [8], and workpiece setup [7], [30], have received much attention from many researchers in the manufacturing community and robotics. When we measure the point set in the machine reference using a computer-controlled coordinate measuring frame machine (CMM) or on-machine touch probe, there are two questions that need to be answered. 1) In order to do a reliable workpiece localization, how many points should we probe on the workpiece surfaces? 2) If a point number is specified, how should we plan the measurement points on the CAD model? For the first question, a reliability-analysis method of workpiece localization [8] can be applied to check whether the localization result can be accurate enough with the measurement point set . The second question is about measurement synthesis on the CAD model. We call it the optimal planning problem. In this paper, a near-optimal probing strategy is proposed. The strategy is a combination of reliability analysis of workpiece localization and near-optimal planning algorithms. With this probing strategy, a near-minimal number of optimally planned points, which can be computed online, will be probed, while the transformation errors of the workpiece are within given error bounds. The remainder of the paper is organized as follows. In Section II, we review existing coordinate-sampling strategies
I. INTRODUCTION
O
VER the last 20 years, a great deal of effort has been dedicated to improving production efficiency in manufacturing through automating production processes, and by eliminating human interaction as much as possible. In many manufacturing processes and robotic applications, accurately localizing geometric features is very important and useful. In the manufacturing literature, workpiece localization is a problem defined as follows. Assuming a rigid workpiece is arbitrarily fixtured to a machine table, determine the position and orientation of the workpiece frame relative to a known machine frame from a set of coordinates measured on the workpiece.
Manuscript received January 16, 2003; revised August 25, 2003. This paper was recommended for publication by Associate Editor F.-T. Cheng and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported in part by the National Science Foundation of China under Grant 50305019 and Grant 50390063, in part by the Research Grant Council (RGC) under Grant HKUST6226/02E and Grant CUHK4376/02E, and in part by the Scientific Research Foundation for Returned Overseas Chinese Scholars (SRF for ROCS), State Education Ministry (SEM). This paper was presented in part at the IEEE International Conference on Robotics and Automation, Taipei, Taiwan, R.O.C., September 2003. Z. Xiong is with the Robotics Institute, School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200030, China (e-mail: [email protected]). M. Y. Wang is with the Chinese University of Hong Kong, Shatin, NT, Hong Kong, on leave from the Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]). Z. Li is with the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TRO.2004.829474
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for reverse engineering or dimension inspection. In Section III, based on an accuracy-analysis model from fixture planning, we present two sequential optimization algorithms for near-optimal planning on workpiece surfaces. In Section IV, we present the near-optimal probing strategy for workpiece localization. Simulation and experimental results of the near-optimal planning and probing strategy are given in Section V. After some discussions in Section VI, we draw conclusions in Section VII. II. LITERATURE REVIEW Minimizing sampling time and cost while maintaining accurate measurement is a key concern in the coordinate metrology of workpiece surfaces. In [11], the authors reviewed error sources of sampling and different sampling strategies. It is noted that, in order to accurately measure part geometry, much higher sampling densities than those in the current practice must be incorporated. Minimal sampling is always desirable in industry practice for efficiency and productivity. Many efforts have been devoted to developing a rapid and accurate sampling strategy. Woo [28], [29] investigated two deterministic sequences of numbers, namely, the Hammersley sequence and the Halton–Zaremba sequence, which give a low level of discrepancy. After comparing with uniform sampling in two dimensions (2-D), it is found that the adoption of either sequence would give a rather significant improvement in the accuracy or savings in sample size, and hence, in time. Later on, in [18] and [19], it is shown that, by modeling a machined surface as a Wiener process, the root mean square (RMS) error of measurement is equivalent discrepancy of the complement of the sampling to the points. The Zaremba sequence, which is optimal in terms of discrepancy, was shown to require quadratically fewer points than does the uniform or random sequence with the same order of accuracy in measurement. In [16], a feature-based methodology which integrates the Hammersley sequence and a stratified sampling method was developed. The method can be used to derive a sampling strategy for multiple feature surfaces with multiple variances. The stratified Hammersley sampling was found to be more robust than the stratified random sampling and the stratified uniform sampling. In [15], experiments were conducted with the measurements of actual surfaces taken to evaluate different types of sampling strategies. Concerning the accuracy of flatness through the total sample size, a systematic random sampling method and the Halton–Zaremba sequence sampling method are found to yield better performance than the Hammersley sequence sampling method and the aligned systematic sampling method. It is noted that the above sampling methods are mostly used in reverse engineering or dimensional inspection applications. The purpose is to reduce the measurement error of sampled geometric features with real surfaces. For workpiece localization application, it is important to accurately recover the position and orientation of the workpiece subject to sampling errors. For this purpose, it is necessary to consider the geometric relations among the measured surfaces. It is known that a different set of sampling points will yield different transformation results
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[20]. Since sampling errors are inevitable, it becomes valuable to use a good sampling plan to ensure satisfactory recovery of Euclidean transformation . In [20], an upper bound of the transformation error was estimated with a normalized sensitivity measure. This measure serves as an index that reflects the joint effect of both the number of measurement points and the geometric attributes of measurement locations. Although such an index has been proposed for a given part geometry, a method for the synthesis of the measurement points has yet to be developed. In [5], a simple example is used to show how the probing points affect the possible displacement region of the object. A method using hitting sets and set covers was developed to obtain near-optimal probe placements for any known polygonal object, while the exact optimal solution is NP-hard. For a sculptured workpiece, it is difficult to apply this sampling method. In [4], an index was used for planning fixture locators based on the variance of the resultant localization error. Nonlinear programming was used to minimize the variance. The method can only deal with one continuous surface. When there are many surfaces in consideration, there exists a combinatorial problem on the point number assignment of different surfaces. For the application of medical image registration in [24], a 3-D model is constructed from images, using a sensor such as a computed tomographic (CT) scanner. In the synthesis procedure, a noise-amplification index is used to automatically generate near-optimal data configurations over a discrete point set [22]. Four methods were applied, namely, steepest-ascent hill climbing (SAH), near-ascent hill climbing (NAH), population-based incremental learning (PBIL), and a hybrid PBIL-hill-climbing approach. It was shown that the planning of 10–75 optimal points may need up to several hundred minutes of computational time. Such a method is suitable only for offline planning. III. SEQUENTIAL OPTIMAL PROBING In this section, we first discuss the general procedure for workpiece localization with CMM probing data. Then, we present a model for localization accuracy analysis originally developed for fixture planning. Finally, two sequential optimization algorithms are given for near-optimal planning of the probing points. A. Workpiece Localization Procedure The objective function given in (1) depends on the Euclidean transformation (Special Euclidean Group), and the , . If either set of home surface point the variables are fixed, the objective function becomes a simple function of the other set of variables. The alternating-variable method [9], [23], in which a function is minimized by changing one variable at a time, and the process continues iteratively until convergence occurs, is particularly suited for this problem. Given an initial Euclidean transformation , the first stage is to compute optimal home surface points. Once the home surface points are found, we then update the Euclidean transformation to . In this stage, there are many different methods, for example, the variational algorithm [14], the tangent algorithm [6],
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In this paper, we further restrict the feasible set of probing points in a discrete point-set domain instead of continuous surfaces, as in [26] and [27]. The advantages of point-set data are obvious. It makes the planning applicable for arbitrary geometric features. It is also noted that a fine point sample should not significantly affect the planning results. In [26], it is noticed that (4)
Fig. 1. Workpiece localization with probing data.
the iterative closest point (ICP) algorithm [3], Menq’s algorithm [20], and the Hong–Tan algorithm [13]. After updating the Euclidean transformation, the updated home surface points then are computed. This procedure will continue until a given precision is reached. Detailed procedures and a performance comparison of the different algorithms can be found in [6]. B. Localization Accuracy Analysis As described above, the position and orientation of the workpiece are determined from the measured coordinate data. At each probe point, there exists a certain measurement error, due to the accuracy of the probing device, as well as the geometric and surface inaccuracy of the workpiece. Therefore, the positioning errors of the coordinate data will result in positional (translational and rotational) errors of the workpiece derived from the measurement data. As shown in Fig. 1, for each measured coordinate point in the world frame , its home point may be specified in the on the workpiece. From the kinematic analysis body frame previously developed for workpiece fixturing [1], [26], we have the following linear model of workpiece localization: (2) where is the unit normal at the th home point, represents the projection of the positioning error of the th measured point defines the along the normal direction, and small perturbation of the workpiece position in the twist coordinate [21]. For a collection of measured points, we can combine the equations at all measurements to obtain (3) where and establishes the linear relation between probing errors workpiece location errors .
. This and the
C. Sequential Near-Optimal Probing Algorithms A goal of probe planning is to find the best set of positions on the workpiece for measurement among a set of virtually infinite feasible probing positions. The globally optimal solution is certainly difficult to find, even for a polygonal workpiece [5]. Therefore, any optimal planning algorithm is usually developed for suboptimal solutions.
is called the information matrix. where the matrix is used as the index of optimization. By maxiThen mizing , one can minimize the variance of the workpiece location error in (3). It is also called D-optimization [2]. One advantage of using the determinant function in designing a sequential optimization algorithm is as follows. Given a dispoints, for an initial set of points, the crete point set of information matrix is (5) Now, if we need to add (or delete) the th point from the set (or ) points, the resulting information matrix of and its inverse are given as (6) (7) where
In fact, is a diagonal element of the so-called prediction matrix , and it is easy to show that . Furthermore (8) These recursive relations lead us to a sequential deletion algorithm. Starting from a discrete point set of size , the candi, is chosen and is subtracted date point , which minimizes from the point set. This procedure is repeated till a prespecified number of measurement points remain. This algorithm represents a “top-down” approach, and its efficiency is quite limited when the initial number is large. An alternative approach of “bottom-up” [27] is a sequential addition algorithm. In an initial step, we select six points out discrete point set, such that matrix is of full rank. of the Then, an interchange step is engaged to improve the initial planning. This is achieved by, at each iteration, selecting a candidate point from the current set and exchanging it with another point from the set of candidate points, so that a maximum increase in is obtained. From (6)–(8), the interchange of a current as point and a candidate point results in a change in (9) . When is greater than 1, the interwhere . By maximizing change will increase the resulting at each iteration, we can maximize until there is no further increase.
XIONG et al.: A NEAR-OPTIMAL PROBING STRATEGY FOR WORKPIECE LOCALIZATION
After the six points, we can add an additional point from the points by maximizing using (8). One by remaining one, we will obtain a set of measurement points with the prespecified number. IV. NEAR-OPTIMAL PROBING STRATEGY With the plan of sequential sampling and a rough estimation of workpiece location, the touch probe can be programmed to collect a set of points on the workpiece surfaces. The set of points will be used to localize the workpiece in the machine with a localization algorithm. In general, reference frame for a least-squares algorithm, more points give a better result. To determine the quality of sampling and whether enough points are sampled, we first perform a reliability analysis of workpiece localization to evaluate the localization results. A. Reliability Analysis Assume that we have determined a workpiece localization algorithm to use, such as the Hong–Tan algorithm [17]. Let and , be the solution, and the value . Because of inevitable errors of the objective function at in the measurement and a finite number of points are to be probed over certain regions of the workpiece, the computed transformation will differ from the (unknown) actual (or true) transforma. Using the -test [12] in statistical analysis, the translation tional error between the computed and the actual transformations [8], by is bounded, with a confidence limit
(10) is the smallest eigenvalue of the translational error where residue matrix
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.. . is the surface normal at , and , where the critical value at the level in statistic analysis [12]. Similarly, assuming that
is
(12) for some unit vector by
, the orientation error
is bounded
(13) is the smallest eigenvalue of the orientation error where residue matrix
.. .
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and , . defined in (10) and (13) give It should be noted that upper bounds on the translation and orientation errors, based on
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and of the geometric transforthe two specific measures mation. While these measures are widely recognized in the current industrial practices, they have undesirable properties related to certain frame-dependent characteristics, as discussed recently in [25]. We will discuss implications of using these measures in the discussion section. B. Near-Optimal Probing Strategy Suppose that the desired accuracy requirement of a particular localization task has been obtained and is specified by . If we assume that the accuracy can be achieved by probing enough points on the workpiece surfaces, we then need to determine the minimal number of points so that the resulting estimaand , respectively. The impact tion errors are bounded by of the choice of measurement points on the accuracy of the computed transformation can not be underestimated. In Sections II and III, we have shown how to incrementally increase the localization accuracy (defined by the determinant of the information matrix) by adding or interchanging a probing point. The reliability analysis allows us to find a sample size that is sufficient to reduce the uncertainty of the localization error to a limited bound. By combining the above-described two tools of localization accuracy and reliability analysis, we now can define a near-optimal probing strategy for finding the best probing locations and a suitable sampling size. It should be pointed out that our probing strategy requires an initial seven points. In the sequential algorithm for optimal probing, at least six initial points are needed to make the information matrix in (5) be of full rank. Furthermore, the reliability analysis requires no less than seven points. Otherwise, the critical value in (10) and (13) for the -test will be infinity. Finally, our proposed probing strategy is described as follows. Near-optimal probing strategy: Input: sampled a) CAD model of the workpiece, with points; b) Surface finishing and sensor accuracy information. Output: a) Estimated transformation that is within given error bounds. Step 0: a) Determine acceptable transformation-error bounds and confidence limit ; b) Probe initial seven points (often manually) (set ); c) Compute the translational and orientational errors. If and , exit. Else, continue; d) Align the CAD model relative to the machine frame on the user screen, manually eliminate unaccessible regions and points of the CAD model. Let be the candidate set available for measurement. Step 1: a) Set , a new set of points to be planned; b) Use the sequential optimization algorithm to genpoints; erate points from the remaining set of . c) Set
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Fig. 2.
Fig. 3.
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Simulation model for the near-optimal probing strategy.
Sampled simulation model.
Step 2: a) Cluster and sequence the previous set of points; b) Generate a path to probe these points; c) Command the touch probe to sample the points. Step 3: a) Compute the transformation using all points; b) Compute the translational and orientational errors. If they are within the error bound, exit. Else, if go to Step 1, and if , report that no satisfactory transformation can be found and exit. Note that in the last case when no satisfactory transformation can be found, either the error bounds have to be relaxed or a finer sample domain has to be generated from the CAD model. In Step 1(b), we can use the reversed sequence by the sequential deletion method or directly use the sequential addition method. V. SIMULATION AND EXPERIMENTAL RESULTS A. Simulation Results Here we show the simulation results, using a model shown in Fig. 2. Assume that we are interested in using surfaces , , and only. We first discretize the three surfaces. A sampled model is shown in Fig. 3. For this simulation, we sample surface with 877 points, surface with 365 points, and surface with 357 points. There are, in total, 1599 points, i.e., . The acceptable transformation-error bounds are set at mm and degrees, and the confidence limit is set as 95%. Since regions near surface boundaries are not desirable for probing, we leave a 5-mm margin from every boundary of the surfaces , , and . Thus, a total of 991 points remain, i.e., . These points are shown in Fig. 4. Then, the sequential deletion algorithm is applied for an optimal initial six points, as shown in Fig. 5. They are generated in 69 s on a Pentium II 400 MHz PC, using MATLAB 6. In this
Fig. 4.
Sampled simulation model for computation.
Fig. 5.
Near-optimal planning scheme with six points on the sculptured model.
case, . In the procedure, we also get a sequence of deleted points. The points that are deleted earlier are less desirable to probe than the points deleted later. and be given and Let computed Euclidean transformations, respectively. The orientation and translation errors are defined as where and . Measurement data is generated based on the 991 sampled points. A known Euclidean transformation is applied to these points. Random noise, which simulates measurement errors and dimensional errors, is added to these points. The random noise is , where the mean assumed to be of normal distribution is the measurement bias, and the variance depends on machining processes and measurement devices. Seven points with an optimal planning scheme are first used to localize the workpiece. The Hong–Tan algorithm is used in this simulation. After each localization, the translation and orientation errors are estimated, using (10) and (13). If both estimated errors are and , we terminate the within the given error bounds simulation and report success. Otherwise, several measurement points are added and the process continues. Fig. 6 shows and . The the simulation results with is set as 95%. In Fig. 6(a), translation confidence limit are shown. In error and estimated translation-error bound Fig. 6(b), rotation error and estimated rotation-error bound are shown. It turns out that when the number of points reaches 95, both error bounds are satisfied. The measurement scheme for 95 points is shown in Fig. 7. If we use the sequential addition algorithm, after several interchanges from a randomly generated six points, we then compare the six points with one generated by the sequential deletion algorithm in Fig. 8. The points planned by sequential addition are marked with triangles, while the points planned by sequential
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Fig. 6.
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Simulation results for deletion sequence with 95% confidence.
are translation error and estimated translation-error bound shown. In Fig. 9(b), rotation error and estimated rotation-error bound are shown. It turns out that when the number of points reaches 225, both error bounds are satisfied. The measurement scheme for 225 points is shown in Fig. 10. B. Experimental Results
Fig. 7. Near-optimal planning scheme with 95 points on the sculptured model.
Fig. 8.
Near-optimal planning scheme comparison on the sculptured model.
deletion are marked with asterisks. In this case, we only need, on average, 0.05 s to get the solution using MATLAB, with a result . Then it takes, on average, 0.066 of s to add an additional optimal point. From the time aspect, the sequential addition algorithm is more desirable, since we need not compute all the point sequences, as in the sequential deletion algorithm. For the point sequence generated with the sequential addition and algorithm, Fig. 9 shows the simulation results with . The confidence limit is set as 95%. In Fig. 9(a),
In the experiment, we use the workpiece shown in Fig. 11. We first discretize the model with 467, 683, and 511 points on the top, front, and side surfaces, respectively. There are, altogether, 1661 points, as shown in Fig. 11(a). For probing safety reasons, we remove the points near the surface boundaries. Since the radius of the probe tip is near 3 mm, points within 3 mm of the boundaries are removed. The candidate point set is shown in Fig. 11(b). Now the point numbers are 286, 484, and 352, which gives a total of 1122 points. is set as 95% in this experiThe confidence limit ment. We define the translation-error bound as 0.1 mm, and as 0.2 degrees. Six points are first the rotation-error bound manually probed to determine the rough transformation for auto probing [30]. Then seven optimally planned points are probed with a computer-aided setup system, which is developed on a HAAS Mini Mill center by adding a probing unit and a master computer, as shown in Fig. 12. If the two error bounds are not satisfied, the probing system probes five more points. By this procedure, finally, when the total probed points reaches 185, the error bounds are both satisfied, with the estimated translation and rotation-error bounds of 0.0900 mm and 0.1988 degree, respectively. These points are shown in Fig. 14. Fig. 13(a) and (b) shows the estimated translation-error bound and rotation-error bound during the probing procedure. VI. DISCUSSION It is known that there are two different components in sampling geometric errors of workpiece surfaces. One is a deterministic error component. The other is a random error component. In [31], two identical sculptured surfaces machined with a ball-end milling process were studied, with one using the one-way cutting
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Fig. 9.
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Simulation results for addition sequence with 95% confidence.
Fig. 10. Near-optimal planning scheme with 225 points on the sculptured model.
Fig. 12. Fig. 11. Sampled points of the experimental model. (a) Sampled model. (b) Candidate point set.
path, and the other using the zigzag cutting path. It turns out that the deterministic errors are usually much more complex than the original geometry of the surface. An iterative approach was proposed to separate the two error components. However, it is still possible in precision machining, such as high-precision datum surfaces, that the deterministic error component is negligibly small, compared with the random error component on the workpiece surfaces [31]. Our proposed near-optimal probing strategy mainly deals with this type of surface.
Computer-aided setup system.
We should further discuss the upper-bound estimates of and , defined in (10) and (13). In the simulation results presented in Section V-A, it is observed that these estimated upper bounds are much larger than the translation and orientation errors specified for the simulated cases. This is caused by two contributing factors. First, these upper bounds are specified based on the theory of -statistics with a conservative measure of the errors. In (10) and (13), the 3-D translational error is represented with the in (11), smallest eigenvalue of the translational-error matrix while the 3-D rotational error is represented with the smallest
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Fig. 13.
Fig. 14.
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Estimated translation and rotation-error bound during probing.
Total probed points in the experiment.
eigenvalue of the rotational-error matrix in (14). The conservative use of the smallest eigenvalues would inevitably yield a rather loose upper-bound estimate of the errors. One may certainly use a less stringent error measure, such as the geometric mean of the eigenvalues, which should result in a tighter upper bound. There is no standard on the error measures in the current industrial practice. Another factor is related to a more fundamental issue on the measures of transformation accuracy. For the translational erin (11), it is widely realized rors represented by error matrix in the robotics literature that the eigenvalues of this matrix are frame invariant with respect to changes of the world and the workpiece reference frames [25]. They can also be explained in terms of the geometry of a level set function that is related to the total variance of the workpiece transformation [25]. Thus, they are appropriate to be used for a measure of the translational accuracy of the workpiece. However, for the rotational-error matrix in (14), it is found to be frame dependent. A measure based for the rotational errors would depend on the eigenvalues of on the choice of coordinate frames. A sampling scheme which
is optimal under one choice of the reference frame may fail to be optimal under another. It is argued in [25] that an appropriate characterization of the rotational errors should use a set of frame-invariant parameters, known as the principal rotational accuracy parameters. Their derivations and geometric interpretation are given in detail in [25]. However, these principal rotational parameters involve and the rotational-error both the translational-error matrix matrix , and their joint effect. A computation of these parameters would require an inverse of a 6 6 matrix. For our online probe planning, this computational requirement may not be practical. Fundamentally, investigations of proper characterizations of the transformation accuracy are still in their early stage. In our implementation presented here, we choose to use the conand in (10) and (13). The distribuventional measures tion pattern of the sample points, determined by the proposed procedure for the cases reported in Section V, reflects the geometric properties of these conventional measures used. See [25] for more details on the geometric interpretation. As a result of these factors, the estimated error bounds and are noticeably large, for both the simulated examples and the experimental results. It should be pointed out that these error bounds are used to serve as a reliability criterion to help determine if additional sample points are needed. Our optimal sampling algorithm remains fundamentally the same if one finds and uses a more accurate estimate of the upper bounds. Another way of making the decision on the additional point is to examine the progressive change in the values of the error bounds. When adding an additional measurement point would not result in any and , one significant decrease in the reliability measures may conclude that no further samples are necessary. This could result in a lower number of measurement points. For our example of the experimental test in Section V-B, this criterion would reduce the number of probed points to roughly 70. VII. CONCLUSION Workpiece localization has many important applications in the manufacturing process. A good planning strategy for
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measuring points on workpiece surfaces would help perform a reliable and economical workpiece localization. In this paper, we address two essential questions of what is a suitable sampling size and what are the best probing locations. Our approach combines two analysis tools. First, with a localization accuracy analysis, we propose a maximum determinant planning strategy. Two sequential optimization algorithms are then introduced, which makes online planning of measurement points possible. Then, a reliability analysis is employed to find a sample size that is sufficient to reduce the uncertainty of the localization error to a given. A combination of these analysis tools results in our proposed near-optimal probing strategy. Simulation and experimental results of different models show that, given the desired translation and orientation-error bounds and desired confidence limit, we can experimentally find the lowest number of near-optimal points to measure.
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
ACKNOWLEDGMENT
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The authors would like to thank the reviewers for their useful suggestions.
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Zhenhua Xiong received the B.E. and M.E. degrees from the Department of Aircraft Design, Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 1995 and 1998, respectively, and the Ph.D. degree from the Hong Kong University of Science and Technology, Kowloon, in 2002. He is currently a Faculty Member at the Robotics Institute, Shanghai Jiaotong University, Shanghai, China. His research interests include workpiece localization, dimensioning inspection, optimal design, and electronics manufacturing. Michael Y. Wang (SM’01) received the B.S. degree in mechanical and manufacturing engineering from Xi’an Jiaotong University, Xi’an City, China, in 1982, the M.S. degree in engineering mechanics from Pennsylvania State University, State College, PA, in 1985, and the Ph.D. degree in mechanical engineering from Carnegie Mellon University, Pittsburgh, PA, in 1989. He is currently with the Chinese University of Hong Kong, Shatin, Hong Kong, on leave from the Department of Mechanical Engineering, University of Maryland, College Park, where he is an Associate Professor. His research interests include precision fixturing and grasping, particle vibration damping for electronics manufacturing, computational design and modeling, and development of 3-D microstructures and microchannels. Zexiang Li (M’83) received the B.S. degree in electrical engineering and economics (with honors) from Carnegie Mellon University, Pittsburgh, PA, in 1983 and the M.A. degree in mathematics and Ph.D. degree in electrical engineering and computer science, both from the University of California, Berkeley, in 1985 and 1989, respectively. He is an Associate Professor with the Electrical and Electronic Engineering Department, Hong Kong University of Science and Technology, Kowloon, Hong Kong. His research interests include robotics, nonlinear system theory, and manufacturing.
